Mechanics Research Communications 85 (27) 5 Contents lists available at ScienceDirect Mechanics Research Communications journal h om epa ge: www.elsevier.com/locate/mechrescom Load s temporal characteristics for annulling forced vibrations of linear elastic plates Arka P. Chattopadhyay, Romesh C. Batra Department of Biomedical Engineering and Mechanics, M/C 29, Virginia Polytechnic Institute and State University, Blacksburg, VA, 246, USA a r t i c l e i n f o Article history: Received 8 June 27 Received in revised form July 27 Accepted 25 July 27 Available online 29 July 27 Keywords: Load duration Fundamental bending frequency Vibration attenuation Linearly elastic structures a b s t r a c t We consider trapezoidal load-time pulses with linearly increasing and affinely decreasing durations equal to integer multiples of the time period of the first bending mode of vibration of a linearly elastic structure. For arbitrary spatial distributions of loads applied to monolithic and laminated orthotropic plates, it is shown through numerical solutions that plates vibrations become miniscule after the load is removed. This phenomenon is independent of the dwell time (i.e., the time duration between the rising and the falling portions) during which the load is kept constant. The primary reason for this response is that for such time-dependent loads, nearly all of plate s strain energy is concentrated in deformations corresponding to the fundamental bending mode of vibration. Thus plate s deformations can be studied by taking the mode shape of the st bending mode as the basis function and reducing the problem to that of solving a single second-order ordinary differential equation. We have verified this postulate by comparing strain energies computed from the 3-dimensional deformations of different plate geometries and boundary conditions with those determined by using the single degree of freedom (DoF) model. Thus for trapezoidal time-dependent loads applied on plates, the DoF model provides reasonably accurate results and saves considerable computational effort. 27 Elsevier Ltd. All rights reserved.. Introduction The 2nd order ordinary differential equation (ODE) with constant coefficients governing the forced motion of a linear springmass system under given initial conditions can be analytically solved. For a time-dependent trapezoidal force that increases linearly in time, stays constant for time t 2, drops affinely to zero in time t 3, and then stays at zero, the analytical solution gives that the mass initially at rest comes to rest for all times greater than the loading time of + t 2 + t 3. This holds if and only if and t 3 are integer multiples of the time-period of the spring-mass system. We refer to this phenomenon as load-dependent vibration attenuation. This observation for the spring-mass system inspired us to investigate if a similar result holds for linearly elastic continuous structures for which and t 3 are integer multiples of the timeperiod of the first bending mode of vibrations of the plate. The plate is loaded either on the entire or on a part of its major surfaces that induce bending dominant deformations. For very thick plates the fundamental mode of vibration may involve only in-plane motions with null transverse displacements, e.g., see [,2]. However, we Corresponding author. E-mail address: rbatra@vt.edu (R.C. Batra). focus on studying problems having bending-dominant deformations. The load-dependent vibration attenuation result has been verified by analyzing 3-dimensional deformations of linearly elastic square and circular monolithic, fiber-reinforced laminated and functionally graded (FG) plates by the finite element method (FEM) using the commercial software, ABAQUS [3]. We also analyze the plate deformations by taking the mode shapes of free bending vibrations of the plate as basis functions that uncouples equations of motion for different mode shapes, and compute the plate strain energy for deformations corresponding to a desired mode shape. For each problem studied with the above-specified load time curve, we found that nearly 95% of the strain energy of deformations of the plate is due to its deformations in the first bending mode of vibrations. Subsequent to the load removal, plate s average acceleration oscillates around zero with a significantly smaller amplitude than that prior to the load removal. The practical significance of the result is that for the load-time variations stipulated, no external stimuli, e.g., dampers, are needed to annul free vibrations of the system after the load is removed. In several books on dynamic problems for discrete and continuous systems, e.g., see [4,5], we have not seen this result. We also have not found any experimental work for the load-time variations envisaged here. http://dx.doi.org/.6/j.mechrescom.27.7.9 93-643/ 27 Elsevier Ltd. All rights reserved.
6 A.P. Chattopadhyay, R.C. Batra / Mechanics Research Communications 85 (27) 5 Non-dimensional forcef(t)/f max By using simple trigonometric identities, we write Eq. (3) as ( ) cos ωt 3 sin(ωt 3/2) sin ωt sin(ωt 3/2) sin ωt 3 cos(ωt 3/2) ( ) t 3 sin ωt 3 sin(ωt 3/2) x (t) = 2Fmax mω 3 cos ωt (4) cos ωt 3 cos(ωt [ 3/2) ] sin + sin(ωt/2) ωt sin(ω/2) + cos ωt cos(ω/2) + t 2 + t 2 + t 3 Time We conclude from Eq. (4) that x (t) = x (t) = x (t) = x (t) = for t > T 3 if and only if Fig.. Trapezoidal load with rise time, dwell time t 2, and fall time t 3. = 2n ω, t 3 = 2n 3 ω (5), 2. Forced vibrations of a linear spring-mass system For a linear spring-mass system of mass m, spring constant k, initially at rest, and subjected to a trapezoidal time-dependent load, F(t), depicted in Fig. and described by F (t) = t [H (t) -H (t T )] + [H (t T ) H (t T 2 )] where n and n 3 are integers. That is, if the linearly increasing loading and the affinely decreasing unloading times are integer multiples of the time period of the fundamental frequency of vibration, then, irrespective of the dwell time, t 2, between the loading and the unloading times, the spring mass-system ceases to vibrate after the load is removed. We call this phenomenon of vanishing of post-load removal vibrations as load-dependent vibration attenuation. The result holds even if and t 3 equal zero provided that the load duration t 2 is given by t 3 (t ( + t 2 + t 3 )) [H (t T 2 ) H (t T 3 )] () t 2 = 2n 2 ω (6) T 3 = + t 2 + t 3, T 2 = + t 2, T = the displacement, x (t), of the spring from its initial unstretched position is given by { } t T 3 H [t T 3 ] t 3 ( { }) ω sin ω t T 3 { } t T 2 x (t) = F max H [t T 2 ] t 3 mω 2 ( { }) ω sin ω t T 2 (2) (t T ) H (t T ) t ω sin (ω (t T )) + ( t ) ω sin (ωt) In Eqs. () and (2) H(y) is the Heaviside step function that equals for y and otherwise, andω = k/m is the natural frequency of the linear spring-mass system in rad/s. After removal of the load, the displacement of the mass in the free vibration state for time t > T 3 is given by x (t) = F max mω 2 ωt 3 + ω ( ) cos ωt 3 sin ωt cos ω (T 3 t 3 ) ( ) sin ωt 3 cos ωt sin ω (T 3 t 3 ) [ ] sin ωt (cos ωt ) cos ωt sin ωt (3) For m = kg, k = 2 N/m, =.2 s, as shown in Fig. 2, the motion of the mass ceases upon removal of the load. Since ω = for this example problem, we have set n = n 2 = in Eqs. (5) and (6). Results plotted in Fig. 2(d) confirm the necessity of conditions (5) since for n =.25 the mass has a steady oscillatory motion after removal of the load. Time histories of the kinetic and the potential energies (not exhibited here) show that they essentially become null upon the load removal implying that there is no source left to drive the system. 3. Vibration attenuation of simply supported and clamped plates We hypothesize that for a linearly elastic plate with the normal traction, g(t) q(x), on its major surfaces with g(t) shown in Fig., and and t 3 given by Eq. (5) where ω equals its first frequency of free bending vibrations, the plate nearly comes to rest upon the load removal. Here q(x) is an arbitrary function of x. We note that for thick plates, the fundamental frequency may correspond to in-plane vibrations for which transverse displacements identically vanish. We exclude these by restricting ourselves to bending-dominant deformations of the plate. We numerically show this for a cantilever Euler-Bernoulli beam, a simply supported square plate (SSP), a clamped square plate (CSP), a clamped circular plate (CCP), a clamped laminated fiberreinforced composite plate (CLP), and a rectangular clamped FG plate (CFGP). Transient 3-dimensional deformations of plates are numerically analyzed by the finite element method (FEM) with the commercial software, ABAQUS [3], using 8-node brick elements with the 2 2 x 2 Gauss integration rule. The FE mesh is successively refined and the time step size decreased to get a converged solution of the problem. The primary reason for the plate response being similar to that of a linear spring-mass system is that for the function g(t) satisfying the hypothesis, nearly all of the strain energy of deformations is concentrated in the first bending mode of vibration.
A.P. Chattopadhyay, R.C. Batra / Mechanics Research Communications 85 (27) 5 7.8.6.4.2.8.6.4.2.8.6.4.2.5 (a).5.5 (b).5.5 (c).5.5 (d).5.5 Fig. 2. Time histories of the displacement of the mass for (a) =.2 s, t 2 = and t 3 =.2 s, (b) =.2 s, t 2 =.5 s and t 3 =.2 s, (c) =.2 s, t 2 =. s and t 3 =.4 s, and (d) =.25 s, t 2 =.2 s and t 3 =.2 s. Taking the mass-normalized N eigenvectors, A (i), of free bending vibrations of a linear elastic plate as base vectors, the 3-D displacements in d (x, x 2, x 3, t) of N nodes can be written as N d (x, x 2, x 3, t) = (i) (t) A (i) (x, x 2, x 3 ) (7) i= Fig. 3. Schematic representation of the plate geometry. where (i) (t) is the time-dependent component of d along the eigenvector A (i). Referring the reader to Hughes text book [6] on the FEM, the 2nd order ODE for the evolution of (i) (t)is (i) + ( ω (i)) 2 (i) = f (i) (t) (8) where f (i) (t) is the component of the external nodal load vector along the eigenvectora (i). For finding the strain energy of deformation corresponding to the i th mode of vibration, we solve ODE (8) for (i), use Eq. (7) to determine nodal displacements without summing on the index i, and then compute the strain energy from the stiffness matrix and the nodal displacements. Results computed by taking N = in Eq. (7) are indicated in Figures as mode. For some example problems, we show that the strain energy of plate s deformations corresponding to the 2nd mode of bending vibrations is negligible as compared to that of its total strain energy. For all example problems studied, the fundamental mode of vibration was found to be bending. 3.. Simply supported plate (SSP) With reference to the coordinate axes and the plate dimensions shown in Fig. 3, we enforce the following boundary conditions on the plate edges on x =, l, u 2 = u 3 =, = on x 2 =, b, u = u 3 =, 22 = For a cm x cm x cm plate with Young s modulus, E = 2 GPa, Poisson s ratio, =.3 and the mass density, = 7.2 g/cc, we have listed in Table the first five converged natural frequencies computed with a uniform mesh of 5,2 (8 8 8) 8-node brick elements. We have also provided the corresponding frequencies from the analytical solution for bending vibrations of Srinivas and Rao [7]. Results from the higher-order plate theory are also listed to show that in-plane modes of vibration exist for thick plates; [] and [8]. It is clear that the presently computed first five natural frequencies are within % of the analytical solutions of Srinivas and Rao [7] and the next two frequencies of in-plane modes of vibration agree well with those from the higher-order plate theory. As pointed out by Batra and Aimmanee [], the in-plane modes of vibration (modes 4 and 5) are absent in the analytical solution of Srinivas and Rao [7] since they tacitly studied only bending vibrations. To study the vibration attenuation phenomena, we consider the SSP and assign the following values to its material parameters: E = 25 GPa, =.25 and = 2.5 g/cc. Hereafter, unless otherwise mentioned, we use these values of material properties in all example problems. The frequency of mode vibrations of the SSP from the FE solution of the 3D problem was obtained as 2.847 khz or equivalently the time-period of the fundamental mode = 35 s. The computed time history of the deflection of the centroid of the top surface and the strain energy of the plate with q(x) =. GPa applied on the top surface and g(t) as a triangular pulse with
8 A.P. Chattopadhyay, R.C. Batra / Mechanics Research Communications 85 (27) 5 Table Non-dimensional frequencies ω n = ωh /E of the SS cm x cm x cm plate (* denotes in-plane mode of vibration). Mode Srinivas and Rao [7] Qian et al. [8] Batra and Aimmanee [] Present 3D FEM.578.578.578.577 2.38.39.39.378 3.38.39.39.378 4.948*.949.948* 5.948*.949.978* u 3 (a/2,b/2,) / h.8.6.4.2 =.8 =.9 = =. =.2 -.2 -.4 2 3 4 t / Fig. 5. Time histories of the centroidal transverse deflection of the CSP for different rise and fall times of the load. Fig. 4. Time histories of the (a) out-of-plane displacement of the centroid of the plate and (b) the strain energy of the plate in modes and 2 of deformations, as well as in all modes of deformations. = 35 s is compared in Fig. 4 with that found by taking only one basis function. These results evince that the entire plate ceases to vibrate after the load removal, most of plate s strain energy is concentrated in the first mode of vibration, and the two centroidal deflections of the plate are very close to each other. 3.2. Clamped square plate (CSP) The converged fundamental frequency of bending vibrations and the corresponding time period of the cm x cm x cm CSP plate were found to be 4.89 khz and 24.5 s, respectively. For q(x) =. GPa on plate s top surface and with the load rise and fall times, = t = 24.5 s, and for ±% and ±2% variations in and t 3, we have exhibited in Fig. 5 time histories of the centroidal deflection found solving the 3-D problem. It is evident that the plate motion ceases upon removal of the load for = t but not for the other load durations. The time along the horizontal axis for each of the curves is normalized by. In Fig. 6 we have plotted time histories of the centroidal deflection and the total strain energy of the CSP for the trapezoidal load pulse, g(t), with = t and dwell times t 2 of.5,.75 and.5 obtained from the 3-D and the mode- deformations. It is clear that the vibration attenuation is independent of the value of t 2, the centroidal deflection is the maximum at the end of the linearly increasing load, stays constant during the dwell time, decreases to Fig. 6. Time histories of (a) the centroidal deflection, and (b) the strain energy of the plate under different trapezoidal loads obtained from the 3-D and the mode deformations.
A.P. Chattopadhyay, R.C. Batra / Mechanics Research Communications 85 (27) 5 9 Fig. 7. Time histories of the centroidal deflection of the CSP with the normal traction applied on different areas, represented by the shaded region in the insets, of the top surface of the plate. Fig. 9. Time histories of (a) the centroidal deflection, and (b) the total strain energy of the CCP under a triangular time variation of the uniform normal traction applied on the plate top surface. 3.3. Clamped circular plate (CCP) For a 5-mm thick clamped circular plate of radius 5 cm, the converged fundamental frequency computed with the FE mesh of 98,275 8-node elements equals 2.97 khz. With the uniformly distributed normal traction on plate s top surface that varies in time as a triangular pulse with = 336.7 s, the transient responses of the plate from the solution of the 3-D and the mode and mode 2 deformations are depicted in Fig. 9. It is clear that plate s vibrations attenuate upon removal of the load, and there is negligible strain energy associated with deformations corresponding to mode 2 vibrations. 3.4. Clamped 4-layered laminated composite plate (CLP) Fig. 8. Time history of the average acceleration of the plate. zero in the unloading phase and subsequent to the load removal it oscillates around zero with negligible amplitude Results displayed in Fig. 7 evince that the plate vibration attenuates even when the triangular load with = t is applied on a part of the plate top surface that is not necessarily symmetrically located around the plate s centroid. The time history of the average acceleration of all nodes of the plate, presented in Fig. 8, evinces that plate s acceleration does not become zero subsequent to the load removal. However, the maximum amplitude of the acceleration is small as compared to that before the load removal. We study transient deformations of a clamped cm cm four-layered laminated plate with each layer 2.5 mm thick and fibers oriented at, 5, 3 and 45 with respect to the global x-axis in the layers starting from the bottom to the top layer. The layer material is assumed to be transversely isotropic with the fiber direction as the axis of transverse isotropy and E L = 72.4 GPa, E T = E L /25, G LT = E T /2, G TT = E T /5, LT = TT =.25, = 2.5, = 2.5 g/cc. Here subscripts L and T denote, respectively, the fiber direction and a direction perpendicular to the fiber and G the shear modulus. Each layer is discretized using 8 8 2 uniform 8-node brick elements. The converged fundamental frequency of the plate equals 4.2 khz. For a spatially uniform pressure with no dwell time and = 248.2 s applied to the plate top surface, time histories of the transverse deflections of the centroid of the plate and of the strain energies obtained from the 3-D, mode and mode 2 deformations are presented in Fig.. It is evident that the vibration
A.P. Chattopadhyay, R.C. Batra / Mechanics Research Communications 85 (27) 5 Fig.. Time histories of (a) the centroidal transverse deflection, and (b) the total strain energy of the CLP for a triangular time variation of the uniform normal traction from 3D and the mode and mode 2 deformations. Fig.. Time history of (a) the centroidal deflection, and (b) the total strain energy of the rectangular CFGP under uniform triangular time pulse obtained from the 3-D, mode and mode 2 deformations. of the plate attenuates once the load is removed, the plate strain energy in the 2nd bending mode is negligible and that in the first mode nearly equals the strain energy found from the solution of the 3-D deformations. 3.5. Clamped functionally graded plate (CFGP) For analyzing the response of non-homogeneous plates, we consider a rectangular (2 cm x cm x cm) CFGP with ( E (z) = + z ) GPa () 2h where the thickness coordinate z varies from h/2 to h/2. We set =.25 and = 2.5 g/cc. The plate is divided into layers of equal thickness with each layer made of a homogeneous material having E deduced from Eq. () at its mid-point. Batra and Jin [9], amongst others, have shown that dividing an inhomogeneous plate into layers of homogeneous materials simulates well its frequencies. Each layer of the rectangular plate is meshed using 2 uniform 8-node brick elements. The fundamental frequency of the rectangular plate equals 6.8 khz. With the spatially uniform peak pressure of 2.76 GPa with = 47 s and t 2 =, time histories of the plate centroidal transverse displacement and the plate strain energy computed using the 3-D, the mode and the mode 2 deformations are presented in Fig.. As for the other problems studied, the displacements essentially become zero and stay at zero upon the load removal, and the strain energy corresponding to mode 2 deformations is negligible as compared to that of mode deformations. The time-history of acceleration of the non-homogenous plate averaged over the domain x [.l,.9l], y [.b,.9b] to eliminate effects of clamped boundaries is exhibited in Fig. 2. It is Fig. 2. Average acceleration of the interior nodes of the clamped nonhomogeneous plate from the 3D FEM solution. evident that after the load removal at 294 s, the plate acceleration sharply decreases and remains small. Thus, even though deformations in higher vibration modes make negligible contributions to the strain energy of the system, they do not cease to exist upon the load removal. 3.6. Cantilever Euler-Bernoulli (EB) beam The fundamental frequency of the 2 cm x mm EB beam made of an isotropic material with E = 25 GPa, =.25, = 2.5 g/cc equals.27 khz. For a triangular time load pulse with = 787.4 s, plots of Fig. 3 show that results computed from the basis function
A.P. Chattopadhyay, R.C. Batra / Mechanics Research Communications 85 (27) 5 affinely decreasing force of duration equal to integer multiples of the time-period of the fundamental frequency of the system, the mass comes to rest as soon as the applied force ceases to act. This motivated us to investigate if a similar result holds for linearly elastic continuous structures with the time-period of the first bending mode of vibration playing the role of the time-period of the linear spring-mass system. By studying free and forced vibration of several plates including laminates, we have found that indeed such a results holds for linearly elastic continuous structures. For all problems studied, we found that nearly all of the strain energy of deformation is due to deformations in the fundamental bending mode of vibration. Contributions from higher vibration modes, although insignificant to the total strain energy of the plate, manifest in the small amplitude acceleration response as indicated by non-zero average acceleration of the plate after the load removal. Acknowledgements This work was supported by the Office of Naval Research Grant N4-6--239 to Virginia Polytechnic Institute and State University with Dr. Y. D. S. Rajapakse as the Program Manager. Views expressed in the article are those of the authors and neither of the funding agency nor of authors institutions. References Fig. 3. Time histories of (a) deflection of free end of the beam, and (b) strain energy of the beam in mode, mode 2, and all modes of deformation. corresponding to the st mode of vibration, the analytical solution using the EB beam theory, and those found by studying plane strain deformations with the FEM agree very well with each other. 3.7. Remarks For a continuous system, deformations in the st mode will cease for t > T 3. However, deformations in other modes can still occur. Since the external force = for t > T 3, the maximum amplitude of vibrations can be such that the maximum kinetic energy of motion in all other modes must be less than the total energy of the system in those modes at t > T 3. [] R. Batra, S. Aimmanee, Missing frequencies in previous exact solutions of free vibrations of simply supported rectangular plates, J. Sound Vib. 265 (4) (23) 887 896. [2] R. Batra, S. Aimmanee, Vibration of an incompressible isotropic linear elastic rectangular plate with a higher-order shear and normal deformable theory, J. Sound Vib. 37 (3) (27) 96 97. [3] Abaqus Analysis User s Manual, Simulia Corp, Providence, RI, USA, 27. [4] L. Meirovitch, Fundamentals of Vibrations, international edition, McGraw-Hill, 2. [5] B. Balachandran, E.B. Magrab, Vibrations, 2nd ed., Cengage Learning, Toronto, Canada, 28. [6] T.J. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Courier Corporation, 22. [7] S. Srinivas, A. Rao, Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates, Int. J. Solids Struct. 6 () (97) 463 48. [8] L. Qian, R. Batra, L. Chen, Free and forced vibrations of thick rectangular plates using higher-order shear and normal deformable plate theory and meshless local Petrov-Galerkin (MLPG) method, Comput. Model. Eng. Sci. 4 (5) (23) 59 534. [9] R. Batra, J. Jin, Natural frequencies of a functionally graded anisotropic rectangular plate, J. Sound Vib. 282 () (25) 59 56. 4. Conclusions Recalling that for a linear spring-mass system subjected to a time-dependent trapezoidal force with linearly increasing and