0
Research Papers

Free and Forced Vibrations of Monolithic and Composite Rectangular Plates With Interior Constrained Points

[+] Author and Article Information
Arka P. Chattopadhyay

Department of Biomedical Engineering
and Mechanics,
M/C 0219 Virginia Polytechnic
Institute and State University,
495 Old Turner Street,
Blacksburg, VA 24061
e-mail: arka@vt.edu

Romesh C. Batra

Honorary Fellow
Department of Biomedical Engineering
and Mechanics,
M/C 0219 Virginia Polytechnic
Institute and State University,
495 Old Turner Street,
Blacksburg, VA 24061
e-mail: rbatra@vt.edu

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 10, 2018; final manuscript received August 9, 2018; published online September 17, 2018. Assoc. Editor: Julian Rimoli.

J. Vib. Acoust 141(1), 011018 (Sep 17, 2018) (15 pages) Paper No: VIB-18-1257; doi: 10.1115/1.4041216 History: Received June 10, 2018; Revised August 09, 2018

The restriction of deformations to a subregion of a system undergoing either free or forced vibration due to an irregularity or discontinuity in it is called mode localization. Here, we study mode localization in free and forced vibration of monolithic and unidirectional fiber-reinforced rectangular linearly elastic plates with edges either simply supported (SS) or clamped by using a third-order shear and normal deformable plate theory (TSNDT) with points on either one or two normals to the plate midsurface constrained from translating in all three directions. The plates studied are symmetric about their midsurfaces. The in-house developed software based on the finite element method (FEM) is first verified by comparing predictions from it with either the literature results or those computed by using the linear theory of elasticity and the commercial FE software abaqus. New results include: (i) the localization of both in-plane and out-of-plane modes of vibration, (ii) increase in the mode localization intensity with an increase in the length/width ratio of a rectangular plate, (iii) change in the mode localization characteristics with the fiber orientation angle in unidirectional fiber reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beats-like phenomenon in a subregion of the plate. Constraining translational motion of internal points can be used to design a structure with vibrations limited to its small subregion and harvesting energy of vibrations of the subregion.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Anderson, P. W. , 1958, “ Absence of Diffusion in Certain Random Lattices,” Phys. Rev., 109(5), p. 1492. [CrossRef]
Hodges, C. , 1982, “ Confinement of Vibration by Structural Irregularity,” J. Sound Vib., 82(3), pp. 411–424. [CrossRef]
Valero, N. , and Bendiksen, O. , 1986, “ Vibration Characteristics of Mistuned Shrouded Blade Assemblies,” ASME J. Eng. Gas Turbines Power, 108(2), pp. 293–299. [CrossRef]
Wei, S.-T. , and Pierre, C. , 1988, “ Localization Phenomena in Mistuned Assemblies With Cyclic Symmetry—Part I: Free Vibrations,” ASME J. Vib., Acoust., Stress, Reliab. Des., 110(4), pp. 429–438. [CrossRef]
Bendiksen, O. O. , 1987, “ Mode Localization Phenomena in Large Space Structures,” AIAA J., 25(9), pp. 1241–1248. [CrossRef]
Pierre, C. , Tang, D. M. , and Dowell, E. H. , 1987, “ Localized Vibrations of Disordered Multispan Beams-Theory and Experiment,” AIAA J., 25(9), pp. 1249–1257. [CrossRef]
Hodges, C. , and Woodhouse, J. , 1983, “ Vibration Isolation From Irregularity in a Nearly Periodic Structure: Theory and Measurements,” J. Acoust. Soc. Am., 74(3), pp. 894–905. [CrossRef]
Pierre, C. , 1990, “ Weak and Strong Vibration Localization in Disordered Structures: A Statistical Investigation,” J. Sound Vib., 139(1), pp. 111–132. [CrossRef]
Herbert, D. , and Jones, R. , 1971, “ Localized States in Disordered Systems,” J. Phys. C, 4(10), p. 1145. [CrossRef]
Kirkman, P. , and Pendry, J. , 1984, “ The Statistics of One-Dimensional Resistances,” J. Phys. C, 17(24), p. 4327. [CrossRef]
Pierre, C. , and Plaut, R. H. , 1989, “ Curve Veering and Mode Localization in a Buckling Problem,” Z. Angew. Math. Phys., 40(5), pp. 758–761. [CrossRef]
Nayfeh, A. H. , and Hawwa, M. A. , 1994, “ Use of Mode Localization in Passive Control of Structural Buckling,” AIAA J., 32(10), pp. 2131–2133. [CrossRef]
Paik, S. , Gupta, S. , and Batra, R. , 2015, “ Localization of Buckling Modes in Plates and Laminates,” Compos. Struct., 120, pp. 79–89. [CrossRef]
Ibrahim, R. , 1987, “ Structural Dynamics With Parameter Uncertainties,” ASME Appl. Mech. Rev., 40(3), pp. 309–328. [CrossRef]
Hodges, C. , and Woodhouse, J. , 1986, “ Theories of Noise and Vibration Transmission in Complex Structures,” Rep. Prog. Phys., 49(2), p. 107. [CrossRef]
Nowacki, W. , 1953, “ Vibration and Buckling of Rectangular Plates Simply-Supported at the Periphery and at Several Points Inside,” Arch. Mech. Stosow., 5(3), pp. 437–454.
Gorman, D. , 1981, “ An Analytical Solution for the Free Vibration Analysis of Rectangular Plates Resting on Symmetrically Distributed Point Supports,” J. Sound Vib., 79(4), pp. 561–574. [CrossRef]
Gorman, D. , and Singal, R. , 1991, “ Analytical and Experimental Study of Vibrating Rectangular Plates on Rigid Point Supports,” AIAA J., 29(5), pp. 838–844. [CrossRef]
Bapat, A. , Venkatramani, N. , and Suryanarayan, S. , 1988, “ A New Approach for the Representation of a Point Support in the Analysis of Plates,” J. Sound Vib., 120(1), pp. 107–125. [CrossRef]
Bapat, A. , Venkatramani, N. , and Suryanarayan, S. , 1988, “ The Use of Flexibility Functions With Negative Domains in the Vibration Analysis of Asymmetrically Point-Supported Rectangular Plates,” J. Sound Vib., 124(3), pp. 555–576. [CrossRef]
Bapat, A. , and Suryanarayan, S. , 1989, “ The Flexibility Function Approach to Vibration Analysis of Rectangular Plates With Arbitrary Multiple Point Supports on the Edges,” J. Sound Vib., 128(2), pp. 209–233. [CrossRef]
Bapat, A. , and Suryanarayan, S. , 1989, “ Free Vibrations of Periodically Point-Supported Rectangular Plates,” J. Sound Vib., 132(3), pp. 491–509. [CrossRef]
Bapat, A. , and Suryanarayan, S. , 1992, “ The Fictitious Foundation Approach to Vibration Analysis of Plates With Interior Point Supports,” J. Sound Vib., 155(2), pp. 325–341. [CrossRef]
Lee, S. , and Lee, L. , 1994, “ Free Vibration Analysis of Rectangular Plates With Interior Point Supports,” J. Struct. Mech., 22(4), pp. 505–538.
Rao, G. V. , Raju, I. , and Amba-Rao, C. , 1973, “ Vibrations of Point Supported Plates,” J. Sound Vib., 29(3), pp. 387–391. [CrossRef]
Raju, I. , and Amba-Rao, C. , 1983, “ Free Vibrations of a Square Plate Symmetrically Supported at Four Points on the Diagonals,” J. Sound Vib., 90(2), pp. 291–297. [CrossRef]
Utjes, J. , Sarmiento, G. S. , Laura, P. , and Gelos, R. , 1986, “ Vibrations of Thin Elastic Plates With Point Supports: A Comparative Study,” Appl. Acoust., 19(1), pp. 17–24. [CrossRef]
Kim, C. , and Dickinson, S. , 1987, “ The Flexural Vibration of Rectangular Plates With Point Supports,” J. Sound Vib., 117(2), pp. 249–261. [CrossRef]
Bhat, R. , 1991, “ Vibration of Rectangular Plates on Point and Line Supports Using Characteristic Orthogonal Polynomials in the Rayleigh-Ritz Method,” J. Sound Vib., 149(1), pp. 170–172. [CrossRef]
Filoche, M. , and Mayboroda, S. , 2009, “ Strong Localization Induced by One Clamped Point in Thin Plate Vibrations,” Phys. Rev. Lett., 103(25), p. 254301. [CrossRef] [PubMed]
Sharma, D. , Gupta, S. , and Batra, R. , 2012, “ Mode Localization in Composite Laminates,” Compos. Struct., 94(8), pp. 2620–2631. [CrossRef]
Batra, R. , and Aimmanee, S. , 2003, “ Missing Frequencies in Previous Exact Solutions of Free Vibrations of Simply Supported Rectangular Plates,” J. Sound Vib., 265(4), pp. 887–896. [CrossRef]
Du, J. , Li, W. L. , Jin, G. , Yang, T. , and Liu, Z. , 2007, “ An Analytical Method for the in-Plane Vibration Analysis of Rectangular Plates With Elastically Restrained Edges,” J. Sound Vib., 306(3–5), pp. 908–927. [CrossRef]
Srinivas, S. , and Rao, A. , 1970, “ Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates,” Int. J. Solids Struct., 6(11), pp. 1463–1481. [CrossRef]
Vel, S. S. , and Batra, R. C. , 1999, “ Analytical Solution for Rectangular Thick Laminated Plates Subjected to Arbitrary Boundary Conditions,” AIAA J., 37(11), pp. 1464–1473.
Shah, P. , and Batra, R. , 2017, “ Stress Singularities and Transverse Stresses Near Edges of Doubly Curved Laminated Shells Using TSNDT and Stress Recovery Scheme,” Eur. J. Mech.-A, 63, pp. 68–83. [CrossRef]
Lo, K. H. , Christensen, R. M. , and Wu, E. M. , 1977, “ A High-Order Theory of Plate Deformation---Part 1: Homogeneous Plates,” ASME. J. Appl. Mech., 44(4), pp. 663–668.
Carrera, E. , 1999, “ A Study of Transverse Normal Stress Effect on Vibration of Multilayered Plates and Shells,” J. Sound Vib., 225(5), pp. 803–829. [CrossRef]
Vidoli, S. , and Batra, R. C. , 2000, “ Derivation of Plate and Rod Equations for a Piezoelectric Body From a Mixed Three-Dimensional Variational Principle,” J. Elasticity, 59(1–3), pp. 23–50. [CrossRef]
Batra, R. C. , and Vidoli, S. , 2002, “ Higher-Order Piezoelectric Plate Theory Derived From a Three-Dimensional Variational Principle,” AIAA J., 40(1), pp. 91–104. [CrossRef]
Mindlin, R. D. , 1951, “ Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” J. Appl. Mech., 18, pp. 31–38.
Alsuwaiyan, A. , and Shaw, S. W. , 2003, “ Steady-State Responses in Systems of Nearly-Identical Torsional Vibration Absorbers,” ASME J. Vib. Acoust., 125(1), pp. 80–87. [CrossRef]
Alsuwaiyan, A. S. , 2013, “ Localization in Multiple Nearly-Identical Tuned Vibration Absorbers,” Int. J. Eng. Technol., 2(3), p. 157. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.882.1873&rep=rep1&type=pdf
Spletzer, M. , Raman, A. , Wu, A. Q. , Xu, X. , and Reifenberger, R. , 2006, “ Ultrasensitive Mass Sensing Using Mode Localization in Coupled Microcantilevers,” Appl. Phys. Lett., 88(25), p. 254102. [CrossRef]
Cosserat, E. , and Cosserat, F. , 1909, “ Theorie des Corps Deformables,” Paris, A. Hermann and Sons.
Srinivas, S. , Rao, C. J. , and Rao, A. , 1970, “ An Exact Analysis for Vibration of Simply-Supported Homogeneous and Laminated Thick Rectangular Plates,” J. Sound Vib., 12(2), pp. 187–199. [CrossRef]
Batra, R. C. , Vidoli, S. , and Vestroni, F. , 2002, “ Plane Wave Solutions and Modal Analysis in Higher Order Shear and Normal Deformable Plate Theories,” J. Sound Vib., 257(1), pp. 63–88. [CrossRef]
Vel, S. S. , and Batra, R. , 2000, “ The Generalized Plane Strain Deformations of Thick Anisotropic Composite Laminated Plates,” Int. J. Solids Struct., 37(5), pp. 715–733. [CrossRef]
Hughes, T. J. , 2012, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Courier Corporation, Mineola, NY.
Qian, L. , Batra, R. , and Chen, L. , 2003, “ Free and Forced Vibrations of Thick Rectangular Plates Using Higher-Order Shear and Normal Deformable Plate Theory and Meshless Petrov-Galerkin (MLPG) Method,” Comput. Model. Eng. Sci., 4(5), pp. 519–534. http://www.techscience.com/doi/10.3970/cmes.2003.004.519.pdf
Meirovitch, L. , 2001, Fundamentals of Vibrations, McGraw-Hill, New York.
Malatkar, P. , and Nayfeh, A. H. , 2003, “ On the Transfer of Energy Between Widely Spaced Modes in Structures,” Nonlinear Dyn., 31(2), pp. 225–242. [CrossRef]
Hirwani, C. K. , Panda, S. K. , and Mahapatra, T. R. , 2018, “ Nonlinear Finite Element Analysis of Transient Behavior of Delaminated Composite Plate,” ASME J. Vib. Acoust., 140(2), p. 021001. [CrossRef]
Xiao, J. , and Batra, R. , 2014, “ Delamination in Sandwich Panels Due to Local Water Slamming Loads,” J. Fluids Struct., 48, pp. 122–155. [CrossRef]
Sapkale, S. L. , Sucheendran, M. M. , Gupta, S. S. , and Kanade, S. V. , 2018, “ Vibroacoustic Study of a Point-Constrained Plate Mounted in a Duct,” J. Sound Vib., 420, pp. 204–226.

Figures

Grahic Jump Location
Fig. 1

Schematic representation of a rectangular plate with the rectangular Cartesian coordinate axes. Points on the normal through thepoint P may be constrained from translating in all three directions.

Grahic Jump Location
Fig. 2

First 100 frequencies, in rad/μs, from the TSNDT and the 3D LET (left), and the relative difference between them (right) for the 80 × 20 × 2 mm SS and clamped plates

Grahic Jump Location
Fig. 3

Total strain energy, in J, from the TSNDT and the 3D LET (left), and the relative error between them (right) for the 80 × 20 × 2 mm SS and clamped plates

Grahic Jump Location
Fig. 4

Frequencies of the first 100 modes of vibration of the e = 16 clamped plate with and without internal points constrained

Grahic Jump Location
Fig. 5

Mode shapes for free vibration of the clamped plate of e = 16 with (right) and without (left) internal constrained points. The red and the blue colors, respectively, represent magnitudes of the maximum positive and the maximum negative transverse displacement. (For references to color in the figure, see the online version.)

Grahic Jump Location
Fig. 6

Mode localization parameter, β1, for the first 100 modes of vibration of a clamped plate with constrained internal points (left) and distribution of modes over different values of the ratio β1 for the first 100 vibration modes (right)

Grahic Jump Location
Fig. 7

Fringe plots of the magnitude of the total displacement for different mode shapes of the SS plate of e = 16 with (right) and without (left) internal constrained points

Grahic Jump Location
Fig. 8

Values of β1 for different modes (top) and the histogram of β1 for the first 100 modes of free vibration of a SS plate of e = 16 with (right) and without (left) constraining internal points

Grahic Jump Location
Fig. 9

Mode shapes of mode 1 (left) and mode 5 (right) of vibration of internally constrained clamped rectangular laminae of e = 20 for fiber angles of 0 deg, 45 deg, and 90 deg

Grahic Jump Location
Fig. 10

Histogram of the distribution of β1 over the first 100 modes of vibration of the internally constrained clamped laminate with different fiber angles

Grahic Jump Location
Fig. 11

Fringe plots of the out-of-plane displacement, u3, for the fundamental mode of vibration of the internally constrained plate with fiber angles of (a) 0 deg, (b) 30 deg, (c) 45 deg, (d) 60 deg, and (e) 90 deg counter-clockwise to the global x1-axis. The fringe colors, in the online version of the paper, represent same levels of u3 (in mm) for each plot.

Grahic Jump Location
Fig. 12

Top view of the mode shapes and fringe plots of the in-plane displacement, u2, for fiber angles of (a) 0 deg, (b) 45 deg, and (c) 90 deg. The fringe colors represent same levels of u2 (in mm) for each plot.

Grahic Jump Location
Fig. 13

Distribution of the energy ratio β1 over the first 100 modes of vibration of the fiber-reinforced lamina with fibers oriented at 0 deg and 90 deg to the global x1-axis (left), and the corresponding histogram (right)

Grahic Jump Location
Fig. 14

Localized mode shapes for the 90 deg composite plate: (a) mode 7, (b) top view of mode 14, and (c) mode17

Grahic Jump Location
Fig. 15

Shapes of modes 1, 2, and 5 showing deformation localization in the SS plate with two (a) symmetrically and (b) asymmetrically located pair of constrained points

Grahic Jump Location
Fig. 16

Three transient impulse loads considered

Grahic Jump Location
Fig. 17

For the three transient loads, time histories of the centroidal deflection and of the strain energy densities of regions R1 and R2 of the plate with internal constrained points

Grahic Jump Location
Fig. 18

Time histories of the centroidal deflection and of the strain energy densities of the two regions of the plate for the mode 1 and the mode 6 excitation frequencies

Grahic Jump Location
Fig. 19

Time histories of the ratio of the total energy of regions R1 and R2 (left) and the ratio of the total energies of each region normalized by the external work done on the entire plate (right)

Grahic Jump Location
Fig. 20

Centroidal displacement history of an internally unconstrained SS plate under mode 6 harmonic loading

Grahic Jump Location
Fig. 21

Mode shapes of transverse vibration for the SS plate of e = 20 without (left) and with (right) the internal constraint points

Grahic Jump Location
Fig. 22

Displacement histories of centroids of regions R1 and R2 under harmonic loads of the two excitation frequencies

Grahic Jump Location
Fig. 23

Time histories of ratio of the total energies of the regions R1 and R2 (left) and the ratio of the total energy of each section of the plate to the cumulative external work done on the entire plate

Grahic Jump Location
Fig. 24

Centroidal displacement histories of the rectangular plate of e = 20 without internal constraints under modes 4 and 8 excitations and the corresponding FFTs of the displacement histories

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In