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Nonlinear vibration analysis of a circular plate–cavity system

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Abstract

Vibration of plate with air cavity has been one of the interesting research fields by many researchers. This topic has many applications in vehicles, airplanes, aircraft, fuselage panels and buildings. In this study, nonlinear vibroacoustic of circular plate with air cavity under harmonic excitation is investigated. The von Karman theory is used to obtain plate equation and solved together with the air pressure equation. First, the nonlinear equation of the plate is converted to ordinary differential equations by using the Galerkin method. Then the method of multiple scales is employed to solve the corresponding nonlinear equations. Frequency response for primary, subharmonic and superharmonic resonances is studied analytically. Using this method, a parametric study is carried out and the effects of different parameters on the frequency response of the plate are investigated. According to the results, jump phenomena are observed for primary and superharmonic resonance cases. Also, with an increase in damping coefficient, the amplitude of the steady-state response increases in the subharmonic resonance case.

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Abbreviations

c :

Depth of acoustic enclosure

c a :

Sound speed

c d :

Structural damping coefficient

D :

Bending stiffness

E :

Young’s modulus of elasticity

h :

Thickness

P i :

Acoustic pressure

P E :

External excitation

φ :

Airy stress function

ρ :

Density

ν :

Poisson’s ratio

σ :

Detuning parameter

Ω :

Excitation frequency

ω n :

Natural frequency

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Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Yazd University, P.O. Box 89195-741, Yazd, Iran

    Fatemeh Sadat Anvariyeh, Mohammad Mahdi Jalili & Ali Reza Fotuhi

Authors
  1. Fatemeh Sadat Anvariyeh
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  2. Mohammad Mahdi Jalili
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  3. Ali Reza Fotuhi
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Corresponding author

Correspondence to Mohammad Mahdi Jalili.

Additional information

Technical Editor: Wallace Moreira Bessa, D.Sc.

Appendices

Appendix 1

Coefficients of Eq. (12)

$$\mu^{mnpq} = \frac{{\left( {\left[ {\varphi_{q} \left( r \right)\acute{\varphi }_{m} \left( r \right)\acute{\psi }_{np} \left( r \right)} \right]_{0}^{a} - \int_{0}^{a} {\acute{\varphi }_{m} \left( r \right)\acute{\varphi }_{q} \left( r \right) \acute{\psi }_{np} \left( r \right){\text{d}}r} } \right)}}{{\int_{0}^{a} {\left( {\rho h + \rho_{\text{air}} \frac{{J_{0} \left( {kr} \right)\cos \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}{{\sqrt {k^{2} + \lambda^{2} } \sin \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}} \right)} r\varphi_{m}^{2} {\text{d}}r}}$$
(39)
$$c_{m} = \frac{{\mathop \smallint \nolimits_{0}^{a} c_{d} r\varphi_{m}^{2} {\text{d}}r}}{{\mathop \smallint \nolimits_{0}^{a} \left( {\rho h + \rho_{\text{air}} \frac{{J_{0} \left( {kr} \right)\cos (\sqrt {k^{2} + \lambda^{2} } c)}}{{\sqrt {k^{2} + \lambda^{2} } \sin (\sqrt {k^{2} + \lambda^{2} } c)}}} \right)r\varphi_{m}^{2} {\text{d}}r}}$$
(40)
$$\hat{\lambda }_{m}^{4} = \frac{{D\mathop \smallint \nolimits_{0}^{a} \left( {\varphi_{m} \frac{{\partial^{4} \varphi_{m} }}{{\partial r^{4} }} + \frac{2}{r}\varphi_{m} \frac{{\partial^{3} \varphi_{m} }}{{\partial r^{3} }} + \frac{1}{{r^{2} }}\varphi_{m} \frac{{\partial^{2} \varphi_{m} }}{{\partial r^{2} }} + \frac{1}{{r^{3} }}\varphi_{m} \frac{{\partial \varphi_{m} }}{\partial r}} \right){\text{d}}r}}{{\mathop \smallint \nolimits_{0}^{a} \left( {\rho h + \rho_{air} \frac{{J_{0} \left( {kr} \right)\cos \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}{{\sqrt {k^{2} + \lambda^{2} } \sin \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}} \right)r\varphi_{m}^{2} {\text{d}}r}}$$
(41)
$$\hat{P}_{Em} = \frac{{\mathop \smallint \nolimits_{0}^{a} rP_{E} \varphi_{m} \left( r \right){\text{d}}r}}{{\mathop \smallint \nolimits_{0}^{a} \left( {\rho h + \rho_{\text{air}} \frac{{J_{0} \left( {kr} \right)\cos \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}{{\sqrt {k^{2} + \lambda^{2} } \sin \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}} \right)r\varphi_{m}^{2} {\text{d}}r}}$$
(42)
$$\hat{P}_{im} = \frac{{\mathop \smallint \nolimits_{0}^{a} rP_{i} \left( {r, - c,t} \right)\varphi_{m} \left( r \right){\text{d}}r}}{{\mathop \smallint \nolimits_{0}^{a} \left( {\rho h + \rho_{\text{air}} \frac{{J_{0} \left( {kr} \right)\cos \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}{{\sqrt {k^{2} + \lambda^{2} } \sin \left( {\sqrt {k^{2} + \lambda^{2} } c} \right)}}} \right)r\varphi_{m}^{2} {\text{d}}r}}$$
(43)

Appendix 2

Coefficients of Eq. (13)

$$\alpha_{1} = \mu^{1111}$$
(44)
$$\alpha_{2} = \mu^{1112} + \mu^{1121} + \mu^{1211}$$
(45)
$$\alpha_{3} = \mu^{1122} + \mu^{1212} + \mu^{1222}$$
(46)
$$\alpha_{4} = \mu^{1222}$$
(47)

Appendix 3

Substitution of Eq. (30) into (29)

$$\begin{aligned} & D_{0}^{2} w_{13} + \omega_{1}^{2} w_{13} = - 2i(\acute{A}_{1} e^{{i\omega_{1} T_{0} }} + \hat{c}_{1} A_{1} e^{{i\omega_{1} T_{0} }} + - ci\omega_{0} A_{1} ) \\ & \quad + \alpha_{1} (A_{1}^{3} e^{{3i\omega_{1} T_{0} }} + \varLambda_{1}^{3} e^{{3i\varOmega T_{0} }} + \bar{\varLambda }_{1}^{3} e^{{ - 3i\varOmega T_{0} }} \bar{A}_{1}^{3} e^{{ - 3i\omega_{1} T_{0} }} \\ & \quad + 3A_{1}^{2} \bar{A}_{1} e^{{i\omega_{1} T_{0} }} + 3A_{1} \bar{A}_{1}^{2} e^{{ - i\omega_{1} T_{0} }} + 3A_{1} \varLambda_{1}^{2} e^{{i(\omega_{1} + 2\varOmega )T_{0} }} + 3A_{1} \bar{\varLambda }_{2}^{3} e^{{i(\omega_{1} - 2\varOmega )T_{0} }} ) \\ & \quad + \alpha_{2} (A_{1}^{2} A_{2} e^{{i(2\omega_{1} + \omega_{2} )T_{0} }} + \bar{A}_{1}^{2} A_{2} e^{{i( - 2\omega_{1} + \omega_{2} )T_{0} }} ) \\ & \quad + A_{2} \varLambda_{1}^{2} e^{{i(\omega_{2} + 2\varOmega )T_{0} }} + A_{2} \bar{\varLambda }_{1}^{2} e^{{i(\omega_{2} - 2\varOmega )T_{0} }} + 2A_{1} \bar{A}_{1} A_{2} e^{{i\omega_{2} T_{0} }} \\ & \quad + 2A_{1} A_{2} \varLambda_{1} e^{{i(\omega_{2} + \omega_{1} + \varOmega )T_{0} }} + 2A_{1} \bar{\varLambda }_{1} A_{2} e^{{i(\omega_{2} + \omega_{1} - \varOmega )T_{0} }} + 2\bar{A}_{1} \bar{\varLambda }_{1} A_{2} e^{{i(\omega_{2} - \omega_{1} - \varOmega )T_{0} }} \\ & \quad + 2\varLambda_{1} \bar{\varLambda }_{1} A_{2} e^{{i(\omega_{2} )T_{0} }} + \bar{A}_{1}^{2} \bar{A}_{2} e^{{i( - 2\omega_{1} - \omega_{2} )T_{0} }} + 2A_{1}^{2} \bar{A}_{2} e^{{i(2\omega_{1} - \omega_{2} )T_{0} }} \\ & \quad + A_{1}^{2} \bar{A}_{2} e^{{i(2\omega_{1} - \omega_{2} )T_{0} }} + \bar{A}_{1}^{2} \bar{A}_{2} e^{{i( - 2\omega_{1} - \omega_{2} )T_{0} }} + \bar{\varLambda }_{1}^{2} \bar{A}_{2} e^{{i( - \omega_{2} - \varOmega )T_{0} }} \\ & \quad + \bar{\varLambda }_{1}^{2} \bar{A}_{2} e^{{i( - \omega_{2} - 2\varOmega )T_{0} }} + 2A_{1} \bar{A}_{1} \bar{A}_{2} e^{{ - i\omega_{2} T_{0} }} + 2A_{1} \varLambda_{1} \bar{A}_{2} e^{{i( - \omega_{2} + \omega_{1} + \varOmega )T_{0} }} \\ & \quad + 2\bar{A}_{1} \bar{\varLambda }_{1} \bar{A}_{2} e^{{i( - \omega_{2} - \omega_{1} - \varOmega )T_{0} }} + 2\varLambda_{1} \bar{\varLambda }_{1} \bar{A}_{2} e^{{ - i(\omega_{2} )T_{0} }} ) + \alpha_{3} (A_{1} A_{2}^{2} e^{{i(2\omega_{2} + \omega_{1} )T_{0} }} \\ & \quad + A_{1} \bar{A}_{2}^{2} e^{{i(2\omega_{1} - \omega_{2} )T_{0} }} + A_{1} A_{2} \bar{A}_{2} e^{{i\omega_{1} T_{0} }} + \bar{A}_{1} A_{2}^{2} e^{{i(2\omega_{2} - \omega_{1} )T_{0} }} + \bar{A}_{1} \bar{A}_{2}^{2} e^{{i( - 2\omega_{2} - \omega_{1} )T_{0} }} \\ & \quad + 2\bar{A}_{1} A_{2} \bar{A}_{2} e^{{ - i\omega_{1} T_{0} }} + A_{2}^{2} \varLambda_{1} e^{{i(2\omega_{2} + \varOmega )T_{0} }} + 2A_{2} \bar{A}_{2} \varLambda_{1} e^{{i(\varOmega )T_{0} }} \\ & \quad + A_{2}^{2} \bar{\varLambda }_{1} e^{{i(2\omega_{2} - \varOmega )T_{0} }} + \bar{A}_{2}^{2} \bar{\varLambda }_{1} e^{{i( - 2\omega_{2} - \varOmega )T_{0} }} + 2A_{2} \bar{A}_{2} \bar{\varLambda }_{1} e^{{i( - \varOmega )T_{0} }} \\ & \quad + \alpha_{4} (A_{2}^{3} e^{{3i\omega_{2} T_{0} }} + \bar{A}_{2}^{3} e^{{ - 3i\omega_{2} T_{0} }} + A_{2}^{2} \bar{A}_{2} e^{{i\omega_{2} T_{0} }} + 3A_{2} \bar{A}_{2}^{2} e^{{ - i\omega_{2} T_{0} }} ) \\ \end{aligned}$$
(48)
$$\begin{aligned} & D_{0}^{2} w_{23} + \omega_{2}^{2} w_{23} = - 2(\acute{A}_{2} e^{{i\omega_{1} T_{0} }} + \hat{c}_{2} A_{2} e^{{i\omega_{1} T_{0} }} + - ci\omega_{0} A_{1} ) \\ & & \quad + \alpha_{5} (A_{1}^{3} e^{{3i\omega_{1} T_{0} }} + \varLambda_{1}^{3} e^{{3i\varOmega T_{0} }} + \bar{\varLambda }_{1}^{3} e^{{ - 3i\varOmega T_{0} }} \bar{A}_{1}^{3} e^{{ - 3i\omega_{1} T_{0} }} + 3A_{1}^{2} \bar{A}_{1} e^{{i\omega_{1} T_{0} }} \\ & \quad + 3A_{1} \bar{A}_{1}^{2} e^{{ - i\omega_{1} T_{0} }} + 3A_{1} \varLambda_{1}^{2} e^{{i(\omega_{1} + 2\varOmega )T_{0} }} + 3A_{1} \bar{\varLambda }_{2}^{3} e^{{i(\omega_{1} - 2\varOmega )T_{0} }} \\ & \quad + \alpha_{6} (A_{1}^{2} A_{2} e^{{i(2\omega_{1} + \omega_{2} )T_{0} }} + \bar{A}_{1}^{2} A_{2} e^{{i( - 2\omega_{1} + \omega_{2} )T_{0} }} + A_{2} \varLambda_{1}^{2} e^{{i(\omega_{2} + 2\varOmega )T_{0} }} \\ & \quad + A_{2} \bar{\varLambda }_{1}^{2} e^{{i(\omega_{2} - 2\Omega )T_{0} }} + 2A_{1} \bar{A}_{1} A_{2} e^{{i\omega_{2} T_{0} }} + 2A_{1} A_{2} \varLambda_{1} e^{{i(\omega_{2} + \omega_{1} + \varOmega )T_{0} }} \\ & \quad + 2A_{1} \bar{\varLambda }_{1} A_{2} e^{{i(\omega_{2} + \omega_{1} - \varOmega )T_{0} }} + 2\bar{A}_{1} \bar{\varLambda }_{1} A_{2} e^{{i(\omega_{2} - \omega_{1} -\Omega )T_{0} }} + 2\varLambda_{1} \bar{\varLambda }_{1} A_{2} e^{{i(\omega_{2} )T_{0} }} \\ & \quad + 2A_{1}^{2} \bar{A}_{2} e^{{i(2\omega_{1} - \omega_{2} )T_{0} }} + \bar{A}_{1}^{2} \bar{A}_{2} e^{{i( - 2\omega_{1} - \omega_{2} )T_{0} }} + A_{1}^{2} \bar{A}_{2} e^{{i(2\omega_{1} - \omega_{2} )T_{0} }} + \bar{A}_{1}^{2} \bar{A}_{2} e^{{i( - 2\omega_{1} - \omega_{2} )T_{0} }} \\ & \quad + \bar{\varLambda }_{1}^{2} \bar{A}_{2} e^{{i( - \omega_{2} - \varOmega )T_{0} }} + \bar{\varLambda }_{1}^{2} \bar{A}_{2} e^{{i( - \omega_{2} - 2\varOmega )T_{0} }} + 2A_{1} \bar{A}_{1} \bar{A}_{2} e^{{ - i\omega_{2} T_{0} }} + 2A_{1} \varLambda_{1} \bar{A}_{2} e^{{i( - \omega_{2} + \omega_{1} + \varOmega )T_{0} }} \\ & \quad + 2\bar{A}_{1} \bar{\varLambda }_{1} \bar{A}_{2} e^{{i( - \omega_{2} - \omega_{1} - \varOmega )T_{0} }} + 2\varLambda_{1} \bar{\varLambda }_{1} \bar{A}_{2} e^{{ - i(\omega_{2} )T_{0} }} ) + \alpha_{7} (A_{1} A_{2}^{2} e^{{i(2\omega_{2} + \omega_{1} )T_{0} }} ) \\ & \quad + A_{1} \bar{A}_{2}^{2} e^{{i(2\omega_{1} - \omega_{2} )T_{0} }} + A_{1} A_{2} \bar{A}_{2} e^{{i\omega_{1} T_{0} }} + \bar{A}_{1} A_{2}^{2} e^{{i(2\omega_{2} - \omega_{1} )T_{0} }} + \bar{A}_{1} \bar{A}_{2}^{2} e^{{i( - 2\omega_{2} - \omega_{1} )T_{0} }} \\ & \quad + 2\bar{A}_{1} A_{2} \bar{A}_{2} e^{{ - i\omega_{1} T_{0} }} + A_{2}^{2} \varLambda_{1} e^{{i(2\omega_{2} + \varOmega )T_{0} }} + 2A_{2} \bar{A}_{2} \varLambda_{1} e^{{i(\varOmega )T_{0} }} + A_{2}^{2} \bar{\varLambda }_{1} e^{{i(2\omega_{2} - \varOmega )T_{0} }} \\ & \quad + \bar{A}_{2}^{2} \bar{\varLambda }_{1} e^{{i( - 2\omega_{2} - \varOmega )T_{0} }} + 2A_{2} \bar{A}_{2} \bar{\varLambda }_{1} e^{{i( - \varOmega )T_{0} }} ) + \alpha_{8} (A_{2}^{3} e^{{3i\omega_{2} T_{0} }} + \bar{A}_{2}^{3} e^{{ - 3i\omega_{2} T_{0} }} \\ & \quad + A_{2}^{2} \bar{A}_{2} e^{{i\omega_{2} T_{0} }} + 3A_{2} \bar{A}_{2}^{2} e^{{ - i\omega_{2} T_{0} }} ) + P_{02} \cos (\varOmega t) \\ \end{aligned}$$
(49)

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Anvariyeh, F.S., Jalili, M.M. & Fotuhi, A.R. Nonlinear vibration analysis of a circular plate–cavity system. J Braz. Soc. Mech. Sci. Eng. 41, 66 (2019). https://doi.org/10.1007/s40430-019-1565-6

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