Vibrations Fundamentals and Practice ch07 Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.
de Silva, Clarence W “Damping” Vibration: Fundamentals and Practice Clarence W de Silva Boca Raton: CRC Press LLC, 2000 Damping Damping is the phenomenon by which mechanical energy is dissipated (usually converted into internal thermal energy) in dynamic systems A knowledge of the level of damping in a dynamic system is important in utilization, analysis, and testing of the system For example, a device having natural frequencies within the seismic range (i.e., less than 33 Hz) and having relatively low damping, could produce damaging motions under resonant conditions when subjected to a seismic disturbance Also, the device motions could be further magnified by low-frequency support structures and panels having low damping This illustrates that a knowledge of damping in constituent devices, components, and support structures is particularly useful in the design and operation of a complex mechanical system The nature and the level of component damping should be known in order to develop a dynamic model of the system and its peripherals A knowledge of damping in a system is also important in imposing dynamic environmental limitations on the system (i.e., the maximum dynamic excitation the system could withstand) under in-service conditions Furthermore, a knowledge of its damping could be useful in order to make design modifications in a system that has failed the acceptance test The significance of knowledge of damping level in a test object, for the development of test excitation (input), is often overemphasized, however Specifically, if the response-spectrum method is used to represent the required excitation in a vibration test, it is not necessary that the damping value used in the development of the required response spectrum specification be equal to actual damping in the test object It is only necessary that the damping used in the specified response spectrum be equal to that used in the test-response spectrum (see Chapter 10) The degree of dynamic interaction between test object and shaker table, however, will depend on the actual level of damping in these systems Furthermore, when testing near the resonant frequency of a test object, it is desirable to have a knowledge of damping in the test object, because it is in this neighborhood that the object response is most sensitive to damping In characterizing damping in a dynamic system, it is important, first, to understand the major mechanisms associated with mechanical-energy dissipation in the system Then, a suitable damping model should be chosen to represent the associated energy dissipation Finally, damping values (model parameters) are determined, for example, by testing the system or a representative physical model, by monitoring system response under transient conditions during normal operation, or by employing already available data 7.1 TYPES OF DAMPING There is some form of mechanical-energy dissipation in any dynamic system In the modeling of systems, damping can be neglected if the mechanical energy that is dissipated during the time duration of interest is small in comparison to the initial total mechanical energy of excitation in the system Even for highly damped systems, it is useful to perform an analysis with the damping terms neglected, in order to study several crucial dynamic characteristics; for example, modal characteristics (undamped natural frequencies and mode shapes) Several types of damping are inherently present in a mechanical system If the level of damping that is available in this manner is not adequate for proper functioning of the system, external damping devices can be added either during the original design or in a subsequent stage of design modification of the system Three primary mechanisms of damping are important in the study of mechanical systems They are: ©2000 CRC Press Internal damping (of material) Structural damping (at joints and interfaces) Fluid damping (through fluid-structure interactions) Internal (material) damping results from mechanical-energy dissipation within the material due to various microscopic and macroscopic processes Structural damping is caused by mechanicalenergy dissipation resulting from relative motions between components in a mechanical structure that has common points of contact, joints, or supports Fluid damping arises from the mechanicalenergy dissipation resulting from drag forces and associated dynamic interactions when a mechanical system or its components move in a fluid Two general types of external dampers can be added to a mechanical system in order to improve its energy dissipation characteristics They are passive dampers active dampers A passive damper is a device that dissipates energy through some motion, without needing an external power source or actuator Active dampers have actuators that need external sources of power They operate by actively controlling the motion of the system that needs damping Dampers may be considered as vibration controllers (see Chapter 12) The present chapter emphasizes damping that is inherently present in a mechanical system 7.1.1 MATERIAL (INTERNAL) DAMPING Internal damping of materials originates from the energy dissipation associated with microstructure defects, such as grain boundaries and impurities; thermoelastic effects caused by local temperature gradients resulting from non-uniform stresses, as in vibrating beams; eddy-current effects in ferromagnetic materials; dislocation motion in metals; and chain motion in polymers Several models have been employed to represent energy dissipation caused by internal damping This variability is primarily a result of the vast range of engineering materials; no single model can satisfactorily represent the internal damping characteristics of all materials Nevertheless, two general types of internal damping can be identified: viscoelastic damping and hysteretic damping The latter term is actually a misnomer, because all types of internal damping are associated with hysteresis-loop effects The stress (σ) and strain (ε) relations at a point in a vibrating continuum possess a hysteresis loop, such as the one shown in Figure 7.1 The area of the hysteresis loop gives the energy dissipation per unit volume of the material, per stress cycle This is termed per-unit-volume damping capacity, and is denoted by d It is clear that d is given by the cyclic integral d= ∫ σdε (7.1) In fact, for any damped device, there is a corresponding hysteresis loop in the displacement-force plane as well In this case, the cyclic integral of force with respect to the displacement, which is the area of the hysteresis loop, is equal to the work done against the damping force It follows that this integral (loop area) is the energy dissipated per cycle of motion This is the damping capacity, which, when divided by the material volume, gives the per-unit-volume damping capacity as before It should be clear that, unlike a pure elastic force (e.g., spring force), a damping force cannot be a function of displacement (q) alone The reason is straightforward Consider a force f(q) that depends on q alone Then, for a particular displacement point q of the component, the force will be the same regardless of the magnitude and direction of motion (i.e., the value and sign of q˙) ©2000 CRC Press FIGURE 7.1 A typical hysteresis loop for mechanical damping It follows that, in a loading and unloading cycle, the same path will be followed in both directions of motion Hence, a hysteresis loop will not be formed In other words, the net work done in a complete cycle of motion will be zero Next consider a force f(q, q˙) that depends on both q and q˙ Then, at a given displacement point q, the force will depend on q˙, as well Hence, even at low speeds, force in one direction of motion can be significantly different from that in the opposite direction As a result, a hysteresis loop will be formed, which corresponds to work done against the damping force (i.e., energy dissipation) One can conclude then that damping force has to depend on a relative velocity q˙ in some manner In particular, Coulomb friction, which does not depend on the magnitude of q˙, does depend on the sign (direction) of q˙ Viscoelastic Damping For a linear viscoelastic material, the stress-strain relationship is given by a linear differential equation with respect to time, having constant coefficients A commonly employed relationship is σ = Eε + E * dε dt (7.2) which is known as the Kelvin-Voigt model In equation (7.2), E is Young’s modulus and E* is a viscoelastic parameter that is assumed to be time independent The elastic term Eε does not contribute to damping, and, as noted before, mathematically, its cyclic integral vanishes Consequently, for the Kelvin-Voigt model, damping capacity per unit volume is dv = E * dε ∫ dt dε (7.3) For a material that is subjected to a harmonic (sinusoidal) excitation, at steady state, one obtains ε = ε max cos ωt ©2000 CRC Press (7.4) When equation (7.4) is substituted in equation (7.3), one obtains dv = πωE * ε 2max Now, ε = εmax when t = in equation (7.4), or when to equation (7.2), is σmax = Eεmax It follows that dv = (7.5) dε = The corresponding stress, according dt πωE * σ 2max E2 (7.6) These expressions for dv depend on the frequency of excitation, ω Apart from the Kelvin-Voigt model, two other models of viscoelastic damping are also commonly used They are, the Maxwell model given by dσ dε = E* dt dt (7.7) dσ dε = Eε + E * dt dt (7.8) σ + cs and the standard linear solid model given by σ + cs It is clear that the standard linear solid model represents a combination of the Kelvin-Voigt model and the Maxwell model, and is the most accurate of the three But, for most practical purposes, the Kelvin-Voigt model is adequate Hysteretic Damping It was noted that the stress, and hence the internal damping force, of a viscoelastic damping material depend on the frequency of variation of the strain (and consequently on the frequency of motion) For some types of material, it has been observed that the damping force does not significantly depend on the frequency of oscillation of strain (or frequency of harmonic motion) This type of internal damping is known as hysteretic damping Damping capacity per unit volume (dh) for hysteretic damping is also independent of the frequency of motion and can be represented by dh = Jσ nmax (7.9) As clear from equation (7.6), a simple model that satisfies equation (7.9), for the case of n = 2, is given by σ = Eε + E˜ dε ω dt (7.10) which is equivalent to using a viscoelastic parameter E* that depends on the frequency of motion in equation (7.2) according to E* = E˜ /ω Consider the case of harmonic motion at frequency ω, with the material strain given by ©2000 CRC Press ε = ε cos ωt (7.11) Then, equation (7.10) becomes σ = Eε cos ωt − E˜ ε sin ωt = Eε cos ωt + E˜ ε cos ωt + π 2 (7.12) Note that the material stress consists of two components, as given by the right-hand side of equation (7.12) The first component corresponds to the linear elastic behavior of a material and is in phase with the strain The second component of stress, which corresponds to hysteretic damping, is 90° out of phase (this stress component leads the strain by 90°) A convenient mathematical representation would be possible, by using the usual complex form of the response according to ε = ε e jωt (7.13) Then, equation (7.10) becomes ( ) σ = E + jE˜ ε (7.14) It follows that this form of simplified hysteretic damping can be represented using a complex modulus of elasticity, consisting of a real part that corresponds to the usual linear elastic (energy storage) modulus (or Young’s modulus) and an imaginary part that corresponds to the hysteretic loss (energy dissipation) modulus By combining equations (7.2) and (7.10), a simple model for combined viscoelastic and hysteretic damping can be given by E˜ dε σ = Eε + E * + ω dt (7.15) in which the parameters E, E *and E˜ are independent of the frequency ω The equation of motion for a system for which the damping is represented by equation (7.15) can be deduced from the pure elastic equation of motion by simply substituting E with the operator E˜ ∂ E + E* + ω ∂t in the time domain Example 7.1 Determine the equation of flexural motion of a non-uniform slender beam whose material has both viscoelastic and hysteretic damping Solution The Bernoulli-Euler equation of bending motion of an undamped beam subjected to a dynamic load of f(x,t) per unit length is given by (see Chapter 6): ©2000 CRC Press ∂2 ∂2 q ∂2 q + = f ( x, t ) ρ A EI ∂x ∂x ∂t (7.16) Here, q is the transverse motion at a distance x along the beam Then, for a beam with material damping (both viscoelastic and hysteretic), one can write, ∂2 ∂2 q ∂2 EI + 2 ∂x ∂x ∂x * E˜ ∂ q ∂2 q E + ω I ∂t∂x + ρA ∂t = f ( x, t ) (7.17) in which ω is the frequency of the external excitation f(x,t) in the case of steady forced vibrations In the case of free vibration, however, ω represents the frequency of free-vibration decay Consequently, when analyzing the modal decay of free vibrations, ω in equation (7.17) should be replaced by the appropriate frequency (ωi) of modal vibration in each modal equation Here, the resulting damped vibratory system possesses the same normal mode shapes as the undamped system The analysis of the damped case is very similar to that for the undamped system, as noted in Chapter Ⅺ 7.1.2 STRUCTURAL DAMPING Structural damping is a result of the mechanical-energy dissipation caused by rubbing friction resulting from relative motion between components and by impacting or intermittent contact at the joints in a mechanical system or structure Energy-dissipation behavior depends on the details of the particular mechanical system in this case Consequently, it is extremely difficult to develop a generalized analytical model that would satisfactorily describe structural damping Energy dissipation caused by rubbing is usually represented by a Coulomb-friction model Energy dissipation caused by impacting, however, should be determined from the coefficient of restitution of the two members that are in contact The common method of estimating structural damping is by measurement The measured values, however, represent the overall damping in the mechanical system The structural damping component is obtained by subtracting the values corresponding to other types of damping, such as material damping present in the system (estimated by environment-controlled experiments, previous data, etc.), from the overall damping value Usually, internal damping is negligible compared to structural damping A large portion of mechanical-energy dissipation in tall buildings, bridges, vehicle guideways, and many other civil engineering structures, and in machinery such as robots and vehicles takes place through the structural-damping mechanism A major form of structural damping is the slip damping that results from the energy dissipation by interface shear at a structural joint The degree of slip damping that is directly caused by Coulomb (dry) friction depends on such factors as joint forces (e.g., bolt tensions), surface properties, and the nature of the materials of the mating surfaces This is associated with wear, corrosion, and general deterioration of the structural joint In this sense, slip damping is time dependent It is common practice to place damping layers at joints, to reduce undesirable deterioration of the joints Sliding will cause shear distortions in the damping layers, causing energy dissipation by material damping and also through Coulomb friction In this way, a high level of equivalent structural damping can be maintained without causing excessive joint deterioration These damping layers should have a high stiffness (as well as a high specific-damping capacity) in order to take the structural loads at the joint For structural damping at a joint, the damping force varies as slip occurs at the joint This is primarily caused by local deformations at the joint, which occur with slipping A typical hysteresis loop for this case is shown in Figure 7.2(a) The arrows on the hysteresis loop indicate the direction ©2000 CRC Press FIGURE 7.2 Some representative hysteresis loops: (a) typical structural damping; (b) Coulomb friction model; and (c) simplified structural damping model of relative velocity For idealized Coulomb friction, the frictional force (F) remains constant in each direction of relative motion An idealized hysteresis loop for structural Coulomb damping is shown in Figure 7.2(b) The corresponding constitutive relation is f = c sgn(q˙ ) (7.18) in which f is the damping force, q is the relative displacement at the joint, and c is a friction parameter A simplified model for structural damping caused by local deformation can be given by f = c q sgn(q˙ ) ©2000 CRC Press (7.19) FIGURE 7.3 A body moving in a fluid medium FIGURE 7.4 Mechanics of fluid damping The corresponding hysteresis loop is shown in Figure 7.2(c) Note that the signum function is defined by sgn(v) = = −1 7.1.3 for v ≥ for v < (7.20) FLUID DAMPING Consider a mechanical component moving in a fluid medium The direction of relative motion is shown parallel to the y-axis in Figure 7.3 Local displacement of the element relative to the surrounding fluid is denoted by q(x,z,t) The resulting drag force per unit area of projection on the x-z plane is denoted by fd This resistance is the cause of mechanical-energy dissipation in fluid damping It is usually expressed as fd = c ρq˙ sgn(q˙ ) d (7.21) in which q˙ = ∂q(x,z,t)/∂t is the relative velocity The drag coefficient cd is a function of the Reynold’s number and the geometry of the structural cross section Net damping effect is generated by viscous drag produced by the boundary-layer effects at the fluid–structure interface, and by pressure drag produced by the turbulent effects resulting from flow separation at the wake The two effects are illustrated in Figure 7.4 Fluid density is ρ For fluid damping, the damping capacity per unit volume associated with the configuration shown in Figure 7.3 is given by ©2000 CRC Press Lx Lz ∫ ∫ ∫ f dzdxdq( x, z, t) d df = 0 (7.22) Lx Lz q0 in which, Lx and Lz are cross-sectional dimensions of the element in the x and y directions, respectively, and q0 is a normalizing amplitude parameter for relative displacement Example 7.2 Consider a beam of length L and uniform rectangular cross section, that is undergoing transverse vibration in a stationary fluid Determine an expression for the damping capacity per unit volume for this fluid–structure interaction Solution Suppose that the beam axis is along the x-direction and the transverse motion is in the y-direction There is no variation in the z-direction, and hence, the length parameters in this direction cancel out Thus, L ∫ ∫ f dxdq( x, t) d df = Lq0 or T L ∫ ∫ f q˙( x, t)dxdt d df = 0 (7.23) Lq0 in which T is the period of the oscillations Assuming constant cd, substitute equation (7.21) into equation (7.23): cd ρ df = Lq0 L T ∫ ∫ q˙ dtdx (7.24) 0 For steady-excited harmonic vibration of the beam at frequency ω and shape function Q(x) (or for free-modal vibration at natural frequency ω and mode shape Q(x)), one has q( x, t ) = qmax Q( x ) sin ωt (7.25) In this case, with the change of variable θ = ωt, equation (7.24) becomes d f = cd ρ ©2000 CRC Press qmax Lq0 π2 L ∫ Q( x ) dxω ∫ cos θdθ 2µg 3µg Use the initial conditions x = − x − and x˙ = at t = π/ωn; then, B1 = and B2 = x − ωn ωn Hence, equation (x) becomes 3µg µg x = x − cos(ω n t ) − ω ω n n (xi) The object will come to rest ( x˙ = 0) next at t = 2π/ωn Hence, the position of the object at the end of the present half cycle would be x r1 = x − 4µg ω 2n (xii) The response for the next cycle is determined by substituting xr1, as given by equation (xii) which is the initial condition, into equation (v) for the left motion; determining the subsequent end point xl2, and using it as the initial condition for equation (x) for the right motion; and soon Then, one can express the general response as: [ ] (xiii) [ ] (xiv) Left motion in cycle i : x = x − ( 4i − 3)∆ cos ω n t + ∆ Right motion in cycle i : x = x − ( 4i − 1)∆ cos ω n t − ∆ where ∆= µg ω 2n (xv) Note that the amplitude of the harmonic part of the response should be positive for that half cycle of motion to be possible Hence, one must have FIGURE 7.17 A typical cyclic response under Coulomb friction ©2000 CRC Press FIGURE 7.18 Frictional characteristics of a pair of spur gears FIGURE 7.19 A friction model for rotatory devices x > ( 4i − 3)∆ for left motion in cycle i x > ( 4i − 1)∆ for right motion in cycle i Also note from equations (xiii) and (xiv) that the equilibrium (central) position for the left motion is +∆, and for the right motion it is –∆ A typical response curve is sketched in Figure 7.17 Ⅺ 7.4.1 FRICTION IN ROTATIONAL INTERFACES Friction in gear transmissions, rotary bearings, and other rotary joints has somewhat similar behavior Of course, the friction characteristics will depend on the nature of the device and also the loading conditions However, experiments have shown that the frictional behavior of these devices can be represented by the interface damping model given here Typically, experimental results are presented as curves of coefficient of friction (frictional force/normal force) versus relative velocity of the two sliding surfaces While in the case of rotary bearings the rotational speed of ©2000 CRC Press the shaft is used as the relative velocity, it is the pitch line velocity that is used for gears An experimental result for a pair of spur gears is shown in Figure 7.18 What is interesting to notice from the result is the fact that, for this type of rotational device, the damping behavior can be approximated by two straight-line segments in the velocity-friction plane — the first segment having a sharp negative slope and the second segment having a moderate positive slope, which represents the equivalent viscous damping constant, as shown in Figure 7.19 7.4.2 INSTABILITY Unstable behavior or self-excited vibrations such as stick-slip and chatter that is exhibited by interacting devices such as metal removing tools (e.g., lathes, drills, and milling machines) can be easily explained using the interface damping model In particular, it is noted that the model has a region of negative slope (or negative damping constant) that corresponds to low relative velocities, and a region of positive slope that corresponds to high relative velocities Consider the singledegree-of-freedom model mx˙˙ + bx˙ + kx = (7.109) without an external excitation force Initially, the velocity is x˙ = But, in this region, the damping constant b will be negative and hence the system will be unstable Thus, a slight disturbance will result in a steadily increasing response Subsequently, x˙ will increase above the critical velocity where b will be positive and the system will be stable As a result, the response will steadily decrease This growing and decaying cycle will be repeated at a frequency that primarily depends on the inertia and stiffness parameters (m and k) of the system Chatter is caused in this manner in interfaced devices PROBLEMS 7.1 7.2 a Give three desirable effects and three undesirable effects of damping b The moment of inertia of a door about its hinges is J kg·m2 An automatic door closer of torsional stiffness K N·m·rad–1 is attached to it What is the damping constant C needed for critical damping with this door closer? Give the units of C a Compare and contrast viscoelastic (material) damping and hysteretic (material) damping b The stress-strain relations for the Kelvin-Voigt, Maxwell, and standard linear solid models of material damping are: σ = Eε + E * 7.3 dε dt σ + cs dσ dε = E* dt dt σ + cs dσ dε = Eε + E * dt dt Sketch spring and dashpot lumped-parameter systems that represent these three damping models a The Kelvin-Voigt model of material damping is represented by the stress-strain model ©2000 CRC Press σ = Eε + E * dε dt and the standard linear solid model of material damping is represented by σ + cs dσ dε = E* dt dt Under what condition could the latter model be approximated by the former? b Damping capacity per unit volume of a material is given by d= ∫ σdε Also, the maximum elastic potential energy per unit volume is given by umax = 1/2 σmaxεmax , or, in view of the relation σmax = Eεmax for an elastic material, by umax = 1/2 Eεmax2, where εmax is the maximum strain in a load cycle Show that the loss factor for a Kelvin-Voigt viscoelastic material is given by η= 7.4 7.5 ωE * E Verify that the loss factors for the material damping models given in Table P7.4 are as given in the last column of the table What would be the corresponding damping ratio expressions? a A damping material is represented by the frequency-dependent standard linear solid model: σ + cs dσ g dε = Eε + + g2 dt dt ω Obtain an approximate expression for the damping ratio of this material TABLE P7.4 Loss Factors for Several Material Damping Models Material Damping Model Stress-Strain Constitutive Relation dε dt Loss Factor (η) ωE * E Viscoelastic Kelvin-Voigt σ = Eε + E * Hysteretic Kelvin-Voigt σ = Eε + Viscoelastic standard linear solid σ + cs dσ dε = Eε + E * dt dt * ωE * − cs E E E + ω cs Hysteretic standard linear solid σ + cs dσ E˜ dε = Eε + dt ω dt ˜ E˜ − ωcs E E E + ω cs ©2000 CRC Press E˜ dε ω dt E˜ E ( ( ( ( ) ) ) ) TABLE P7.6 Lumped-Parameter Models of Damping Damping Type 7.6 Damping Force d(x, x˙ ) per Unit Mass Hysteretic c x˙ ω Structural c x sgn( x˙ ) Structural Coulomb c sgn( x˙ ) b A thin cantilever beam that is made of this material has a mode of transverse vibration with natural frequency 15.0 Hz Also, the following parameter values are known: E = 1.9 × 1011 Pa, g1 = 6.2 × 109 Pa, g2 = 8.6 × 107 Pa·s, and cs = 1.6 × 10–4 s Estimate the modal damping ratio for this mode of vibration (Note: Pa = N·m–2) Consider the single-degree-of-freedom damped system given by the dynamic equation x˙˙ + d ( x, x˙ ) + ω 2n x = ω 2n u(t ) where x u(t) ωn d 7.7 7.8 = = = = response of the lumped mass normalized excitation undamped natural frequency damping force per unit mass Three possible cases of damping are given in Table P7.6 Determine an expression for the equivalent damping ratio in each of these three cases of lumped-parameter models a A load cycle is applied at low speed during axial testing of a test specimen The applied force and the corresponding deflection are measured and the area Af of the resulting hysteresis loop is determined The longitudinal stiffness of the specimen is k and the amplitude of the deflection during the loading cycle is x0 Obtain an expression for the loss factor η of the material Comment on the accuracy of this expression A hysteresis loop that was obtained from a cyclic tensile test on a specimen is shown in Figure P7.7 Estimate the damping ratio of the material a A torque cycle is applied during torsional testing of a shaft The applied torque and the corresponding angle of twist are measured and the area At of the resulting hysteresis loop is determined The torsional stiffness of the shaft is K and the amplitude of the angle of twist is θ0 Show that the loss factor η of the shaft material is given by η= 7.9 At πθ 20 K b A hysteresis loop obtained from a low-speed, cyclic torsional test on a shaft is shown in Figure P7.8 Estimate the damping ratio of the material a A cyclic tensile test was carried out at low speed on a specimen of metal and the stress versus strain hysteresis loop was obtained The area of the hysteresis loop was found to be As The Young’s modulus of the specimen was E and the amplitude of the axial strain was ε0 Show that the loss factor η of the material can be expressed as ©2000 CRC Press FIGURE P7.7 A force vs deflection hysteresis loop obtained from a cyclic tensile test FIGURE P7.8 A torque versus angle of twist hysteresis loop obtained from a cyclic torsional test ©2000 CRC Press FIGURE P7.9 A stress-strain hysteresis loop obtained from a cyclic tensile test η= As πε 20 E b A hysteresis loop that was obtained from a low-speed, cyclic stress-strain test is shown in Figure P7.9 Estimate the damping ratio of the material 7.10 a Consider a single-degree-of-freedom system with the displacement coordinate x If the damping in the system is of hysteretic type, the damping force can be given by fd = h x˙ ω where h is the hysteretic damping constant and ω is the frequency of motion Now, for a harmonic motion given by x = x0sin(ωt), show that the energy dissipation per cycle is ∆Uh = πx 02 h Also, if the stiffness of the system is k, show that the loss factor is given by η= h k b Consider a uniform cylindrical rod of length l, area of cross section A, and Young’s modulus E that is used as a specimen of tensile testing What is its longitudinal stiffness k? Suppose that for a single cycle of loading, the area of the stress-strain ©2000 CRC Press FIGURE P7.11 A damped lumped-parameter system hysteresis loop is As and the amplitude of the corresponding strain is ε0 Show that, for this rod, h= 7.11 7.12 As A πε 20 l What is the damping ratio of the rod in axial motion? What is proportional damping? What is its main advantage? A two-degree-of-freedom lumped-parameter system is shown in Figure P7.11 Using the influence coefficient approach, determine the damping matrix of this system An automated wood-cutting system contains a cutting unit that consists of a DC motor and a cutting blade, which are linked by a flexible shaft and coupling The purpose of the flexible shaft is to locate the blade unit at any desirable configuration, away from the motor itself A simple, lumped-parameter dynamic model of the cutting unit is shown in Figure P7.12 The following parameters and variables are shown in the figure: Jm bm k Jc bc Tm θm ωm Tk θc ωc TL = = = = = = = = = = = = axial moment of inertia of the motor rotor equivalent viscous damping constant of the motor bearings torsional stiffness of the flexible shaft axial moment of inertia of the cutter blade equivalent viscous damping constant of the cutter bearings magnetic torque of the motor motor angle of rotation motor speed torque transmitted through the flexible shaft cutter angle of rotation cutter speed load torque on the cutter from the workpiece (wood) In comparison with the flexible shaft, the coupling unit is assumed rigid, and is also assumed light The cutting load is given by TL = c ω c ω c The parameter c, which depends on factors such as the depth of cut and the material properties of the workpiece, is assumed to be constant in the present problem a Comment on the suitability of the damping models used in this problem ©2000 CRC Press FIGURE P7.12 A wood-cutting machine b Using Tm as the input, TL as the output, and [ωm Tk ωc]T as the state vector, develop a complete (nonlinear) state model for the system shown in Figure P7.12 What is the order of the system? c Using the state model derived in part (a), obtain a single input-output differential equation for the system, with Tm as the input and ωc as the output d Consider the steady operating conditions, where Tm = Tm, ωm = ωm, Tk = Tk , ωc = ωc , and TL = TL are all constants Express the operating point values ωm, Tk, ωc, and TL in terms of Tm and model parameters only You must consider both cases, Tm > and Tm < e Now consider an incremental change Tˆ in the motor torque and the corresponding m changes ωˆ m, Tˆk, ωˆ c, and TˆL in the system variables Determine a linear state model (A, B, C, D) for the incremental dynamics of the system in this case, using x = [ ωˆ , Tˆ , ωˆ ]T as the state vector, u = [ Tˆ ]T as the input and y = [ Tˆ ]T as the output m k c m L f In the nonlinear model (see part (b)), if the twist angle of the flexible shaft (i.e., θm – θc) is used as the output, what would be a suitable state model? What is the system order then? g In the nonlinear model, if the angular position θc of the cutter blade is used as the output variable, explain how the state model obtained in part (b) should be modified What is the system order in this case? h For vibration analysis of the wood-cutting machine, the damped natural frequencies and the associated damping ratios are required How many natural frequencies and damping ratios would you expect for this problem? How would you determine them? Hint for Part (e): ( ) d ω ω = ω c ω˙ c dt c c ( ) d2 ˙˙ c + 2ω˙ c2 sgn(ω c ) ωc ωc = ωc ω dt ©2000 CRC Press FIGURE P7.13(a) A machine with an active suspension FIGURE P7.13(b) The stability region for Mathieu equation 7.13 A machine with an active suspension system is schematically shown in Figure P7.13(a) The mass of the machine is m The active suspension system provides a variable stiffness k(t) by means of a hydraulic actuator The vertical displacement of the machine is denoted by y Under steady conditions, it was found that the suspension stiffness fluctuates about an average value k0 according to the relation k (t ) = k0 − k1 cos ωt It is suspected that this is due to an error in the current amplifier that provides the drive signal to the actuator The frequency of the fluctuation is in fact the line frequency (of ©2000 CRC Press the AC supply) and is 60 Hz Also, k1 = 2.84 × 107 N·m–1 and m = 1000 kg Determine the range of k0 for which the system will be stable Hint: For a system given by the Mathieu equation: d2y + (a − 2b cos 2t ) y = dt the stability depends on the values of a and b, as given by the stability curves of Figure P7.13(b) 7.14 a Prepare a table to compare and contrast the following methods of damping measurement: Logarithmic decrement method Step-response method Hysteresis loop method Magnification-factor method Bandwidth method with regard to the following considerations: Domain of analysis (time or frequency?) Whether it can measure several modes simultaneously Accuracy restrictions Cost Speed Model limitations b A machine with its suspension system weighs 500 kg The logarithmic decrement of its free decay under an initial-condition excitation was measured to be 0.63, and the corresponding frequency was 8.0 Hz i Compute the undamped natural frequency and the damping ratio of the system ii Suppose that the machine, under normal operating conditions, generates an unbalance force f = f0cosωt with the force amplitude f0 = 4.8 × 104 N and the frequency ω = 15.0 × 2π rad·s–1 What is the amplitude of the steady-state vibration of the machine under this excitation force? iii Estimate the resonant frequency and the half-power bandwidth of the system 7.15 A commercial fish processing machine (known as the “Iron Butcher”) has a conveyor belt with holding pockets The fish are placed in the holding pockets and are held from the top using a stationary belt, as schematically shown in Figure P7.15(a) It was found that, under some conditions, the fish undergo stick-slip type vibratory motion during conveying A model that can be used to analyze this unstable behavior is shown in Figure P7.15(b) The model parameters are: m = mass of a fish k = stiffness of a fish b = equivalent damping constant of dissipation between the stationary holding belt and a fish v = velocity of the conveyor x = absolute displacement of a fish Using this model, explain the stick-slip motion of a fish 7.16 a Compare the free decay response of a system under linear viscous damping, with that under Coulomb friction b An object of mass m is restrained by a spring of stiffness k and slides on a surface against a constant Coulomb frictional force F Obtain expressions for the peak motion ©2000 CRC Press FIGURE P7.15 (a) Conveying of fish in a fish processing machine, and (b) dynamic model for analyzing the stick-slip response of a fish in the i+rth cycle in terms of that of the ith cycle, separately, for the two directions of motion 7.17 a Consider an object in cross flow of fluid with velocity v as shown in Figure P7.17 Suppose that the object vibrates in the direction transverse to the fluid flow, at cyclic frequency f The representative transverse dimension of the object is d A nondimenv sional velocity is known to determine the nature of vibration of the object In df particular, v i For small (in the range of 1.0 to 10.0), vortex shedding predominates (e.g., for df large d and f or for stationary fluid) v ii For intermediate (in the range of 10 to 100), galloping predominates (e.g., for df cylindrical objects at reasonably high v, as in transmission lines) v iii For large (in the range of 100 to 1000), flutter predominates (e.g., thin objects df such as aerofoils at high fluid flow speeds) What are other factors that determine the nature of vibration of the object? b Suppose that the object is stationary in a fluid flowing at speed v Then there will be a drag force fd acting on the body in the direction of v, and a lift force fl acting on the body in the transverse direction Thus, fd = c v2 d fl = cv l where cd = drag coefficient cl = lift coefficient are nonlinear parameters and will vary with the direction of the flow ©2000 CRC Press FIGURE P7.17 A vibrating object in a cross flow of fluid Now consider the system shown in Figure P7.17 where the object has a transverse speed y˙ , and the fluid flows at a steady speed v Show that the equivalent linear viscous damping constant for the fluid-structure interaction is given by b= ∂c (0) 1 cd (0) − cl (0) − l v 2 ∂θ where θ = angle of attack = tan–1 y˙ /v and cd(0), cl(0), and ∂cl (0) are the values of ∂θ ∂cl when the object is stationary (i.e., y˙ = or θ = 0) ∂θ The Bernoulli-Euler beam equation is given by cd, cl, and 7.18 ∂2 ∂2v ∂2v + = f ( x, t ) EI ρ A ∂x ∂x ∂t where v(x,t) f(x,t) E I ρ A = = = = = = beam response applied force per unit length of the beam Young’s modulus second moment of area of beam cross-section, about the neutral axis bending mass density area of cross section Derive the corresponding beam equation with material damping represented by the i Kelvin-Voigt model σ = Eε + E * dε dt ii Standard linear solid model σ + cs where ©2000 CRC Press dσ dε = Eε + E * dt dt E* = viscoelastic damping parameter cs = standard linear solid model parameter 7.19 Consider a nonuniform rod of length l, area of cross section A, mass density ρ, and Young’s modulus E Assume that the ends are free and the rod executes longitudinal vibrations of displacement u(x,t) at location x (see Figure P7.19) The equation of free motion is known to be −ρA ∂2u ∂ ∂u + EI =0 ∂t ∂x ∂x with boundary conditions ∂u (0, t ) = ∂x ∂u (l , t ) = ∂x The corresponding modal motions are, for the ith mode, u( x, t ) = Yi ( x ) sin ω i t with the mode shapes Yi ( x ) = cos iπx x and the natural frequencies ωi = iπ l E ρ Consider the following two cases of damping: i External damping of linear viscous type given by a damping force per unit length along the beam: −b ∂u ∂t ii Material damping of the Kelvin-Voigt type, given by the stress-strain equation ∂ σ = E 1 + c ε ∂t 7.20 For each case, determine the modal loss factor and modal damping ratio Compare/contrast these results for the two cases of damping Consider a vibrating system with damping, given by the normalized equation ˙˙ y + 2ζω n y˙ + ω 2n y = u(t ) ©2000 CRC Press FIGURE P7.19 Damped longitudinal vibration of a rod where u(t) y ωn ζ = = = = forcing function response undamped natural frequency damping ratio Suppose that the system is excited by a harmonic force so that at steady state, the response is given by y = y0 sin ωt a Derive the shape of the u vs y curve for this motion b What is the energy dissipation per cycle of motion? ©2000 CRC Press ... (7.8) σ + cs and the standard linear solid model given by σ + cs It is clear that the standard linear solid model represents a combination of the Kelvin-Voigt model and the Maxwell model, and is the... the magnitude and direction of motion (i.e., the value and sign of q˙) ©2000 CRC Press FIGURE 7.1 A typical hysteresis loop for mechanical damping It follows that, in a loading and unloading... structures and panels having low damping This illustrates that a knowledge of damping in constituent devices, components, and support structures is particularly useful in the design and operation