Dynamic Behavior and Vibration Suppression of a Generally Restrained Pre-pressure Beam Structure Attached with Multiple Nonlinear Energy Sinks

Abstract

Beam structures are extensively used in many engineering branches. For marine engineering, the ship shafting system is generally simplified as a vibration model with single or multiple beam structures connected by the coupling stiffness. In engineering, multiple nonlinear energy sinks (NESs) can be arranged on the premise of sufficient installation space to ensure their vibration suppression effect. Considering engineering practice, this study investigates the dynamic behavior and vibration suppression of a generally restrained pre-pressure beam structure with multiple uniformly distributed NESs, where the pre-pressure is typically caused by thrust bearings, installation ways, and others. System governing equations are derived through the generalized Hamiltonian principle and the variational procedure. Dynamic responses of the pre-pressure beam structure are predicted by the Galerkin truncation method. The effect of NESs on dynamic responses and vibration suppression of the pre-pressure beam structure is studied and discussed. Suitable parameters of NESs have a beneficial effect on the vibration suppression at both ends of the pre-pressure beam structure. NESs can modify the vibration frequency and energy transmission characteristics of the vibration system. For different boundary conditions, the optimized parameters of NESs significantly suppress the vibration energy of the pre-pressure beam structure.

Appendices

Appendix A

$$ T_{\text{B}} = \int_{0}^{L} {\frac{1}{2}\rho S} \left( {\frac{\partial u}{{\partial t}}} \right)^{2} {\text{d}} t $$

(A-1)

$$ T_{\text{NES}} = \sum\limits_{z = 1}^{\text{Z}} {\left( {\frac{1}{2}m_{\text{NES}} \frac{{{\text{d}} u_{z} }}{{{\text{d}} t}}} \right)^{2} } $$

(A-2)

$$ V_{{\text{B}}} = \int_{0}^{L} {\frac{1}{2}EI} \left( {\frac{{\partial^{2} u}}{{\partial x^{2} }}} \right)^{2} {\text{d}} x $$

(A-3)

$$ V_{{\text{P}}} = \int_{0}^{L} {\frac{1}{2}P} \left( {\frac{\partial u}{{\partial x}}} \right)^{2} {\text{d}} x $$

(A-4)

$$ \begin{aligned} V_{\text{Boundary}} & = \frac{1}{2}k_{\text{L}} \left[ {u\left( {0,t} \right)} \right]^{2} + \frac{1}{2}k_{\text{R}} \left[ {u\left( {L,t} \right)} \right]^{2} \\ & \quad + \frac{1}{2}K_{\text{L}} \left[ {\frac{{\partial u\left( {0,t} \right)}}{\partial x}} \right]^{2} + \frac{1}{2}K_{\text{R}} \left[ {\frac{{\partial u\left( {L,t} \right)}}{\partial x}} \right]^{2} \\ \end{aligned} $$

(A-5)

$$ V_{\text{NES}} = \frac{1}{4}k_{\text{NES}} \sum\limits_{\text{z} = 1}^{Z} {\left[ {u_{z} - u\left( {x_{z} ,t} \right)} \right]^{4} } $$

(A-6)

$$ \delta W_{\text{B}} = - \int_{0}^{L} {C_{\text{B}} } \frac{\partial u}{{\partial t}}\delta u{\text{d}} x $$

(A-7)

$$ \delta W_{\text{NES}} = - C_{{\text{NES}}} \sum\limits_{z = 1}^{Z} {\left\{ {\left[ {\frac{{{\text{d}} u_{z} }}{{{\text{d}} t}} - \frac{{\partial u\left( {x_{z} ,t} \right)}}{\partial t}} \right]\delta \left[ {u_{z} - u\left( {x_{z} ,t} \right)} \right]} \right\}} $$

(A-8)

$$ \delta W_{\text{F}} = - \int_{0}^{L} {{\text{Dirca}} \left( {x - x_{\text{F}} } \right)F_{0} \sin \left( {\omega t} \right)} \delta u{\text{d}} x $$

(A-9)

Appendix B

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta T_{\text{B}} {\text{d}} t = } - \int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{L} {\rho S\frac{{\partial^{2} u}}{\partial t}} } \delta u{\text{d}} x{\text{d}} t $$

(B-1)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta T_{\text{NES}} {\text{d}} t} = - \sum\limits_{z = 1}^{Z} {\left( {\int_{{t_{1} }}^{{t_{2} }} {m_{\text{NES}} \frac{{{\text{d}}^{2} u_{z} }}{{{\text{d}} t^{2} }}\delta u_{z} } {\text{d}} t} \right)} $$

(B-2)

$$ \begin{aligned} \int_{{t_{1} }}^{{t_{2} }} {\delta V_{{\text{B}}} {\text{d}} t} & = \int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{L} {EI\frac{{\partial^{4} u}}{{\partial x^{4} }}} \delta u{\text{d}} x{\text{d}} t} \\ & \quad + \int_{{t_{1} }}^{{t_{2} }} {\left[ \begin{gathered} - EI\frac{{\partial^{2} u\left( {0,t} \right)}}{{\partial x^{2} }}\delta \left( {\frac{\partial u}{{\partial x}}} \right) + EI\frac{{\partial^{3} u\left( {0,t} \right)}}{{\partial x^{3} }}\delta u \hfill \\ + EI\frac{{\partial^{2} u\left( {L,t} \right)}}{{\partial x^{2} }}\delta \left( {\frac{\partial u}{{\partial x}}} \right) - EI\frac{{\partial^{3} u\left( {L,t} \right)}}{{\partial x^{3} }}\delta u \hfill \\ \end{gathered} \right]} {\text{d}} t \\ \end{aligned} $$

(B-3)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta V_{\text{P}} {\text{d}} t} = P\frac{{\partial u\left( {0,t} \right)}}{\partial x}\delta u - P\frac{{\partial u\left( {L,t} \right)}}{\partial x}\delta u + \int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{L} {P\frac{{\partial^{2} u}}{{\partial x^{2} }}\delta u{\text{d}} x{\text{d}} } t} $$

(B-4)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta V_{\text{Boundary}} {\text{d}} t} = \int_{{t_{1} }}^{{t_{2} }} {\left[ \begin{gathered} k_{\text{L}} u\left( {0,t} \right)\delta u + K_{\text{L}} \frac{{\partial u\left( {0,t} \right)}}{\partial x}\delta \left( {\frac{\partial u}{{\partial x}}} \right) \hfill \\ \quad + k_{\text{R}} u\left( {L,t} \right)\delta u + K_{\text{R}} \frac{{\partial u\left( {L,t} \right)}}{\partial x}\delta \left( {\frac{\partial u}{{\partial x}}} \right) \hfill \\ \end{gathered} \right]{\text{d}} t} $$

(B-5)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta V_{\text{NES}} {\text{d}} t} = \sum\limits_{z = 1}^{Z} {\left\{ {\int_{{t_{1} }}^{{t_{2} }} {k_{\text{NES}} \left[ {u_{z} - u\left( {x_{z} ,t} \right)} \right]^{3} \delta \left[ {u_{z} - u\left( {x_{z} ,t} \right)} \right]{\text{d}} t} } \right\}} $$

(B-6)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta W_{\text{B}} {\text{d}} t} = - \int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{L} {C_{\text{B}} } \frac{\partial u}{{\partial t}}\delta u{\text{d}} x{\text{d}} t} $$

(B-7)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta W_{\text{NES}} {\text{d}} t} = - \sum\limits_{z = 1}^{{Z}} {\int_{{t_{1} }}^{{t_{2} }} {C_{\text{NES}} \left[ {\frac{{{\text{d}} u_{z} }}{{{\text{d}} t}} - \frac{{\partial u\left( {x_{z} ,t} \right)}}{\partial t}} \right]\delta \left[ {u_{z} - u\left( {x_{z} ,t} \right)} \right]{\text{d}} t} } $$

(B-8)

$$ \int_{{t_{1} }}^{{t_{2} }} {\delta W_{\text{F}} {\text{d}} t} = - \int_{{t_{1} }}^{{t_{2} }} {\int_{0}^{L} {\delta \left( {x - x_{\text{F}} } \right)F_{0} \sin \left( {\omega t} \right)} \delta u{\text{d}} x{\text{d}} t} $$

(B-9)

Appendix C

$$ R_{m1} = \int_{0}^{L} {\rho S\left[ {\sum\limits_{i = 1}^{{N}} {\varphi_{i} \left( x \right)\frac{{{\text{d}}^{2} q_{i} \left( t \right)}}{{{\text{d}} t^{2} }}} } \right]\psi_{m} \left( x \right){\text{d}} x} $$

(C-1)

$$ R_{m2} = \int_{0}^{L} {C_{\text{B}} \left[ {\sum\limits_{i = 1}^{{N}} {\varphi_{i} \left( x \right)\frac{{{\text{d}} q_{i} \left( t \right)}}{{{\text{d}} t}}} } \right]\psi_{m} \left( x \right){\text{d}} x} $$

(C-2)

$$ R_{m3} = \int_{0}^{L} {EI\left[ {\sum\limits_{i = 1}^{{N}} {\frac{{{\text{d}}^{4} \varphi_{i} \left( x \right)}}{{{\text{d}} x^{4} }}q_{i} \left( t \right)} } \right]\psi_{m} \left( x \right){\text{d}} x} $$

(C-3)

$$ R_{m4} = \int_{0}^{L} {P\left[ {\sum\limits_{i = 1}^{{N}} {\frac{{{\text{d}}^{2} \varphi_{i} \left( x \right)}}{{{\text{d}} x^{2} }}q_{i} \left( t \right)} } \right]\psi_{m} \left( x \right){\text{d}} x} $$

(C-4)

$$ R_{m5} = \psi_{m} \left( {x_{F} } \right)F_{0} \sin \left( {\omega t} \right) $$

(C-5)

$$ R_{m6} = \sum\limits_{z = 1}^{Z} {\left\{ {C_{\text{NES}} \left[ {\sum\limits_{{{\text{i}} = 1}}^{{N}} {\varphi_{i} \left( {x_{z} } \right)\frac{{{\text{d}} q_{i} \left( t \right)}}{{{\text{d}} t}}} - u_{z} \left( t \right)} \right]\psi_{m} \left( {x_{z} } \right)} \right\}} $$

(C-6)

$$ R_{{m{7}}} = \sum\limits_{z = 1}^{{Z}} {\left\{ {k_{\text{NES}} \left[ {\sum\limits_{i = 1}^{{N}} {\varphi_{i} \left( {x_{z} } \right)q_{i} \left( t \right)} - u_{z} \left( t \right)} \right]^{3} \psi_{m} \left( {x_{z} } \right)} \right\}} $$

(C-7)