Vibrations and stability of complex beam systems

Introduction

Mechanical systems often consist of complex structures formed by the interaction of two or more basic systems, particularly those connected by elastic layers, which are prevalent in the mechanical, construction, and aeronautical industries These systems exhibit complex vibrations and possess a higher number of natural frequencies due to their elastic connections The increased number of natural frequencies heightens the risk of resonance conditions, leading to potential breakage and damage To bridge theoretical research and engineering practice, numerous linear dynamic models have been developed to describe system motion, providing initial approximations and insights into dynamic behavior under slight motion For further exploration of system behavior, especially concerning non-linearity effects, these models serve as a robust foundation for continued research.

Mechanical structures subjected to cyclical loads often experience damage, altering their dynamic behavior and necessitating a thorough understanding to avoid resonance and its consequences In damaged structures, increased displacement of deformable elements leads to more pronounced geometric non-linearity, differing significantly from non-damaged counterparts Identifying conditions that trigger resonance due to interactions between vibration modes is crucial Mathematical models for both elastically connected and damaged beams have incorporated effects of rotary inertia and transverse shear For decades, researchers have focused on the vibrations and stability of these systems, driven by the need to prevent breakage and permanent damage in mechanical and construction applications This interest has spurred the development of analytical and numerical methods for analyzing vibrations in both linear and non-linear mechanical systems.

The vibrations of beams connected by a Winkler elastic layer have garnered significant attention from researchers, particularly regarding the dynamics of two elastically linked beams This study aims to establish conditions under which the system functions effectively as a dynamic absorber in practical applications Seelig and Hoppman developed a mathematical model to analyze the impact of impulse loads on beams, resulting in a system of partial differential equations that characterize their vibrations Their theoretical and experimental findings validated the analytical solutions derived for slender beams experiencing small transverse motions, utilizing the Euler-Bernoulli theory.

Oniszczuk analyzed the free and forced vibration of two elastically connected Euler-Bernoulli beams, providing analytical solutions for eigenfrequencies, amplitude functions, and vibration modes He examined how the stiffness of the elastic interlayer influences the system's frequencies and amplitudes Additionally, he identified the conditions necessary for resonance and explored the system's behavior as a dynamic absorber.

Zhang et al [4, 5] conducted an analysis of a system comprising two elastically connected Euler-Bernoulli beams, focusing on free and forced vibrations influenced by axial compression forces They provided analytical solutions for the natural frequencies of the system, highlighting the impact of axial compression on vibration amplitude Their findings established a relationship between the system's critical force and the Euler critical load, demonstrating the interdependence of axial forces in the connected beams.

Vu et al [6] investigated the forced and damped vibrations in a system of two elastically connected Euler-Bernoulli beams, presenting analytical solutions to the associated partial differential equations that incorporate a damping factor Their research highlighted the impact of system damping on vibration characteristics and amplitude responses under harmonic forcing The analytical findings demonstrated a strong agreement with experimental data from Seelig and Hoppmann [1] regarding dynamic absorbers.

Li and Hua [7] employed the spectral finite element method to obtain numerical solutions for the natural frequencies of two elastically connected Timoshenko beams with varying support conditions Their research focused on identifying the vibration modes and analyzing the amplitude-frequency relationship during forced vibrations of the system.

The study of system vibration in complex structures, such as reinforced models, coupled support systems in mechanical engineering, and multistory buildings in civil engineering, has garnered significant attention from researchers Recent investigations have focused on structures composed of three or more elastically connected beams The advancements in computer technology and the application of numerical mathematics to these models have further fueled interest among scientists since the 20th century.

Li et al [8] investigated the dynamics of three elastically connected Timoshenko beams, focusing on the numerical determination of natural frequencies, mode shapes, and the impact of elastic inter-layer stiffness on system vibrations Kelly and Srinivas [9] explored multiple elastically connected Euler beams, utilizing the Rayleigh-Ritz method to identify natural frequencies and mode shapes for beams with uniform characteristics Ariaei et al [10] examined a movable body interacting with multiple elastically connected Timoshenko beams on an elastic foundation, concluding that maximum deflection decreases when the body moves closer to the surface Mao [11] employed the Adomian decomposition method to ascertain the mode shapes for the first ten modes of multiple elastically connected Euler beams on an elastic foundation.

Stojanović et al [12] investigated the free vibration and static stability of two elastically connected beams, considering the impacts of rotary inertia and transverse shear that cause cross-section rotation Their analysis provided analytical solutions for the natural frequencies and identified the critical force for the coupled beam system.

Stojanović and Kozić explored the forced vibration of elastically connected beams, examining how axial compression affects vibration amplitude under various external forces, establishing conditions for resonance and dynamic vibration absorption In a subsequent study, Stojanović et al analyzed the static stability of three elastically connected Timoshenko beams on an elastic foundation, deriving expressions for critical force influenced by elastic Winkler layers Their research on multiple Timoshenko and Reddy-Bickford beams highlighted the analytical forms of natural frequencies and their variations due to axial compression The vibration of damaged beams is a crucial area of research, as damage alters structural integrity and increases breakage risk Christides and Barr identified that damage reduces natural frequencies in simply supported beams, while Sinha et al developed a model to represent changes in beam geometry due to damage, confirming findings through finite element analysis Pandey et al focused on how damage affects natural mode shapes, employing deviations in these shapes for damage detection, while Panteliou et al utilized changes in damping factors for localization The field has seen significant interest in non-linear vibrations of damaged beams, with Bikri et al demonstrating differences in mode shapes between linear and non-linear models Andreaus et al studied non-linear vibrations of brackets under harmonic excitation, modeling damage as a bilinear oscillator Building on this, Stojanović et al introduced a new p-version finite element method for non-linear vibrations of damaged Timoshenko beams, accounting for changes in both stiffness and weight, allowing for the determination of natural frequencies with fewer motion functions, revealing longitudinal vibrations in doubly clamped beams solely due to damage.

Numerical experiments on the Timoshenko beam model revealed the influence of rotary inertia and transverse shear in elastically connected beams The study demonstrated that the interaction between transverse vibrations and cross-section component vibrations led to asymmetry in non-linear vibration time modes To address the non-linear vibration region, the Newmark method was employed for direct integration of non-linear partial differential equations related to the forced vibration of damaged Timoshenko beams Additionally, the continuation method was utilized to identify bifurcation points in the free non-linear vibration model, as referenced by Stojanović and Ribeiro A novel p-version of the finite element method was developed to analyze the effects of cross-section rotation, rotary inertia, and transverse shear on the geometrically non-linear vibrations of beams with varying thicknesses.

Vibrations of Euler, Rayleigh, Timoshenko and Reddy-Bickford

This section of the chapter provides an overview of beam theories, focusing on the application of partial differential equations of motion essential for analyzing complex beam systems The Euler–Bernoulli theory, established in the 18th century, initially overlooked rotary inertia, which was later addressed by Rayleigh in 1894 Timoshenko's 1921 theory improved upon this by incorporating shear effects, making it a more comprehensive model than Euler–Bernoulli Additionally, the Reddy-Bickford theory offers a more precise mathematical interpretation of stress and deformation, yielding more accurate approximations crucial for understanding the vibrations of thick beams.

The Euler-Bernoulli theory for homogeneous elastic slender prismatic beams suggests that during vibration, the beams' cross-sections remain perpendicular to the neutral axis during deformation, disregarding the influences of rotary inertia and transverse shear This theory effectively approximates solutions for small transverse motions in slender beams.

(a) Fig 1.2.1 (a) A simply supported beam under the influence of axial compression forces and continuous loads ( , ) and ( , ) (b) Deformation in various beam theories

Fig 1.2.2 An elementary part of a Euler beam

In the context of the elementary beam illustrated in Figure 1.2.1a, the functions of longitudinal and transverse motions are designated as ( , , ) and ( , , ), respectively Consequently, specific relationships can be established concerning the angles depicted in Figure 1.2.2.

The longitudinal and transverse motions of a beam's point on the axis are represented by the equations ( , ) and ( , , ), respectively The deformation in the direction of motion and the stress-deformation ratio, in accordance with Hooke’s law, are defined by the relationship (1.2.1).

, (1.2.2) ( , , ) = ( , , ), (1.2.3) where is Young’s modulus The equations are derived based on the principle of virtual work to which the following applies

+ + = 0, (1.2.4) where , and represent virtual work of inertial, internal and external forces respectively Virtual work of inertial forces is

Having regard to the relation of longitudinal motion, ( , , ) = ( , ) −

The effect of rotary inertia is represented by a member in the integrand of the wave equation for an Euler beam However, since the mathematical model of the Euler beam does not consider rotary inertia, it can be excluded from the derivation of the wave equation The virtual work of internal forces is articulated as follows.

Virtual work of external forces acting transversally is given by

( , ) = ( , ) − ( , ), (1.2.7) where δ stands for variation operator By substituting the equations (1.2.5-1.2.7) into the equation (1.2.4), we get

By successive application of Green’s theorem on the equation (1.2.8), the partial equation for transverse vibration of the Euler beam is obtained

The Rayleigh theory of homogeneous elastic slender prismatic beams posits that the motion of beam points ensures the cross-sections remain perpendicular to the neutral axis during deformation, similar to the Euler model, while accounting for the effects of rotary inertia Although the constitutive relations mirror those of a Euler beam, the dynamic equation is modified to incorporate the member ( , , ) δ ( , , ) in the virtual work expression for inertial forces By designating the functions of longitudinal and transverse motions as ( , , ) and ( , , ), the relationships governing these motions for the Euler beam can be established based on the beam's point position ( , , ).

The longitudinal and transverse motion of a beam point along the axis can be represented as (1.2.10) The deformation occurring in the specified direction is influenced by the motion, and according to Hooke's law, the relationship between stress and deformation is established.

, (1.2.11) ( , , ) = ( , , ) (1.2.12) Hence, virtual work of inertial forces is given by

= − ( , , ) δ ( , , ) + ( , , ) δ ( , , ) (1.2.13) Virtual work of internal forces is

Virtual work of external forces is expressed as

, ( , ) = ( , ) − ( , ), (1.2.15) Substituting the equations (1.2.13-1.2.15) into the equation (1.2.4), yields

By applying Green’s theorem to the equation (1.2.16), we get the equation of transverse vibration of the Rayleigh beam in the following form

= ( , ), (1.2.17) where ρ denotes mass density, A the area of the cross-section, Young’s modulus, moment of inertia of the cross-section area for x –axis and F the axial compression force

The theory of homogeneous elastic prismatic Timoshenko beams provides a mathematical model that incorporates the effects of rotary inertia and accounts for shear forces affecting the rotation of the beams' cross-sections This model introduces a function for cross-section rotation, which is derived from the elementary deformed beam part illustrated in Figure 1.2.3.

In a Timoshenko beam, the elementary part illustrated in Figure 1.2.3 demonstrates the longitudinal and transverse motion of a beam point located at (0, , ) on the –axis The angle between the deformed beam's cross-section and the y-axis is also represented The deformations occurring in both the direction and tangentially as a result of shear forces exhibit a specific form.

Assuming the material is isotropic and elastic, the Hooke’s law on the relation between stress and deformation applies

0 , (1.2.21) where G is the shear modulus and is the shear factor The equations of motion are derived based on the principle of virtual work for which the following applies

+ + = 0, (1.2.22) where , and represent virtual work of inertial, internal and external forces respectively Virtual work of inertial forces is given by

Virtual work of external forces is expressed as

, ( , ) = ( , ) − ( , ) (1.2.25) Substituting the equations (1.2.23-1.2.25) into an equation (1.2.22) yields

By applying Green’s theorem on the equation (1.2.26), we get the equations of vibration for Timoshenko beam in the following form

Through the application of Timoshenko theory [27], the equations of transverse vibration of beams were derived taking into account the rotation of cross-sections caused by shear forces

Timoshenko theory does not consider that shear stress at the endpoints of cross-sections is zero; instead, it utilizes a shear factor, k The Reddy-Bickford beam mathematical model, as referenced in studies by Reddy [29, 30] and Wang et al., effectively illustrates this concept.

The deformation of a beam's cross-section in the yz plane can be approximated along the shear stress value points, leading to a more accurate mathematical interpretation of stress and deformation in beam point motion This approach eliminates the need for the shear factor k, as outlined in reference [31].

In the context of beam mechanics, we define the longitudinal movement of a beam point at (0, y) along the y-axis as \( u = \frac{4}{3}h \) and the transversal movement at (0, y) along the x-axis as \( v = 3 \) The angle of the beam’s cross-section at its intersection with the neutral line relative to the y-axis is denoted as \( \theta \), as illustrated in Figure 1.2.4 Under the assumption that the material is isotropic and elastic, Hooke’s law is applicable.

0 , (1.2.32) where G stands for shear modulus The equations of motion are derived based on virtual work principle like in the previous examples for which the following applies

+ + = 0, (1.2.33) where , and represent virtual work of inertial, internal and external forces respectively Virtual work of inertial forces is given by

Fig 1.2.4 An elementary part of a Reddy-Bickford beam

Virtual work of internal forces is expressed as

Virtual work of external forces is

, ( , ) = ( , ) − ( , ) (1.2.36) where δ stands for variation operator By substituting the equations (1.2.34- 1.2.36) into the equation (1.2.33), we have

By applying the Green’s theorem on the equation (1.2.37), we get the equation of vibration for Reddy-Bickford beam in the following form

The Timoshenko and Reddy-Bickford models, which incorporate the effects of rotary inertia and transverse shear, provide a more accurate approximation for analyzing thick beams where cross-section rotation occurs.

Fig 1.2.5 An elementary part of a cross-sectioned Euler, Rayleigh, Timoshenko and Reddy-Bickford beam

The mathematical models for Euler and Rayleigh beams assume that the beams' cross-sections remain perpendicular to the neutral axis, a limitation not applicable to thick beams To address this, the Timoshenko and Reddy-Bickford models incorporate the effects of shear forces on the rotation of the beams' cross-sections, with the Reddy-Bickford model specifically satisfying shear stress criteria in boundary regions, unlike the Timoshenko model Figure 1.2.5 illustrates the deformation of fibers in the beams' cross-section based on the applied theory.

V Stojanović and P Kozić, Vibrations and Stability of Complex Beam Systems,

17 Springer Tracts in Mechanical Engineering, DOI: 10.1007/978-3-319-13767-4_2

Free Vibrations and Stability of an Elastically Connected Double-Beam System

Chapter 2 explores the free oscillations and static stability of two elastically connected beams, presenting analytical results that highlight the influence of various mechanical parameters on natural frequencies and amplitudes The findings are validated through comparisons with established classical models, introducing a novel double-beam model defined by newly derived equations of motion that incorporate rotational inertia effects and transverse shear (Rayleigh’s model, Timoshenko’s model, Reddy-Bickford's model) The chapter formulates static stability conditions for different types of elastically connected beams, providing analytical expressions for critical force values Numerical experiments further affirm the analytical results, demonstrating the necessity of considering rotational inertia and transverse shear in thick beam models, as neglecting these factors leads to increasing errors with higher vibration modes.

Free Vibration of Two Elastically Connected Rayleigh Beams

This article examines the impact of rotary inertia on the free transverse vibrations of an elastically connected double-beam system, as studied by Stojanović et al The two beams, which are of equal length, are linked by an elastic layer characterized by a specific stiffness modulus Additionally, axial compression forces are applied at the ends of the beams, as depicted in Figure 2.1.1.

Fig 2.1.1 An elastically connected double-beam system

Let the functions of longitudinal and transverse motion of the Rayleigh-beam system be as follows ( , , ), ( , , ), ( , , ) and ( , , ) Analogous to the relation (1.2.10), the following applies to the double-beam system

Deformation in the - direction in the function of motion and the stress- deformation relation according to the Hooke’s law are as follows

, (2.1.3) ( , , ) = ( , , ), ( , , ) = ( , , ) (2.1.4) Virtual work of inertial forces is expressed as

Virtual work of external forces is given by

Based on the principle of virtual work + + = 0, 1,2 and the equations (2.1.5-2.1.9), we have

By successive application of Green’s theorem on the expressions (2.1.10-2.1.11), we obtain the equations of transverse vibration for elastically connected Rayleigh- beam system in the following form

The equation (2.1.13) represents the relationship where ρ denotes mass density, A indicates the cross-sectional area, Young’s modulus is a material property, and the second moment of inertia pertains to the beam's cross-section along the x-axis The system under consideration is an elastically connected double-beam configuration, specifically made up of simply supported Rayleigh beams, which are subject to defined initial and boundary conditions.

Considering the harmonic motion of beams’ points, we assume the solutions to the equations (2.1.12) and (2.1.13) as the product of functions

(2.1.16) where T in (t) is the unknown time function and Z n (z) is the mode shape function

By substituting the proposed solutions into the equations governing the transverse vibration of the elastically connected Rayleigh double-beam system, we derive a set of two differential equations.

+ ( − ) − = 0, (2.1.19) where the following substitutions = 1,2 have been introduced

The solutions to differential equations (2.1.18) and (2.1.19) can be assumed in the following form

= , = , = √−1, (2.1.20) where denotes the natural frequency of the system If we substitute the expression (2.1.20) into equations (2.1.18) and (2.1.19), we get the set of homogeneous algebraic equations

The algebraic equations (2.1.21) and (2.1.22) yield non-trivial solutions when the determinant of the matrix is zero This condition leads to the derivation of the characteristic system equation, represented as a fourth-degree polynomial.

The characteristic equation (2.1.23) has two different positive real roots

The amplitude ratio of vibration mode for each frequency , is given by

The general solution to the system of differential equations (2.1.18) and (2.1.19) is in the form of

The solutions to the system of partial differential equations (2.1.12) and (2.1.13) describe the free transverse vibration of two elastically connected simply supported beams, which are influenced by rotary inertia The equations are represented as ( ) = [sin( ) + cos( )], with unknown constants denoted by (i = 1, 2).

Unknown constants and (i = 1, 2) are determined from initial conditions (2.1.14) by using the orthogonality condition of functions expressed by

= = [sin( )] d 2, where is the Kronecker delta function If we enter the initial conditions (2.1.14) into expressions (2.1.31) and (2.1.32), we obtain

By multiplying the expressions (2.1.34) and (2.1.35) by the eigenfunction , then integrating them with respect to z from 0 to l and using the orthogonality condition (2.1.33), we have

After solving the system of equations (2.1.36) and (2.1.37), we determine the unknown constants in the following form

Free Vibrations of Two Elastically Connected Timoshenko

This article examines the effects of rotary inertia and transverse shear on the free transverse vibration of a double-beam system that is elastically connected by a Winkler layer, as discussed by Stojanović et al The analysis assumes that, similar to the Rayleigh model, both beams have the same length (l) and are linked by an elastic layer characterized by a specific stiffness modulus.

At the ends of the beams, axial compression forces are applied, as illustrated in Figure 2.1.1 The functions representing longitudinal and transversal motion, along with the angle of cross-section rotation, are denoted as ( , , ), ( , , ), ( , ), ( , , ), ( , , ), and ( , ) Corresponding to relation (1.2.18), the equations of motion for the double-beam system are established.

Deformation in the function of motion and the stress-strain ratio according to the Hooke’s law are

Virtual work of inertial forces is

Virtual work of external forces is given by

Substituting the equations (2.2.6-2.2.11) into a general equation of virtual work principle + + = 0, = 1,2 , yields

By applying Green’s theorem repeatedly to the expressions (2.2.12-2.2.13), we derive the vibration equations for a system composed of two elastically connected Timoshenko beams.

The equation (2.2.17) describes the relationship between mass density (ρ), cross-sectional area (A), Young's modulus, shear modulus, shear factor, and the moment of inertia for the x-axis of a beam This framework applies to two elastically connected simply supported Timoshenko beams, which are analyzed under specific initial and boundary conditions.

Considering the harmonic motion of beams’ points, we assume the solutions to the equations (2.2.14-2.2.17) as the product of functions

(2.2.21) where ( ) and ( ) ( ) are the unknown time functions, while ( ) and ( ) are the mode shapes functions

By substituting the proposed solutions into the transverse vibration equations of a system comprising two elastically connected Timoshenko beams, we derive a set of four second-degree differential equations.

We assume the solutions of the equation system (2.2.23) in the following form

If we substitute the assumed solutions (2.2.24) into the system of equations (2.2.23), we get a homogeneous set of algebraic equations which has non-trivial solutions when the system determinant equals zero

The characteristic system equation (2.2.25) is derived from the criteria for the existence of non-trivial solutions represented as an eighth-degree polynomial By applying the substitution =, we can express the frequency equation accordingly.

+ ̃ + ̃ + ̃ + ̃ = 0 (2.2.26) Constants ̃ , ̃ , ̃ and ̃ are given in the Appendix 2.2.1 The polynomial of the fourth degree may be factored in the following form

In the expressions presents one of the roots of the third-degree equation

After factoring the equation (2.2.26), the solutions may be written as a square of natural frequencies

4 − (2.2.29) The three roots of the equation (2.2.28) are

By selecting one of the three roots from expression (2.2.30), we obtain distinct solutions for equation (2.2.26), resulting in four real positive natural frequencies: = , = , = , = The amplitude ratio for each frequency is calculated using equation (2.2.25).

Appendix 2.2.2 General solution to the system of differential equations (2.2.23) is of the following form

The equation ( )( ) = sin + cos represents the unknown constants for the free transverse vibrations of a system made up of two elastically connected simply supported beams This analysis takes into account the effects of rotary inertia and transverse shear, leading to a set of partial differential equations that describe the system's behavior.

The unknown constants , , and (i = 1, 2; g=3, 4) are determined based on initial conditions (2.2.18) using the orthogonality condition for trigonometric functions

2 (2.2.43) where is the Knocker delta function If we substitute the initial conditions (2.2.18) into the expressions (2.2.39 − 2.2.42), we obtain

By multiplying the expression (2.2.44) by appropriate eigenfunctions , i.e then integrating it with respect to z from 0 to l and using the orthogonality condition (2.2.43), we have

After solving the system of equations (2.2.45-2.2.48), the unknown constants are obtained in the following form

Free Vibrations of Two Elastically Connected Reddy-Bickford

2.3 Free Vibratio ns of Two Elastical ly Co nnected Reddy-Bic kford Beams

In the Timoshenko model, we consider beams of equal length connected by an elastic layer with stiffness modulus K These beams experience axial compression forces at their ends, as illustrated in Figure 2.1.1 We denote the functions of longitudinal and transverse motion, along with the rotation angle of the beam's cross-section on a neutral line, as ( , , ), ( , , ), ( , ), ( , , ), ( , , ), and ( , ) respectively Correspondingly, the equations of motion for the double-beam system are established in a manner analogous to relation (1.2.29).

Deformation in the function of motion and the stress-strain ratio according to the Hooke’s law are

0 (2.3.7) Virtual work of inertial forces is given by

= − ( , )δ ( , ) + ( , )δ ( , ) (2.3.11) irtual work of external forces is expressed as

By substituting the equations (2.3.8-2.3.11) into a general equation based on the virtual work principle + + = 0, = 1,2 , we get

By successive application of Green’s theorem on the expressions (2.3.14-2.3.15), we get the equations of vibration for the system composed of elastically connected Reddy-Bickford beams in the following form

In the expressions (2.3.20-2.3.21), ρ denotes mass density, A the area of the cross-section, Young’s modulus, and is the shear modulus

Reddy-Bickford’s beam theory, a sixth-order theory, differs from the fourth-order Euler and Timoshenko beam theories by incorporating six boundary conditions, with three conditions applied at each end of the beam By applying variations to the constitutive relations and considering the virtual work of internal forces, we can derive significant results related to beam behavior.

(2.3.23) are the bending moment, shear force, resulting stress in normal direction and the resulting tangent stress respectively By substituting the relations (2.3.3-2.3.7) into (2.3.23), we get

For simply supported beams, the moments and deflections at both ends must be zero This leads to three essential conditions for each beam end: moment equals zero, deflection equals zero, and a third condition ensuring the moment remains zero Consequently, the boundary conditions align with those established for the Timoshenko beam, following the same fundamental criteria.

We define initial conditions as

Considering the harmonic motion of the beams’ points, we assume the solutions to the equations (2.3.16-2.3.19) as the product of functions

(2.3.28) where in (t) and ( ) are unknown time functions whereas Z n (z) and ( ) are mode shape functions

By substituting the assumed solutions (2.3.28) into equations (2.3.16-2.3.19), we get a system of four differential equations of the second degree as follows

We assume the solutions to the system of equations (2.3.30) in the following form

If we substitute the assumed solutions (2.3.28) into the system of equations (2.3.16 − 2.3.19), we get a homogeneous system of algebraic equations which has non-trivial solutions when system determinant equals zero

The frequency system equation (2.3.32) is derived from the condition for the existence of non-trivial solutions, where the determinant equals zero, resulting in an eighth-degree polynomial By introducing a substitution, the polynomial can be simplified to a more manageable form.

Constants ̃ , ̃ , ̃ and ̃ are given in the Appendix 2.3.1 The fourth degree polynomial can be factored as

In the expressions presents one of the roots of a third degree equation

After factoring the equation (2.3.33), the solutions may be written as the square of natural frequencies

4 − (2.3.36) The three roots of the equation (2.3.35) are

By choosing one of the three roots in the expression (2.3.37), we get unique solutions to the equation (2.3.33) where four natural frequencies of the system are real positive roots = , = , = , =

Amplitude ratios are determined from the equation (2.3.31) for each frequency, Appendix 2.3.2 The general solution to differential equations (2.3.30) is given as

The solutions to the partial differential equations governing free transverse vibrations in a system of two elastically connected simply supported beams, which account for the effects of rotary inertia and transverse shear, are expressed in the form ( )( ) = sin + cos, where the constants , , and (i = 1, 2; g = 3, 4;) remain unknown.

Unknown constants , , and (i = 1, 2; g=3,4) are determined based on initial conditions (2.3.27) using the orthogonality condition for trigonometric functions

= = [sin( )] d = 2 , = = [cos( )] d = 2 (2.3.50) where is the Kronecker delta function If we substitute initial conditions (2.3.27) and include them in the expressions (2.3.46 − 2.3.49), we get

By multiplying the expression (2.3.51) by appropriate eigenfunctions , i.e then integrating it with respect to z from 0 to l and using the orthogonality condition (2.3.50), we have

After solving the system of equations (2.3.52-2.3.55), we get the unknown constants in the following form

Critical Buckling Force of the Two Elastically Connected Beams

2.4 Critical Buckling Force of the Two Elastically Connected Beams with Numerical Analysis

2.4 Critical Buc kli ng Force of the Tw o El asticall y Co nnected Beams

The static stability conditions for a system comprising two elastically connected beams, specifically of Timoshenko or Reddy-Bickford types, will be established under the assumption that the beams share identical material properties and transverse shear characteristics.

As compression forces increase, the natural frequency of vibration decreases, reaching zero when the forces equal a critical value, resulting in the beam system achieving a state of indifferent equilibrium Introducing a new variable, ζ, which represents the ratio of axial compression force on the second beam to that on the first beam (where 0 ≤ ζ ≤ 1), allows us to solve the frequency equations under specific conditions The solutions derived indicate that the lower value corresponds to the critical buckling force of the system, as referenced in [12].

The critical buckling force for a system of two elastically connected Reddy-Bickford beams is detailed in Appendix 2.4.1 This analysis focuses on the impact of rotary inertia and transverse shear concerning the system parameters outlined in references [2-5].

The natural frequencies, considering the impact of rotary inertia and transverse shear in a system of two elastically connected beams, are presented in Tables 2.4.1 to 2.4.3 The numerical experiment utilized two sets of values for shear modulus and shear factor, as referenced in Kaneko [24].

= ( ) , = , which best corresponds to experimental results, and based on the beams’ material properties for the example used in reference [12] = 0.417 ∗

The differences in approximate solutions are illustrated through models presented in Tables 2.4.1 to 2.4.3 and Figures 2.4.1 to 2.4.7 The values for the shear factor, shear modulus, and stiffness modulus of the Winkler layer were sourced from Table 2.4.1.

Table 2.4.1 The effects of rotary inertia and transverse shear on natural frequencies ω [ ] of the system composed of two elastically connected beams

Table 2.4.2 Differences in approximate solutions for the system of beams with the thickness of h=5h 1 on natural frequencies of the system ω [ ]

Table 2.4.1 reveals minor differences in the approximations of natural frequency solutions across varying shear factor values in the Timoshenko and Reddy-Bickford models However, as shown in Table 2.4.2, thicker beams exhibit a more pronounced discrepancy, with the Euler and Rayleigh models providing unsatisfactory approximations.

Table 2.4.3 Differences in approximate solutions of the second natural frequency of the system ω [ ] for the beam system joined by a Winkler layer with different stiffness modulus K

The thickness of the beam significantly impacts the natural frequencies of the system, particularly for the first mode This relationship highlights the importance of beam thickness in determining the vibrational characteristics of the structure.

The thickness factor of the beam significantly impacts the natural frequencies (ω) of the system, particularly evident in the third mode This relationship highlights the importance of beam thickness in determining the vibrational characteristics of the structure.

The impact of beam thickness on the natural frequencies ω of the system is significant, particularly in the fifth mode Analyzing how variations in beam thickness affect these frequencies provides crucial insights into the system's dynamic behavior.

The analyzed models provide similar approximate solutions for lower modes, but discrepancies increase notably in thicker beams Figures 2.4.1-2.4.3 illustrate the variations in natural frequencies based on the beam thickness factor (ℎ/) It is evident that when this factor exceeds 0.2, a significant divergence arises between the results from the Euler and Rayleigh models, while the Timoshenko and Reddy-Bickford models yield considerably better approximations.

Fig 2.4.4 Transverse motion of beam’s centre lines for the sum of the first seven members of the order with the given initial conditions = 10 , = 0.1 , = 0.2 , 1 ⁄ , = 2 ⁄

The set of Figures 2.4.1-2.4.3 show changes in the frequency up to the value

In cases where the dimensions of elastic bodies are nearly equal, it is possible to analyze their frequency behavior, particularly in multi-span beams where the distances between supports are minimal compared to the beam's thickness, or in the vibration of hulls under short wavelengths Figures 2.4.4-2.4.6 illustrate the transverse motion of the beams' center lines across various models and time intervals, focusing on the first seven members of the order under specified initial conditions The findings indicate that different model types exhibit varying transverse motions of the beam's center lines, with these differences becoming more pronounced as the number of members included in the analysis increases.

Fig 2.4.8 The influence of axial compression forces ζ on the critical force ratio ⁄ in modes 1, 2 and 3 (ℎ = ℎ , Timoshenko model)

Fig 2.4.9 The influence of axial compression forces ζ on the critical force ratio ⁄ in modes 1, 2 and 3 (ℎ = ℎ , Reddy-Bickford model)

Fig 2.4.10 The influence of axial compression forces ζ on the critical force ratio ⁄ in modes 1, 2 and 3 (ℎ = 3ℎ ,Timoshenko model)

Fig 2.4.11 The influence of axial compression forces ζ on the critical force ratio ⁄ in modes 1, 2 and 3 (ℎ = 3ℎ , Reddy-Bickford model)

Fig 2.4.12 The influence of axial compression forces ζ on the critical force ratio ⁄ for different beam thickness (Timoshenko model)

Fig 2.4.13 The influence of axial compression forces ζ on the critical force ratio ⁄ for different beam thickness (Reddy-Bickford model)

Figures 2.4.8-2.4.13 illustrate the critical force ratio in relation to the axial compression force on the second beam ζ The analysis reveals a slight difference in the critical force ratio between the Timoshenko and Reddy-Bickford models, indicating that both theories are equally effective regardless of beam thickness or the mode of critical force consideration It can be concluded that only the Timoshenko and Reddy-Bickford theories are applicable to thicker beams, with the Reddy-Bickford model having the advantage of determining shear stress without requiring a shear factor As the vibration mode increases, the Reddy-Bickford beam theory provides the most accurate solutions The differences in approximate solutions among the theories can guide the selection of the appropriate mathematical model for the dynamic analysis of two elastically connected beams.

Effects of Axial Compression Forces, Rotary Inertia and Shear on Forced Vibrations of the

System of Two Elastically Connected Beams

Effects of Axial Co mpress ion Forces, Rotary Inertia a nd Shear on F orced Vibrati ons

Forced Vibrations of Two Elastically Connected Rayleigh

3.1 Forced Vibrations of Two Elastically Connected Rayleigh Beams

This article examines the impact of axial compression forces and rotary inertia on the forced transverse vibrations of a system made up of two elastically connected Rayleigh beams, as studied by Stojanović and Kozić The beams, which are of equal length \( l \), are linked by an elastic layer characterized by a stiffness modulus \( K \) They experience forced loads and axial compression forces at their ends, as illustrated in Figure 3.1.1.

Let the functions of longitudinal and transverse motion of the system composed of Rayleigh beams ( , , ), ( , , ), ( , , ) and ( , , ) be

Fig 3.1.1 The system of two elastically connected beams under the influence of forced loads

Deformation in direction in the function of motion and the relation between stress and deformation according to Hooke’s law are

(3.1.3) ( , , ) = ( , , ), ( , , ) = ( , , ), (3.1.4) Virtual work of inertial forces is given by

= − ( , , ) δ ( , , ) + ( , , ) δ ( , , ) (3.1.6) Virtual work of internal forces is expressed as

Based on the principle of virtual work + + = 0, = 1,2 and the equations (3.1.5-3.1.9), we have

By successive application of Green’s theorem on expressions (3.1.10-3.1.11) we get the equations of transverse vibration for the system of elastically connected Rayleigh beams in the form

The equation ( , ) − ( , ) = 2 ( , ) (3.1.13) defines the relationship between mass density (ρ), cross-sectional area (A), Young’s modulus, and the moment of inertia of the beam's cross-section concerning the x-axis The initial and boundary conditions for the system of two elastically connected simply supported Rayleigh beams are established to analyze their behavior under specific constraints.

Considering the harmonic motion of beams’ points, the solutions to the equations (3.1.12) and (3.1.13) are assumed in the following form

In the Rayleigh model, the unknown time functions, denoted as ( ) with frequencies ( = 1,2), are expressed in equations (3.1.16) and (3.1.17) By substituting these assumed solutions into the non-homogeneous system of partial differential equations presented in (3.1.12) and (3.1.13), we derive significant results that further our understanding of the model's dynamics.

Multiplying the equations (3.1.18) and (3.1.19) by eigenfunction , then integrating them with respect to z from 0 to and using the orthogonality condition, we have

Using the amplitude ratios determined in the previous chapter (2.1.26) and the equations (3.1.20) and (3.1.21), we get

From the equations (3.1.16), (3.1.17), (3.1.25) and (3.1.26) and the solution to the equation (3.1.24), ref [5], the analytical forms of forced vibration of two elastically connected Rayleigh beams follow

General solutions for forced vibration (3.1.27) and (3.1.28) may be applied in analyzing the system under the influence of different types of forced loads.

Forced Vibrations of Two Elastically Connected Timoshenko

Let us analyze the effects of axial compression forces, rotary inertia and transverse shear on forced transverse vibration of the system of two elastically connected

Timoshenko beams exhibit both longitudinal and transverse motion in their points, along with a defined angle of rotation for the beam's cross-section The functions representing these motions are denoted as ( , , ), ( , , ), ( , ), ( , , ), ( , , ), and ( , ) The equations governing the motion of Timoshenko beams are established accordingly.

Deformation in the function of motion and the relation between stress and deformation according to the Hooke’s law are

0 (3.2.5) Virtual work of inertial forces is given by

, (3.2.11) where δ represents variation operator Substituting the equations (3.2.6-3.2.11) into a general equation of the virtual work principle + + = 0,

By successive application of Green’s theorem on the expressions (3.2.12-3.2.13), we get the equations of forced vibration for the system of elastically connected Timoshenko beams as

The equation (3.2.17) indicates that the material and geometrical properties of the beams retain the same labels as previously discussed The initial conditions for the system of two elastically connected beams will be defined using the main coordinates Additionally, the boundary conditions for this system are specified as follows.

Considering the harmonic motion of beam’s points, the solutions to the equations (3.2.14-3.2.17) are assumed as the product of the following functions

(3.2.21) where ( ) and ( ) ( ) are unknown time functions, while Z n (z) and ( ) are mode shape functions

By substituting the assumed solutions into the transverse vibration equations of the system and applying eigenfunctions, we integrate from 0 to l along the beam Utilizing the orthogonality condition, we derive the results from this process.

The solutions to non-homogeneous systems of differential equations can be effectively determined through modal analysis, as outlined by Kelly [25] The equations can be conveniently expressed in matrix form, facilitating a clearer understanding of the underlying mathematical relationships.

The squares of the system's natural frequencies correspond to the eigenvalues in the matrix A non-homogeneous system of differential equations with generalized coordinates can be transformed into a system of main coordinates by utilizing modal matrices, where the columns represent normalized mode shapes.

Transformation (3.2.25) is equivalent to linear transformation between generalized and main coordinates of the system

By substituting the equation (3.2.26) into the system (3.2.24), we get

If we multiply both sides of the expression by a scalar { }for arbitrary r=1,2,3,4, we get:

By utilizing the mode-shape orthogonality condition, we identify a unique non-zero member in each sum when r equals j Given that the mode shapes are normalized, equation (3.2.28) can be expressed in a simplified form.

( ) + ( ) = ( ), = 1,2,3,4., (3.2.29) where ( ) = , { } If the initial conditions are defined by main coordinates (0) = 0, (0) = 0, the solutions may be written in the form of convolution integral

By substituting the obtained solutions (3.2.30) into the equation of coordinate transformation (3.2.25), we get

The time functions thus obtained may now be substituted into assumed solutions (3.2.21) from where general solutions for forced transverse vibration of two elastically connected Timoshenko beams follow

General solutions for forced vibration are essential for analyzing resonance conditions in dynamic vibration absorption and assessing system amplitudes under varying axial compression forces The system's modal matrix is detailed in Appendix 3.2.1.

Forced Vibrations of Two Elastically Connected Reddy-Bickford

In the context of the Timoshenko model, we consider two beams of equal length \( l \) connected by an elastic layer characterized by a stiffness modulus \( K \) These beams are subjected to axial forces and variable continuous loads at their ends We denote the functions representing the longitudinal and transverse motions, as well as the angle of the tangent of the deformed beam's cross-section along the neutral line, as \( (u, v, \theta) \), \( (x, y, z) \), \( (t) \), \( (s, t, u) \), \( (p, q, r) \), and \( (k) \) The equations governing the motion of this double-beam system are established accordingly.

Deformation in the motion function and the relations between stress and deformation according to the Hooke’s law are

Virtual work of inertial forces is given by

, (3.3.13) where δ is the variation operator By substituting the equations (3.3.10-3.3.13) into a general equation of virtual work principle + + = 0, 1,2, we get

By successive application of Green’s theorem to the expressions (3.2.14-3.2.15), we get the equations of vibration for the system of two elastically connected Reddy-Bickford beams in the following form

Boundary conditions for the system of two elastically connected simply supported Reddy-Bickford beams based on (2.3.22 − 2.3.24) are given by

Considering the harmonic motion of beam’s points, we assume the solutions to the equations (3.3.16-3.3.19) as the product of functions

(3.3.25) where ( ) and ( ) ( ) are the unknown time functions, whereas Z n (z) and ( ) are mode shape functions

By substituting the proposed solutions into the equations governing transverse system vibrations and applying eigenfunctions, we integrate from 0 to l along the beam Utilizing the orthogonality condition, we derive the results effectively.

Matrixes and are given in the Appendix 3.3.1 The squares of system’s natural frequencies , , and represent eigenvalues of the matrix

A non-homogeneous system of differential equations can be transformed into a system of primary coordinates by utilizing modal matrices, where the columns represent normalized mode shapes This approach facilitates the analysis of the system's dynamics through the introduction of generalized coordinates.

Transformation (3.3.28) is equivalent to linear transformation between generalized and main system coordinates

By substituting the equations (3.3.28) into the system (3.3.27), we get

If both sides of the expression are multiplied by a scalar { } for arbitrary r=1,2,3,4, we get

By applying the mode-shape orthogonality condition we get a single member in each sum that is different from zero, namely for r=j As mode shapes are normalized, the equation (3.3.31) becomes

( ) + ( ) = ( ), = 1,2,3,4., (3.3.32) where ( ) = , { } If initial conditions are defined by main coordinates

(0) = 0, (0) = 0, the solutions can be written in the form of convolution integral

By substituting the obtained solutions (3.3.33) into the equation of coordinate transformation (3.3.29), we get

The time functions thus obtained can now be substituted into assumed solutions (3.3.25) from where general solutions for forced transverse vibration of two elastically connected Timoshenko beams follow

General solutions for forced vibration are essential for analyzing the system's behavior as a dynamic absorber and assessing its amplitude responses to axial compression forces and various types of external forcing The modal matrix of the system is presented in analytical form in Appendix 3.3.1.

Particular Solutions for Special Cases of Forced Vibrations

Particular Solutions for Forced Vibration of the System

By substituting the equations (3.4.1), (3.4.2) and (3.4.3) into general solutions (3.1.25-3.1.26), we get

Substituting the equations (3.4.4) and (3.4.5) into general solutions to the equations (3.1.27) and (3.1.28), yields

Depending on the type of forcing in the general solutions, only the parameter for which the following applies, changes case 1) = 2 ( ) sin( ) , = 1,2,3, case 2) = 4

Hence, the forced vibration of the mechanical system is of the following form

( , ) = sin( ) sin( ) , = 1,2,3, , (3.4.9) where , represent the amplitudes of the normal vibration mode The conditions of resonance and dynamic vibration absorption are as follows

Particular Solutions for Forced Vibration for the System

We will identify specific solutions for the forced vibrations of the coupled Timoshenko beam system influenced by various excitation types By substituting equations (3.4.1), (3.4.2), and (3.4.3) into the general forced vibration equations (3.2.32) and (3.2.35), we derive the necessary results.

Particular solutions for forced vibration of mechanical system are given as

= 1,3,5, , (3.4.17) where represents the amplitudes of the normal vibration mode The conditions of resonance and dynamic vibration absorption are as follows

Forced frequency at which the system acts as a dynamic absorber is obtained from the condition that = 0 After factorizing the equation +

+ + = 0, we get a bicubic polynomial equation for unknown forced frequencies

+ + + from where three solutions for forcing frequencies for which the amplitude of the first beam equals zero, follow

3 cos θ 3 cos θ+ 2π 3 cos θ+ 4π 3 , (3.4.19) on condition that = + < 0 and where

Particular Solutions for Forced Vibration for the System

of Two Elastically Connected Reddy-Bickford Beams

The article explores specific solutions for forced vibrations in coupled Reddy-Bickford beams subjected to various types of external forces This is achieved by substituting relevant equations into the general forced vibration equations, resulting in a comprehensive analysis of the system's behavior under different forcing conditions.

Particular solutions for forced vibration of mechanical system are expressed as

= 1,2,3, , (3.4.27) where are the amplitudes of the normal vibration mode The conditions of resonance and dynamic vibration absorption are

The forcing frequency at which the system functions as a dynamic absorber is derived similarly to the Timoshenko beam system, leading to three distinct solutions for the forcing frequencies.

Numerical Analysis

Considering the effects of rotary inertia and transverse shear is essential for understanding the variations in amplitude experienced by thicker beams under axial forces These factors significantly impact the accuracy of solution approximations, particularly when the cross-sections of beams rotate due to shear forces A numerical experiment was conducted using a two-beam model with identical material and geometric properties to analyze these effects.

The axial force factor on the first beam is introduced as a critical value, while the axial force factor on the second beam is dependent on the axial force of the first beam and the amplitude ratios, both with and without the influence of axial forces.

= , 0 ≤ ≤ 1, =ζ , 0 ≤ζ≤ 1, , , , , ref [51] Euler, Rayleigh: = , ref [52] Timoshenko: 1 + , ref [31] Reddy − Bickford: = π 840 + π

In the case of uniformly continuous harmonic excitation, the amplitudes of normal forced vibrations in a beam system can be determined using established analytical expressions based on system parameters Figures 3.5.1 to 3.5.16 illustrate amplitude ratios for an excitation characterized by = 0.6 and = 1 The results were compared to those for forced vibrations in elastically connected Euler beams, as referenced in Zhang et al [5] Notably, the amplitude ratio for the second beam consistently exceeds that of the first Additionally, when comparing results across the same vibration mode for varying levels of axial compression (ζ= 0.1 and ζ= 0.9), it can be concluded that the axial force has a minimal impact on the difference in vibration amplitude ratios between the two beams.

In the analysis of doubly thicker beams, it is observed that as the vibration mode increases, the amplitude ratio differences between various models escalate more rapidly, with the Timoshenko and Reddy-Bickford models yielding significantly better approximations This suggests that the analytically determined results presented in this chapter are suitable for the two elastically connected thicker beam models Figures 3.5.17-3.5.18 illustrate the amplitude-frequency characteristics of the vibration in these beams, highlighting the points of resonance and the system's behavior as a dynamic absorber The variations in the solutions underscore the necessity of employing theories that incorporate rotary inertia and transverse shear effects, which become more pronounced with higher forced vibration modes.

Fig 3.5.18 h=h 1, n=5 ; ( = 5, = 0.9) Fig 3.5.16 ; ( = 5, = 0.9) © Springer International Publishing Switzerland 2015

Static and Stochastic Stability of an Elastically Connected Beam System on an Elastic

Chapter 4 examines the static and stochastic stability of three elastically connected beams on an elastic foundation A new set of partial differential equations is derived for analyzing deflections and critical buckling forces in this complex mechanical system The chapter presents a comparative study of static stability across systems with one, two, and three beams on an elastic foundation, analytically determining the critical buckling force for each configuration The findings conclude that the system exhibits the highest stability when configured as a single beam on the elastic foundation.

This article investigates the stochastic stability of three elastically connected beams on an elastic foundation, modeled as a system of three coupled oscillators Stochastic stability is characterized by the Lyapunov exponent and moment Lyapunov exponents A new transformation set is established to derive Ito differential equations applicable to any system of three coupled oscillators Utilizing the method of regular perturbation, asymptotic expressions for these exponents are determined in the presence of small intensity noises Analytical results are provided for both almost sure and moment stability of the stochastic dynamical system, with applications to assess the moment stability of complex structures influenced by white noise excitation from axial compressive stochastic loads.

Critical Buckling Force of Three Elastically Connected Timoshenko

4.1 Critical Buc kl ing Force of Three Ela stically C onnected Ti mo she nko Beams

This article examines a system of three elastically connected Timoshenko beams of equal length (l) situated on a Winkler-type foundation characterized by a stiffness modulus (K) and subjected to identical axial loads (F) The motion of the system is analyzed through the derivation of partial differential equations, which are established using the principle of virtual work, considering beams that possess identical geometric and material properties.

Fig 4.1.1 a) Three-beam system on an elastic foundation b) An elementary deformed beam part

This article discusses the functions of longitudinal and transverse motions, as well as the angle between a beam's cross-section and its axis Specifically, it defines various parameters related to these motions and angles, including ( , , ), ( , ), ( ), ( , ), ( , ), ( ), ( , ), ( , ) and ( ) The motion of beam points is expressed in relation to a clockwise angle of the cross-section, referencing sources [14, 26].

Deformation in the function of motion and the relation between stress and deformation according to the Hooke’s law are

0 , = 1,2,3 (4.1.3) Virtual work of internal and external forces is given by

By integrating equations (4.1.4-4.1.7) into the general equation of the virtual work principle, excluding inertial forces, we derive a system of equations represented as + = 0 for states 1, 2, and 3 Utilizing Green's theorem, we can then formulate the partial differential equations that describe the point motion of a beam within a static region.

By eliminating from the equations (4.1.8 - 4.1.10), we get the system of three partial differential equations of the fourth degree

As the beams are simply supported, we apply the boundary conditions for which the following applies

We can assume the deflection of the beam as the order

(4.1.15) where the function Z n (z) is given by

By substituting the assumed solutions (4.1.15) into equations (4.1.11-4.1.13), we obtain a homogeneous system of algebraic equations that can be written in the matrix form

The existence of non-trivial solutions requires that the matrix determinant in the equation (4.1.17) equals zero, which yields a characteristic third degree equation expressed as

The solutions of the third degree polynomial equation (4.1.19) are

If we substitute possible solutions (4.1.20) into the expression (4.1.18), we will obtain three values for force , the lowest of which represents the critical buckling force and has the following form

1 + π (4.1.21) For = 0 in the equation (4.1.21), we have

1 + π , (4.1.22) which represents the critical buckling force of the n-th mode for the Timoshenko beam [27] For = 1 , the equation (4.1.22) is reduced to

1 + π (4.1.23) which is the lowest value of the force affecting the beam in the stability region

We will verify the solutions to the equation (4.1.19) using the trigonometric method [28] Let us assume the solutions to the system of homogeneous algebraic equations (4.1.17) are non-trivial

By substituting the assumed solutions (4.1.24) into the equation system (4.1.17), we get

By solving the equation (4.1.25) for variable x, we have

By substituting the assumed solution (4.1.24) and x in the expression (4.1.26) into the first equation of the system (4.1.17), we get the following identity

( + 2 )⋅ − ⋅ ≡ 0, ⇒ (2 ϕ − 2 + 2 )⋅ ϕ − 2ϕ = 0, (4.1.27) whereas from the third equation of the system (4.1.17), we get

− ⋅ + ( + )⋅ = 0 ⇒ − ⋅ 2ϕ + (2 ϕ − )⋅ 3ϕ = 0 (4.1.28) Expansion and transformation of the equation (4.1.28) yields

The solutions to the trigonometric equation (4.1.29) are ϕ

The equation (4.1.17) may be rewritten in the following form for different values of ϕ

Buckling forces in the equation (4.1.32) are determined as

1 + π , = 1,2,3,… , (4.1.32) from where by substituting ϕ = π for = 1,2,3 into the expression (4.1.33) for respective buckling forces follow

Critical Buckling Force of Two Elastically Connected Timoshenko

4.2 Critical Buckling Force of Two Elastically Connected

Timoshenko Beams on an Elastic Foundation

Let us analyze the case of two elastically connected Timoshenko beams on the elastic foundation Both beams are under the influence of equal axial compression forces, reference [14]

Fig 4.2.1 Two-beam system on an elastic Winkler layer

The equation (4.1.17) in the matrix form is now reduced to the following

By expanding the determinant (4.2.1), we get the equation

The lower value represents the critical buckling force for the n-th mode of system vibration and is given by

Critical Buckling Force of Timoshenko Beams on an Elastic

In case of a single beam on an elastic surface, the equation (4.1.17) in the matrix form is reduced to the following

Fig 4.3.1 The beam on an elastic Winkler-type foundation

By solving the equation (4.3.1) for the n-th vibration mode, we obtain the expression for critical buckling load in the form

Numerical Analysis

The values of the physical parameters for the system used in the numerical experiment

Introducing the non-dimensional parameter ξ, defined as the ratio of cross-section thickness to beam length (ξ = h/L), allows for the expression of both the cross-sectional area and the moment of inertia in a simplified form.

Figure 4.4.1 demonstrates that the relative ratio of critical forces decreases as beam thickness increases, while the critical force ratio also diminishes with the addition of beams at lower Winkler layer stiffness This indicates that a single beam on an elastic foundation offers the highest stability, with stability declining as the number of beams increases Similar trends in the relative critical force ratio for various modes are illustrated in Figure 4.4.2 Additionally, Figure 4.4.3 presents the changes in the system's static stability concerning beam number, Winkler layer stiffness, and beam thickness.

In Figure 4.4.4, the marked points illustrate the physical parameters at which the system exhibits identical critical force values across different modes Notably, point A12 indicates the bifurcation point for a single-beam system, representing the beam thickness-to-length ratio where the first and second modes have the same critical force As the beam thickness increases from A12, the system's stability in the second mode enhances Similarly, point A13 denotes the ratio where the first and third modes share the same critical force This pattern is also observed in double-beam (points B12, B13) and three-beam systems (points C12, C13) Additionally, as the number of beams increases, the system's sensitivity to elastic layer stiffness grows, necessitating thinner beams to achieve the same critical force across different modes.

Fig 4.4.1 The influence of beam thickness on critical force ratio / ( =0.5 , , 1.5 )

Fig 4.4.2 The influence of thickness and the number of beams on critical force ratio / (n=1,2,3)

Fig 4.4.3 The influence of thickness and the number of beams on critical force (

Fig 4.4.4 The influence of thickness and the number of beams on critical force (n=1,2,3)

Stochastic Stability of Three Elastically Connected Beams on an

4.5 Stochastic Stability of Three Elastically Connected Beams on an Elastic Foundation

4.5 Stochastic Stability of Three Elastical ly Co nnected Beams

The investigation of transverse vibration instability in complex systems on elastic foundations, subjected to stochastic compressive axial loading, reveals significant physical challenges in engineering This study focuses on a system comprising three beams experiencing stochastic excitation, while considering negligible rotary inertia and shear deformation The motion of these beams is governed by specific partial differential equations, based on the assumption that the plane cross-sections remain unchanged during flexure, and the radius of curvature exceeds the beam’s depth This theory is applicable when the depth-to-length ratio of the beams is minimal The general equations for the transverse vibrations of elastically connected beams can be derived, as illustrated in the accompanying figure.

The equation (4.5.3) describes the relationship between transverse beam deflections, denoted as positive when downward, and incorporates parameters such as the second moments of inertia, cross-sectional areas, Young’s modulus, and mass density of the beams It also defines damping coefficients per unit axial length and the stiffness modulus of a Winkler elastic layer, while considering stochastically varying static loads The boundary conditions for simply supported beams of equal length are established at (0, ) and ( , ) = (0, )⁄ = ( , )⁄ = 0 for n = 1,2,3 Utilizing the Galerkin method, only fundamental modes are addressed, with boundary conditions met by substituting ( , ) = ( ) sin( ⁄ ) into the equations of motion, allowing for the expression of unknown time functions.

= , = , = , = , = , = , and assume that the compressive axial forces are stochastic white-noise processes with small intensity ( ) = ( ) = ( ) = √ ( ) we have oscillatory system in form d d + + 2d d −d d + (2 − ) − √ ( ) = 0, d d + + 2d d −d d −d d + (2 − − ) − √ ( ) = 0, d d + + d d −d d + ( − ) − √ ( ) = 0 (4.5.7)

The system is characterized by unknown generalized coordinates that are functions of time, along with natural frequencies and viscous damping coefficients The stochastic term √( ) represents a white-noise process with low intensity To assess the dynamic stability of an oscillatory system, it is essential to determine the maximal Lyapunov exponent and the moment Lyapunov exponent, which are key indicators of stability.

The solution process of a linear dynamical system is influenced by the maximal Lyapunov exponent, which determines the almost-sure stability of the system A negative maximal Lyapunov exponent indicates almost-sure stability, while a non-negative value suggests instability The exponential growth rate of the solution in a randomly perturbed dynamical system is characterized by the moment Lyapunov exponent.

In the context of stochastic systems, moment stability is defined by the condition where the moment Lyapunov exponents approach zero as time approaches zero, particularly when ( ) < 0 Despite their significance in analyzing dynamic stability, evaluating moment Lyapunov exponents presents considerable challenges Furthermore, the equilibrium state characterized by = = 0 in Eq (32) highlights the complexities associated with almost-sure and moment stability assessments Utilizing the transformation = , researchers can further explore these stability concepts.

= , = , = , = , = the equation (1) can be represented in the first-order form by Stratonovich differential equations

⁄ ( ) is the white-noise process with zero mean and autocorrelation function

( , ) = [ ( ) ( )] = ( − ), (4.5.9) where is the intensity of the random process ( ), () is the Dirac delta function and [ ] denotes expectation Using corresponding transformation

= cos cos cos , = − cos cos sin , = cos sin cos ,

= − cos sin sin , = sin cos , = − sin sin ,

Ito's rule provides a framework for deriving equations related to the th power of the norm of response and phase variables In this context, a trigonometric transformation is utilized to express the norm of the response, with the angles of the three oscillators represented as , while and illustrate the coupling or energy exchange among the oscillators.

= ( , , , , ) + ( , , , , ) ( ) (4.5.10) where ( ) is the standard Weiner process and

− 8 (cos3 sin + 2 sin 2 + cos sin 3 ), (4.5.11)

( , , , , ) = − sin 2 − cos cot sin − cos sin tan sin

+ ℎ 1 + cos 2 − cos cos cot −tan sin cos sin

− 8 (cos3 sin + 2 sin 2 + cos sin 3 ), (4.5.13)

− 8 3 sin 2 + tan (1 + cos 2 ) − 4 sin cot sin cos + 1 cos − tan

2 1 + cos 2 − cos cos cot sin −cos cos cot sin +cos cot sin sin tan )

− 8 (cos 3 sin + 2 sin 2 + cos sin 3 ), (4.5.15)

4 [cos sin sin 2 sin 2 − sin 2 (cos 2 sin 2 + sin 2 )] +4 sin 2 (1 − cos 2 ) (1 + cos 2 ) − sin sin sin 2

4 [sin 2 (cos 2 sin 2 − sin 2 + sin 2 cos sin ) + 2 sin cos sin (1 − cos 2 )] +4 sin 2 (1 − cos 2 ) (1 − cos 2 ) −2

4 [sin 2 sin 2 − 2 cos sin sin (1 + cos 2 )] +4 sin 2 (cos 2 − 1) +2 sin sin sin 2 (1 + cos 2 )

+ 64 {2 cos [sin 4 sin 2 cos − 8 cos sin 2 (cos − cos 2 sin )] +2 sin [sin 4 sin 2 sin −8 cos sin 2 (cos + cos 2 sin )] +16 sin 2 cos (cos 2 + cos 2 sin ) + sin 2 sin 2 sin 2 (4 sin 2 + sin 4 )

−4 cos sin 4 sin 2 sin 2 − 4 sin sin 4 sin 2 sin2 } ,

8 sin 2 [ sin 2 (1 + cos 2 ) + sin 2 (1 − cos 2 ) − 2 sin 2 sin2 ], (4.5.18)

( , , , , ) = ℎ [sin sin (cos sin − 2 cos cos )] + [sin sin (2 cos sin − sin sin )]

−ℎ sin cos (cos cos − 2 cos sin + cos tan )] + sin cos sin

−sin sin 2 + cos sin tan )] −

16 {[4 cos sin 2 (cos 2 − cos 2 sin )]

−[4 cos × sin 2 (cos 2 + cos2 sin )] + [ sin 2 sin 2 sin4 ] } ,(4.5.19)

4 sin 2 [ sin 2 − sin2 ], (4.5.20) ( , , , , ) = {ℎ [cos cos sin (2 cos cos − cos sin )] + [cos cos sin × sin sin − 2 cos sin )]

2 [sin sin (− cos sin 2 − 2 cos (cos cos − 2 cos × sin ))] +

2 sin sin sin sin 2 + 2 cos cos sin − 2 sin sin ))]

+ 128 cos (2cos cos 2 (10 + 3 + 4(2 + ) − ( − 2)cos2 (3 + 4 cos 2 )) + ( − 2) sin 2

+ 128 cos (2 cos cos 2 (10 + 3 − 4(2 + ) + ( − 2) cos 2 ) × (4 cos 2 − 3)) + (p − 2) sin 2 (cos 4 + 8 cos 4 sin ))

+ 64 (2 cos (4 cos 2 − 3) cos 2 × ( − 2) + 3 + 2 − 4 cos 2 ) + ( − 2) cos 4 sin 2 +

2 ( sin 2 cos cos + sin2 cos sin + sin 2 sin ) (4.5.22) Applying a linear stochastic transformation

On April 5, 2023, a new norm process has been introduced utilizing the scalar function defined on the stationary phase process The Itô equation for this new norm process can be derived using Itô's Lemma.

When the transformation function is both bounded and non-singular, the stability behavior of the processes remains consistent Consequently, the transformation function is selected to ensure that the drift term in the Itô differential equation is independent of the phase process.

Transformation function ( , , , , ) is given by the following equation

[ + ] ( , , , , ) = ( ) ( , , , , ) (4.5.26) Where and are the differential operators in the forms

Moment Lyapunov Exponents

First Order Perturbation

Equation given from first-order perturbation is

= ( ) ( , , , , ) + ( ) ( , , , , ), (4.6.5) and Eq (4.6.5) has a periodic solution if and only if

The first-order perturbation equation can be rewritten as

It is important to take into consideration commensurable frequencies where exists a relation of the form of = = where , and are integers and expressions for second and third frequency are = and

= respectively The function ( , , , , ) can be written in the form

+ ( , , ) cos 4 + ( , , ) cos 4 + ( , , ) sin 2 + ( , , ) cos 2 cos 2 + ( , , ) cos 2 sin 2 + ( , , ) cos 2 cos 4 + ( , , ) cos 4 cos 2

Functions ( , , ) are periodic in , and and given as

( , , ) = { }, = 0,1, … ,11, (4.6.10) Where is the vector in the form

The combination of coefficients and suggests that function ( , , , , ) can be written as

+ ( , , ) cos 2 cos 2 + ( , , ) cos 2 sin 2 + ( , , ) cos 2 cos 4+ ( , , ) cos 4 cos 4 + ( , , ) sin 2 sin + ( , , ) cos 4 cos 2

Equating terms of the equal trigonometry function to give set of partial differential equations

(4.6.11) The functions ( , , ) can be written as

Here ( − , − ) are the same arbitrary functions of three variables for which we make assumptions as follows

As each function ( , , ), = 0,1, … ,11 must be positive and periodic, unknown constants , , , , can be determined using conditions

Second Order Perturbation

The second-order perturbation equation must satisfied the condition of periodic function ( , , , , ) in , and given as

( ) can be obtained symbolically After integrating, the solution has the following form

= ( ) + ( ) sin 4 + ( ) cos 4 + ( ) cos 3 sin + ( ) cos sin 3

The weak noise expansion of the moment Lyapunov exponent in the second- order perturbation for the stochastic system is determined in the form

The Lyapunov exponent can be obtained by using a property of the moment Lyapunov exponent

+ε + sin 4 + cos 4 + cos 3 sin + cos sin 3

Stochastic Stability Conditions

The values used for the parameters of the stochastic system in the calculations for determining Moment Lyapunov exponent and Lyapunov exponent are given as follows

(4.6.21) where we assume for the simplicity that

We analytically established the moment stability boundary for first-order perturbation across various values of the moment (1, 2, and 4), ensuring that the moment stability condition is met with ( ) < 0 By applying the findings from the first-order perturbation to the moment Lyapunov exponent, we derived the relationship ( ) = ( ) + ( ).

The stability boundary of an oscillatory system can be assessed through first-order perturbation analysis An oscillatory stochastic system is considered asymptotically stable if the Lyapunov exponent is less than zero This relationship is expressed through the equation ε + (ε), highlighting the conditions necessary for stability in such systems.

Using the natural frequencies of 19.739, 24.674, and 29.194 derived from the system parameters, we established the moment stability boundary through second-order perturbation equations.

We determined the almost-sure stability boundary in the second-order perturbation by the same procedure applied to =ε +ε + (ε ) and we have condition

Fig 4.6.1 Moment Lyapunov exponent ( ) for = √200 × 10 , = = = 0.01, thin lines - double beam [53], thick lines – three beam system on elastic foundation, dashed lines – second perturbation of the three beam system

Fig 4.6.2 Stability regions for almost-sure (a-s) and pth moment stability for = 0.002

The variation of the moment Lyapunov exponent for both double and three beam systems on an elastic foundation is illustrated in Fig 4.6.1 This study compares the numerical results of the double beam system with those of the three beam system, revealing that the stochastic stability of the three beam system is superior to that of the double beam system without a foundation Our analysis indicates that adding an additional beam and incorporating an elastic foundation significantly enhances the stochastic stability region, despite increasing the system's degrees of freedom The findings demonstrate that the current system exhibits greater stability, as evidenced by the negative values of the moment Lyapunov exponent across a broader range of parameters Negative values signify system stability, while Fig 4.6.2 presents the moment stability boundaries concerning the damping coefficients for the first perturbation.

We can conclude that the stability region is increasing with increasing the damping coefficient what was expected © Springer International Publishing Switzerland 2015

The Effects of Rotary Inertia and Transverse Shear on the Vibrations and Stability of the

The Effects of Rotary Inertia and Tra nsverse Shear on the Vi bration and Stability

Chapter 5 analyzed free vibration of the multiple elastically connected beams of Timoshenko’s type on an elastic foundation under the influence of axial forces Analytical solutions for the natural frequencies and the critical buckling forces are determined by the trigonometric method and verified numerically It is shown that the fundamental natural frequency in the first mode of the multiple beam system tends to the value of the natural frequency of the system with one beam resting on an elastic foundation with the tendency of increasing the number of connected beams with the same stiffness of the layers between.

Free Vibration of Elastically Connected Timoshenko Beams

The study examines a system of m Timoshenko beams of equal length l, which are elastically interconnected by Winkler-type layers characterized by a stiffness modulus These beams are subjected to axial compression forces with an intensity denoted as F, as referenced in source [15].

Fig 5.1.1 a) The system of elastically connected beams on an elastic foundation b) Elementary deformed beam part

The motion of beam points is defined by the functions of longitudinal and transverse motion, as well as the angle between the beam's cross-section and the axis, represented as ̃(, ,), ̃(, ), and ̃(,) for ̃ = 1, 2, …, n For a clockwise angle of the cross-section, as illustrated in Figure 5.1.1 b), we derive the motion equations ̃(, ,) = ̃(, ) - ̃(, ) and ̃(, ,) = ̃(, ) Additionally, the deformation in relation to motion and the correlation between stress and deformation, as per Hooke’s law, is expressed as ̃(, ,) = ̃(,).

0 ̃ ̃ ̃ , ̃ = 1,2, … , (5.1.3) Virtual work of inertial forces is given by ̃ = − ̃ ̃ ̃ ( , , ) δ ̃ ( , , ) + ̃ ( , , ) δ ̃ ( , , ) ̃ ̃ (5.1.4)

Virtual work of internal forces is ̃= − ̃ ̃ ( , )δ ̃( , ) + ̃( , )δ ̃( , ) (5.1.5)

Virtual work of external work is expressed as

By substituting equations (5.1.4-5.1.8) into the general equation of the virtual work principle, represented as ̃ + ̃ + ̃ = 0 for ̃ = 1, 2, …, and applying Green's theorem, we derive a coupled system of partial differential equations that describe vibration phenomena.

The system of coupled differential equations (5.1.9-5.1.14) is reduced to the following form after the elimination of the variable ψ

Initial and boundary conditions for simply supported beams are ̃ ( , 0) = ̃ ( ), ̃ ( , 0) = ̃ ( ), ψ ̃ ( , 0) = ψ ̃ ( ), ψ ̃ ( , 0) = ω ̃ ( ), (5.1.18) ̃(0, ) = ̃( , ) = ̃(0, ) = ̃( , ) = 0, ̃ = 1,2,3 ⋯ (5.1.19)

Assuming harmonic beam motion, the transverse beam motions can be represented as the product of time-dependent functions and mode shape functions Specifically, the equation ̃( , ) = ( ) ̃ ( ) illustrates this relationship, where ̃ represents unknown time functions and Z n (z) denotes the mode shape function that adheres to the boundary conditions for simply supported beams.

If we substitute the assumed transverse motions (5.1.20) into equations (5.1.15-5.1.17), we get

In this analysis, we propose solutions to the equations (5.1.22 - 5.1.24) in the form of ̃ = ̃, where ̃ represents the natural frequency of the system and takes values 1, 2, 3, and so on By substituting these solutions into the differential equations, we derive a homogeneous system of algebraic equations for the unknown variables, resulting in the expression ω − 1 + 1 + + 1 ∗ This formulation allows us to explore the underlying dynamics of the system effectively.

For determining analytical solutions, we assume the beams have identical material and geometric properties The homogeneous system of algebraic equations (5.1.26-5.1.28) can then be written as a matrix equation

The solutions to the homogeneous algebraic equations (5.1.26 - 5.1.28) are non-trivial only when the determinant of the matrix in equation (5.1.29) is zero Under this condition, the resulting solution is expressed as a polynomial.

4 -th degree, the solutions to which can be determined only numerically for specific data

For connected beam systems with over three beams, solutions can be derived using the trigonometric method previously discussed in Chapter 4 This method, as outlined by Raskovic [28], involves assuming the unknowns in the form of ̃= sin( ̃ϕ) for ̃ = 1, 2, 3, and so on.

From substituting (5.1.30) into ̃-th equation of the system (5.1.29), for ̃ = 1,2,3 ⋯ , it follows that

( − 2 cosϕ) sin( ̃ϕ) = 0, (5.1.32) the equation (5.1.32) is satisfied when ≠ 0 and sin( ̃ϕ) ≠ 0 as the unknowns ̃ ≠ 0, hence it must follow that

By substituting variables into equation (5.1.33), we derive a fourth-degree polynomial frequency equation for the unknown Additionally, replacing = sinϕ and = sin2ϕ in the first equation of system (5.1.29) yields the equation sinϕ − sin2ϕ = 0, which simplifies to ( − 2 cosϕ) sinϕ ≡ 0.

The unknown ϕ is determined from the last equation of the system (5.1.29) By substituting ( ) and , we get:

2 cos ϕ = 0 (5.1.36) The solutions to the equation (5.1.36) are ϕ

2 = 0 ∨ ϕ = 0 ⇒ ϕ = π, = 1,2,3, ⋯ , (5.1.37) as ≠ 0, ≠ 0 and ϕ ≠ 0 By substituting the solution (5.1.37) into (5.1.33), we get a frequency equation in the following form ω ,

The equation (5.1.39) is biquadratic so the squares of natural frequencies follow in the form ω , =1

The static stability of the system is influenced by the stiffness of elastic layers, as described by equation (5.1.39) When subjected to axial forces, the system reaches a state of indifferent equilibrium when its natural frequency, represented by ω, equals zero Applying this condition (ω = 0) allows for further analysis based on the aforementioned equation.

The minimum solution value for the unknown axial force F is identified as the critical force of a system comprising m elastically connected beams, specifically corresponding to the n-th mode, as referenced by Stojanović et al [15].

For = 0 from the equation (5.1.43) it follows that

1 + π (5.1.44) is the critical buckling load of a single Timoshenko beam corresponding to the n-th mode Its minimum value at which buckling takes place is obtained for n=1

Numerical Analysis in the Frequency Domain of the System of

of Elastically Connected Timoshenko Beams

5.2 Numerical A nal ysis in t he Freque ncy Do mai n of the Syste m

Numerical analysis compares the results obtained for systems composed of 3, 5, 7 and 9 elastically connected Timoshenko beams with identical material properties

Table 5.2.1 presents the natural frequencies of a system comprising various elastically connected Timoshenko beams, derived through trigonometric methods and numerical analysis The findings indicate that as the number of beams increases, the lowest natural frequency approaches that of a single beam without elastic layers.

Table 5.2.1 Natural frequencies [s ] of elastically connected Timoshenko beams for

= 1, = 0 i = 3, 5, 7 i 9 ref.[15] , Trigonometric method Numerical solution

The results presented in Table 5.2.1 indicate that the discrepancy between the numerical and analytical solutions derived from the trigonometric method is minimal We will now examine how the axial compression force and the stiffness of elastic layers (0.5, 1.0, and 1.5) affect the lowest natural frequency of a system consisting of one, three, or five elastically connected Timoshenko beams The analytical expressions for the frequencies, based on equation (5.1.40), are provided for the first mode with configurations of 1, 3, and 5 beams.

Fig 5.2.1 The influence of elastic layer stiffness on the lowest natural frequency a) = 1; b) = 3; c) = 5

An increase in the number of beams leads to a decrease in natural frequencies, as illustrated in Figure 5.2.1 In systems with a higher number of beams but consistent stiffness in the elastic layers, natural frequencies tend to converge, which is crucial for analyzing the geometric non-linear vibrations of elastically connected beams The analytical expressions derived for these systems provide a foundational understanding of how vibrations are affected by variations in geometric and material properties This approach eliminates the need for constant mathematical modeling, allowing for straightforward adjustments to physical parameters and the number of elastically connected beams to obtain results efficiently.

Fig 5.2.2 The influence of beam number on the lowest natural frequency in the function of axial compression force for K=K 0

Numerical Analysis in the Static Stability Region for the System

5.3 Numerical Analysis in the Static Stability Region for the System of Elastically Connected Timoshenko Beams

By substituting values of = 1, 3, and 5 into equation (5.1.42) for minimum values of = 1, we derive analytical expressions for the critical buckling force in a system composed of one, three, or five elastically connected Timoshenko beams.

Fig 5.3.1 The influence of beam number on critical buckling force in the function of stiffness of Winkler elastic layers

Figure 5.3.1 illustrates the static stability regions for systems comprising one, three, or five elastically connected Timoshenko beams The data indicates that the system demonstrates maximum stability with a single beam on an elastic foundation, while stability decreases as the number of elastically connected beams increases.

The Effects of Rotary Inertia and Transverse Shear on Vibrations and Stability of the System of Elastically Connected Reddy-Bickford Beams on Elastic Foundation

The Effects of Rotary Inertia and Transverse Shear on Vibration

Chapter 6 explores the free vibration of a multiple elastically connected beam system of Reddy-Bickford's type on an elastic foundation, considering the effects of axial forces It compares the theoretical research on frequency and stability across four different beam theories The analytical solutions for natural frequencies and critical buckling forces are derived using the trigonometric method and validated numerically, consistent with the previous chapter's approach The study reveals that the Reddy-Bickford model yields the most accurate approximation for natural frequencies.

Free Vibration of the System of Elastically Connected Reddy-Bickford Beams

Let us examine the system of m elastically connected Reddy-Bickford beams of the same length l, under the influence of axial compression forces of the same intensity F, reference [15] a) b)

Fig 6.1.1 a) The system of elastically connected beams on an elastic foundation b) Elementary deformed beam part

The functions governing longitudinal and transverse motion, along with the angle between the tangent of the deformed cross-section of the beam along the neutral line of the axis, are denoted as ̃( , , ), ̃( , ), ̃( ), and ̃ for indices 1, 2, … This angle is illustrated in Figure 6.1.1 b) and is derived from the equations referenced in the literature.

[15], [29-31], we define the functions of motions as ̃( , , ) = ̃ ( , ) + ̃ ( , ) − ̃ ( , ) + ̃ ( , )

Deformation in the function of motions and the relation between the stress and deformation according to Hooke’s law are ̃( , , ) = ̃ ( , )

0 ̃ ̃ ̃ , ̃ = 1,2, … , (6.1.4) Virtual work of inertial forces is ̃ = − ̃ ̃ ̃ ( , , ) δ ̃ ( , , ) + ̃ ( , , ) δ ̃ ( , , ) ̃ ̃ (6.1.5)

Virtual work of internal forces is given by ̃= − ̃ ̃( , )δ ̃( , ) + ̃( , )δ ̃( , ) (6.1.6) Virtual work of external forces is expressed as

By substituting equations (6.1.5-6.1.9) into the general virtual work principle equation, we derive a coupled system of partial differential equations for vibration This process involves applying Green's theorem to obtain the resulting equations in the specified form.

To eliminate the variable, we utilized an application developed in Mathematica 9, as detailed in Appendix 6.1.1 This process allowed us to reduce the original system of 2m partial differential equations (6.1.10-6.1.15) to a simpler system consisting of m partial differential equations.

The simplified form of the coefficients ( ̃) ∗,∗ ̃ , ̃ = 1,2, … , is given in Appendix 6.1.2

Let us assume that, as in the previous chapter for the case of the Timoshenko connected beam system, the beams have identical material and geometric properties

Initial and boundary conditions of simply supported beams based on (2.3.22 − 2.3.24) are ̃( , 0) = ̃ ( ), ̃( , 0) = ̃ ( ), ψ ̃ ( , 0) =ψ ̃ ( ), ψ ̃ ( , 0) =ω̃ ( ), ̃(0, ) = ̃ ( , ) = ̃ (0, ) = ̃ ( , ) = 0, ̃ = 1,2,3 ⋯ (6.1.21)

If we assume harmonic beam movement, transverse beam motions can be written as the product of mode shape functions and unknown time functions in the following form ̃ ( , ) = ( ) ̃ ( ), ̃ = ̃ , ̃ = 1,2,3, ⋯ , = √−1 (6.1.22)

The function meets boundary conditions for simply supported beams if expressed as

If we substitute the assumed transverse motions (6.1.22) into equations (6.1.17- 6.1.19), we obtain the homogeneous system of algebraic equations for unknowns , , , ⋯ in the form

The solutions to the homogeneous system (6.1.24) are non-trivial only when the determinant of the matrix in the equation equals zero In such instances, a fourth-degree polynomial solution is derived, which can only be numerically determined based on specific data The solution obtained for m elastically connected Timoshenko beams will be utilized to formulate a trigonometric frequency equation from the subsequent relations.

− 2 cosϕ= 0 ⇒ = 2 cosϕ, ϕ( , ) = π, = 1,2,3, ⋯ , (6.1.26) where m is the number of elastically connected beams while s represents the ordinal number of frequency From the relations (6.1.25) and (6.1.26) this follows

( ) + ( ) + ( ) = −2 ( ) + ( ) cos π , (6.1.27) i.e the biquadratic equation in the form

From the expression (6.1.27) follow the solutions given as the squares of natural frequencies ω , =− ( ) + 2 ( ) cos π

The static stability of a system with elastic layers can be analyzed using the equation (6.1.27), which relates to the natural frequencies of the system The system reaches a state of indifferent equilibrium when axial forces are applied, resulting in a natural frequency of zero (ω = 0) Under this condition, the minimum solution for the unknown axial force F represents the critical force of a system composed of m elastically connected beams, corresponding to the n-th mode.

1 − cos π , = 1,2,3 ⋯ (6.1.31) For = 0, this follows from the equation (6.1.30)

The moment of inertia for the beam's cross-section along the x-axis and the critical buckling load for a single Reddy-Bickford beam, corresponding to the n-th mode, are represented in expression (6.1.31) The minimum buckling load occurs at n=1, as referenced in [31].

Numerical Analysis and the Results in the Static and Frequency

6.2 Numerical Analysis and the Results in the Static and

Frequency Domain for the System of Elastically Connected Reddy-Bickford's Beams

6.2 Numerical A nal ysis and the Res ults i n t he Static a nd Frequency D o mai n

In the numerical analysis, we compared the results of 3, 5, 7, and 9 elastically connected beams with identical material properties This comparison highlights the importance of considering rotary inertia and transverse shear effects, especially in thicker beams.

Tables 6.2.1-6.2.3 present the natural frequency values of a system based on the number of elastically connected beams, their thickness, the stiffness of Winkler layers, and various vibration modes By disregarding the impacts of transverse shear and rotary inertia from equation system (5.1.29), and assuming uniform material properties and identical stiffness in the elastic Winkler layers, we derive the expressions for the natural frequencies of elastically connected Rayleigh and Euler beams.

Using analytical expressions for the frequencies of Timoshenko beams and Reddy-Bickford beams, we derive the natural frequency values presented in tables 6.2.1 to 6.2.3.

Table 6.2.1 Natural frequencies [ ] of elastically connected beams = 1, ℎ = ℎ , , = 0 ω −

Table 6.2.2 Natural frequencies [ ] of elastically connected beams = 1, ℎ = 5ℎ ,

Table 6.2.3 Natural frequencies [ ] of elastically connected beams = 1, ℎ = ℎ , 2 , = 0 ω −

Figure 6.2.1 shows the change in natural frequencies of the system according to the influence of axial compression forces defined by a non-dimensional parameter

The study investigates the impact of different types of elastically connected beams—Euler, Rayleigh, Timoshenko, and Reddy-Bickford—on the system's natural frequency It is observed that as the number of beams increases, the natural frequency decreases Notably, the beam type significantly influences the approximate solutions, particularly in thicker beams where rotary inertia and transverse shear effects are pronounced For three elastically connected beams, the Rayleigh model offers reasonable approximations due to minimal transverse shear influence However, with more than three beams, Timoshenko and Reddy-Bickford models yield better approximations as transverse shear effects become more significant Additionally, as axial compression forces rise, the differences in approximate solutions diminish In scenarios where a single beam on an elastic foundation is subjected to axial forces 70% above the critical threshold, the effects of transverse shear on natural frequencies become negligible, allowing the Rayleigh model to be applicable for thicker beams.

Figures 6.2.2a-b illustrate how the natural frequencies of Reddy-Bickford beams are affected by axial compression forces, which vary based on the number of beams Additionally, Figures 6.2.3 and 6.2.4 demonstrate the relationship between natural frequencies and beam thickness, represented by the non-dimensional parameter = ℎ/ Notably, in Figure 6.2.3 a), all theories yield similar approximate results up to a certain value.

As the thickness of cross-sections increases, the discrepancies in approximate solutions also grow However, when multiple beams are involved, the Timoshenko and Reddy-Bickford models yield comparable results Notably, for a single beam on an elastic foundation, Reddy-Bickford’s model offers the most accurate solution approximation.

Fig 6.2.1 The influence of axial compression change on the lowest natural frequency of the beam with a thickness of h=15h 1 a) = 1; b) = 3; c) = 5 d) = 7;

Fig 6.2.2 The influence of compressive axial forces on the lowest natural frequency of the beam with a thickness of h=h 1 a) = 1, 3, 5, 7; b) = 9, 11, 15, 20

Fig 6.2.3 The influence of the parameter on the beam’s lowest natural frequency a) = 1; b) = 3; c) = 5; d) = 7

The analysis presented in Figure 6.2.4a highlights the beam thickness region where various theories yield accurate approximations for three elastically connected beams across different vibration modes Notably, as the vibration mode increases, the discrepancies in frequency approximations become more pronounced, leading to a reduction in the beam thickness range where all theories align effectively This trend is particularly evident in the Euler and Reddy-Bickford models.

Fig 6.2.4 The influence of parameter on the beam’s lowest natural frequency for different modes n a) = 1; b) = 3; c) = 5; d) = 7

Figure 6.3.1 illustrates the static stability regions for a system with one, three, and five elastically connected beams of various types While the differences in solution approximations among the model types are minimal, it is important to highlight that the system exhibits the greatest stability when consisting of a single beam on an elastic foundation.

Fig 6.2.5 The influence of beam number on critical buckling force / in the function of the Winkler’s elastic layer stiffness

Appendix 6.1.2- Coefficients in partial differential equations (6.1.17-6.1.19)

Geometrically Non-linear Vibrations of

Timoshenko Damaged Beams Using the New p –Version of Finite Element Method

7 Geom etrically Non-linear Vibrations of Tim oshen ko Dam aged Beam s

Chapter 7 explores the geometrically nonlinear forced vibrations of damaged Timoshenko beams, introducing an innovative p-version of the finite element method specifically designed for these damaged structures This new approach offers significant advantages over the traditional p-version, delivering improved solution approximations while utilizing fewer degrees of freedom in numerical analysis, thereby enhancing computational efficiency and accuracy.

This article presents significant advancements in computational mechanics and non-linear vibrations of beams, highlighting that traditional methods fail to provide accurate solutions for cases with minimal damage width The study compares its findings with results from the commercial software Ansys, demonstrating the advantages of a newly proposed p-version finite element tailored for geometrically non-linear vibrations of damaged Timoshenko beams This innovative p-element utilizes new displacement shape functions that are dependent on the damage location, leading to more efficient modeling with enhanced accuracy and reduced computational costs Numerical tests reveal a coupling between cross-sectional rotation and longitudinal vibrations, characterized by abrupt changes in longitudinal displacements and cross-sectional rotation at the damage site Additionally, the investigation of geometrically nonlinear forced vibrations in the time domain using Newmark’s method uncovers further couplings between displacement components.

Development of the New p–Version of Finite Element Method

The advancement of supercomputers and numerical methods has enabled researchers to explore non-linear mathematical models with greater accuracy Notably, the finite element method (FEM) is widely utilized for developing non-linear discretized structures of beams and plates This method involves approximating solutions by solving relevant polynomial functions, as highlighted by Petyt Typically, the structure is segmented into smaller elements that define local function forms, using shape functions that are simple polynomials of a fixed lower degree p The precision of the approximate solution improves as the number of elements in the model increases, which reduces their width h Additionally, maintaining a constant network while increasing the degree p of shape functions enhances approximation accuracy, leading to what is known as the p-version of the finite element method, as referenced by Ribeiro.

The widespread use of the p-version of the finite element method is attributed to several key advantages over traditional models Firstly, it enhances solution accuracy without requiring changes to the existing network Secondly, it employs a matrix with non-linear lower-degree polynomial members, facilitating improved approximations for derivatives Additionally, the method utilizes diagonal linear member matrices, making it simpler than the conditioned matrices found in classic finite element approaches The ease of joining polynomial elements of varying degrees allows for straightforward incorporation of additional degrees of freedom Furthermore, simple structures can be effectively modeled with a single element, resolving issues related to internal continuity and connections This method also enables the selection of various shape functions for different displacements, providing deeper insights into the interactions of component vibrations Finally, the p-version achieves greater solution accuracy by increasing the polynomial degrees of the mode shapes without necessitating an increase in the number of elements within the structure.

The p-version of the finite element method significantly reduces modeling and analysis time compared to conventional methods, offering a key advantage in non-linear analysis where iterative calculations are necessary This method demonstrates quicker convergence to accurate solutions, as evidenced by studies on static linear analysis of beams, plates, and shells (Szabó et Sahrmann), non-linear geometric static analysis of composite plates (Han et al.), and the analysis of free and forced vibrations in various structures.

The new p-version of the finite element method emerged from research into the geometric non-linear vibration of damaged beams, addressing limitations of the conventional method that failed to detect small discontinuities in the beam's cross-section To overcome this challenge, innovative displacement shape functions were introduced, which are tailored to the boundary conditions and the specific size and location of damage on the beam A model illustrated in Figure 7.1.1 depicts a homogeneous, elastic, isotropic beam characterized by its length, width, and thickness The existing damage is represented as a cut through the rectangular cross-section along the entire beam width, with a depth defined by the local coordinate axis, indicating the start and end points of the discontinuity.

Fig 7.1.1 The model of a damaged beam with local and global coordinate system

The p-version of the finite element method differs from the classical approach by varying the number of elements based on the geometry of the analyzed structure, as demonstrated in Ribeiro's study of a frame beam, which utilized only three elements In contrast, the classical method enhances accuracy in areas with geometric changes by increasing the surrounding element count Despite these fundamental differences, it is possible to achieve an accurate model even with geometric alterations due to damage by incorporating new displacement shape functions, as shown by Stojanovic et al Additionally, the relationship between local and global coordinates is defined by a specific expression.

If we label the functions of longitudinal and transverse displacement corresponding to the global coordinate system as ( , , ) and ( , , ), then the following relations apply

The relationship between the longitudinal and transverse displacements of a beam's point on the axis can be expressed as ( , , ) = ( , ) + ( , ) The cross-section rotation, denoted as ( , ), is determined by the angle change between the z-axis and the beam's cross-section in the plane around the -axis This analysis follows von Karman's principles to understand the beam's behavior under various conditions.

Kármán model of geometric non-linearity, deformation along the x axis ( , ) and the deformation caused by transverse shear ( , ) are given by

In each element, vector ( , ), comprised of the motion component can be expressed as a combination of hierarchical shape functions

In the model, the vector of longitudinal generalized displacement, the vector of transverse generalized displacement, and the vector of generalized rotation around the -axis are crucial components The matrix shape function, which includes these vectors, plays a significant role in enhancing model accuracy Increasing the number of shape functions within the model leads to improved discretization accuracy Ultimately, the total count of shape functions utilized in a single element model reflects the system's degree of freedom.

The introduction of two new shape functions into the p-version of the finite element method enhances the system of shape functions, enabling better recognition of increased beam flexibility in areas of damage These new shape functions, denoted by "d," are influenced by the location of the damage center, as illustrated in Figure 7.1.1 Consequently, the orders of the vectors representing longitudinal, transverse, and rotational displacement of the beam's cross-sections with discontinuity are now reformulated.

(7.1.6) where , and represent the number of longitudinal, transverse and rotation shape functions to which the functions ( , , ) and ( , , )have been added The first four functions are given by

4 (7.1.7) The remaining shape functions are obtained by way of a formula, ref Petyt [32], p.298

In the analysis of doubly clamped beams, shape functions are defined to satisfy boundary conditions, ensuring that both the functions and their first derivatives meet specific criteria at the boundaries These functions, represented as third-degree polynomials known as Hermite polynomials, are essential for modifying boundary conditions in various beam support types Newly implemented analytical functions enhance the construction and formulation of these shape functions, facilitating more accurate modeling in structural applications.

In a non-dimensional damage location example, where the ratio of global to local coordinate systems is set at = 0.5, the newly introduced mode shape functions are depicted in Figure 7.2.1 The function \( f(x, y, z) \) reaches its maximum at the damage center, while the function \( g(x, y, z) \) is defined to have a zero at the same location Incorporating these two functions into the element maintains the continuity condition.

We assume the beam’s is elastic and isotropic hence the Hooke’s law applies

In the context of elastic materials, the matrix of elastic coefficients relates stress and deformation, with Young’s modulus and shear modulus defined as /[2(1 + )], where represents Poisson’s ratio For the numerical experiment, a shear factor of k=(5+5 )/(6+5 ) was utilized to achieve optimal alignment with experimental results, as referenced in [24] Longitudinal deformation can be expressed accordingly.

The deformation in equation (7.1.12) is composed of both linear and non-linear components, where 'z' denotes the linear longitudinal deformation resulting from buckling, and the geometric non-linear longitudinal deformation is also included These elements can be expressed within the given model.

In equation (7.1.13), the first derivative of the variable is denoted by ", " which is crucial for understanding stress behavior By integrating the normal and shear stress, along with the moment of normal stress from expressions (7.1.11), we can derive the forces acting in both perpendicular and tangential directions, as well as the corresponding moment, represented in the form of dz.

By inserting the specified quantities from expressions (7.1.13) into equations (7.1.14-7.1.16), we establish the relationship between the forces and moments, as well as their impact on deformation.

The equation (1, , ) dz, (7.1.17) represents the relationship between the coefficient of extension, the coupling coefficient of bending and extension deformation, and the bending coefficient In cases of beam damage, this coupling coefficient is non-zero, unlike classic beams where it typically equals zero.

Mode Shapes of Component Longitudinal and Transverse Vibration

7.2 Mode Shapes of Component Longitudinal and Transverse Vibration and Component Vibration Mode Shapes of

7.2 Mo de Sha pes of Co mpone nt Lo ngit udina l a nd Transverse Vibratio n

The p-version model of the finite element method was applied to a doubly clamped beam whose characteristics have been given in the papers [17], [45],

[46] Depending on the damage depth in the numerical experiment, the following cases were considered:

= 1330 mm, = 25.3 mm, ℎ = 25.3 mm, = 203.91 GNm , = 7800 kgm Damage position: = 222.5 mm, = 247.5 mm, = 1.87%

Damage depth: a) 4 mm b) 8 mm c) 12 mm.

= 406 mm, = 20 mm, ℎ = 2 mm, = 7.172 GNm , = 2800 kg m

Damage depth: a) 0.3 mm b) 0.6 mm c) 0.9 mm.

= 406 mm, = 20 mm, ℎ = 20.3 mm, = 7.172 GNm , = 2800 kg m Damage position: = 68 mm, = 75.5 mm, = 1.7%.

In the numerical experiment, the relative damage length is expressed as a percentage The selected beams possess specific geometric and material characteristics to validate the results against previously published experimental findings These experimental results are based on the work of Sinha.

In this study, we compared experimental results of a non-damaged beam, referenced from Wolf [46], with numerical results from a non-linear vibration analysis of the same beam, as detailed by Ribeiro [45] We validated existing numerical and experimental findings using the new p-version of the finite element method, assuming zero damage depth in the beam model The analysis in paper [45] focused on a thicker, non-damaged beam (L/h), which provided insights into the vibration characteristics of damaged thicker beams Additionally, the Ansys software package [47] was utilized to compare the models based on their degrees of vibration freedom.

The results presented in Tables 7.2.1, 7.2.2, and 7.2.3 illustrate the transverse natural beam frequencies corresponding to various damage depths For comparison purposes, the Ansys software utilized the h-version of the “BEAM189” element, which is grounded in Timoshenko’s theory and features three binding points with six degrees of vibrational freedom each The findings were derived from a model comprising 300 elements, aimed at verifying the p-version of the finite element method, although convergence can be achieved with just thirty elements Additionally, the tables display results from the traditional p-version of the finite element method, employing shape functions referenced in paper [49], without the inclusion of the implemented functions f 1 d and f 1 d.

Table 7.2.1 Natural frequencies [Hz] of clamped-clamped beam for case 1.1

Table 7.2.2 Natural frequencies [Hz] of clamped-clamped beam for case 2.1

Table 7.2.3 Natural frequencies [Hz] of a clamped-clamped beam for case 2.2

The new p-version of the finite element method demonstrates superior accuracy with fewer vibration degrees of freedom compared to the h-version of commercial Ansys software, as shown in Tables 7.2.1-7.2.3 The alignment with experimental data validates the effectiveness of this innovative approach In contrast, the traditional finite element method struggles to produce reliable results for damaged beams, often yielding frequencies that exceed those of non-damaged beams.

Changes in natural frequencies alone do not fully capture the impact of damage on structural vibrations, as different damage locations can yield identical natural frequencies Therefore, it is essential to analyze additional vibration characteristics, such as the independent components of mode shapes For undamaged doubly clamped beams, only two vibration modes exist: transverse displacement of the beam's centerline and cross-section rotation However, damage introduces anti-symmetry in the beam's geometry, resulting in the coupling of transverse rotation and longitudinal displacement, which is reflected in altered mass and stiffness submatrices This damaged beam model facilitates the determination of longitudinal mode shape components Notably, sudden changes in longitudinal vibration are observed at the damage site, with maximum longitudinal displacement shifting based on the mode In the first and sixth modes, maximum displacement occurs farther from the clamp, while in higher modes, it is closer to the clamp This behavior is a critical dynamic feature of the model, arising solely from the beam's damage.

Fig 7.2.1 Longitudinal components of mode shapes

Figures 7.2.2 and 7.2.3 illustrate the transverse and rotational components of beam mode shapes for case 1.1, with normalized amplitudes for comparison at the first local extreme of the non-damaged beam The natural frequencies listed in Table 7.1.1 correspond to these component mode shapes The figures reveal a noticeable deviation in mode shapes at the damage location, which becomes increasingly pronounced in higher modes.

Transverse component – mode 1 Transverse component – mode 2

Transverse component – mode 3 Transverse component – mode 4 (c) (d)

Fig 7.2.2 Transverse component of mode shapes: Case 1.1 ▬▬ non-damaged beam; - 8mm damage; ▬▬ 12mm damage

The mutual coupling of longitudinal and rotational vibrations in cross-section results in new mode shapes, as illustrated in Figure 7.2.3 Notably, as damage depth increases, the amplitude of transverse vibration in the damaged area also rises Interestingly, the cross-section rotation angle at the center of the damage remains constant regardless of the damage depth, while the rotation angle in the region extending from the damage center to its ends increases, eventually aligning with the angles of a non-damaged beam This phenomenon is consistent across various vibration modes However, experimentally determining the changes in the cross-section angle is challenging If achievable, these component mode shapes could significantly enhance the accuracy of damage detection in beams.

Rotational component of cross-section - mode 1

Rotational component of cross - section - mode 2

Fig 7.2.3 Rotational components of beams’ cross-sections mode shapes: Case 1.1 ▬▬ non- damaged beam; - 8mm damage; ▬▬ 12mm damage

7.3 Geom etrically Non-linear Vibration s of a Dam aged Tim oshenko Beam

7.3 Geometrically Non-linear Vibrations of a Damaged

Timoshenko Beam in the Time Domain

7.3 Geom etrically Non-linear Vibration s of a Dam aged Tim oshenko Beam

The geometric non-linear vibration analysis of a damaged beam is crucial for practical engineering applications, emphasizing the significance of understanding the dynamic behavior of such structures over time This involves assessing the displacement of the beam's centerline based on its location and the extent of damage The non-linear partial differential equations were addressed using the Newmark method, with a numerical experiment applying harmonic concentrated external excitation The new p-version of the finite element method was evaluated on a non-damaged beam, utilizing 15 shape functions to model its behavior in a non-linear context Results indicated that the transverse displacement of the center point of the non-damaged beam closely matched experimental data, validating the method's effectiveness Specifically, at an excitation frequency of ω = 1.02 ω, the calculated maximum amplitude of transverse displacement was 0.425, aligning closely with the experimental value of 0.43, thus demonstrating the accuracy of the new modeling approach.

Fig 7.3.1 Amplitude-time diagram, = 0, Case - 2.1.1, external excitation - 0.134 N, ω =ω , ▬▬ non-damaged beam; ■■■ 2.1.1 b); ▲▲▲ 2.1.1 c)

The analysis of diagram 7.3.1 reveals minimal differences in the amplitude of the beam's center lines when subjected to external excitation at the center point However, as the measurement points approach the damage location, the amplitude of beam vibrations shows significant variations Figure 7.3.2 further illustrates the amplitudes of two symmetrically selected beam points along the vertical axis z, indicating that the damage has led to asymmetrical vibrations This asymmetry is evident when comparing the results from the point nearest to the damage center.

The impact of a constant member on a beam's deflection is illustrated in Fourier's spectrum diagram 7.3.5 c) When combined with non-linear members, this effect amplifies square non-linearities and prevents the neglect of the second harmonic during vibrations.

Fig 7.3.2 Amplitude-time diagram of beam’s symmetric points case 2.1.1 c), external concentrated excitation at beam’s centre with the amplitude of 0.134 N, ω = ω ;── −78mm; ••• = 78 mm; ▬▬ = −15 mm;●●● = 15 mm

Fig 7.3.3 Amplitude-time diagram = 62.4 mm for case 2.1.1 c) external concentrated excitation = 62.4mm, 0.134 N, ω = ω ,(a) linear mode (b) non-linear mode ▬▬ non- damaged beam; ▲▲▲2.1.1 c)

Diagrams 7.3.3 a) and b) illustrate the amplitudes resulting from external concentrated excitation in both linear and non-linear models Notably, the non-linear model reveals a symmetry in vibration relative to the horizontal axis x, a feature absent in the linear model This results in increased vibration amplitude on the damage side, specifically the beam’s top side in this context This phenomenon arises from the coupling of transverse and rotational displacements within the stiffness matrix of non-linear members, as described in equation 7.1.24.

Fig 7.3.4 Amplitude –time diagram = 62.4 mm, case 2.1.1, amplitude of forcing 0.134 N, ω = ω ,▬▬ non-damaged beam; ■■■ 2.1.1 b); ▲▲▲ 2.1.1 c)

Figures 7.3.4-7.3.5 illustrate the impact of geometric non-linearity on transverse displacement amplitudes in beams with varying damage depths As damage depth increases, the amplitudes of displacement at the beam's center line also rise, particularly near the damaged areas The phase diagram in Figure 7.3.5 a) highlights the asymmetry in vibrations resulting from damage and geometric non-linearity Additionally, the Poincaré diagram in Figure 7.3.5 b) emphasizes the differences in vibration patterns between non-damaged beams and those with varying degrees of damage.

Fig 7.3.5 Case 2.1.1 amplitude of forcing x = 0, 0.134N, ω = ω (a) Phase diagram

= 62.4mm;(b) Poincaré diagram = 62.4 mm; (c) Fourier spectrum = 62.4 mm

The amplitude difference of a beam with varying damage depths increases with higher forcing modes At the external excitation frequency matching the linear frequency in the third mode, a notable asymmetry in vibration occurs, causing the beam to shift towards the damaged side As the damage depth increases, the asymmetry in beam vibrations becomes more pronounced at elevated vibration modes.

Fig 7.3.6 Case 2.1.1 amplitude of forcing 0 4N, x= 101.5mm, ω = ω Amplitude-time diagram ; (b) Phase diagram; (c) Poincaré diagram; (d) Fourier spectrum; ( 62.4 mm)▬▬, ● non-damaged beam; ■■■ 2.1.1 b); ▲▲▲ 2.1.1 c)

The new p-version of the finite element method is effective for analyzing thicker beams Results indicate that the behavior of thicker beams exhibits similar qualitative effects of geometric linearity and damage as observed in slender beams.

Fig 7.3.7 Case 2.2 amplitude of forcing 2000N,x= 0, ω = ω (a) Amplitude-time diagram; (b) Phase diagram; (c) Poincaré diagram; ( = 62.4 mm) ▬▬, ● non-damaged beam; ■■■ 2.2 a); ▲▲▲ 2.2 b)

7.4 Free Geom etric Non-linear Vibration of the Dam aged Tim oshen ko Beam

Free Geometric Non-linear Vibrations of the Damaged Timoshenko

7.4 Free Geom etric Non-linear Vibrations of t he Dam aged Tim oshenko Beam

The harmonic balance method and the Continuation method are effective tools for identifying bifurcation points commonly found in nonlinear mechanics In the case of a damaged beam system without external forces, applying the virtual work principle yields significant insights into its behavior.

The products [ ]{ }, [ ]{ } and T { } represent small quantities and do not affect the solutions in a non-linear mode as shown in ref [44] and can be disregarded

The vector of generalized coordinates and its second derivative, or acceleration, can be expressed as the sum of the first three terms of a trigonometric series, as these initial terms adequately represent the findings from the Fourier spectrum analysis.

( ) = − cos( ) − 4 cos(2 ) − 9 cos(3 ) (7.4.4) where represents the system’s natural frequency The vector of new unknowns is now

If we substitute the expressions (7.4.3) and (7.4.4) into the initial set of equations of motion (7.4.2), we get

Amplitude-frequency characteristic of the model’s first and third harmonic is shown in Figure 7.4.1

Fig 7.4.1 Amplitude-frequency diagram for case 2.1.1

Figure 7.4.1 illustrates that the interaction between higher vibration modes at various damage positions on the beam leads to bifurcations, as noted by Stojanovic et Ribeiro [23] These bifurcations indicate that the beam enters a state of internal resonance, making it crucial to identify the amplitude-frequency relationship at which this occurs The diagram also highlights the locations of potential dual solutions for the first and third harmonics in the vibration of a damaged beam.

Stojanović et al [50] demonstrate that damage-induced changes in beam geometry lead to new couplings between component mode shapes in both linear and non-linear vibration modes Deviations in these mode shapes can serve as a basis for developing a damage detection model Additionally, findings related to longitudinal vibrations and asymmetry in transverse vibrations reveal new insights into the displacement dynamics of damaged beam points, which can be applied in the analysis of real-world structures.

Appendix 7.1.1 - Mass and stiffness matrixes of linear and non-linear members in the expression (7.1.24)

Beam vibration issues are fundamental in mechanical, civil, and aeronautical engineering, as well as in the transportation industry Achieving greater accuracy in solutions is crucial, especially when analyzing the motion of complex systems like deformable bodies with varying degrees of vibrational freedom More precise approximations help mitigate cumulative errors in solving complex dynamic systems This research focuses on a model involving elastically connected beams, considering the effects of rotary inertia and transverse shear It specifically analyzes the analytical solutions for linear vibrations in thicker elastically connected beams with indefinite vibrational freedom The dynamics of two elastically connected beams are of particular interest for their potential as dynamic absorbers In systems with multiple connected beams, the study emphasizes stability and the analytical determination of natural frequencies, which tend to increase with the number of connections, raising the risk of resonance The widespread application of these mechanical systems in civil engineering, such as multistory buildings and reinforcement grids, has led researchers to incorporate additional physical influences into mathematical models for improved solution approximations.

Dynamic systems in civil and mechanical engineering, as well as the aeronautical and transportation industries, often encounter complex problems involving elastic bodies that exhibit intricate motion patterns These complexities arise particularly in non-linear vibration modes, with damaged beams representing a critical category of mechanical systems Traditionally, the oscillatory behavior of these systems is determined through experimental methods However, advancements in mathematical modeling, numerical methods, and software tools have enhanced the analysis of damaged beams, allowing for more accurate solutions regarding their vibrations while reducing calculation time for mechanical system motions in non-linear conditions This research focuses on the non-linear oscillatory motion of damaged beams, employing a newly developed finite element method to derive numerical solutions that account for rotary inertia and transverse shear effects.

This article explores the dynamics of elastically connected and damaged beams, incorporating factors such as rotary inertia and transverse shear to provide more accurate solutions for both thicker and slender beams It is structured into seven chapters, starting with an introduction that reviews existing research on elastically connected and damaged structures and outlines the methods used to derive partial differential equations for mechanical system movement Chapters 2 through 6 focus on the linear vibration analysis of elastically connected beams, while Chapter 7 addresses the geometric non-linear vibration of damaged beams through a novel finite element method.

Chapter 2 explores the free vibration and static stability of two elastically connected beams, referencing Stojanovic et al [12] It presents various examples demonstrating how mechanical parameters impact natural frequencies and vibration amplitudes, with analytical results verified against the classical Euler-Bernoulli beam theory The chapter formulates equations for free vibration in beams connected by a Winkler layer, incorporating rotary inertia through Rayleigh’s model and Timoshenko’s model It also analyzes the static stability of different beam types, providing analytical expressions for critical force values Numerical experiments validate the analytical findings by comparing them with existing literature The chapter concludes that rotary inertia and transverse shear effects are crucial for thicker beams, as ignoring them leads to significant errors in higher vibration modes It highlights changes in natural frequencies and stability regions based on mechanical system parameters, affirming that higher-order deformation theory offers the most accurate solutions.

Chapter 3 investigates the forced vibration of two elastically connected Rayleigh, Timoshenko, and Reddy-Bickford beams under axial forces, presenting analytical solutions for three types of external excitations: arbitrarily continuous harmonic, uniformly continuous harmonic, and harmonic concentrated excitation Utilizing modal analysis, partial differential equations were derived for the forced vibrations of these beam models influenced by axial compression forces The chapter also provides general solutions for the forced vibrations of a system with two connected beams, considering rotary inertia and transverse shear effects It details analytical solutions for forced vibrations under various excitations and discusses resonance conditions and dynamic vibration absorption Notably, the increase in axial compression forces up to their critical value leads to a rise in beam vibration amplitude under continuous uniform harmonic excitation The precision of solutions varies by beam model, with Reddy-Bickford and Timoshenko models yielding more accurate results than Rayleigh and Euler models Additionally, increased vibration modes exacerbate solution discrepancies, highlighting the importance of accounting for rotary inertia and transverse shear effects.

Chapter 4 explores the static and stochastic stability of two or three elastically connected beams, as well as a single Timoshenko beam on an elastic foundation It presents derived partial differential equations governing the motion of points along the beams' center lines during deformation and identifies the critical buckling force for each configuration The findings indicate that the system exhibits the highest stability when a single beam is placed on an elastic foundation.

Chapters 5 and 6 explore the free vibration of elastically connected Timoshenko and Reddy-Bickford beams on an elastic foundation subjected to axial compression forces Analytical solutions for natural frequencies and critical forces, as referenced by Stojanovic et al [15], were derived using a trigonometric method from Raskovic [28] and verified numerically The findings indicate that the Reddy-Bickford beam model offers the most accurate approximations, yielding lower natural frequencies compared to the Timoshenko model This difference is particularly significant in beams with large cross-sections, where employing a higher-order deformation theory is most appropriate.

Chapter 7 explores the forced geometric non-linear vibration of a damaged doubly clamped Timoshenko beam, utilizing the innovative p-version of the finite element method to effectively address small-width damages This advanced method offers superior solution approximations with fewer degrees of freedom compared to traditional methods, which struggle to accurately model small-width damage even with increased polynomial degrees The analysis reveals that newly developed shape functions, influenced by damage location, can be applied to non-linear analysis of undamaged beams An open crack beam model was established through geometric alterations, leading to significant coupling between longitudinal and rotational displacements in the beam's mass and stiffness matrices In section 7.1, non-linear partial differential equations for the forced vibration of the damaged beam were derived, highlighting the emergence of longitudinal vibrations that facilitate easier damage detection compared to changes in mode shapes The study found that maximum longitudinal motions occur at boundary damage areas, with mode shape deviations increasing alongside damage depth and vibration mode The numerical experiments corroborated the theoretical findings, demonstrating strong alignment with experimental results In section 7.2, the Newmark method was employed to solve the non-linear equations, revealing that the amplitudes of the damaged beam escalate with increased damage depth, particularly in the affected region Additionally, vibration asymmetry was noted, especially in higher forced vibration modes, where beams exhibit vertical movement toward the damage side Frequency domain analysis identified bifurcation points and amplitude-frequency characteristics for the first and third harmonics, indicating internal resonance at specific damage locations Ultimately, the study underscores the necessity of conducting amplitude-frequency beam analysis for each damage scenario due to the interactions of frequencies among higher modes, which can only be resolved numerically.

Recent research has developed mathematical models that account for geometric non-linearity and the changing stiffness of elastic inter-layers during component motions under damping This work provides a comprehensive framework for analyzing the non-linear vibrations of damaged dynamic systems, enabling the conditions for such systems to be treated as undamaged by incorporating elastic foundations with variable stiffness This is particularly beneficial for thicker beams acting as dynamic absorbers in bridge construction, which are susceptible to aero-elastic vibrations The findings lay the groundwork for further exploration of non-linear free and forced vibrations in elastic bodies, addressing the interactions of longitudinal, transverse, torsional, and rotary vibrations in elastically connected beams with three-dimensional damage Such advancements could enhance the design of active two-layered bridges with innovative suspension systems As technology in bridge construction and monitoring continues to evolve, ongoing research and development could expand these boundaries Additionally, the potential to construct diverse dynamic continuous systems from composite materials in aerospace, military, and robust bridge applications could be realized through further scientific inquiry, new vibration theories, software advancements, numerical methods, and experimental validations.

This author will concentrate on advancing new theories in non-linear mechanics of deformable bodies, creating innovative numerical methods for their solutions, and developing software to analyze the stress-deformation states of damaged mechanical systems, building on initial progress toward a novel finite element method.

[1] Seelig, J.M., Hoppmann, W.H.: Impact on an elastically connected double-beam system J Appl Mech 31, 621–626 (1964)

[2] Oniszczuk, Z.: Free transverse vibrations of elastically connected simply supported double-beam complex system J Sound and Vib 232, 387–403 (2000)

[3] Oniszczuk, Z.: Forced transverse vibrations of an elastically connected complex simply supported double-beam system J Sound and Vib 264, 273–286 (2003)

[4] Zhang, Y.Q., Lu, Y., Wang, S.L., Liu, X.: Vibration and buckling of a double-beam system under compressive axial loading J Sound and Vib 318, 341–352 (2008)

[5] Zhang, Y.Q., Lu, Y., Ma, G.W.: Effect of compressive axial load on forced transverse vibrations of a double-beam system Int J of Mech Sci 50, 299–305

[6] Vu, H.V., Ordonez, A.M., Karnopp, B.H.: Vibration of a double-beam system J Sound and Vib 229, 807–822 (2000)

[7] Li, J., Hua, H.: Spectral finite element analysis of elastically connected double-beam systems Fin Elem Analysis Des 15, 1155–1168 (2007)

[8] Li, J., Chen, Y., Hua, H.: Exact dynamic stiffness matrix of a Timoshenko three- beam system Int J Mech Sci 50, 1023–1034 (2008)

[9] Kelly, S.G., Srinivas, S.: Free vibrations of elastically connected stretched beams J Sound and Vib 326, 883–893 (2009)

[10] Ariaei, A., Ziaei-Rad, S., Ghayour, M.: Transverse vibration of a multiple- Timoshenko beam system with intermediate elastic connections due to a moving load Arch Appl Mech 81, 263–281 (2011)

[11] Mao, Q.: Free vibration analysis of elastically connected multiple-beams by using the Adomian modified decomposition method J Sound and Vib 331, 2532–2542

[12] Stojanović, V., Kozić, P., Pavlović, R., Janevski, G.: Effect of rotary inertia and shear on vibration and buckling of a double beam system under compressive axial loading Arch Appl Mech 81, 1993–2005 (2011)

[13] Stojanović, V., Kozić, P.: Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load Int J Mech Sci 60, 59–

[14] Stojanović, V., Kozić, P., Janevski, G.: Buckling instabilities of elastically connected Timoshenko beams on an elastic layer subjected to axial forces J Mech Materials Struct 7, 363–374 (2012)

[15] Stojanović, V., Kozić, P., Janevski, G.: Exactclosed-form solutions for the natural frequencies and stability of elastically connected multiple beam system using Timoshenko and high-order shear deformation theory J Sound and Vib 332, 563–

[16] Christides, S., Barr, A.D.S.: One dimensional theory of cracked Bernoulli-Euler beams Int J Mech Sci 26, 639–648 (1984)

[17] Sinha, J.K., Friswell, M.I.: Edwards S Simplified models for the location of cracks in beam structures using measured vibration data J Sound and Vib 251, 13–38 (2002)

[18] Pandey, A.K., Biswas, M., Samman, M.M.: Damage detection from change in curvature mode shapes J Sound and Vib 145, 321–332 (1991)

[19] Panteliou, S.D., Chondros, T.G., Argyrakis, V.C., Dimarogonas, A.D.: Damping factors as an indicator of crack severity J Sound and Vib 241, 235–245 (2001)

[20] Bikri, E.I., Benamar, R., Bennouna, M.M.: Geometrically non-linear free vibrations of clamped–clamped beams with an edge crack Comput Struct 84, 485–502 (2006)

[21] Andreaus, U., Casini, P., Vestroni, F.: Non-linear dynamics of a cracked cantilever beam under harmonic excitation Int JNon-Linear Mech 42, 566–575 (2007)

[22] Stojanović, V., Ribeiro, P., Stoykov, S.: Non-linear vibration of Timoshenko damaged beams by a new p-version finite element method Comput Struct 120, 107–119 (2013)

[23] Stojanović, V., Ribeiro, P.: Modes of vibration of damaged beams by a new p- version finite element Paper presented at the 23rd International Congress of Theoretical and Applied Mechanics, Beijing, China, August 19-24 (2013)

[24] Kaneko, T.: On Timoshenko’s correction for shear in vibrating beams J Physics D 8, 1927–1936 (1975)

[25] Kelly, S.G.: Mechanical Vibrations: Theory and Applications Cengage, Stamford

[26] De Rosa, M.: A Free vibrations of Timoshenko-beams on two-parameter elastic foundation Comput Struct 57, 151–156 (1995)

[27] Timoshenko, P.S., Gere, M.J.: Theory of Elastic Stability McGraw-Hill, New York

[28] Rašković, D.: Theory of Oscillations Naučnaknjiga, Belgrade (1965)

[29] Reddy, J.N.: Energy and Variational Methods in Applied Mechanics John Wiley, New York (1984)

[30] Reddy, J.N.: Mechanics of Laminated Composite Plates: Theory and Analysis CRC Press, Boca Raton (1997)

[31] Wang, C.M., Reddy, J.N., Lee, K.H.: Shear deformable beams and plates Elsevier, Oxford (2000)

[32] Petyt, M.: Introduction to Finite Element Vibration Analysis Cambridge University Press, Cambridge (1990)

[33] Ribeiro, P.: A p-version, first order shear deformation, finite element for geometrically non-linear vibration of curved beams Int J Numer Methods Eng 61, 2696–2715 (2004)

[34] Szabó, B.A., Sahrmann, G.J.: Hierarchic plate and shell models based on p- extension Int J Numer Methods Eng 26, 1855–1881 (1988)

[35] Han, W., Petyt, M., Hsiao, K.M.: An investigation into geometrically nonlinear analysis of rectangular laminated plates using the hierarchical finite element method Fin Elem Analysis Design 18, 273–288 (1994)

[36] Meirovitch, L., Baruh, H.: On the inclusion principle for the hierarchical finite element method Int J Numer Methods Eng 19, 281–291 (1983)

[37] Meirovitch, L.: Elements of Vibration Analysis McGraw-Hill, New York (1986)

[38] Zhu, D.C.: Development of hierarchical finite element methods at BIAA In: Conference on Computational Mechanics Springer, Tokyo (May 1986)

[39] Houmat, A.: Hierarchical finite element analysis of the vibration of membranes J Sound and Vib 201, 465–472 (1997)

[40] Houmat, A.: An alternative hierarchical finite element formulation applied to plate vibrations J Sound and Vib 206, 201–215 (1997)

Tiêu đề	Vibrations and Stability of Complex Beam Systems
Tác giả	Vladimir Stojanović, Predrag Kozić
Người hướng dẫn	Seung-Bok Choi, Board Editor, Haibin Duan, Board Editor, Yili Fu, Board Editor, Jian-Qiao Sun, Board Editor
Trường học	University of Niš
Thể loại	thesis
Năm xuất bản	2015
Thành phố	Niš

Định dạng
Số trang	170
Dung lượng	18,14 MB