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Dynamic Response and Vibration Power Flow Analysis of Rectangular Isotropic Plate Using Fourier Series Approximation and Mobility Approach

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Abstract

Purpose

In this paper, a unified and numerically efficient method is proposed to analyse the dynamic response and vibration power flow characteristics of flat isotropic rectangular plate subjected to a harmonic loading.

Method

The boundary of the plate system has been assumed to be elastic boundary and expressed as a combination of translational and rotational spring stiffness, making the system near to real system. The plate displacement function has been derived using combination of trial beam functions, expressed as modified Fourier cosine series, in x and y direction. Finally, Rayleigh-Ritz method is employed to the Lagrangian function to derive the frequency matrix.

Results

The derived frequency matrix can be universally applied to analyse the dynamic response of the plate system and power flow distribution on the plate surface for any condition of boundary condition and force application.

Conclusion

The method is numerically superior to other method because of omission of complex hyperbolic trigonometric functions and appreciable reduction of the size of the frequency matrix, which has been established through comparison of results with the existing literature

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Defence Institute of Advanced Technology (Deemed University), Girinagar, Pune, Maharashtra, 411 025, India

    Kavikant Mahapatra & S. K. Panigrahi

Authors
  1. Kavikant Mahapatra
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  2. S. K. Panigrahi
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Corresponding author

Correspondence to S. K. Panigrahi.

Appendix A

Appendix A

$$G_{11} = \, \left( {1 + \eta i} \right) - \hat{K}_{Tx0} \left( {\cos \left( {2a} \right)/\left( {6\sin \left( {2a} \right)} \right) - \cos \left( a \right)/\left( {3\sin \left( a \right)} \right)} \right)$$
$$G_{12} = - \hat{K}_{Tx0} \left( {1/\left( {3\sin \left( a \right)} \right) - 1/\left( {6\sin \left( {2a} \right)} \right)} \right)$$
$$G_{13} = - \hat{K}_{Tx0} \left( {\cos \left( {2a} \right)/\left( {6\sin \left( {2a} \right)} \right) - \left( {4\cos \left( a \right)} \right)/\left( {3\sin \left( a \right)} \right)} \right)$$
$$G_{14} = - \hat{K}_{Tx0} \left( {4/\left( {3\sin \left( a \right)} \right) - 1/\left( {6\sin \left( {2a} \right)} \right)} \right)$$
$$G_{21} = - \hat{K}_{Txa} (\sin (2a)/6 - \sin (a)/3 + \cos (2a)^{2} /(6\sin (2a)) - \cos (a)^{2} /(3\sin (a)))$$
$$G_{22} = \hat{K}_{Txa} \left( {\cos \left( {2a} \right)/\left( {6\sin \left( {2a} \right)} \right) - \cos \left( a \right)/\left( {3\sin \left( a \right)} \right)} \right){-}1 - \eta i$$
$$G_{23} = - \hat{K}_{Txa} (\sin (2a)/6 - (4\sin (a))/3 + \cos (2a)^{2} /(6\sin (2a)) - (4\cos (a)^{2} )/(3\sin (a)))$$
$$G_{24} = \hat{K}_{Txa} \left( {\cos \left( {2a} \right)/\left( {6\sin \left( {2a} \right)} \right) - \left( {4\cos \left( a \right)} \right)/\left( {3\sin \left( a \right)} \right)} \right)$$
$$G_{31} = \left( {1 + \eta i} \right) \, \left( {\cos \left( a \right)/\left( {3\sin \left( a \right)} \right) \, - \, \left( {2\cos \left( {2a} \right)} \right)/\left( {3\sin \left( {2a} \right)} \right)} \right)$$
$$G_{32} = \, \left( {1 + \eta i} \right) \, \left( {2/\left( {3\sin \left( {2a} \right)} \right) \, - \, 1/\left( {3\sin \left( a \right)} \right)} \right)$$
$$G_{33} = \hat{K}_{Ry0} - \left( {1 + \eta i} \right)\left( {\left( {2\cos \left( {2a} \right)} \right)/\left( {3\sin \left( {2a} \right)} \right) + \left( {4\cos \left( a \right)} \right)/\left( {3\sin \left( a \right)} \right)} \right)$$
$$G_{34} = \left( {1 + \eta i} \right) \, \left( {2/\left( {3\sin \left( {2a} \right)} \right) - 4/\left( {3\sin \left( a \right)} \right)} \right)$$
$$G_{41} = (1 + \eta i)((2\sin (2a))/3 - \sin (a)/3 + (2\cos (2a)^{2} )/(3\sin (2a)) - \cos (a)^{2} /(3\sin (a)))$$
$$G_{42} = \left( {1 + \eta i} \right) \, \left( {\cos \left( a \right)/\left( {3\sin \left( a \right)} \right) - \left( {2\cos \left( {2a} \right)} \right)/\left( {3\sin \left( {2a} \right)} \right)} \right)$$
$$G_{{43}} = (1 + \eta i)((2\sin \,(2a))/3 - (4\sin \,(a))/3 + (2\cos \,(2a)^{2} )/(3\sin \,(2a)) - (4\cos \,(a)^{2} )/(3\sin \,(a)))$$
$$G_{44} = \hat{K}_{Ryb} - \left( {1 + \eta i} \right) \, \left( {\left( {2\cos \left( {2a} \right)} \right)/\left( {3\sin \left( {2a} \right)} \right) + \left( {4\cos \left( a \right)} \right)/\left( {3\sin \left( a \right)} \right)} \right)$$

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Mahapatra, K., Panigrahi, S.K. Dynamic Response and Vibration Power Flow Analysis of Rectangular Isotropic Plate Using Fourier Series Approximation and Mobility Approach. J. Vib. Eng. Technol. 8, 105–119 (2020). https://doi.org/10.1007/s42417-018-0079-3

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