INTRODUCTION
Motivation
Cantilever beams can represent many structures in mechanical, civil, and electronics engineering (for example, MEMS structure) For details, the cantilever beams also can be found in high-speed applications such as fast manipulators, micro CNC milling machines, coordinate measuring machines (CMM), atomic force microscopies (AFM)
It is shown that the cantilever beam structures yield is a viable solution for the case of lightweight and compact size However, it is well known that the lightweight structures may result in vibrations when the cantilever beam moves, especially in the case of high- speed motion Therefore, the vibrations of the cantilever beam need to be analyzed during motions of the cantilever beam In this case, the beam is considered flexible
(a) Atomic Force Microscope (b) Coordinate-measuring machine
(c) Micro CNC milling machines (d) Gantry system
In the development of dynamic models of the flexible cantilever beam, Timoshenko beam theory [1] and Euler Bernoulli beam theory [2] have been employed The dynamic models based on Timoshenko beam theory are complex due to the assumption of the existence of the shear deformation, which results in the non-orthogonality of the neutral axis and cross-sectional area of the beam In contrast, in the case of Euler-Bernoulli beam theory used, the dynamic models of the cantilever beam system are developed by using the extended Hamilton’s principle [3], and the partial differential equations (PDE) governing the vibrations of the beam are derived by the Hamiltonian differential equation In the case of the flexible cantilever beam fixed on the moving hub, the PDEs describing the vibrations of the beam are nonlinear [4] because of the effects of the motions of the hub The nonlinearity of the PDEs may yield complexity for dynamic analysis of the flexible beam To overcome this problem, approximation methods such as the Galerkin decomposition [5] and the finite element method [6] are used These methods convert the PDEs with infinite order into a set of ordinary differential equations (ODEs) with finite order The accuracy of models using the ODEs depends on the order of ODEs
Vibration control methods to reduce oscillations of the cantilever beam systems with the moving hubs have been developed based on PDE models [7] and ODE models In terms of the PDE based control methods, in the work [8], a boundary control scheme is proposed for a flexible articulated wing on a robotics aircraft, where the control design is performed by the backstepping technique Dadfarnia et al [7] developed a control method for a translational Euler-Bernoulli beam, in which the control algorithm is designed to drive the moving hub and to suppress the vibration of the flexible beam The advantage of the ODE models is to facilitate the applications of the control theories for ODE systems, which are well established In the research [9], an ODE model developed by the Galerkin method was used to design nonlinear control laws for a two-link flexible manipulator
Experimental techniques, such as free-vibration decay, forced vibration, rotating-beam deflection, and pulse propagation, have been widely used to measure the dynamic elastic modulus and damping of materials These methods are now accompanied by numerical simulations, which play an important role in modern vibration analysis These solutions give reliable answers if the experiments have practical problems, are too expensive or it is impossible to analyze the problem with analytical methods [10]
In particular, models based on beam-like elements, with different boundary conditions, can be used to simulate the response of structures in engineering applications For example, we can model the vibrational response of spacecraft antennae, robot arms, building components, bridge structures, and parts of musical instruments
Experimental testing and system identification play a key role because they help the structural dynamicist to reconcile numerical predictions with experimental investigations The term ‘system identification’ is sometimes used in a broader context in the technical literature and may also refer to the extraction of information about the structural behavior directly from experimental data, i.e., without necessarily requesting a model (e.g., identification of the number of active modes or the presence of natural frequencies within a certain frequency range) In the scope of the dissertation, system identification refers to the development (or the improvement) of structural models from input and output measurements performed on the real structure using vibration sensing devices
Linear system identification is a discipline that has evolved considerably during the last
30 years [11] Modal parameter estimation termed modal analysis is indubitably the most popular approach to performing linear system identification in structural dynamics The model of the system is known to be in the form of modal parameters, namely the natural frequencies, mode shapes, and damping ratios The popularity of modal analysis stems
4 from its great generality; modal parameters can describe the behavior of a system for any input type and any range of the input
The fact that the structure under different conditions could display nonlinear distortions phenomena In structural dynamics, typical sources of nonlinearities are [11]:
Geometric nonlinearity results when a structure undergoes large displacements and arises from the potential energy
Inertia nonlinearity derives from nonlinear terms containing velocities and/or accelerations in the equations of motion, and takes its source in the kinetic energy of the system (e.g., convective acceleration terms in a continuum and Coriolis accelerations in motions of bodies moving relative to rotating frames)
A nonlinear material behavior may be observed when the constitutive law relating stresses and strains is nonlinear
Damping dissipation is essentially a nonlinear and still not fully modeled and understood phenomenon The modal damping assumption is not necessarily the most appropriate representation of the physical reality, and its widespread use is to be attributed to its mathematical convenience
Nonlinearity may also result due to boundary conditions (for example, free surfaces in fluids, vibro-impacts due to loose joints or contacts with rigid constraints, clearances, imperfectly bonded elastic bodies), or certain external nonlinear body forces (e.g., magnetoelastic, electrodynamic, or hydrodynamic forces).
Objectives and Scope of The Dissertation
The primary goal of the present dissertation is to improve the state of knowledge regarding the structural dynamic response of flexible cantilever beams In order to achieve this goal experimental, numerical, and analytical methods are used to characterize the time-dependent strain and displacement fields of flexible cantilever beams
The remainder of the dissertation is organized in the following manner First, in Chapter
2 a comprehensive flexible cantilever beam modeling is provided The details of the model of structural dynamics of a cantilever beam attached to a moving base are addressed thoroughly
In Chapter 3, the overview of the identification method is explored A brief description of the continuous-time system identification (CONTSID) toolbox for MATLAB, which supports continuous-time (CT) transfer function and state-space model identification directly from regularly or irregularly time-domain sampled data, without requiring the determination of a discrete-time (DT) model is provided
In Chapter 4, a comprehensive description of the experimental setup is provided The details of the experimental procedure and numerical simulation are given, the associated difficulties are addressed thoroughly The numerical solution consists of Galerkin method for spatial discretization and a high-order time-spectral method for temporal discretization The experimental setup consists of the cantilever beam attached to a moving base and a vision camera system as well
While numerical simulation can provide detail and fidelity, oftentimes analytical solutions can provide insight into parameter dependence which is unattainable through simulation In addition, approximate analytical solutions can be used to improve simulation efforts by uncovering possible scaling laws, thereby providing information to reduce the number of simulations needed Furthermore, analytical solutions can provide guidance on how to improve numerical methods for solving the problem in question Finally, a summary of conclusions and suggestions for future works are given in Chapter
5 In particular, future work is given
FLEXIBLE BEAM MODELING
Equations of Motion of The Moving Cantilever Beam
Figure 1 shows a flexible cantilevered beam of constant length L One end of the beam is clamped into a translational hub with the mass m moved in the 1D plane by an external force f(t) Moreover, shear deformation, rotary inertia, and the effect of axial force are neglected w(z, t)
(a) 1 st Mode (b) 2 nd Mode (c) Forced Vibration α(t) u(z, t)
Figure 2.1 Configuration of vibration cantilever beam
For a displacement of the hub α and an elastic deflection 𝑤, the total displacement 𝑢(𝑧, 𝑡) of a point along the beam at distance z from the hub can be described:
𝑢(𝑧, 𝑡) = 𝛼(𝑡) + 𝑤(𝑧, 𝑡) (2.1) The equations of motion for flexible Euler-Bernoulli beam will be derived using
7 where T, P, and W are the kinetic energy, potential energy, and work done by external forces, respectively
The total kinetic energy includes the kinetic energy of the hub, of the beam, and of the tip load
The kinetic energy of the hub
The kinetic of the beam
In this case, the length of the beam is considered as a constant
(2.6) Where 𝑚, 𝜌 are the mass of hub, the mass density per unit length
The total potential energy (generate by bending moment of the beam) can be found as
Work Done by external forces to the beam is:
𝑊 = 𝑓(𝑡)𝛼(𝑡) (2.8) where the force f is applied to the hub to generate the motion (t).
Derivation of Equation of Motion
2.2.1 The variation with respect to variable
The variation of the kinetic energy with respect to variable 𝛼 can be written as:
The variation of the potential energy with respect to variable 𝛼 can be written as:
The variation of the work done with respect to variable 𝛼 can be written as:
Substituting the Eqs (2.9), (2.10) and (2.11) into Hamilton’s principle, we have:
Because 𝛿𝛼 vanishes at 𝑡 = 0 and 𝑡 = 𝑡 𝑓 , 𝛿 𝛼 can be rewritten as:
The virtual displacement 𝛿𝛼is arbitrary and independent Hence, we must have:
2.2.2 The variation with respect to variable 𝒘
The variation of the kinetic energy with respect to the variable w can be written as:
(2.15) The variation of the potential energy with respect to variable 𝑤 can be written as:
The variation of the work done with respect to variable 𝑤 can be written as:
Substituting the above Eqs into Hamilton’s principle, we have
Because 𝛿𝑤 vanishes at 𝑡 = 0 and 𝑡 = 𝑡 𝑓 , 𝛿 𝑤 can be rewritten as:
The displacement 𝛿𝑤 is arbitrary Hence, we must have:
0 = 0 Moreover, 𝛿𝑤 and 𝛿𝑤 ′ vanish at the point𝑧 = 0, 𝛿𝑤and 𝛿𝑤 ′ ≠ 0 at the point𝑧 = 𝐿
We obtain boundary conditions: at 𝑧 = 0, 𝑤 = 0, 𝑤 ′ = 0 at z = 𝐿, 𝐸𝐼𝑤 ‴ = 0, 𝑤 ″ = 0 (2.21)
The Galerkin Decomposition
Using Eq (2.14), (2.20) and (2.21), the dynamic equation of the flexible beam can be determined:
𝑧 = 𝐿, 𝐸𝐼𝑤 ‴ = 0, 𝑤 ″ = 0 (2.24) Based on the Galerkin procedure, the elastic deflection w is approximated by a series of time-varying coefficients 𝑞 𝑖 (𝑡) and linear undamped mode shape function𝜑 𝑖 (𝑧)
𝑖=1 (2.25) where the linear undamped mode shape function of the cantilever beam is assumed as follows
𝜑 𝑖 (𝑧) = 𝐴 𝑖 𝑐𝑜𝑠ℎ𝜆 𝑖 𝑧 + 𝐵 𝑖 𝑠𝑖𝑛ℎ𝜆 𝑖 𝑧 + 𝐶 𝑖 𝑐𝑜𝑠 𝜆 𝑖 𝑧 + 𝐷 𝑠𝑖𝑛 𝜆 𝑖 𝑧 (2.26) The derivative of the mode shape function is:
Substitute (2.27) into the boundary condition 𝑤(0, 𝑡) = 0, we have
Substitute (2.27) into the boundary condition𝑤 ′ (0, 𝑡) = 0, we have
Substitute (2.27) into the boundary condition 𝑤 ″ (𝐿, 𝑡) = 0, we have
From the boundary condition 𝐸𝐼𝑢 ‴ = 0 and equation (2.1), we have:
The boundary conditions must be satisfied for all time (i.e., arbitrary choice of t) Hence, we must have
In order to obtain a nontrivial solution, the determinant of the coefficient matrix (2.11) must is zero, i.e.,
1 + 𝑐𝑜𝑠ℎ(𝜆𝐿) 𝑐𝑜𝑠(𝜆𝐿) = 0 (2.34) The solution of the above equation is the free vibration frequencies of the system For each free vibration frequency, we can assume that any of the term in four terms
𝐴 𝑖 , 𝐵 𝑖 , 𝐶 𝑖 , 𝐷 𝑖 is constant If we set 𝐴 𝑖 = 1, then 𝐶 𝑖 = −1 The equations (2.33) is redundant, and it becomes:
By solving this, 𝐵 𝑖 and 𝐷 𝑖 are obtained The linear undamped mode shape function of the cantilever beam is determined as follows:
Hence, equation (2.23) can be rewritten:
The finite-dimensional dynamical system is obtained by integrating the multi Eq (2.22) and the weighting function 𝜑 𝑗 (𝑥)
The above integer can be rewritten:
When a system is influenced by viscous damping, it is possible to write the generalized forces corresponding to this as:
𝜉 is the distributed viscous-damping coefficient
The n-mode description of the motion equation of the beam and the boundary condition (2.3) can be presented in the following matrix form
The term 𝑝 contains external forces other than the viscous damping forces
Equation (2.40) can be expressed in the state-space form:
The natural frequencies and modes of the undamped system satisfy the algebraic eigenproblem
Giving the eigenpairs (𝜔 𝑟 2 , 𝜙 𝑟 ), 𝑟 = 1,2, … , 𝑁 the modes 𝜙 𝑟 are assumed to have been normalized and the modal parameters (𝑀 𝑟 , 𝐶 𝑟 , 𝐾 𝑟 ) are calculated by:
IDENTIFICATION METHOD
Introduction to Experimental Modal Analysis
The experimental determination of natural frequencies, mode shapes, and damping ratios is called experimental modal analysis (EMA) and is based on vibration measurements that fall within the general designation of modal testing The objective of this form of vibration testing is to acquire sets of frequency response functions (FRFs) that are sufficiently accurate and extensive, in both the frequency and spatial domains, to enable analysis and extraction of the dynamic properties for all the required modes of vibration of the structure Prior knowledge of areas such as vibration analysis, instrumentation, signal processing, and modal identification, to state just a few, is required to understand modal testing
In this section, we emphasize the important role played in the experimental modal analysis by frequency-response functions (FRFs) Although real structures may experience various forms of damping and may exhibit nonlinear behavior, this introduction to EMA is restricted to the assumption that the structure under consideration can be modeled as a finite-DOF, linearly elastic, viscously damped system Therefore, in the present discussion, the N-DOF mathematical model as equation (2.40)
The solution for the steady-state response 𝑢, based on the modes 𝜙 𝑟 of the undamped system, is given by
(3.2) Where 𝜔 is the forcing frequency in rad/s, which is varied over the frequency range of interest in the modal test
The steady-state displacement response at coordinate i due to harmonic (force) excitation of unit magnitude only at coordinate j is called the frequency-response function for the response at i due to excitation at j When the FRF deals with displacement per unit force, it is called a receptance FRF However, it is far more common to measure acceleration and use accelerance FRFs The (complex) receptance FRF has the form
In modal testing, the forcing frequency is usually given as f hertz = 𝜔/2𝜋, so the forcing frequency ratio for the r th mode is 𝑟 𝑟 = 2𝜋/𝜔 𝑟 Obviously, FRFs contains information about natural frequencies (in 𝜔 𝑟 ), damping factors (in 𝜉 𝑟 ), and mode shapes (in 𝜙 𝑟 ) and these are directly related to the mass, damping, and stiffness properties of the structure Modal testing is the procedure that is employed to measure FRFs, and from them to estimate these physical properties of the structure being tested
Some observations about the information that can be obtained from the FRFs:
- At every forcing frequency f, the FRF matrix is symmetric That is, 𝐻 𝑖𝑗 (𝑓) 𝐻 𝑗𝑖 (𝑓)
- For this simple structure, the natural frequencies are widely separated
- Since the modes are lightly damped and the natural frequencies are widely separated, when the forcing frequency is equal to one of the undamped natural frequencies (i.e., when 𝑓 = 𝑓 𝑟 ), the FRFs are dominated by that r th mode Therefore, peaks occur in the FRFs at or very near every natural frequency
Figure 3.1 Interrelation among dynamic models.
Estimation of Frequency-Response Functions
The mathematical representation of a single-input-single-output frequency response function 𝐻(𝑓), is given by
We need to consider procedures for estimating FRFs from the measured signal Figure 3.1 shows a model that is frequently used to assess how best to calculate FRFs from
“noisy” input and output signals
The auto-power spectrum of a time-domain signal 𝑎(𝑡) is
𝐺 𝐴𝐴 = 𝐴(𝑓)𝐴 ∗ (𝑓) (3.5) and the cross-power spectrum of time-domain signals 𝑎(𝑡) and 𝑏(𝑡) is
20 where 𝐴(𝑓) and 𝐵(𝑓) are the Fourier transforms of signals 𝑎(𝑡) and 𝑏(𝑡), respectively, and ( ) ∗ denotes the complex conjugate Note that the auto-power spectrum is a real function of frequency, but the cross-power spectrum is a complex function that carries both magnitude and phase information
If 𝐻(𝑓) represents a linear system and 𝑃(𝑓) and 𝑈(𝑓) are, respectively, the transforms of the true input and output signals, we can form
𝑈(𝑓)𝑃 ∗ (𝑓) = 𝐻(𝑓)𝑃(𝑓) ∗ 𝑃(𝑓) 𝑎𝑛𝑑 𝑈(𝑓)𝑈 ∗ (𝑓) = 𝐻(𝑓)𝑃(𝑓)𝑈 ∗ (𝑓) (3.7) by post multiplying Eq 3.4 by 𝑃 ∗ (𝑓) and 𝑈 ∗ (𝑓), respectively Therefore, 𝐻(𝑓) can be calculated from either
However, we do not have true input and output signals but must use the measured signals, which contain added noise
To minimize the effect of the noise, we obtain the following averaged auto-power spectrum from different segments ( ) 𝑛 of the time-domain signal 𝑎(𝑡):
(3.7) and the following averaged cross-power spectrum of time-domain signals 𝑎(𝑡) and 𝑏(𝑡):
If the input noise m(t) and output noise n(t) are not correlated with each other or with the true input signal or true output signal, then with sufficient averaging, the respective auto- and cross-spectra become
Figure 3.2 Measurement system with noise sources
Finally, using averaged spectra of measured input and output signals, and using the two- equation for H(f) in Eq 3.6, we define the following three FRF estimators
From Eqs 3.11 and 3.12, we can see that:
• 𝐻 1 (𝑓) ≤ 𝐻 0 (𝑓) ; that is, at every frequency 𝑓, 𝐻 1 (𝑓) is a lower bound to the true FRF
• 𝐻 1 (𝑓) in affected only by input noise m(t) Therefore, 𝐻 1 is the poorest near resonances, where the input is small
• 𝐻 2 (𝑓) ≥ 𝐻 0 (𝑓) that is, at every frequency f, 𝐻 2 (𝑓) is an upper bound to the true FRF
• 𝐻 2 (𝑓) in affected only by output noise n(t) Therefore, 𝐻 2 (𝑓) is the poorest near antiresonances, where the output is small
• 𝐻 𝑣 (𝑓) is an average of 𝐻 1 (𝑓) and 𝐻 2 (𝑓)
A measure of the amount of the output signal that is due to the input signal is the ordinary coherence, called 𝛾 2 , which is given by
It can be shown that 𝛾 2 ≤ 1.0, with 𝛾 2 = 1.0 when there is no noise on the input or output Most modal analyzers provide for calculating and displaying the coherence along with the associated FRF
Reference works on modal testing should be consulted for corresponding discussions of the effects of noise on the calculation of FRFs and coherence from tests that involve multiple inputs
From Eq 3.3, the magnitude as a function of the forcing frequency is given by
(3.14) and the phase (lag) angle 𝛼 𝑟 (i.e., ∠𝐻 𝑢𝑟 = −𝛼 𝑟 ) is given by tan 𝛼 𝑟 (𝑓) = 2𝜉 𝑟 𝑟 𝑟
In modal testing, the forcing frequency is usually given as f hertz = 𝜔/2𝜋, so the forcing frequency ratio for the r th mode is 𝑟 𝑟 = 2𝜋/𝜔 𝑟 The receptance FRF can also be expressed in terms of its real part,
(3.17) For the systems with modal damping, the mobility FRF is given by
(3.18) and the accelerance FRF (i.e., acceleration output at i per unit force input at j) is given by
Finally, the partial-fraction (pole-residue) form for the receptance FRF of an underdamped MDOF system is
(3.20) where 𝐴 𝑖𝑗𝑟 , 𝐴 𝑖𝑗𝑟 ∗ are the (complex conjugate) modal residues, and
24 are the corresponding poles By equating the expressions for 𝐻 𝑖𝑗 (𝑓) in Eqs 3.3 and 3.18 on a mode-by-mode basis, it can be shown that the analytical expressions for the residues of an MDOF system with modal damping are the pure imaginary quantities given by
However, in modal testing practice, the mass matrix is seldom known, and the modal damping factor can be estimated only roughly Therefore, the following expression is the pole-residue equation that is most appropriate for representing the real-normal-mode form of the receptance FRF matrix in experimental modal analysis:
(3.22) where the values of the 𝑄 are determined in the process of fitting this expression to measured FRF data.
Modal Parameter Estimation
The first phase of an experimental modal analysis is the acquisition of analog time- domain records of inputs and responses The second phase is digital signal processing, involving sampling, filtering, calculation of Fourier transforms, calculation of frequency response functions, and so on The third phase consists of estimating and validating the modal parameters of the test article, including the complex-valued poles (𝜆 𝑟 = 𝜎 𝑟 +
𝑖𝜔 𝑑𝑟 , the modal vectors 𝜃 𝑟 , the modal scaling (modal mass 𝑀 𝑟 or modal 𝑎 𝑟 ), and the modal participation vectors This phase is called modal parameter estimation
Modal parameter estimation procedures can be identified according to several categories, including the following:
- Modal parameter versus direct parameter Direct-parameter methods are used to compute an MCK model (Eq 2.40) or a state-space model (Eq 2.41) from which modal parameters are then calculated
- Frequency domain versus time domain Frequency-domain methods use FRFs or auto- and cross-spectra; time-domain methods use impulse-response functions
- SDOF versus MDOF SDOF methods identify one mode at a time on one FRF at a time; MDOF methods identify several modes on one or more FRFs at a time
- Single input versus multiple input Multiple-input (poly reference) testing is required when a high order model is required or when the test structure has closely spaced frequency modes
From frequency-response functions that have been calculated from measured input and vibration response data by the-procedures discussed above provide the primary data that are used in an experimental modal analysis to calculate estimates of the modal parameters Expressions for FRFs for MDOF systems were developed in the previous section Based on Eq 3.22 we can write the following expression for the receptance FRF matrix
(3.23) where, for generality, the complex-mode form has been selected In practice, in a modal test there will be 𝑁 𝑖 inputs, 𝑁 𝑜 outputs, and 𝑁 active modes Then, for parameter identification purposes, Eq 3.23 is written in the form
Typically, modal parameter estimation is concentrated on a limited-frequency band containing 𝑁 𝜆 observable modes All modes having natural frequencies below this band are approximated by a single term, called the lower residual term or residual inertia term All of the modes above the band of interest are represented by a single term called the upper residual or residual flexibility Then Eq 3.24 takes the form
The residual terms in Eq 3.25, especially the residual flexibility term, make it possible to match the measured FRFs more closely over the frequency range of interest, particularly near antiresonances.
Measurement Techniques
In order to perform modal testing, a number of hardware components must be available These components may be interfaced with a host computer allowing for coordination of the operation of the overall system and enhancing the data-processing capabilities if adequate software is available Basically, there are three main measurement mechanisms:
- The data acquisition and processing mechanism
The excitation mechanism is constituted by a system which provides the input motion to the structure under analysis, generally under the form of a driving force f(t) applied at a given coordinate There are many variants for this system, their choice depending on several factors such as the desired input, accessibility, and physical properties of the test structure The excitation signals, in these cases, are generated by a signal generator and can be chosen from a variety of different possibilities (stepped-sine, swept sine, impulse, random, etc.), to match the requirements of the structure under test
The sensing mechanism is, basically, constituted by sensing devices known as transducers There is a large variety of such devices The transducers generate electrical signals that are proportional to the physical parameters one wants to measure A common way is to use a contactless motion transducer, such as a laser sensor or camera The laser sensor is based on the detection of the shift of a laser beam light scattered from the moving surface The camera can capture the edge of the marked position and extract the digital value in the coordinate of the image
The basic objective of the data acquisition and processing mechanism is to measure the signals developed by the sensing mechanism and to ascertain the magnitudes and phases of the excitation forces and responses Due to the rapid development of computers, there is nowadays a tendency for the computer to replace the analyzer, provided it incorporates adequate software and a data acquisition board More sophisticated systems are based on personal computers (PCs) or workstations with an acquisition front-end module These solutions have the additional advantage of allowing for storage and data processing without having to go through the intermediate steps of data transfer If based on a portable computer (lap-top), these systems may easily be carried to the test site Understanding the principles behind signal acquisition and processing is very important for anyone involved with the use of modal analysis test equipment The validity and
28 accuracy of the experimental results may strongly depend on the knowledge and experience of the equipment user
Various excitation signals are commonly used in experimental modal analysis They may be classified as single-frequency and broadband excitation signals, and each technique incorporates several different methods Although each method has specific advantages and disadvantages, the selection of a testing technique is often based (unfortunately) on the type of equipment and expertise available rather than on its suitability for a particular job
Stepped sine and swept sine are the two types of single-frequency excitation signals commonly used In the stepped sine testing technique, a shaker is used to excite the structure sinusoidally at a single, precisely controlled frequency The structure is allowed to settle under this excitation, to remove any transient effects Steady-state measurements are then made of the magnitude and phase relationship between the input force and the response at the precise excitation frequency for any desired response location The division of the response by the force input gives the value of the FRF at that particular frequency The excitation frequency is then changed by a small increment and the measurement process is repeated once the structure has settled at the new frequency Thus, the FRF can be constructed for any frequency range
One of the advantages of the stepped-sine testing technique is the large signal-to-noise ratio for all the force and response measurements Furthermore, the input ranges for the force and response signals can be adjusted automatically for each excitation frequency point, and this allows the best possible use of the measuring equipment This technique is specially adequate to investigate nonlinear behavior
In the swept-sine testing technique, a shaker is used to excite the structure with a sine signal whose frequency varies slowly and continuously along with the test frequency range Sets of simultaneous measurements of the excitation signal and response signals are then taken at a given rate The frequency of the sine signal allows a low rate of change, assuming that each set of measurements corresponds to virtually steady-state characteristics, i.e., that the technique sinusoidally excites the structure for each set of measurements at virtually one single frequency
Though similar, swept sine techniques are faster than stepped sine techniques However, since there is a continuous rate of frequency shift, accuracy is affected
In broadband testing, the structure is excited with a signal containing energy over a wide range of frequencies simultaneously The time-domain force and response signals are filtered, digitized, and then passed through a Fourier analysis process to transform the time domain information to frequency domain spectra By appropriate combination of the force and response spectra, the required FRFs for the structure can be derived
Broadband excitation techniques may be classified as nonperiodic (purely random), periodic (pseudorandom, periodic random, and periodic chirp) and transient (burst random, burst chirp, and impact) Random, periodic random, and impact are probably the most commonly used types of broadband excitation signals
Periodic random excitation is a special form of periodic excitation, similar to pseudo- random, which has several benefits for signal processing Although called ‘random’ it is not random in the true sense of the word Within a time period equal to the analysis time frame, the signal is random, but the same signal is then repeated continuously Usually, the signals are generated inside the data acquisition equipment in the frequency domain The spectrum of a periodic random signal consists of discrete frequencies at integer multiples of the frequency resolution used by the DFT Once the measurement frequency
30 range and number of frequency lines have been selected, a flat excitation frequency spectrum may be generated by setting the magnitude of all spectral lines to the same value The phases of the spectral components are then randomized By use of the inverse Fourier transform (IFT), this frequency domain spectrum is transformed into the time domain to produce a random-like excitation signal in the analysis time frame As a consequence of generating the signal in this way, it is exactly periodic in the analysis time frame, and both the force and response signals will be periodic in the analysis time frame also
The values measured by the test equipment represent electrical voltages and therefore it is necessary to obtain a calibration factor that translates these values into units of acceleration and force
Though transducer manufacturers generally provide reliable calibration information, the use of their quoted sensitivities may not be accurate enough since they can change with time and environmental conditions In addition, the transducers may have suffered some kind of damage due to rough handling or other extreme conditions and, although still working, may have lost their response linearity Finally, the remaining units in the measurement chain (amplifiers, filters, signal conditioners, cable lengths, etc.) may change, albeit slightly, the overall sensitivity It is, therefore, good practice to recalibrate the transducers before performing a test, preferably using the same measuring set-up that will be used in the test program
Various transducer calibration techniques are available to the test engineer The simplest and most common calibration procedure is the classical back-to-back method that compares the accelerometer to be calibrated with a reference accelerometer This entails keeping a special reference transducer that offers a high level of linearity and stability Another simple calibration procedure is based on the use of small hand-held calibrators
31 that most manufacturers commercialize In this case, the calibration is performed at one frequency only and, therefore, a flat frequency response of the transducer over the frequency band of interest is assumed
NUMERICAL SIMULATION AND EXPERIMENTAL RESULTS
Numerical Simulation Result
In order to obtain the resonant frequencies of the flexible beam described in Fig 1.1, the flexible cantilever beam is considered with the system parameters listed in Table 4.1 These parameters are used in both numerical simulations and experiments
Beam height of area h 28.6 mm
Beam width of area b 0.88 mm
Beam linear density ρ 0.00806 kg.m -1 Beam moment of inertia I 1.624210 -12 M 4 Beam Young modulus E 20010 9 N/m 2
Numerical simulations to determine the resonant frequencies corresponding to the first and second modes are performed by MATLAB R2016a software Eq (2.41) with a chirp signal, the external force is solved using function ode45 of MATLAB The chirp signal with a frequency varies from 0 to 30 Hz was applied to the moving hub to have the chance of exciting the first two modes of the structure
The amplitude vibration of the beam corresponds to the chirp is depicted in the following Fig 4.1 and Fig 4.2 The resonant frequency corresponding to the first mode is 3.535
Hz, and the one for the second mode is 21.99 Hz
Figure 4.1 Time response of the chirp excitation and output signal
Figure 4.2 Simulation frequencies response of the system
The result obtained through experiments and simulations is shown in figure 4.3 The simulation data shows in Fig 4.3 correspond to the beam response taken at the first part of the figure, where the first mode provides more impact Due to the uncertainty information of the damping, the simulation result could not show clearly the effect of the second mode
Figure 4.3 Comparison of the response obtained by the experiment and the numerical simulation
Experimental Results
The experiment set-up (as shown in Fig 4.4) is used to verify the studied cantilever beam An aluminum beam with the parameters as shown in Table 4.1 is installed in the linear motor of 2 DOF positioning system The external force is generated by the linear motor Yokogawa The linear motor system is controlled by a motion controller UMAC The motion program controlling the whole linear motor system is generated from accompanying software with the UMAC controller (PeWin32Pro) The measured signals
35 are collected via a PC with a NI PCI motion card and processed in the LabVIEW program The input signal is encoder feedback pulse from the linear motor drive and the beam vibration’s data in pixels is produced by a high-speed camera, then transferred to the computer through an Ethernet cable The laser sensor is used for calibration procedure in order to convert pixel value to mm unit The whole system is synchronized with the clock generated from the UMAC controller
Figure 4.4 Schematic diagram of the experimental set-up
The vision method of measurement of beam vibrations is based on the recording of data of images that contains the marked point’s displacement The purpose of this analysis is to determine vibration parameters describing the studied beam The amplitude of vibrations is determined for each of the characteristic points of the beam by detecting the position’s edge of the interesting points in the beam Measurements using the vision method are carried out in two stages:
- Configuration of the vision system, calibrate the measured value
- Apply the excitation signal to the system, gathering input/output data, analyzing the experimental data
Figure 4.5 The studied beam with a dark background
In the first stage, the optical system is configured The image is set up so that the entire cantilever beam is visible in the capturing space of the camera The parameters of the vision system are chosen to enable image recording with a sampling rate of 100 frames/second Images are processed inside the vision controller (Keyence CX420) with a resolution of 1600×1000 pixels The sampling rate is selected with regard to the resolution of the capturing area of the camera, the size of the studied object, and the lighting conditions at the experimental set-up These parameters are selected on the basis of tests; their selection is dictated by the necessity of obtaining appropriate contrast between elements of the scene: the color of the marked points in the beam and the background Fig 4.5 presents a prospect in which the beam and background, enabling
37 observation of entire the beam visible To carry out the experiment, the excitations are applied to the beam at two interesting point points as presented in Fig 4.6 The so-called y10 and y225 will stand for two measured outputs respectively in Fig 4.6 These variables will be continuously used in other results
Captured by Camera at 100hz
Output Measured input Measured output
Figure 4.6 Measured points in the beam
Before gathering the measured output signals, a calibration procedure is taken to convert the digital value extracted from the vision controller to mm unit In this procedure, the moving hub attached in the linear motor will make some equal movement segments back and forth A laser sensor is used to measure the actual movement on the beam, while the vision controller returns the movement of the interesting point in pixel to the computer when the beam completes a move and keep stable Several couple values will be used to find the curve fitting to convert pixels value to mm value by using Curve Fitting tool in MATLAB 2016a Then, the coefficients achieved from the Curve Fitting tool will be programmed in a gathering data program to covert the pixel data to displacement in mm
38 data automatically during the experiment Fig 4.7 shows the steps for this procedure, Fig 4.8 shows the parameters for this curve fitting line The equation form of the interpolation curve fitting of two displacement data is constructed as
𝑓(𝑥) = 𝑝 1 𝑥 9 + 𝑝 2 𝑥 8 + 𝑝 3 𝑥 7 + 𝑝 4 𝑥 6 + 𝑝 5 𝑥 5 + 𝑝 6 𝑥 4 + 𝑝 7 𝑥 3 + 𝑝 8 𝑥 2 + 𝑝 9 𝑥 Where 𝑓(𝑥) is the displacement after converting from pixel data and 𝑥 is the return displacement from the camera in pixel The precision of the data transformation depends on the RMSE coefficient (root mean square), then in this calibration procedure, the process finishes when the RMSE coefficient is less than 0.1
Figure 4.7 Calibration procedure set-up
Figure 4.8 Curve fitting line converting the pixels value (from camera) to mm value
The second stage of measurement using the vision method involves applying the excitations and data gathering and data processing to achieve the natural frequencies and transfer functions The measured input and output signals are sampled at 0.005 sec and the signal duration is 60 sec
Three types of applied excitation signals are:
- Chirp signal (to obtain the resonance frequencies), Fig 4.10 shows the two resonance frequencies with chirp signal input are 3.567Hz and 22.15Hz The chirp signal has the basic form of the equation that generates the array is:
2𝑡 2 ) The signal is scaled with amplitude 𝐴 = 0.4𝑚𝑚 to obtain the beam vibration displacement magnitude in the captured area of the vision system The interested frequencies are between 0-30 Hz in 60 sec in total interval length with 0.005 sec selected interval
- Multi-sine wave signal (for finding transfer functions and validating the obtained model), Fig 4.11 shows the measured input and output signal Fig 4.12 shows signal spectrum analysis The multi-sine input signals are constructed according to:
Where l is the total number of frequencies and
𝑙 ; 2 ≤ 𝑖 ≤ 𝑙 With 𝜙 1 = 0 The amplitude of 𝑢(𝑡) is then scaled such that the signal kept between 1 and -1
- Step signal (to validate the estimated model)
There are six datasets of multi-sine excitation, three of which are used for estimation, and the other three are used for model validation
The frequencies used for estimation (labeled as E1, E2 and E3) are given by
The accuracy of the estimated model is then validated based on unseen data Hence, three datasets are independent of the estimation datasets are also provided The frequency content of the three multi-sine inputs used for validation (labeled as V1, V2, and V3) is given by
These signals are applied to the moving hub in the manner described by the variable a(t) in the direction perpendicular to the beam axis In this way, an excitation is affected The applied signals are generated from MATLAB 2016a, then the value will be converted to the motion program that the UMAC controller running During the experiment, a running LabVIEW program will gather the measured input signal from the encoder feedback pulse of the linear motor drive and the measured output signal from the camera, it also converts the pixels value to mm unit itself After the experiment finishes, the collected data will be extracted to the MATLAB for detailed analysis purposes From input/output data, we can achieve the natural frequencies, transfer functions, and damping ratios In this section, the transfer functions extracted from input/output signals are achieved by using the CONTSID Toolbox (software support for data-based continuous-time modeling)
The estimated transfer functions of two interesting points have the form:
𝑛=1 where 𝑘 𝑛 is the gain of the open-loop, n is the mode number, 𝜉 𝑛 and 𝜔 𝑛 are the damping ratio and natural frequency of the n th mode, respectively However, it is impractical to identify the infinite number of vibration modes In practice, only a certain bandwidth is of interest, and dynamic response of the flexible structures is often dominated by the vibrations of the lower modes Therefore, the higher modes’ parameters are neglected
In this case, we will consider significantly the first two modes’ parameters of the system, which is given by:
CONCLUSIONS
In this dissertation, we investigated the vibrations of a flexible cantilever beam subjected to a moving hub In particular, we studied both experimentally and theoretically the behaviors of the modeling cantilever beam and the real system
By Euler-Bernoulli beam theory, the mode shape frequencies are theoretically calculated for the cantilever beam subjected to a moving hub The flexible cantilever beam modeling was investigated to obtain the natural frequencies and their mode shapes
An experimental parametric identification technique to estimate the linear damping coefficients and to obtain the natural frequencies was also deployed We carried out an experimental study of the response of a metallic flexible cantilever beam A system including a motion system and vision system was developed to support the experiment
We observed various dynamic phenomena, like two vibration modes, the modulation frequency; which depends on various parameters like the amplitude and frequency of excitation, damping factors, etc The fact that flexible cantilever beam under different excitation could display nonlinear distortions phenomena, which depend on the amplitude of the excitation, the interested frequencies of the excitation, and also the surrounding environment (humidity, thermal noise…), the initial geometric shape of the beam But since these usual distortions have been observed in real engineering structures and mechanical systems, there is a need to study such phenomena in more detail
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APPENDIX: The CONTSID Toolbox: A Software Support for Data-based
The CONtinuous-Time System IDentification (CONTSID) toolbox provides Matlab functions for estimating continuous-time black-box models of dynamical systems from measured data without having to fully characterize the mathematics governing the system behavior The toolbox includes tools for standard identification of linear continuous-time models such as simple process, transfer functions and state-space models The toolbox also provides algorithms for more advanced identification such as errors-in-variable (EIV) and closed-loop model estimation or to capture nonlinear system dynamics This paper presents an overview of the main features of the latest release of the CONTSID toolbox and outlines some recent developments for on-line parameter and time-delay system estimation
The CONTSID toolbox to be run with Matlab was the first toolbox entirely dedicated to continuous-time (CT) model identification from sampled data It was first released in
1999 (Garnier and Mensler, 1999) at a time where discrete-time model identification was the classical approach Fortunately, things have recently changed and continuous-time model identification has now taken over discrete-time model identification as exemplified by the more pronounced role of continuous-time model in the System identification toolbox (Ljung and Singh, 2012) One of the clear reasons is coming from the fact that control scientists and engineers have a better understanding and every-day practice of continuous time models, while they are less familiar with input/output polynomial black-box models such as discrete-time ARX, ARMAX or Box-Jenkins models
The CONTSID toolbox includes tools for basic identification of linear black-box continuous-time models such as:
- Identification of simple (low-order) process models;
- Identification of transfer function models;
- Identification of input/output black-box polynomial models such as autoregressive (CARX), output-error (COE) and Box-Jenkins (CBJ) models;
- Identification of state-space models with free or canonical parametrizations;
- Identification from time-domain response data;
- Identification from frequency-domain response data
The CONTSID toolbox also includes tools for more advanced identification such as:
- Identification from irregularly sampled data;
- Identification of errors-in-variables (EIV) models;
- Identification of nonlinear block-oriented (Hammerstein and Hammerstein- Wiener) models;
- Identification of linear parameter varying (LPV) input/output models;
- On-line identification for tracking time-varying system dynamics
In practice, the common system identification workflow is iterative as shown in Figure 6.1 (Ljung, 1999) It includes several tasks Starting from measured input/output data, a set of candidate models is estimated by using suitable identification algorithms The identified model which produces the best results according to the chosen validation criterion is finally selected The system identification work flow is general and pragmatic It is independent of the chosen discrete-time or continuous-time model parametrization used, although the latter can present many advantages
Figure 6.1 The system identification procedure
The latest version 7.3 of the CONTSID toolbox offers a variety of parametric model estimation methods for common linear and nonlinear model structures Tables 6.1 and 3.2 summarize the main CONTSID toolbox commands for standard linear model identification and for more advanced identification respectively
Input/output polynomial models lssvf (CARX models) ivsvf (CARX models) coe (COE models) srivc (COE models) rivc (CBJ models)
State-space models sidgpmf ssivgpmf