Damped Vibration Absorbers for Multi-mode Longitudinal Vibration Control of a Hollow Shaft

Abstract

Objective

The present work adopted damped vibration absorbers (DVAs) to attenuate multi-mode longitudinal vibration of the hollow shaft. The dynamic behavior of a shaft is analyzed for the guidance to the initial parameter design of DVAs and the parameters of DVAs are optimized. Subsequently, specific structures of DVAs are designed taking into account the hollow characteristic of the shaft. Furthermore, finite element stimulation of solid modeling has been performed.

Methods

In this work, with the aid of the transfer matrix method (TMM) in conjunction with the substructure synthesis method (SSM), a dynamic model of the shaft incorporating multiple DVAs is established. The proposed method is developed for the calculation of dynamics character of the coupled system and estimation of absorption effectiveness of DVAs.

Simulation

To demonstrate the validity of theoretical result obtained from the TMM proposed in this paper, a simulation of the shaft vibration by means of the finite element method (FEM) is also carried out.

Conclusion

Theoretical and stimulation results both demonstrate that the resonance peaks of the shaft longitudinal vibration are suppressed obviously with the application of those DVAs, which verify the effectiveness of the absorption performance of DVAs.

Appendix

The transfer matrices of various elements in Sect. 3 are given as

$$ {\mathbf{U}}_{{{\mathbf{0,}}i}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{i} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{i} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{i} } \right)} {\left( {{\text{EA}}_{i} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{i} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{i} \frac{\omega }{a}\sin \frac{\omega }{a}l_{i} } & \quad {\cos \frac{\omega }{a}l_{i} } & \quad 1 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right] $$

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$$ {\mathbf{U}}_{{{\mathbf{0,}}k}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad k = 1,2, \ldots ,i - 1,i + 1, \ldots ,p $$

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$$ {\mathbf{U}}_{{{\mathbf{F}}_{0} {\mathbf{,}}j}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{j} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{j} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{j} } \right)} {\left( {{\text{EA}}_{j} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{j} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{j} \frac{\omega }{a}\sin \frac{\omega }{a}l_{j} } & \quad {\cos \frac{\omega }{a}l_{j} } & \quad {F_{0} } \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right] $$

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$$ {\mathbf{U}}_{{{\mathbf{F}}_{{\mathbf{0}}} {\mathbf{,}}k}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad \left( {k = 1,2, \ldots ,j - 1,j + 1, \ldots ,p} \right) $$

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$$ {\mathbf{U}}_{k} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\,\left( {k = 1,2, \ldots ,i} \right) $$

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$$ {\mathbf{U}}_{{{\mathbf{P,}}i}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{i} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{i} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{i} } \right)} {\left( {{\text{EA}}_{i} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{i} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{i} \frac{\omega }{a}\sin \frac{\omega }{a}l_{i} } & \quad {\cos \frac{\omega }{a}l_{i} } & \quad P \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right] $$

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$$ {\mathbf{U}}_{{{\mathbf{P,}}k}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {EA_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {EA_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad \left( {k = 1,2, \ldots ,i - 1,i + 1, \ldots ,p} \right) $$

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$$ {\mathbf{U}}_{{{\mathbf{P}}_{{\mathbf{j}}} {\mathbf{,}}k}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad \left( {k = 1,2, \ldots ,i_{{P_{j - 1} }} ,i_{{P_{j + 1} }} , \ldots ,p} \right) $$

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$$ {\mathbf{U}}_{{{\mathbf{F}}_{0} {\mathbf{,}}t}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{t} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{t} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{t} } \right)} {\left( {{\text{EA}}_{t} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{t} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{t} \frac{\omega }{a}\sin \frac{\omega }{a}l_{t} } & \quad {\cos \frac{\omega }{a}l_{t} } & \quad {F_{0} } \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]_{{}} $$

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$$ {\mathbf{U}}_{{{\mathbf{F}}_{{\mathbf{0}}} {\mathbf{,}}k}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad \left( {k = 1,2, \ldots ,t - 1,t + 1, \ldots ,p} \right) $$

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$$ {\mathbf{U}}_{{{\mathbf{P,}}i_{{P_{k} }} }} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{{i_{{P_{k} }} }} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{{i_{{P_{k} }} }} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{{i_{{P_{k} }} }} } \right)} {\left( {{\text{EA}}_{{i_{{P_{k} }} }} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{{i_{{P_{k} }} }} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{{i_{{P_{k} }} }} \frac{\omega }{a}\sin \frac{\omega }{a}l_{{i_{{P_{k} }} }} } & \quad {\cos \frac{\omega }{a}l_{{i_{{P_{k} }} }} } & \quad {P_{{i_{{P_{k} }} }} } \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad \left( {k = 1,2, \ldots ,n} \right) $$

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$$ {\mathbf{U}}_{{{\mathbf{P,}}k}} = \left[ {\begin{array}{*{20}c} {\cos \frac{\omega }{a}l_{k} } & \quad {{{\left( {\sin \frac{\omega }{a}l_{k} } \right)} \mathord{\left/ {\vphantom {{\left( {\sin \frac{\omega }{a}l_{k} } \right)} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} \right. \kern-0pt} {\left( {{\text{EA}}_{k} \frac{\omega }{a}} \right)}}} & \quad 0 \\ { - {\text{EA}}_{k} \frac{\omega }{a}\sin \frac{\omega }{a}l_{k} } & \quad {\cos \frac{\omega }{a}l_{k} } & \quad 0 \\ 0 & \quad 0 & \quad 1 \\ \end{array} } \right]\quad \left( {k = 1,2, \ldots ,p\;{\text{and}}\;k \ne i_{{P_{1} }} ,i_{{P_{2} }} , \ldots ,i_{{P_{n} }} } \right). $$

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