Nonlinear vibro-acoustic analysis of a double-panel structure with an enclosure cavity

Abstract

An analytical study on the nonlinear vibro-acoustic behaviors of a double-panel structure with an acoustic cavity under harmonic excitation is presented. The flexible panels are modeled by the von-Karman plate theory. The solution of the corresponding nonlinear equations is obtained using the Galerkin method in conjunction with the multiple scales method (MMS). Primary, subharmonic and superharmonic resonances are then stated, and the frequency responses of different harmonics are obtained by MMS. The effects of excitation level and damping coefficient as well as the cavity depth on the frequency responses are investigated. According to the results jump phenomena is observed for primary and superharmonic resonance cases. Also, with increase in cavity depth, the steady state amplitude of the vibration increases in the sub harmonic resonance cases.

Appendix

Coefficients of Eqs. (15) and (16)

$$ P_{1} \left( t \right) = {\text{Re}} \left[ {\frac{{ - B_{1} }}{{A_{2} B_{2} - A_{1} B_{1} }}\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} 2p_{i} \exp \left( {j\Omega t - jk_{0} x\sin \varphi \cos \theta - jk_{0} y\sin \theta \sin \varphi } \right)\sin \frac{m\pi x}{a}\cos \frac{n\pi y}{b}dxdy} \right] = P_{01} \cos \Omega t $$

(42)

$$ P_{2} \left( t \right) = {\text{Re}} \left[ {\frac{{B_{2} }}{{A_{2} B_{2} - A_{1} B_{1} }}\mathop \int \limits_{0}^{a} \mathop \int \limits_{0}^{b} 2p_{i} \exp \left( {j\Omega t - jk_{0} x\sin \varphi \cos \theta - jk_{0} y\sin \theta \sin \varphi } \right)\sin \frac{\pi x}{a}\cos \frac{\pi y}{b}dxdy} \right] = P_{02} \cos \Omega t $$

(43)

$$ \lambda_{1}^{4} = \frac{{B_{1} \overline{\lambda }_{1}^{4} }}{{A_{1} B_{1} - A_{2} B_{2} }} $$

(44)

$$ \lambda_{2}^{4} = \frac{{A_{1} \overline{\lambda }_{2}^{4} }}{{A_{1} B_{1} - A_{2} B_{2} }} $$

(45)

$$ \alpha_{1} = \frac{{B_{1} - A_{2} }}{{A_{2} B_{2} - A_{1} B_{1} }}\mu $$

(46)

$$ \alpha_{2} = \frac{{A_{2} \overline{\lambda }_{2}^{4} }}{{A_{2} B_{2} - A_{1} B_{1} }} $$

(47)

$$ \alpha_{3} = \frac{{A_{1} - B_{2} }}{{A_{2} B_{2} - A_{1} B_{1} }}\mu $$

(48)

$$ \alpha_{4} = \frac{{B_{2} \overline{\lambda }_{1}^{4} }}{{A_{2} B_{2} - A_{1} B_{1} }} $$

(49)

where

$$ A_{1} = 1 + \frac{{\rho_{1} \cosh k_{022} c}}{{k_{022} \sinh k_{022} c}} $$

(50)

$$ A_{2} = - \frac{{\rho_{2} }}{{k_{022} \sinh k_{022} c}} $$

(51)

$$ B_{1} = 1 + \frac{{\rho_{2} \cosh k_{022} c}}{{k_{022} \sinh k_{022} c}} $$

(52)

$$ B_{2} = - \frac{{\rho_{1} }}{{k_{022} \sinh k_{022} c}} $$

(53)

and

$$ \overline{\lambda }_{1}^{4} = \frac{{\rho_{1} h_{1} \hat{\omega }_{1}^{2} }}{{D_{1} }} $$

(54)

$$ \overline{\lambda }_{2}^{4} = \frac{{\rho_{2} h_{2} \hat{\omega }_{2}^{2} }}{{D_{2} }} $$

(55)

in which

$$ \hat{\omega }_{1} = \pi^{2} \left[ {\left( \frac{1}{a} \right)^{2} + \left( \frac{1}{b} \right)^{2} } \right]\left( {\frac{{D_{1} }}{{\rho_{1} h_{1} }}} \right)^{2} $$

(56)

$$ \hat{\omega }_{2} = \pi^{2} \left[ {\left( \frac{1}{a} \right)^{2} + \left( \frac{1}{b} \right)^{2} } \right]\left( {\frac{{D_{2} }}{{\rho_{2} h_{2} }}} \right)^{2} $$

(57)