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 哈尔滨工程大学学报  2018, Vol. 39 Issue (10): 1727-1732  DOI: 10.11990/jheu.201712025 0

### 引用本文

SUN Min, CHEN Jianen, CHEN Huanlin. Comparison on vibration absorption efficiency of parallel and series nonlinear energy sinks[J]. Journal of Harbin Engineering University, 2018, 39(10), 1727-1732. DOI: 10.11990/jheu.201712025.

### 文章历史

1. 天津城建大学 理学院, 天津 300384;
2. 天津理工大学 天津市先进机电系统设计与智能控制重点实验室, 天津 300384;
3. 天津理工大学 机电工程国家级实验教学示范中心, 天津 300384

Comparison on vibration absorption efficiency of parallel and series nonlinear energy sinks
SUN Min1, CHEN Jianen2,3, CHEN Huanlin1
1. School of Science, Tianjin Chengjian University, Tianjin 300384, China;
2. Tianjin Key Laboratory of the Design and Intelligent Control of the Advanced Mechatronical System, Tianjin University of Technology, Tianjin 300384, China;
3. National Demonstration Center for Experimental Mechanical and Electrical Engineering Education, Tianjin University of Technology, Tianjin 300384, China
Abstract: The comparative study on the vibration absorption efficiency of parallel and series nonlinear energy sinks is carried out to guide the selection and optimization of the two kinds of absorbers. The equations of motion for the systems including the two energy sinks are derived under the consideration of the influence of temperature effect on the dynamic property of beam. The dissipation rates of the two energy sinks are calculated by numerical method. The effects of stiffness coefficient, damping coefficient and instillation location on the vibration absorption efficiency are analyzed. The variation law of the vibration absorption efficiency of the two energy sinks along with shock amplitude and temperature is investigated. The results show that the two energy sinks can both achieve good vibration suppressing effect, however, the principle for parameter setting of the two energy sinks are quite different. Relatively speaking, the variation amplitude of the vibration absorption efficiency of the series nonlinear energy sink, with the change of external factors, is larger.
Keywords: nonlinear energy sink    energy dissipation rate    shock load    vibration absorption efficiency    vibration control    pure nonlinear    parameter optimization    dynamics    temperature

1 动力学方程

 $u\left( {x, z, t} \right) = - z\frac{{\partial {w_0}\left( {x, t} \right)}}{{\partial x}}, w\left( {x, z, t} \right) = {w_0}\left( {x, t} \right)$ (1)

 ${\varepsilon _x} = - z\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}}, {\sigma _x} = E({\varepsilon _x} - \alpha \Delta T)$ (2)

 $\begin{array}{*{20}{c}} {I\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}} + {\mu _0}\frac{{\partial {w_0}}}{{\partial t}} + D\frac{{{\partial ^4}{w_0}}}{{\partial {x^4}}} + {N^T}\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} + }\\ {[{k_1}{{({u_1} - {v_1})}^3} + {\mu _1}({{\dot u}_1} - {{\dot v}_1})]\delta (x - {a_1}) + }\\ {[{k_2}{{({u_2} - {v_2})}^3} + {\mu _2}({{\dot u}_2} - {{\dot v}_2})]\delta (x - {a_2}) = F\delta \left( {x - b} \right), }\\ {{m_1}{{\ddot v}_1} + {k_1}{{({v_1} - {u_1})}^3} + {\mu _1}({{\dot v}_1} - {{\dot u}_1}) = 0, }\\ {{m_2}{{\ddot v}_2} + {k_2}{{({v_2} - {u_2})}^3} + {\mu _2}({{\dot v}_2} - {{\dot u}_2}) = 0} \end{array}$ (3)

 $\begin{array}{*{20}{c}} {I\frac{{{\partial ^2}{w_0}}}{{\partial {t^2}}} + {\mu _0}\frac{{\partial {w_0}}}{{\partial t}} + D\frac{{{\partial ^4}{w_0}}}{{\partial {x^4}}} + {N^T}\frac{{{\partial ^2}{w_0}}}{{\partial {x^2}}} + }\\ {[{k_3}{{({u_3} - {v_3})}^3} + {\mu _3}({{\dot u}_3} - {{\dot v}_3})]\delta (x - {a_3}) = F\delta \left( {x - b} \right), }\\ {{m_3}{{\ddot v}_3} + {k_3}{{({v_3} - {u_3})}^3} + {\mu _3}({{\dot v}_3} - {{\dot u}_3}) + }\\ {{k_4}{{({v_3} - {v_4})}^3} + {\mu _4}({{\dot v}_3} - {{\dot v}_4}) = 0, }\\ {{m_4}{{\ddot v}_4} + {k_4}{{({v_4} - {v_3})}^3} + {\mu _4}({{\dot v}_4} - {{\dot v}_3}) = 0} \end{array}$ (4)

 $D = \smallint _{ - h/2}^{h/2}\mathit{\boldsymbol{E}}{z^2}{\rm{d}}z, \mathit{\boldsymbol{I}} = \smallint _{ - h/2}^{h/2}\rho {\rm{d}}z, {\mathit{\boldsymbol{N}}^{\rm{T}}} = \smallint _{ - h/2}^{h/2}E\alpha T{\rm{d}}z,$ (5)

 $\left\{ \begin{array}{l} {{\bar w}_0} = \frac{{{w_0}}}{h}, {{\bar v}_i} = \frac{{{v_i}}}{h}, \bar x = \frac{x}{l}, \bar F = \frac{{{l^4}}}{{E{h^4}}}F, {{\bar k}_i} = \frac{{{l^4}}}{{Eh}}{k_i}, \\ \bar t = \frac{h}{{{l^2}}}{\left( {\frac{E}{\rho }} \right)^{1/2}}t, {{\bar \mu }_i} = \frac{l}{h}{\left( {\frac{1}{{\rho E}}} \right)^{1/2}}{\mu _i}, {{\bar a}_i} = \frac{{{a_i}}}{l}, \\ \bar b = \frac{b}{l} \end{array} \right.$ (6)

 ${w_0}\left( {x, t} \right) = \mathop \sum \limits_{n = 1}^N {w_n}\left( t \right){q_n}\left( x \right)$ (7)

 $\begin{array}{*{20}{c}} {{{\ddot w}_m} + {\gamma _0}{{\dot w}_m} + \omega _m^2{w_m} + \{ {k_1}{{[\mathop \sum \limits_{n = 1}^N {w_n}{q_n}({a_1}) - {v_1}]^3}} + }\\ {\gamma [\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}({a_1}) - {{\dot v}_1}]\} {q_m}({a_1}) + }\\ {\{ {k_2}{{[\mathop \sum \limits_{n = 1}^N {w_n}{q_n}({a_2}) - {v_2}]^3}} + }\\ {{\gamma _2}[\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}({a_2}) - {{\dot v}_2}]\} {q_m}({a_2}) = F{q_m}\left( b \right), }\\ {{\varepsilon _1}{{\ddot v}_1} + {k_1}{{[{v_1} - \mathop \sum \limits_{n = 1}^N {w_n}\left( t \right){q_n}({a_1})]^3}} + }\\ {{\gamma _1}[{{\dot v}_1} - \mathop \sum \limits_{n = 1}^N {{\dot w}_n}\left( t \right){q_n}({a_1})] = 0, }\\ {{\varepsilon _2}{{\ddot v}_2} + {k_2}{{[{v_2} - \mathop \sum \limits_{n = 1}^N {w_n}\left( t \right){q_n}({a_2})]^3}} + }\\ {{\gamma _2}[{{\dot v}_2} - \mathop \sum \limits_{n = 1}^N {{\dot w}_n}\left( t \right){q_n}({a_2})] = 0} \end{array}$ (8)

 $\begin{array}{*{20}{c}} {{{\ddot w}_m} + {\gamma _0}{{\dot w}_m} + \omega _m^2{w_m} + \{ {k_3}{{[\mathop \sum \limits_{n = 1}^N {w_n}{q_n}({a_3}) - {v_3}]^3}} + }\\ {{\gamma _3}[\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}({a_3}) - {{\dot v}_3}]\} {q_m}({a_3}) = F{q_m}\left( b \right), }\\ {{\varepsilon _3}{{\ddot v}_3} + {k_3}{{[{v_3} - \mathop \sum \limits_{n = 1}^N {w_n}\left( t \right){q_n}({a_3})]^3}} + }\\ {{\gamma _3}[{{\dot v}_3} - \mathop \sum \limits_{n = 1}^N {{\dot w}_n}\left( t \right){q_n}({a_3})] + }\\ {{k_4}{{({v_3} - {v_4})}^3} + {\gamma _4}({{\dot v}_3} - {{\dot v}_4}) = 0, }\\ {{\varepsilon _4}{{\ddot v}_4} + {k_4}{{({v_4} - {v_3})}^3} + {\gamma _4}({{\dot v}_4} - {{\dot v}_3}) = 0} \end{array}$ (9)
2 吸振效能对比研究

 $F = \left\{ {\begin{array}{*{20}{l}} {A\sin \left( {2{\rm{ \mathsf{ π} }}t/T} \right)}&{0 \le t \le T/2}\\ {0, }&{t > T/2} \end{array}} \right.$ (10)

 $\begin{array}{*{20}{c}} {{\eta _1} = }\\ {\frac{{\smallint _0^t({\gamma _1}{{[\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}({a_1}) - {{\dot v}_1}]^2}} + {\gamma _2}{{[\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}({a_2}) - {{\dot v}_2}]^2}}){\rm{d}}\tau }}{{\smallint _0^TF\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}\left( b \right){\rm{d}}t}}}\\ \end{array}$

 $\begin{array}{*{20}{c}} {{\eta _2} = }\\ {\frac{{\smallint _0^t({\gamma _3}{{[\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}({a_3}) - {{\dot v}_3}]^2}} + {\gamma _4}{{({{\dot v}_3} - {{\dot v}_4})}^2}){\rm{d}}\tau }}{{\smallint _0^TF\mathop \sum \limits_{n = 1}^N {{\dot w}_n}{q_n}\left( b \right){\rm{d}}t}}} \end{array}$

 Download: 图 2 并联非线性能量阱的吸振效能 Fig. 2 Vibration absorption efficiency of the parallel NES

 Download: 图 3 串联非线性能量阱的吸振效能 Fig. 3 Vibration absorption efficiency of the series NES

 Download: 图 4 非线性能量阱吸振效能与刚度系数、阻尼系数的关系 Fig. 4 The relation between vibration absorption efficiency of NESs and stiffness coefficient, damping coefficient

 Download: 图 5 非线性能量阱吸振效能随冲击幅值和温差的变化 Fig. 5 The variation of vibration absorption efficiency of NESs with shock amplitude and temperature difference

 Download: 图 6 含并联非线性能量阱系统的位移响应 Fig. 6 The displacements of the system with parallel NES
 Download: 图 7 含串联非线性能量阱系统的位移响应 Fig. 7 The displacements of the system with series NES
3 结论

1) 两种非线性能量阱的参数设置原则差异较大。并联非线性能量阱中的两个子装置的参数可设置的较为接近。串联非线性能量阱中，直接连接于主振结构的子能量阱的刚度系数应数倍于另一个子能量阱的刚度系数。

2) 随着冲击幅值的增加，并联与串联非线性能量阱吸振效能的变化规律均为先增大后减小，随着温度的增加，两种非线性能量阱的吸振效能均逐渐降低。相对而言，串联非线性能量阱的吸振效能容易受到外部因素的影响，并联非线性能量阱显示了相对较好的鲁棒性。

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