DYNAMIC BEHAVIOR OF ELASTIC BEAM SYSTEM COUPLED BY NONLINEAR ELEMENT WITH END
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摘要: 建立了端部非线性单元耦合弹性梁系统物理模型, 从能量角度, 根据广义哈密顿原理和变分法建立弹性梁系统的振动控制方程. 首先, 运用伽辽金法展开弹性梁系统的横向振动位移并建立其残差方程, 运用龙格库塔算法求解数值结果. 然后, 在保证计算数值结果正确性的基础上, 深入研究端部耦合非线性单元对弹性梁系统频率响应的影响规律, 探索端部耦合非线性单元在单频激励下对弹性梁系统振动响应的影响规律并掲示弹性梁系统非线性动力学行为的生成机理. 最后, 研究了弹性梁系统在复杂非线性振动响应下的振动能量传递特性. 研究结果表明, 合理利用端部非线性单元, 能够表现出主动梁振动能量传递给从动梁的现象, 降低主动梁振动, 从动梁类似于吸振器. 在复杂非线性振动状态下, 弹性梁系统出现靶向能量传递现象, 准周期振动状态是出现该现象的标志. 靶向振动能量传递现象的出现为从时域角度单向控制弹性梁系统的振动水平提供可能.Abstract: In this paper, a physical model of elastic beam system with end nonlinear elements coupling is established. From the energy perspective, the vibration-governing equations of the elastic beam system are established based on the generalized Hamilton principle and variational procedure. Firstly, the Galerkin method is employed to expand the transverse vibration displacements of the elastic beam system and establish its residual equation, and the Runge-Kutta algorithm is used to solve the numerical results. Then, on the basis of ensuring the correctness of the calculated numerical results, the influence of the end-coupled nonlinear element on the frequency response of the elastic beam system is investigated in depth, and the influence of the end-coupled nonlinear element on the vibration response of the elastic beam system under the single-frequency excitation is explored, and the generating mechanism of the nonlinear dynamics of the elastic beam system is revealed. Finally, the vibration energy transfer characteristics of the elastic beam system under complex nonlinear vibration response are investigated. The results show that the reasonable utilization of end nonlinear element can exhibit the phenomenon of vibration energy transfer from the active beam to the passive beam, reducing the active beam vibration, and the passive beam is similar to an absorber. Under the complex nonlinear vibration state, the elastic beam system appears the phenomenon of targeted energy transfer, and the quasi-periodic vibration state is the sign of the phenomenon. The emergence of the targeted vibration energy transfer phenomenon provides a possibility to unidirectionally control the vibration level of the elastic beam system from the time domain perspective.
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Keywords:
- double-beam system /
- end-coupled nonlinearity /
- Galerkin method /
- nonlinear response
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引 言
弹性梁系统广泛应用于桥梁、工业机械、风力发电、轴系、铁路轨道和航空航天等不同领域的多种结构中. 弹性梁系统在工作过程中通常受到动力设备引入的振动激励, 易发生振动问题. 考虑到复杂工程系统通常可简化为简单结构的组合, 深入研究简单结构的振动特性是开展复杂系统振动控制的前提. 杨晓东等[1]研究了带有扭转弹簧两端铰支轴向运动梁的固有频率和模态函数. 周渤等[2]利用改进傅里叶级数和瑞利-里兹法研究了多跨梁结构的振动特性. 赵雨皓等[3]利用改进傅里叶级数和瑞利-里兹法研究了轴向载荷条件下弹性边界约束梁结构的振动特性. 张大鹏等[4]利用传递函数法研究非局部黏弹性地基梁的自由振动特性问题. 鲍四元等[5]利用谱几何方法和瑞利-里兹法分析了任意弹性边界下欧拉-伯努利梁的横向振动特性. 李炀等[6]运用广义哈密顿原理和Galerkin截断法分析了两端带有弹簧支撑的轴向运动梁的横向振动特性.
国内外专家学者对弹性梁的振动控制进行了大量的研究. Kerboua等[7]运用被动压电振动分流控制技术研究了与PZT粘接悬臂梁的振动控制. Cheung等[8]证明了在受到谐波激励的振动梁中, 如果阻尼可以忽略不计, 平移吸振器和旋转吸振器组合使用可以吸收和隔离梁的一部分平移运动和旋转运动. 为提高传统吸振器的工作频带及吸振效果, 一种无线性正刚度的非线性吸振器近年来受到国内外学者的广泛关注, 即非线性能量阱[9]. 在此基础上, 众多学者将非线性能量阱引入弹性梁系统, 并研究了非线性能量阱在弹性梁系统振动控制中的潜在应用. Chen等[9]研究了并行非线性能量阱对简支梁振动抑制的影响, 发现在一定条件下并行非线性能量阱相较于单非线性能量阱能够更有效降低弹性梁系统的不利振动. Zhang等[10]提出了一种通过边界NES抑制弹性梁结构弯曲振动的新方法. Zhang等[11]研究了经优化后的边界惯性非线性能量阱可以有效抑制几何非线性梁的振动. Zhang等[12]研究了在热冲击作用下用非线性能量阱抑制轴向运动梁的振动. Wang等[13]提出了一种时变边界方法, 能同时有效地抑制梁的多模态共振. 王亮等[14]采用LQR方法研究了主动振子和主动力对轴向运动悬臂梁的横向振动控制. 王战等[15]提出输入约束下的鲁棒自适应边界振动控制器, 实现了对边界扰动下欧拉-伯努利梁系统的振动控制. 叶光伟等[16]提出了一种仿激光位移传感器的视觉测量方法, 并结合自抗扰算法实现了对柔性梁实时振动控制.
大量学者在弹性梁系统广泛研究的基础上, 进一步研究双梁系统. Vu等[17]提出了一种求解阻尼双梁系统在简谐激励下振动的精确方法, 该方法通过简单的变量变换和模态分析, 可以对控制微分方程解耦和求解. Hao等[18]运用改进的Fourier-Ritz方法研究了一般边界条件下双梁连接系统的自由振动, 讨论了弹簧约束刚度和弹性层刚度对模态特性的影响. Oniszczuk[19-20]研究了弹性连接简支双梁系统的自由和受迫横向振动. Li等[21]研究了黏弹性层阻尼和Winkler层对双梁系统动力学响应的影响, 提出了一种半解析法来分析由黏弹性层连接双梁系统的固有频率和振型. Zhao等[22]利用GTM预测非线性支撑双梁系统的受迫横向非线性振动, 并研究了非线性支撑对双梁系统幅频响应和单频响应的影响. Zhou等[23]在离散化技术和传递矩阵法的基础上提出了一种近似离散化方法, 可用于变截面部分黏弹性连接双梁系统的振动分析. Li等[24]提出了一种新的状态空间法, 研究了夹层连接双梁系统的横向振动. Han等[25]采用改进Wittrick-Williams算法求解双梁结构的复杂超越频率方程并精确分析了其动态特性. Mao等[26]采用Adomian改进分解法研究了在轴向压缩和随动载荷的组合作用下悬臂双梁系统的振动和稳定性. Fang等[27]建立了由曲梁、Winkler弹性层和预紧梁组成的双梁系统, 研究了该系统的振动抑制特性并设计双稳态约束用于增强阻尼和抑制稳态响应. Zhao[28]推导出了双梁系统在经典、非经典和混合边界条件下的自由振动和受迫振动的一般解, 提出的形函数法能够可靠地确定在一般边界条件下双梁系统横向振动的模态振型和固有频率. Kakanda等[29]提出一种基于同调分析法的新算法, 研究了在横向载荷和纵向载荷作用下具有线性和非线性内层简支双梁系统的受迫横向振动. Han等[30]提出一种基于动态刚度法的双梁系统精确模态分析方法, 可以在不求解频率方程和特征方程的情况下, 快速得到系统的频率和模态振型信息. Zhao等[31]运用形函数法和非耦合模态振型法研究了带有任意中间支撑和轴向载荷的双梁系统在一般边界条件下受到任意激振力作用时的自由振动和强迫振动. Palmeri等[32]提出了一种用于双梁系统振动分析的Galerkin新型状态空间形式, 该方法准确且通用, 在频域和时域分析中均有效. Rahman等[33]提出了一种新改进多级残余谐波平衡方法, 研究了轴向载荷双梁系统的非线性受迫振动分析, 该方法得的结果与经典谐波平衡方法结果吻合良好. Zhao等[34]系统地研究了双曲梁系统在任意边界条件下的受迫振动, 并基于格林函数方法得到双曲梁系统受迫振动的一般解析解. 上述研究主要基于线性振动理论研究了由弹性层、弹簧阻尼等连接的平行双梁系统振动特性及响应规律. 根据现有研究可知, 非线性能量阱实质是利用非线性弹簧连接运动质量和主结构, 其本质是将三次非线性刚度作为一种连接单元进行使用. 与此同时, 现有研究鲜有将三次非线性刚度作为耦合单元以连接弹性结构系统, 且相关研究主要集中于讨论耦合三次非线性刚度在多层耦合结构内部连接的情况. 在工程中, 耦合单元被广泛应用于各个领域中, 部分复杂工程结构可简化为由耦合元件连接的多梁系统, 例如, 通过弹性联轴器连接的船舶推进系统, 通过耦合单元连接的双层桥梁等. 现有研究尚未针对通过非线性耦合单元连接的弹性结构开展系统性研究, 导致端部非线性耦合单元对弹性梁系统振动响应的影响规律尚不清晰, 限制了非线性耦合单元在弹性系统振动控制中的工程应用.
针对现有研究的不足, 为探索三次非线性刚度作为端部耦合单元时对耦合弹性梁系统动力学行为的影响规律及其在弹性结构振动控制中的潜在应用价值, 本研究将三次非线性刚度考虑为一种非线性耦合单元, 并建立了端部非线性单元耦合弹性梁系统物理模型. 基于所建立的振动分析模型, 从能量角度, 根据广义哈密顿原理和变分法, 推导得到弹性梁系统的控制方程. 之后, 采用伽辽金法展开弹性梁的横向振动位移并建立其残差方程. 在保证数值计算结果正确的基础上, 深入研究端部耦合非线性单元对弹性梁系统频率响应的影响规律, 探索端部耦合非线性单元在单频激励下对弹性梁系统振动响应的影响规律, 并掲示了弹性梁系统非线性现象的生成机理. 最后, 研究了弹性梁系统在复杂非线性振动响应下的振动能量传递特性.
1. 理论模型
本文研究端部非线性单元耦合弹性梁系统的动力学响应. 图1是端部非线性单元耦合弹性梁系统的物理模型. 如图1所示, 该物理模型是由通过端部非线性元件耦合的两根梁、4个平动约束弹簧和两个旋转约束弹簧组成的. 其中平动弹簧kI和kS起支撑作用, mL和mR是端部非线性单元相对梁1和梁2引入的附加质量. 此外, u1(x1,t)和u2(x2,t)分别为梁1和梁2的横向振动位移. 在本研究中, F(x2,t)为谐波激励, 是作用在梁2上的外部激励载荷, 其具体形式为
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图 1 通过端部非线性单元耦合弹性梁系统物理模型Figure 1. Physical model of elastic beam system coupled by end nonlinear element$$ F\left( {{x_{\text{2}}},t} \right) = \delta \left( {{x_{\text{2}}} - {L_{\text{2}}}} \right){F_{\text{0}}}{\text{sin}}\left( {\omega t} \right) $$ (1) 式中, δ(·)为Dirca函数; x2 = L2为谐波激励作用位置; F0为谐波激励幅值; ω为谐波激励角频率.
根据振动理论, 该物理系统的动能为
$$ {T_{{\text{System}}}} = {T_{{\text{Beam1}}}} + {T_{{\text{Beam2}}}} + {T_1} + {T_2} + {T_3} $$ (2a) 式中, TBeam1和TBeam2是梁1和梁2的动能; T1, T2和T3分别是质量块mA, mL和mR的动能. 物理系统的势能为
$$ \begin{split} & {V_{{\text{System}}}} = {V_{{\text{Beam1}}}} + {V_{{\text{Beam2}}}} + {V_{\text{1}}} + {V_{\text{2}}} + {V_{\text{3}}} + \\ &\qquad {V_{\text{4}}} + {V_{\text{5}}} + {V_{\text{6}}} + {V_{\text{7}}} \end{split} $$ (2b) 式中, VBeam1和VBeam2分别是梁1和梁2的势能; V1, V2, V3和V4分别是平动约束弹簧kL, kI, kS和kR的势能; V5和V6分别是旋转约束弹簧KL和KR的势能; V7是端部非线性单元的势能. 物理系统的虚功为
$$ \delta {W_{{\text{System}}}} = \delta {W_{\text{1}}} + \delta {W_{\text{2}}} + \delta {W_{\text{3}}} + \delta {W_{\text{4}}} $$ (2c) 式中, δW1是谐波激励所做虚功; δW2和δW3分别是梁1和梁2黏性阻尼所做虚功; δW4是端部非线性单元黏性阻尼所做虚功. 该物理系统的动能、势能和虚功的表达式见附录A.
本文从能量角度, 根据广义哈密顿原理和变分法, 推导得到端部非线性单元耦合弹性梁系统的振动控制方程和边界控制方程, 其中CE, kE和knE分别是端部非线性单元的端部耦合黏性阻尼、端部耦合线性刚度和端部耦合非线性刚度; I1和I2分别是梁1和梁2的惯性矩. 梁1和梁2的振动控制方程和边界控制方程推导见附录B. 梁1的振动控制方程为
$$\begin{split} & {\rho _{\text{1}}}{S_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{1}}}){m_{\text{A}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + {E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{4}}}{u_{\text{1}}}}}{{\partial x_{\text{1}}^{\text{4}}}} + {C_{{\text{B1}}}}\frac{{\partial {u_{\text{1}}}}}{{\partial t}}+ \\ &\qquad \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} = 0 \end{split} $$ (3a) 梁1在x1 = 0和x1 = L1处的边界控制方程为
$$ {x_{\text{1}}} = 0 :\left\{\begin{aligned} & {{E_{\text{1}}}{I_1}\frac{{{\partial ^{\text{3}}}{u_{\text{1}}}({\text{0}},t)}}{{\partial x_{\text{1}}^{\text{3}}}} + {k_{\text{L}}}{u_{\text{1}}}(0,t) = {\text{0}}} \\ & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}({\text{0}},t)}}{{\partial x_{\text{1}}^{\text{2}}}} - {K_{\text{L}}}\frac{{\partial {u_{\text{1}}}({\text{0}},t)}}{{\partial {x_{\text{1}}}}} = {\text{0}}} \end{aligned} \right. $$ (3b) $$ {x_{\text{1}}} = {L_{\text{1}}}:\left\{ \begin{aligned} & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{3}}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{\partial x_{\text{1}}^{\text{3}}}} - {k_{\text{E}}}[{u_{\text{1}}}({L_{\text{1}}},t) - {u_{\text{2}}}({\text{0}},t)]} -\\ &\qquad { k{n_{\text{E}}}{{\left[{u_{\text{1}}}({L_{\text{1}}},t) - {u_{\text{2}}}({\text{0}},t)\right]}^{\text{3}}} - {k_{\text{I}}}{u_{\text{1}}}({L_{\text{1}}},t)}- \\ &\qquad { {C_{\text{E}}}\left[\frac{{{\text{d}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{{\text{d}}t}} - \frac{{{\text{d}}{u_{\text{2}}}({\text{0}},t)}}{{{\text{d}}t}}\right] = {\text{0}}} \\ & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{\partial x_{\text{1}}^{\text{2}}}} = {\text{0}}} \end{aligned} \right. $$ (3c) 梁2的振动控制方程为
$$\begin{split} & {\rho _{\text{2}}}{S_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}}){m_{\text{R}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}} - {x_{\text{S}}}){k_{\text{S}}}{u_{\text{2}}} + \\ &\qquad {E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{4}}}{u_{\text{2}}}}}{{\partial x_{\text{2}}^{\text{4}}}} + {{\text{C}}_{{\text{B2}}}}\frac{{\partial {u_{\text{2}}}}}{{\partial t}} + \delta ({x_{\text{2}}} - {L_{\text{2}}}){F_{\text{0}}}{\text{sin}}(\omega t) = {\text{0}}\\ \end{split} $$ (4a) 梁2在x2 = 0和x2 = L2处的边界控制方程为
$$ {x_{\text{2}}} = {\text{0}}:\left\{\begin{aligned} & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{3}}}{u_{\text{2}}}({\text{0}},t)}}{{\partial x_{\text{2}}^{\text{3}}}} + {k_{\text{E}}}[{u_{\text{2}}}({\text{0}},t) - {u_{\text{1}}}({L_{\text{1}}},t)]} +\\ &\qquad { k{n_{\text{E}}}{{[{u_{\text{2}}}({\text{0}},t) - {u_{\text{1}}}({L_{\text{1}}},t)]}^{\text{3}}}} +\\ &\qquad { {C_{\text{E}}}\left[\frac{{{\text{d}}{u_{\text{2}}}({\text{0}},t)}}{{{\text{d}}t}} - \frac{{{\text{d}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{{\text{d}}t}}\right] = {\text{0}}} \\ & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}({\text{0}},t)}}{{\partial x_{\text{2}}^{\text{2}}}} = {\text{0}}} \end{aligned} \right. $$ (4b) $$ {x_{\text{2}}} = {L_{\text{2}}}:\left\{ \begin{aligned} & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{2}}}({L_{\text{2}}},t)}}{{\partial x_{\text{2}}^{\text{2}}}} + {K_{\text{R}}}\frac{{\partial {u_{\text{2}}}({L_{\text{2}}},t)}}{{\partial {x_{\text{2}}}}} = {\text{0}}} \\ & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{3}}}{u_{\text{2}}}({L_{\text{2}}},t)}}{{\partial x_{\text{2}}^{\text{3}}}} - {k_{\text{R}}}{u_{\text{2}}}({L_{\text{2}}},t) = {\text{0}}} \end{aligned} \right. $$ (4c) 通过观察方程(3c)和式(4b)可以发现, 在梁1和梁2的边界控制方程出现含端部非线性单元参数的复杂项, 直接求解较困难. 因此, 本文采用狄拉克函数将边界控制方程的复杂项代入弹性梁系统的振动控制方程. 弹性梁系统的振动控制方程和边界控制方程被重建. 重建得到的梁1和梁2的振动控制方程具体表达式为
$$\begin{split} & {\rho _{\text{1}}}{S_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_1}}}{{\partial {t^{\text{2}}}}} + {E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{4}}}{u_1}}}{{\partial x_1^{\text{4}}}} + \delta ({x_{\text{1}}} - {L_{\text{1}}})\\ &\qquad \Bigg\{ {k_{\text{E}}}[{u_{\text{1}}} - {u_{\text{2}}}(0,t)] + k{n_{\text{E}}}{{[{u_{\text{1}}} - {u_{\text{2}}}(0,t)]}^{\text{3}}} +\\ &\qquad {C_{\text{E}}}\left[\frac{{\partial {u_{\text{1}}}}}{{\partial t}} - \frac{{{\text{d}}{u_{\text{2}}}(0,t)}}{{{\text{d}}t}}\right] \Bigg\} + \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \\ &\qquad {C_{{\text{B1}}}}\frac{{\partial {u_{\text{1}}}}}{{\partial t}} + \delta ({x_{\text{1}}}){m_{\text{A}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} = 0 \end{split} $$ (5a) $$ \begin{split} & {\rho _{\text{2}}}{S_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}}){m_{\text{R}}}\frac{{{\partial ^2}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}} - {x_{\text{S}}}){k_{\text{S}}}{u_{\text{2}}} +\\ &\qquad {E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{4}}}{u_{\text{2}}}}}{{\partial x_{\text{2}}^{\text{4}}}} + \delta ({x_{\text{2}}} - {L_{\text{2}}}){F_0}{\text{sin}}(\omega t)+ {{\text{C}}_{{\text{B}}2}}\frac{{\partial {u_2}}}{{\partial t}} +\\ &\qquad \delta ({x_{\text{2}}}) \Bigg\{ {k_{\text{E}}}[{u_{\text{2}}} - {u_{\text{1}}}({L_{\text{1}}},t)] + k{n_{\text{E}}}{{[{u_{\text{2}}} - {u_{\text{1}}}({L_{\text{1}}},t)]}^{\text{3}}} +\\ &\qquad {C_{\text{E}}}\left[\frac{{\partial {u_{\text{2}}}}}{{\partial t}} - \frac{{{\text{d}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{{\text{d}}t}}\right] \Bigg\} = 0 \end{split} $$ (5b) 重建得到的梁1和梁2的边界控制方程为
$$ {x_{\text{1}}} = {\text{0}} :\left\{\begin{aligned} & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{3}}}{u_{\text{1}}}(0,t)}}{{\partial x_{\text{1}}^{\text{3}}}} + {k_{\text{L}}}{u_{\text{1}}}({\text{0}},t) = {\text{0}}} \\ & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}(0,t)}}{{\partial x_{\text{1}}^{\text{2}}}} - {K_{\text{L}}}\frac{{\partial {u_{\text{1}}}({\text{0}},t)}}{{\partial {x_{\text{1}}}}} = {\text{0}}} \end{aligned} \right. $$ (6a) $$ {x_{\text{1}}} = {L_{\text{1}}}:\left\{ \begin{aligned} & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{3}}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{\partial x_{\text{1}}^{\text{3}}}} - {k_{\text{I}}}{u_{\text{1}}}({L_{\text{1}}},t) = {\text{0}}} \\ & {{E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{\partial x_{\text{1}}^{\text{2}}}} = {\text{0}}} \end{aligned} \right. $$ (6b) $$ {x_{\text{2}}} = {\text{0}}:\left\{ \begin{aligned} & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{3}}}{u_{\text{2}}}({\text{0}},t)}}{{\partial x_{\text{2}}^{\text{3}}}} = {\text{0}}} \\ & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}({\text{0}},t)}}{{\partial x_{\text{2}}^{\text{2}}}} = {\text{0}}} \end{aligned} \right. $$ (6c) $$ {x_{\text{2}}} = {L_{\text{2}}}:\left\{\begin{aligned} & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{2}}}({L_{\text{2}}},t)}}{{\partial x_{\text{2}}^{\text{2}}}} + {K_{\text{R}}}\frac{{\partial {u_{\text{2}}}({L_{\text{2}}},t)}}{{\partial {x_{\text{2}}}}} = {\text{0}}} \\ & {{E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{3}}}{u_{\text{2}}}({L_{\text{2}}},t)}}{{\partial x_{\text{2}}^{\text{3}}}} - {k_{\text{R}}}{u_{\text{2}}}({L_{\text{2}}},t) = {\text{0}}} \end{aligned}\right. $$ (6d) 根据式(6)可知, 重建得到的梁1和梁2的边界控制方程是线性的, 可以作为伽辽金法的权函数和试函数. 在下一节将用伽辽金法来建立端部非线性单元耦合弹性梁系统的残差方程.
2. 模型求解
本节采用伽辽金法来预报端部非线性单元耦合弹性梁的动力学响应. 选取无阻尼和端部非线性单元的模态振型函数作为伽辽金法的权函数与试函数, 模态振型函数的表达式见附录C. 运用伽辽金法的权函数来离散弹性梁系统的振动控制方程, 并用伽辽金法的试函数来展开弹性梁的横向振动位移. 此外无阻尼和端部非线性单元的模态振型函数满足重新构建梁1和梁2边界控制方程的边界条件.
根据模态叠加原理, 将梁1和梁2的横向振动位移分别展开, 其表达形式如下
$$ {u_{\text{1}}}\left( {{x_{\text{1}}},t} \right) = \sum\limits_{i = {\text{1}}}^{{{N} _{\text{1}}}} {{\varphi _{{\text{1}}i}}\left( {{x_{\text{1}}}} \right){q_{{\text{1}}i}}\left( t \right)} $$ (7) $$ {u_{\text{2}}}\left( {{x_{\text{2}}},t} \right) = \sum\limits_{i = {\text{1}}}^{{{N} _{\text{2}}}} {{\varphi _{{\text{2}}i}}\left( {{x_{\text{2}}}} \right){q_{{\text{2}}i}}\left( t \right)} $$ (8) 式中, φ1i(x1)和φ2i(x2)分别为梁1和2的横向振动位移的第i阶模态振型函数, 同时φ1i(x1)和φ2i(x2)也可定义为伽辽金法的试函数; q1i(t)和q2i(t)为对应的第i阶时间项; N1和N2分别为梁1和梁2的位移截断数. 将方程(7)和方程(8)分别代入方程(5a)和方程(5b)中, 根据伽辽金法的离散条件, 可得到梁1和梁2的残差方程. 梁1的第m阶残差方程为
$$ \begin{split} &\int_0^{{L_{\text{1}}}} \left\{ {\rho _{\text{1}}}{S_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + {E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{4}}}{u_{\text{1}}}}}{{\partial {x_{\text{1}}}^{\text{4}}}} + \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} +\right.\\ &\qquad {C_{{\text{B1}}}}\frac{{\partial {u_{\text{1}}}}}{{\partial t}} + \delta ({x_{\text{1}}}){m_{\text{A}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{1}}} - {L_{\text{1}}})\cdot\\ &\qquad \Bigg\{ {k_{\text{E}}}[{u_{\text{1}}} - {u_{\text{2}}}(0,t)] + k{n_{\text{E}}}{[{u_{\text{1}}} - {u_{\text{2}}}(0,t)]^{\text{3}}} +\\ &\qquad \left.{C_{\text{E}}}\left[\frac{{\partial {u_{\text{1}}}}}{{\partial t}} - \frac{{{\text{d}}{u_{\text{2}}}(0,t)}}{{{\text{d}}t}}\right] \right\} \Bigg\} {\psi _{{\text{1}}m}}{\text{(}}{x_{\text{1}}}{\text{)d}}{x_{\text{1}}}{\text{ = 0}}\end{split} $$ (9) 式中, ψ1m(x1)为梁1的第m阶权函数.
梁2的第n阶残差方程为
$$\begin{split} &\int_0^{{L_{\text{2}}}} \left\{ {\rho _{\text{2}}}{S_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + {{\text{C}}_{{\text{B2}}}}\frac{{\partial {u_{\text{2}}}}}{{\partial t}} + {E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{4}}}{u_{\text{2}}}}}{{\partial {x_{\text{2}}}^{\text{4}}}} +\right.\\ &\qquad \delta ({x_{\text{2}}}){m_{\text{R}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}} - {x_{\text{S}}}){k_{\text{S}}}{u_{\text{2}}} + \delta ({x_2} - {L_2})\cdot\\ &\qquad {F_0}{\text{sin}}(\omega t) + \delta ({x_2})\Bigg\{ {k_{\text{E}}}[{u_2} - {u_1}({L_1},t)] + \\ &\qquad \left.\left.k{n_{\text{E}}}{[{u_{\text{2}}} - {u_{\text{1}}}({L_{\text{1}}},t)]^{\text{3}}} + {C_{\text{E}}}\left[\frac{{\partial {u_{\text{2}}}}}{{\partial t}} - \frac{{{\text{d}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{{\text{d}}t}}\right] \right\} \right\}\cdot \\ &\qquad {\psi _{{\text{2}}n}}\left( {{x_{\text{2}}}} \right){\text{d}}{x_{\text{2}}} = 0 \end{split}$$ (10) 式中, ψ2n(x2)为梁2的第n阶权函数.
将含有加速度项的式子放在等式一边, 可将残差方程简化为
$$\begin{split} & \int_0^{{L_{\text{1}}}} {\left[ {\rho _{\text{1}}}{S_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{1}}}){m_{\text{A}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{{{\mathrm{L}}}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} \right]}\cdot\\ &\qquad {\psi _{{\text{1}}m}}{\text{(}}{x_{\text{1}}}{\text{)d}}{x_{\text{1}}} = - \int_0^{{{{L}}_{\text{1}}}} \left\{ {E_{\text{1}}}{I_{\text{1}}}\frac{{{\partial ^{\text{4}}}{u_{\text{1}}}}}{{\partial {x_{\text{1}}}^{\text{4}}}} + {C_{{\text{B}}1}}\frac{{\partial {u_{\text{1}}}}}{{\partial t}} + \right.\\ &\qquad \delta ({x_{\text{1}}} - {L_{\text{1}}})\Bigg\{ {k_{\text{E}}}[{u_{\text{1}}} - {u_{\text{2}}}(0,t)] + k{n_{\text{E}}}{[{u_1} - {u_2}(0,t)]^3} +\\ &\qquad \left.\left.{C_{\text{E}}}\left[\frac{{\partial {u_1}}}{{\partial t}} - \frac{{{\text{d}}{u_{\text{2}}}(0,t)}}{{{\text{d}}t}}\right]\Bigg\} \right\} \right. {\psi _{{\text{1}}m}}{\text{(}}{x_{\text{1}}}{\text{)d}}{x_{\text{1}}} = - {{{R}}_{{\text{1}}m}} \end{split} $$ (11) $$ \begin{split} & \int_0^{{L_{\text{2}}}} {\left[ {{\rho _{\text{2}}}{S_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}}){m_{\text{R}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}}} \right]} {\psi _{{\text{2}}n}}\left( {{x_{\text{2}}}} \right){\text{d}}{x_{\text{2}}} = \\ &\qquad - \int_0^{{L_{\text{2}}}} \left\{ {E_{\text{2}}}{I_{\text{2}}}\frac{{{\partial ^{\text{4}}}{u_{\text{2}}}}}{{\partial {x_{\text{2}}}^{\text{4}}}} + {{{C}}_{{\text{B2}}}}\frac{{\partial {u_{\text{2}}}}}{{\partial t}} + \delta ({x_{\text{2}}} - {x_{\text{S}}}){k_{\text{S}}}{u_{\text{2}}} +\right.\\ &\qquad \delta ({x_{\text{2}}} - {L_{\text{2}}}){F_0}{\text{sin}}(\omega t) \delta ({x_{\text{2}}})\Bigg\{ {k_{\text{E}}}[{u_{\text{2}}} - {u_{\text{1}}}({L_{\text{1}}},t)] +\\ &\qquad \left.\left.k{n_{\text{E}}}{[{u_{\text{2}}} - {u_{\text{1}}}({L_{\text{1}}},t)]^{\text{3}}} + {C_{\text{E}}}\left[\frac{{\partial {u_{\text{2}}}}}{{\partial t}} - \frac{{{\text{d}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{{\text{d}}t}}\right] \right\}\right\}\cdot\\ &\qquad {\psi _{{\text{2}}n}}\left( {{x_{\text{2}}}} \right){\text{d}}{x_{\text{2}}} = - {{{R}}_{{\text{2}}n}}\end{split} $$ (12) 方程(11)和式(12)又可改写为
$$\begin{split} & \int_0^{{L_1}} {\left[{\rho _{\text{1}}}{S_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{1}}}){m_{\text{A}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}} \right]}\cdot \\ &\qquad {\psi _{{\text{1}}m}}\left( {{x_{\text{1}}}} \right){\text{d}}{x_{\text{1}}} = {C_{m{\text{1}}}}\frac{{{{\text{d}}^2}{q_{{\text{11}}}}}}{{{\text{d}}{t^{\text{2}}}}} + \cdots + {C_{mi}}\frac{{{{\text{d}}^2}{q_{{\text{1}}i}}}}{{{\text{d}}{t^{\text{2}}}}} + \cdots +\\ &\qquad {C_{m{{{N}}_1}}}\frac{{{{\text{d}}^2}{q_{{\text{1}}{{{N}}_{\text{1}}}}}}}{{{\text{d}}{t^{\text{2}}}}} = - {R_{{\text{1}}m}} \end{split} $$ (13) $$\begin{split} & \int_0^{{L_{\text{2}}}} {\left[ {{\rho _{\text{2}}}{S_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}} + \delta ({x_{\text{2}}}){m_{\text{R}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}}} \right]} {\psi _{{\text{2}}n}}\left( {{x_{\text{2}}}} \right){\text{d}}{x_{\text{2}}} = \\ &\qquad {{{d}}_{n1}}\frac{{{{\text{d}}^{\text{2}}}{q_{{\text{21}}}}}}{{{\text{d}}{t^{\text{2}}}}} + \cdots + {{{d}}_{nj}}\frac{{{{\text{d}}^2}{q_{{\text{2}}j}}}}{{{\text{d}}{t^{\text{2}}}}} + \cdots + {{{d}}_{n{{{N}}_{\text{2}}}}}\frac{{{{\text{d}}^{\text{2}}}{q_{{\text{2}}{{{N}}_{\text{2}}}}}}}{{{\text{d}}{t^{\text{2}}}}} = - {R_{{\text{2}}n}} \end{split}$$ (14) 式中, N1和N2分别是梁1和梁2的截断数; R1m, R2n, Cmi和dnj的表达式见附录D.
进一步将方程(13)和式(14)整理为矩阵形式并根据矩阵运算得到
$$ \left[ {\begin{array}{*{20}{c}} {\dfrac{{{{\text{d}}^{\text{2}}}{q_{{\text{11}}}}}}{{{\text{d}}{t^2}}}} \\ {\cdots} \\ {\dfrac{{{{\text{d}}^{\text{2}}}{q_{{\text{1}}i}}}}{{{\text{d}}{t^{\text{2}}}}}} \\ {\cdots} \\ {\dfrac{{{{\text{d}}^{\text{2}}}{q_{{\text{1}}{{N}_{\text{1}}}}}}}{{{\text{d}}{t^{\text{2}}}}}} \end{array}} \right] = - {\left[ {\begin{array}{*{20}{c}} {{c_{{\text{11}}}}}&{\cdots}&{{c_{1i}}}&{\cdots}&{{c_{{\text{1}}{{N}_{\text{1}}}}}} \\ {\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots} \\ {{c_{m{\text{1}}}}}&{\cdots}&{{c_{mi}}}&{\cdots}&{{c_{m{{N}_{\text{1}}}}}} \\ {\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots} \\ {{c_{{{M1}}}}}&{\cdots}&{{c_{{{M}}i}}}&{\cdots}&{{c_{{{M}}{{N}_{\text{1}}}}}} \end{array}} \right]^{ - {\text{1}}}}\left[ {\begin{array}{*{20}{c}} {{R_{{\text{11}}}}} \\ {\cdots} \\ {{R_{{\text{1}}m}}} \\ {\cdots} \\ {{R_{{{1M}}}}} \end{array}} \right] $$ (15) $$ \left[ {\begin{array}{*{20}{c}} {\dfrac{{{{\text{d}}^{\text{2}}}{q_{{\text{21}}}}}}{{{\text{d}}{t^2}}}} \\ {\cdots} \\ {\dfrac{{{{\text{d}}^{\text{2}}}{q_{{\text{2}}j}}}}{{{\text{d}}{t^{\text{2}}}}}} \\ {\cdots} \\ {\dfrac{{{{\text{d}}^{\text{2}}}{q_{{\text{2}}{{N}_{\text{2}}}}}}}{{{\text{d}}{t^{\text{2}}}}}} \end{array}} \right] = - {\left[ {\begin{array}{*{20}{c}} {{{\text{d}}_{{\text{11}}}}}&{\cdots}&{{{\text{d}}_{{\text{1}}j}}}&{\cdots}&{{{\text{d}}_{{\text{1}}{{N}_{\text{2}}}}}} \\ {\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots} \\ {{{\text{d}}_{n{\text{1}}}}}&{\cdots}&{{{\text{d}}_{ni}}}&{\cdots}&{{{\text{d}}_{n{{N}_{\text{2}}}}}} \\ {\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots} \\ {{{\text{d}}_{{{N1}}}}}&{\cdots}&{{{\text{d}}_{{N}i}}}&{\cdots}&{{{\text{d}}_{{N}{{N}_{\text{2}}}}}} \end{array}} \right]^{ - {\text{1}}}}\left[ {\begin{array}{*{20}{c}} {{R_{{\text{21}}}}} \\ {\cdots} \\ {{R_{{\text{2}}n}}} \\ {\cdots} \\ {{R_{{{2N}}}}} \end{array}} \right] $$ (16) 其中, M和N分别为梁1和梁2对应的残差方程截断数.
方程(15)和式(16)可以用数值方法去求解. 本文用龙格库塔算法去求解相应的等式并将计算结果代入方程(7)和式(8), 得到弹性梁系统的横向振动位移, 进而可得到相应的振动动能.
3. 数值结果与分析
基于第2节推导的振动系统残差方程, 本节采用数值仿真软件计算端部非线性单元耦合弹性梁系统的振动响应. 在本节研究中, 截断数满足N1 = N2 = M = N的关系. 首先, 研究伽辽金法在预报端部非线性单元耦合弹性梁系统振动响应时的稳定性与可靠性. 在保证数值结果正确性的基础上, 研究端部非线性单元耦合参数对弹性梁系统振动响应的影响规律并揭示系统非线性动力学行为生成机理.
为从全局角度研究弹性梁系统的振动响应, 本文选取梁结构振动动能作为目标函数. 此外, 本文研究端部非线性单元耦合弹性梁系统的振动响应时, 时域计算区间选定为500个外部激励周期, 并以最后100个周期作为稳定响应区间. 本文弹性梁系统中的杨氏模量E1 = E2取值为6.89 × 1010 Pa; 密度ρ1 = ρ2取值为2.8 × 103 kg/m3; 长度L1 = L2取值为1 m; 直径D1和D2取值为0.016 m和0.02 m; 黏性阻尼CB1 = CB2取值为5 N·s/m; 集中质量mA取值0.01 kg; 内部平动弹簧的位置xS取值0.2 m; 激励力的幅值F0取值10 N; 平动约束弹簧kL = kI = kR的刚度取值为5.0 × 103 N/m, 1.0 × 103 N/m; 旋转约束弹簧KL = KR的刚度取值为103 N·m/rad; 附加质量mL = mR取值为0.005 kg; 端部耦合黏性阻尼取值为1 N·s/m; 端部耦合线性刚度取值为1.0 × 103 N/m3; 端部耦合非线性刚度取值为1.0 × 108 N/m3.
考虑到实际工程应用, 工程师们更关注非线性耦合单元应用于弹性梁系统时的潜在振动控制效果及其参数影响规律, 本文后续将主要集中研究端部耦合非线性单元对弹性梁系统振动响应及能量传递特性的影响规律. 此外, 考虑到混沌振动在工程中通常会对系统的稳定性产生不利影响, 工程师们通常希望能够避免系统发生混沌运动以提高其鲁棒性. 为使本研究数值仿真分析更贴近工程, 在后续研究中将主要讨论稳态振动的情况.
3.1 模型验证
本节研究伽辽金法在预报端部非线性单元耦合弹性梁系统振动响应时的可靠性与稳定性. 首先, 通过对比分析不同截断数下振动响应曲线来研究截断数对伽辽金法稳定性的影响. 在确定伽辽金法稳定性的基础上, 对比伽辽金法、拉格朗日法与谐波平衡法得到的振动响应以验证伽辽金法的可靠性.
首先, 研究伽辽金法在预报端部非线性单元耦合弹性梁系统振动响应时的稳定性. 图2为不同截断数下梁1和梁2的振动响应曲线. 在本研究中, 弹性梁系统振动动能参考值为10−12 J. 由图2可知, 截断数影响伽辽金法的稳定性. 当截断数超过4后弹性梁系统的振动响应曲线趋于稳定. 因此可以认为伽辽金法在预报端部非线性单元耦合弹性梁系统的动能响应时具有稳定性. 本文在后续研究中, 均选取截断数为4来研究端部非线性单元耦合弹性梁系统的振动响应.
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图 2 不同截断数下梁1和梁2的振动响应曲线Figure 2. Vibration energy response curves of beam 1 and beam 2 under different truncation numbers然后, 研究伽辽金法在预报端部非线性单元耦合弹性梁系统振动响应时的可靠性. 图3是在伽辽金法、拉格朗日法和谐波平衡法下得到的梁1和梁2振动响应曲线对比. 蓝色线代表伽辽金法计算结果, 红色线代表拉格朗日法计算结果, 绿色线代表谐波平衡法计算结果. 值得注意的是伽辽金法和拉格朗日法的建模方法不同, 伽辽金法从弹性梁系统的控制方程来建立残差方程, 拉格朗日法从拉格朗日函数来建立弹性梁系统的矩阵方程. 谐波平衡法和伽辽金法求解的目标方程(弹性梁系统的残差方程(15)和式(16))是相同的. 此外, 伽辽金法和拉格朗日法是从时域角度来求解系统的振动响应, 而谐波平衡法是从频域角度来求解系统的振动响应. 由图3可知, 采用伽辽金法、拉格朗日法和谐波平衡法得到的振动响应曲线吻合良好. 综合考虑上述3种建模方法的差异与数值结果的吻合程度, 即本文通过不同建模方法得到相同的数值仿真结果, 说明伽辽金法能够正确预报端部非线性单元耦合弹性梁系统的振动响应. 综上所述, 伽辽金法在预报端部非线性单元耦合弹性梁系统振动响应时具有可靠性与稳定性.
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图 3 在伽辽金法、拉格朗日法和谐波平衡法下得到的梁1和梁2振动响应曲线对比Figure 3. Comparison of vibration response curves of beam 1 and beam 2 obtained by Galerkin method, Lagrange method, and harmonic balance method3.2 端部耦合非线性单元对弹性梁系统振动响应的影响
为有效研究端部耦合非线性单元对弹性梁系统振动响应的影响, 考虑端部耦合非线性单元包含非线性刚度和黏性阻尼, 开展其关键参数研究有利于端部非线性单元在弹性梁系统振动控制的工程应用. 为此, 本节首先研究端部耦合非线性刚度和端部耦合黏性阻尼对弹性梁系统振动响应的影响规律. 之后, 进一步解释端部非线性耦合单元对弹性梁系统振动响应的影响机理.
首先, 研究端部耦合非线性刚度对弹性梁系统振动响应的影响规律. 端部耦合非线性刚度knE分别取0, 1.0 × 108, 2.0 × 108和4.0 × 108 N/m3. 图4是梁1和梁2在不同端部耦合非线性刚度下的振动响应曲线. 由图4可知, 在弹性梁系统振动响应主共振区内, 端部耦合非线性刚度变化显著影响其动力学行为. 随着端部耦合非线性刚度的增加, 弹性梁系统主共振区均向高频偏移. 当端部耦合非线性刚度增加至一定值时(knE取2.0 × 108和4.0 × 108 N/m3), 弹性梁系统振动响应在主共振区处出现峰值跳跃现象(对应于图4中红色和蓝色曲线). 当端部耦合非线性刚度增加至4.0 × 108 N/m3时, 弹性梁系统出现复杂非线性振动响应, 即单一激励频率作用下, 系统存在多个振动响应幅值. 为研究弹性梁系统复杂非线性振动状态, 绘制了弹性梁系统观测点处(0.4 m)复杂非线性振动响应的相轨迹图(4.0 × 108 N/m3与100 Hz), 并在相轨迹图中绘制庞加莱点. 从图4中可以看到相轨迹图稳定, 庞加莱点具备形成封闭曲线的趋势, 表明复杂非线性振动响应状态是准周期振动.
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图 4 梁1和梁2在不同端部耦合非线性刚度下的振动响应曲线Figure 4. Vibration response curves of beam 1 and beam 2 under different end-coupled nonlinear stiffnesses综上所述, 端部非线性刚度具备激发弹性梁系统复杂非线性振动的潜力. 在本研究中, 考虑到谐波激励作用于梁2上, 将梁2定义为主动梁, 梁1定义为从动梁. 弹性梁系统通过端部耦合非线性刚度的引入, 使得梁2的振动能量降低, 梁1的振动能量增加. 端部耦合非线性刚度有利于系统振动能量从主动梁传递给从动梁. 值得注意的是, 此时从动梁有类似于吸振器的作用.
然后, 研究端部耦合黏性阻尼对弹性梁系统振动响应的影响规律. 端部耦合黏性阻尼分别取1, 5, 10和20 N·s/m. 图5是梁1和梁2在不同端部耦合黏性阻尼下的振动响应曲线. 由图5可知, 端部耦合黏性阻尼变化对弹性梁系统振动响应影响显著. 端部耦合黏性阻尼为1 N·s/m时, 梁1振动响应在第4主共振区出现峰值跳跃现象, 梁2振动响应峰值跳跃现象较不明显; 弹性梁系统在该黏性阻尼下出现复杂非线性振动响应. 为研究弹性梁系统复杂非线性振动状态, 绘制了弹性梁系统观测点处(0.4 m)复杂非线性振动响应的相轨迹图(1 N·s/m与100 Hz), 并在相轨迹图绘制庞加莱点. 从图5中可以看出相轨迹图稳定, 庞加莱点具备形成封闭曲线的趋势, 表明复杂非线性振动响应状态是准周期振动. 端部耦合黏性阻尼的增加会使弹性梁系统振动能量耗散减小, 因此端部耦合黏性阻尼的增加会抑制弹性梁系统非线性现象的出现.
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图 5 梁1和梁2在不同端部耦合黏性阻尼下的振动响应曲线Figure 5. Vibration response curves of beam 1 and beam 2 with different end-coupled viscous damping综上所述, 端部耦合黏性阻尼的增加, 对弹性梁系统的振动控制起积极影响作用. 弹性梁系统在低阻尼下会出现复杂非线性振动响应.
以上是从全局角度去观测端部非线性单元耦合弹性梁系统的能量特性和规律, 可以发现当端部耦合非线性刚度和端部耦合黏性阻尼增大至一定值时, 弹性梁系统振动响应出现了非线性现象. 考虑到与线性系统相比, 本文弹性梁系统所引入的额外量是端部耦合非线性恢复力, 为深入分析图4和图5非线性现象的生成原因, 需深入研究端部耦合非线性刚度和端部耦合黏性阻尼变化时端部耦合非线性恢复力与非线性现象的关系.
图6为端部耦合非线性恢复力在不同端部耦合非线性刚度和端部耦合黏性阻尼下的响应曲线, 其中图6(a)是端部耦合非线性恢复力在不同端部耦合非线性刚度影响下的响应曲线, 图6(b)是端部耦合非线性恢复力在不同端部耦合黏性阻尼影响下的响应曲线. 对照图6(a)和图4可发现, 随着端部耦合非线性刚度的增加, 弹性梁系统振动响应在主共振区处出现峰值跳跃现象(对应于图4中红色和蓝色曲线), 端部耦合非线性恢复力同样在主共振区处出现峰值跳跃现象(对应于图6(a)中红色和蓝色曲线). 当端部耦合非线性刚度增加至4.0 × 108 N/m3时, 弹性梁系统和端部耦合非线性恢复力均出现复杂非线性振动响应. 以上现象表明端部耦合非线性恢复力是激发图4非线性现象出现的原因.
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图 6 端部耦合非线性恢复力在不同端部耦合非线性刚度和端部耦合黏性阻尼下的响应曲线Figure 6. Response curves of end-coupled nonlinear restoring force under different end-coupled nonlinear stiffness and end-coupled viscous damping对照图6(b)和图5发现, 随着端部耦合黏性阻尼的减小, 弹性梁系统振动响应在主共振区处出现峰值跳跃现象(对应于图5中红色和蓝色曲线), 端部耦合非线性恢复力同样在主共振区处出现峰值跳跃现象(对应于图6(b)中红色和蓝色曲线). 当端部耦合黏性阻尼减小至1 N·s/m时, 弹性梁系统和端部耦合非线性恢复力均出现复杂非线性振动响应. 以上现象表明端部耦合非线性恢复力是激发图5非线性现象出现的原因. 另外在低阻尼下弹性梁系统能够出现非线性现象和发生准周期振动. 由3.4节可知, 系统发生准周期振动时振动能量可以单向传递即出现靶向能量传递的现象, 在工程中有从时域角度单向地控制振动能量传递的可能, 从而达到减小主动梁振动能量的效果. 在低阻尼下改变端部耦合非线性刚度有更好地调控系统在准周期状态下振动能量单向控制至从动梁实现减小主动梁振动能量的可能, 同时也可以调控系统的振动能量在非准周期振动状态下从主动梁传递至从动梁, 进一步达到减小主动梁振动能量的效果.
综上所述, 端部耦合非线性恢复力是激发弹性梁系统非线性现象出现的原因. 此外, 在低阻尼下端部耦合非线性刚度对弹性梁系统振动响应具有良好的效果.
3.3 端部耦合非线性单元在单频激励下对弹性梁系统振动响应的影响
实际工程中, 动力设备如发动机、柴油机等通常在额定工况下工作, 其产生的谐波激励处于稳定状态. 动力设备引入的谐波激励会引起结构振动响应. 因此研究单频激励下端部耦合非线性单元对弹性梁系统振动响应的影响具有重要工程意义. 在3.2节的基础上, 本节将进一步研究端部耦合非线性刚度和端部耦合黏性阻尼在单频激励下对弹性梁系统振动响应影响机理.
首先, 研究单频激励下端部耦合非线性刚度对弹性梁系统振动响应的影响规律. 图7是梁1和梁2在单频激励下随端部耦合非线性刚度变化的振动响应曲线, 端部耦合非线性刚度取值范围在108 ~ 1010 N/m3. 根据弹性梁系统振动状态, 将图7划分为3个区域, 其中区域1和区域3是单周期振动状态, 区域2中弹性梁系统出现复杂非线性振动响应. 为研究弹性梁系统复杂非线性振动状态, 绘制了弹性梁系统观测点处(0.4 m)复杂非线性振动响应的相轨迹图(109.4 N/m3与50 Hz), 并在相轨迹图中绘制庞加莱点. 从图7中可以看到相轨迹图稳定, 庞加莱点具备形成封闭曲线的趋势, 表明复杂非线性振动响应状态是准周期振动. 由图7可发现, 在单频激励下弹性梁系统随着端部耦合非线性刚度的增加出现先单周期振动状态后准周期振动状态再单周期振动状态的现象. 生成上述现象的原因是当弹性梁系统处于额定工况时, 其振动能量保持稳定没有产生突变, 系统的振动位移基本保持不变. 由图4可知, 增加端部耦合非线性刚度会使主共振区向高频偏移. 随着端部耦合非线性刚度的增加, 系统非线性恢复力增大, 当主共振区偏移至激励频率为50 Hz时会激发系统复杂非线性振动响应, 即系统振动状态由单周期振动改变为准周期振动. 继续增加端部耦合非线性刚度使得共振区向高频偏移, 此时系统处于非共振区, 振动幅值减小, 系统振动状态由准周期振动改变为单周期振动. 此外, 在单频激励下端部耦合非线性刚度的增加, 使得主动梁的振动能量减小, 从动梁的振动能量增加, 表明单频激励下端部耦合非线性刚度有益于主动梁的振动控制, 不利于从动梁的振动控制.
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图 7 梁1和梁2在单频激励下随端部耦合非线性刚度变化的振动响应曲线Figure 7. Vibration response curves of beam 1 and beam 2 with end-coupled nonlinear stiffness variation under single frequency excitation综上所述, 单频激励下端部耦合非线性刚度具备激发弹性梁系统复杂非线性振动响应的潜力. 端部耦合非线性刚度有利于系统振动能量在单频激励下由主动梁传递至从动梁. 值得注意的是, 此时从动梁有类似吸振器的作用.
然后, 研究在单频激励下端部耦合黏性阻尼对弹性梁系统振动响应的影响规律. 图8是梁1和梁2在单频激励下随端部耦合黏性阻尼变化的振动响应曲线. 端部耦合黏性阻尼取值范围为1 ~ 100 N·s/m. 根据弹性梁系统振动状态, 将图8划分为两个区域, 其中区域2为单周期振动状态, 区域1中弹性梁系统出现复杂非线性振动响应. 为研究弹性梁系统复杂非线性振动状态, 绘制了弹性梁系统观测点处(0.4 m)复杂非线性振动响应的相轨迹图(2 N·s/m与50 Hz), 并在相轨迹图中绘制庞加莱点. 从图8中可以看到相轨迹图稳定, 庞加莱点具备形成封闭曲线的趋势, 表明复杂非线性振动响应状态是准周期振动. 由图8可以发现, 在单频激励下弹性梁系统随着端部耦合黏性阻尼的增加出现先准周期振动状态后单周期振动状态的现象. 生成上述现象的原因是增加端部耦合黏性阻尼会耗散弹性梁系统振动能量, 系统振动状态由准周期振动会改变为单周期振动. 随着端部耦合黏性阻尼的增加, 弹性梁系统的振动能量增加, 不利于系统振动控制. 综上所述, 在单频激励下低阻尼对弹性梁系统振动响应具有良好的效果.
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图 8 梁1和梁2在单频激励下随端部耦合黏性阻尼变化的振动响应曲线Figure 8. Vibration response curves of beam 1 and beam 2 with end-coupled viscous damping variation under single frequency excitation3.4 准周期振动状态的能量传递特性
由3.2节和3.3节的分析可知, 端部非线性单元对弹性梁系统振动响应影响显著, 合适的端部非线性单元参数有利于弹性梁系统的振动控制. 但是, 在一定参数下端部非线性单元具备激发弹性梁系统复杂非线性振动响应和改变弹性梁系统振动状态的能力. 为了深入研究复杂非线性振动响应下系统的振动能量传递特性, 本节研究梁1和梁2在准周期振动状态下的能量传递特性.
图9是梁1和梁2在准周期振动状态下的动能响应和动能响应峰值, 其中图9(a)是梁1在准周期振动状态下的动能响应, 图9(b)是梁2在准周期振动状态下的动能响应, 图9(c)是梁1和梁2在准周期振动状态下的动能响应峰值. 由图9(a)和图9(b)可知, 梁1和梁2的动能在准周期振动状态下呈现周期性振荡变化. 梁2在谐波激励作用下其动能呈现周期性振荡变化. 梁1通过一定参数下端部非线性单元与梁2连接呈现周期性振荡变化. 在图9(c)中, 红色线是梁1的动能响应峰值曲线, 蓝色线是梁2的动能响应峰值曲线. 由图9(c)可知, 梁1和梁2的动能峰值周期性振荡变化规律不同. 在灰色区域中, 梁1的动能峰值单调增大, 梁2的动能峰值单调减小, 此时振动能量由梁2靶向传递至梁1. 在白色区域中, 梁1和梁2的动能峰值同时单调减小或同时单调增大, 此时振动能量在梁1和梁2之间没有靶向传递. 在青色区域中, 梁1的动能峰值单调减小, 梁2的动能峰值单调增大, 此时振动能量由梁1靶向传递至梁2. 由于梁1和梁2在灰色区域和青色区域中出现单向振动能量靶向传递现象, 因此可称这两个区域为靶向传递区. 由于弹性梁系统在灰色区域和青色区域中发生单向振动能量靶向传递, 而在中间的白色区域中不发生单向振动能量靶向传递, 因此可称白色区域为能量传递区. 梁1和梁2在周期性振荡中存在时域动能峰值的相位变化, 导致梁1和梁2在靶向传递区出现靶向能量传递的现象. 由于单周期时域动能峰值不存在相位差, 振动能量是稳定地传递, 因此在弹性梁系统中单向振动能量靶向传递现象仅出现于准周期运动.
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图 9 梁1和梁2在准周期振动状态下的动能响应和动能响应峰值Figure 9. Kinetic energy response and peak kinetic energy response of beam 1 and beam 2 under quasi-periodic vibration state综上所述, 在准周期振动状态下, 弹性梁系统振动能量出现靶向能量传递现象. 靶向振动能量传递现象的出现为从时域角度单向控制弹性梁系统的振动水平提供可能.
4. 结 论
为探讨端部耦合非线性单元对由其耦合弹性梁系统振动响应的影响, 本文建立了端部非线性单元耦合弹性梁系统的物理模型. 根据广义哈密顿原理和变分法, 推导弹性梁系统的振动控制方程和边界控制方程. 基于所建立的振动分析模型, 从能量角度, 根据广义哈密顿原理和变分法, 推导得到弹性梁系统的控制方程. 之后, 采用伽辽金法展开弹性梁的横向振动位移并建立其残差方程. 在保证数值结果正确性的基础上, 深入研究端部耦合非线性单元对弹性梁系统频率响应的影响规律, 探索端部耦合非线性单元在单频激励下对弹性梁系统振动响应的影响规律, 揭示弹性梁系统复杂非线性现象的生成机理. 此外还研究了在准周期振动状态下的弹性梁系统振动能量传递特性. 本文结论如下.
(1)伽辽金法能够准确预报端部非线性单元耦合弹性梁系统的振动响应. 合适的截断数能够保证弹性梁系统振动响应的稳定性.
(2)端部非线性单元对弹性梁系统振动响应影响显著, 适当选取端部非线性单元参数可以有效实现对弹性梁系统的振动控制, 有助于弹性梁系统在工程实际中的应用. 端部非线性单元引入非线性恢复力的变化是影响弹性梁系统非线性振动响应出现的根本原因.
(3)在复杂非线性振动状态下, 弹性梁系统出现靶向能量传递现象, 准周期振动状态是出现该现象的标志. 靶向振动能量传递现象的出现为从时域角度单向控制弹性梁系统的振动水平提供可能.
(4)合理利用端部非线性单元, 能够表现出主动梁振动能量传递至从动梁的现象, 降低主动梁振动能量. 此时, 从动梁类似于吸振器.
附录A
$$ {T_{{\text{Beam1}}}} = \int_0^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} {\rho _{\text{1}}}{S_{\text{1}}}{\left(\frac{{\partial {u_{\text{1}}}}}{{\partial t}}\right)^{\text{2}}}{\text{d}}{x_{\text{1}}} $$ (A1) $$ {T_{{\text{Beam2}}}} = \int_0^{{L_{\text{2}}}} {\frac{{\text{1}}}{{\text{2}}}} {\rho _{\text{2}}}{S_{\text{2}}}{\left(\frac{{\partial {u_{\text{2}}}}}{{\partial t}}\right)^{\text{2}}}{\text{d}}{x_{\text{2}}} $$ (A2) $$ {T_{\text{1}}} = \int_0^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{1}}}){m_{\text{A}}}{\left(\frac{{\partial {u_{\text{1}}}}}{{\partial t}}\right)^{\text{2}}}{\text{d}}{x_{\text{1}}} $$ (A3) $$ {T_{\text{2}}} = \int_0^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}{\left(\frac{{\partial {u_{\text{1}}}}}{{\partial t}}\right)^{\text{2}}}{\text{d}}{x_{\text{1}}} $$ (A4) $$ {T_{\text{3}}} = \int_0^{{L_{\text{2}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{2}}}){m_{\text{R}}}{\left(\frac{{\partial {u_{\text{2}}}}}{{\partial t}}\right)^{\text{2}}}{\text{d}}{x_{\text{2}}} $$ (A5) $$ {V_{{\text{Beam1}}}} = \int_0^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} {E_{\text{1}}}{I_{\text{1}}}{\left(\frac{{{\partial ^{\text{2}}}{u_1}}}{{\partial {x_{\text{1}}}^{\text{2}}}}\right)^{\text{2}}}{\text{d}}{x_{\text{1}}} $$ (A6) $$ {V_{{\text{Beam2}}}} = \int_0^{{L_{\text{2}}}} {\frac{{\text{1}}}{{\text{2}}}} {E_{\text{2}}}{I_{\text{2}}}{\left(\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {x_{\text{2}}}^{\text{2}}}}\right)^{\text{2}}}{\text{d}}{x_{\text{2}}} $$ (A7) $$ {V_{\text{1}}} = \int_0^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{1}}}){k_{\text{L}}}u_{\text{1}}^{\text{2}}{\text{d}}{x_{\text{1}}} $$ (A8) $$ {V_{\text{2}}} = \int_0^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{1}}} - {{\text{L}}_{\text{1}}}){k_{\text{I}}}u_{\text{1}}^{\text{2}}{\text{d}}{x_{\text{1}}} $$ (A9) $$ {V_{\text{3}}} = \int_0^{{L_{\text{2}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{2}}} - {x_{\text{S}}}){k_{\text{S}}}u_{\text{2}}^{\text{2}}{\text{d}}{x_{\text{2}}} $$ (A10) $$ {V_{\text{4}}} = \int_0^{{L_{\text{2}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{2}}} - {L_{\text{2}}}){k_{\text{R}}}u_{\text{2}}^{\text{2}}{\text{d}}{x_{\text{2}}} $$ (A11) $$ {V_{\text{5}}} = \int_{\text{0}}^{{L_{\text{1}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{1}}}){K_{\text{L}}}{\left(\frac{{\partial {u_{\text{1}}}}}{{\partial {x_{\text{1}}}}}\right)^{\text{2}}}{\text{d}}{x_{\text{1}}} $$ (A12) $$ {V_{\text{6}}} = \int_{\text{0}}^{{L_{\text{2}}}} {\frac{{\text{1}}}{{\text{2}}}} \delta ({x_{\text{2}}} - {L_{\text{2}}}){K_{\text{R}}}{\left(\frac{{\partial {u_{\text{2}}}}}{{\partial {x_{\text{2}}}}}\right)^{\text{2}}}{\text{d}}{x_{\text{2}}} $$ (A13) $$ \begin{split} &{V_{\text{7}}} = \int_{\text{0}}^{{{{L}}_{\text{1}}}} {\int_{\text{0}}^{{{{L}}_{\text{2}}}} {\delta ({x_{\text{1}}} - {L_{\text{1}}})} } \delta \left( {{x_{\text{2}}}} \right)\left[\frac{{\text{1}}}{{\text{2}}}{k_{\text{E}}}{({u_{\text{1}}} - {u_{\text{2}}})^{\text{2}}} +\right.\\ &\qquad \left.\frac{{\text{1}}}{{\text{4}}}k{n_{\text{E}}}{({u_{\text{1}}} - {u_{\text{2}}})^{\text{4}}} \right]{\text{d}}{x_{\text{1}}}{\text{d}}{x_{\text{2}}}\end{split} $$ (A14) $$ \delta {W_{\text{1}}} = - \int_{\text{0}}^{{{{L}}_{\text{2}}}} \delta ({x_{\text{2}}} - {L_{\text{2}}}){F_{\text{0}}}{\text{sin}}(\omega t)\delta {u_{\text{2}}}{\text{d}}{x_{\text{2}}} $$ (A15) $$\delta {W_{\text{2}}} = - \int_{\text{0}}^{{L_{\text{1}}}} {{C_{{\text{B}}1}}} \frac{{\partial {u_{\text{1}}}}}{{\partial t}}\delta {u_{\text{1}}}{\text{d}}{x_{\text{1}}} $$ (A16) $$ \delta {W_{\text{3}}} = - \int_{\text{0}}^{{L_{\text{2}}}} {{C_{{\text{B2}}}}\frac{{\partial {u_{\text{2}}}}}{{\partial t}}\delta {u_{\text{2}}}} {\text{d}}{x_{\text{2}}} $$ (A17) $$ \begin{split} & \delta {W_{\text{4}}} = - \int_{\text{0}}^{{L_{\text{1}}}} {\int_{\text{0}}^{{L_{\text{2}}}} {\delta ({x_{\text{1}}} - {L_{\text{1}}})} } \delta ({x_{\text{2}}}){C_{\text{E}}}\left(\frac{{\partial {u_{\text{1}}}}}{{\partial t}} - \frac{{\partial {u_{\text{2}}}}}{{\partial t}}\right) \cdot \\ &\qquad \delta ({u_{\text{1}}} - {u_{\text{2}}}){\text{d}}{x_{\text{1}}}{\text{d}}{x_{\text{2}}} \end{split} $$ (A18) 附录B
$$ \delta \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {{T_{{\text{Beam1}}}}} {\text{d}}t = - \int_{{t_{\text{1}}}}^{{{{t}}_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{1}}}} {{\rho _{\text{1}}}} } {S_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}}\delta {u_{\text{1}}}{\text{d}}{x_{\text{1}}}{\text{d}}t $$ (B1) $$ \delta \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {{T_{{\text{Beam2}}}}} {\text{d}}t = - \int_{{t_{\text{1}}}}^{{{{t}}_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{2}}}} {{\rho _{\text{2}}}} } {S_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}}\delta {u_{\text{2}}}{\text{d}}{x_{\text{2}}}{\text{d}}t $$ (B2) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{T_{\text{1}}}{\text{d}}t = } - \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{1}}}} {\delta ({x_{\text{1}}}){m_{\text{A}}}} } \frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}}\delta {u_{\text{1}}}{\text{d}}{x_{\text{1}}}{\text{d}}t $$ (B3) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{T_{\text{2}}}{\text{d}}t = } - \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{1}}}} {\delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}} } \frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {t^{\text{2}}}}}\delta {u_{\text{1}}}{\text{d}}{x_{\text{1}}}{\text{d}}t $$ (B4) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{T_{\text{3}}}{\text{d}}t = } - \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{2}}}} {\delta ({x_{\text{2}}}){m_{\text{R}}}} } \frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {t^{\text{2}}}}}\delta {u_{\text{2}}}{\text{d}}{x_{\text{2}}}{\text{d}}t $$ (B5) $$ \begin{split} & \delta \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {{V_{{\text{Beam1}}}}} {\text{d}}t = \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{E_{\text{1}}}} {I_{\text{1}}}\frac{{{\partial ^{\text{2}}}{u_{\text{1}}}}}{{\partial {x_{\text{1}}}^{\text{2}}}}\delta \left(\frac{{\partial {u_{\text{1}}}}}{{\partial {x_{\text{1}}}}}\right)\Bigg| _{0}^{L_1} {\text{d}}t - \\ & \qquad \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {{E_{\text{1}}}} {I_{\text{1}}}\frac{{{\partial ^{\text{3}}}{u_{\text{1}}}}}{{\partial x_{\text{1}}^{\text{3}}}}\delta {u_{\text{1}}}\Bigg| _{0}^{L_1}{\text{d}}t + \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{1}}}} {{E_{\text{1}}}} } {I_{\text{1}}}\frac{{{\partial ^{\text{4}}}{u_{\text{1}}}}}{{\partial x_{\text{1}}^{\text{4}}}}\delta {u_{\text{1}}}{\text{d}}{x_{\text{1}}}{\text{d}}t \end{split} $$ (B6) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{V_{\text{1}}}} {\text{d}}t = \int_{{{{t}}_{{1}}}}^{{t_{\text{2}}}} {{k_{\text{L}}}} {u_{\text{1}}}({\text{0}},t)\delta {u_{\text{1}}}{\text{d}}t $$ (B7) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{V_{\text{2}}}} {\text{d}}t = \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{k_{\text{I}}}} {u_{\text{1}}}({L_{\text{1}}},t)\delta {u_{\text{1}}}{\text{d}}t $$ (B8) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{V_{\text{5}}}} {\text{d}}t = \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{K_{\text{L}}}} \frac{{\partial {u_{\text{1}}}({\text{0}},t)}}{{\partial {x_{\text{1}}}}}\delta \left(\frac{{\partial {u_{\text{1}}}}}{{\partial {x_{\text{1}}}}}\right){\text{d}}t $$ (B9) $$ \begin{split} & \delta \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {{V_{{\text{Beam2}}}}} {\text{d}}t = \int_{{{\text{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{E_{\text{2}}}} {I_{\text{2}}}\frac{{{\partial ^{\text{2}}}{u_{\text{2}}}}}{{\partial {x_{\text{2}}}^{\text{2}}}}\delta \left(\frac{{\partial {u_{\text{2}}}}}{{\partial {x_{\text{2}}}}}\right)\Bigg| _{0}^{L_2}{\text{d}}t - \\ &\qquad \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {{E_{\text{2}}}} {I_{\text{2}}}\frac{{{\partial ^{\text{3}}}{u_{\text{2}}}}}{{\partial x_{\text{2}}^{\text{3}}}}\delta {u_{\text{2}}}\Bigg| _{0}^{L_2}{\text{d}}t + \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_0^{{L_{\text{2}}}} {{E_{\text{2}}}} } {I_{\text{2}}}\frac{{{\partial ^{\text{4}}}{u_{\text{2}}}}}{{\partial x_{\text{2}}^{\text{4}}}}\delta {u_{\text{2}}}{\text{d}}{x_{\text{2}}}{\text{d}}t \end{split} $$ (B10) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{V_{\text{4}}}} {\text{d}}t = \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{k_{\text{R}}}} {u_{\text{2}}}({L_{\text{2}}},t)\delta {u_{\text{2}}}{\text{d}}t $$ (B11) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{V_{\text{3}}}} {\text{d}}t = \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{2}}}} {\delta ({x_{\text{2}}} - {x_{\text{S}}})} } {k_{\text{S}}}{u_{\text{2}}}\delta {u_{\text{2}}}{\text{d}}{x_{\text{2}}}{\text{d}}t $$ (B12) $$ \delta \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{V_{\text{6}}}} {\text{d}}t = \int_{{{{t}}_{\text{1}}}}^{{t_{\text{2}}}} {{K_{\text{R}}}} \frac{{\partial {u_{\text{2}}}({L_{\text{2}}},t)}}{{\partial {x_{\text{2}}}}}\delta \left(\frac{{\partial {u_{\text{2}}}}}{{\partial {x_{\text{2}}}}}\right){\text{d}}t $$ (B13) $$\begin{split} &\delta \int_{{t_1}}^{{t_2}} {{V_7}} {\text{d}}t = \int_{{t_1}}^{{t_2}} \Big\{ {k_{\text{E}}}[{u_{\text{1}}}({L_1},t) - {u_{\text{2}}}(0,t)] + k{n_{\text{E}}}[{u_{\text{1}}}({L_1},t) -\\ &\qquad {u_{\text{2}}}(0,t)]^{\text{3}} \Big\} \delta ({u_1} - {u_2}){\text{d}}t\end{split} $$ (B14) $$ \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\delta {W_{\text{1}}}{\text{d}}t = \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{{{L}}_{\text{2}}}} \delta ({x_{\text{2}}} - {L_{\text{2}}}){F_{\text{0}}}{\text{sin}}(\omega t)\delta {u_{\text{2}}}{\text{d}}{x_{\text{2}}}} {\text{d}}t} $$ (B15) $$ \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\delta {W_{\text{2}}}{\text{d}}t = - \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{1}}}} {{C_{{\text{B1}}}}} \frac{{\partial {u_{\text{1}}}}}{{\partial t}}\delta {u_{\text{1}}}{\text{d}}{x_{\text{1}}}} {\text{d}}t} $$ (B16) $$ \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\delta {W_3}} {\text{d}}t = - \int_{{t_{\text{1}}}}^{{t_{\text{2}}}} {\int_{\text{0}}^{{L_{\text{2}}}} {{C_{{\text{B2}}}}\frac{{\partial {u_{\text{2}}}}}{{\partial t}}\delta {u_{\text{2}}}} {\text{d}}{x_{\text{2}}}{\text{d}}t} $$ (B17) $$ \int_{{t_1}}^{{t_2}} {\delta {W_{\text{4}}}{\text{d}}t = - \int_{{t_1}}^{{t_2}} {{C_{\text{E}}}\left[\frac{{{\text{d}}{u_{\text{1}}}({L_1},t)}}{{{\text{d}}t}} - \frac{{{\text{d}}{u_{\text{2}}}(0,t)}}{{{\text{d}}t}}\right]\delta ({u_{\text{1}}} - {u_{\text{2}}}){\text{d}}t} } $$ (B18) 附录C
本文建立了弹性梁系统无阻尼和端部非线性单元的模态振型函数通用分析模型, 如图C1所示. 通过边界光滑傅里叶级数[3]可以得到梁1和梁2的模态振型函数, 具体推导过程如下.
通用分析模型的横向振动位移可以表达为
$$ u({x_i},t) = w({x_i}){{\mathrm{e}}^{{\text{j}}\omega {\text{t}}}} $$ (C1) 式中, w(xi)为模态振型函数. 当i = 1时为梁1的模态振型函数; 当i = 2时为梁2的模态振型函数.
拉格朗日方程为
$$ L = {V_{{\text{system}}}} - {T_{{\text{system}}}} $$ (C2) 式中, Vsystem为通用分析模型的总势能, Tsystem为通用分析模型的总动能.
通过边界光滑傅里叶级数表示模态振型函数w(xi)
$$ w({x_i}) = \sum\limits_{n = 0}^{\infty } {{a_n}\cos \left({\lambda _n}{x_i}\right) + {b_1}{\xi _1}({x_i}) + {b_2}} {\xi _2}({x_i}) + {b_3}{\xi _3}({x_i}) + {b_4}{\xi _4}({x_i}) $$ (C3) 式中, ${\lambda _n} = \dfrac{{n\text{π} }}{L}$, n为从0开始的正整数, ${\xi _1}({x_i}),$ ${\xi _2}({x_i}),$ ${\xi _3}({x_i})$和${\xi _4}({x_i})$为通用分析模型的边界光滑辅助函数
$$ \left. \begin{split} & {\xi _1}({x_i}) = \frac{{9L}}{{{{4\text{π} }}}}{{{\mathrm{sin}}}}\left(\frac{{\text{π} {x_i}}}{{2L}}\right) - \frac{L}{{{{12\text{π} }}}}{{{\mathrm{sin}}}}\left(\frac{{3\text{π} {x_i}}}{{2L}}\right) \\ & {\xi _2}({x_i}) = - \frac{{9L}}{{{{4\text{π} }}}}{{{\mathrm{cos}}}}\left(\frac{{\text{π} {x_i}}}{{2L}}\right) - \frac{L}{{{{12\text{π} }}}}{{{\mathrm{cos}}}}\left(\frac{{3\text{π} {x_i}}}{{2L}}\right) \\ & {\xi _3}({x_i}) = \frac{{{L^3}}}{{{{{\text{π}}}^3}}}{{{\mathrm{sin}}}}\left(\frac{{\text{π} {x_i}}}{{2L}}\right) - \frac{{{L^3}}}{{{{3}}{{{\text{π} }}^3}}}{{{\mathrm{sin}}}}\left(\frac{{3\text{π} {x_i}}}{{2L}}\right) \\ & {\xi _4}({x_i}) = - \frac{{{L^3}}}{{{{{\text{π} }}^3}}}{{{\mathrm{cos}}}}\left(\frac{{\text{π} {x_i}}}{{2L}}\right) - \frac{{{L^3}}}{{{{3}}{{{\text{π} }}^3}}}{{{\mathrm{cos}}}}\left(\frac{{3\text{π} {x_i}}}{{2L}}\right)\end{split} \right\} $$ (C4) 进一步将通用分析模型的横向振动位移写为矩阵相乘的形式
$$ u({x_i},t) = {\boldsymbol{\varPhi }}({x_i}){\boldsymbol{A}}{{\mathrm{e}}^{{\text{j}}\omega {\text{t}}}} $$ 式中, ${\boldsymbol{\varPhi }}({x_i})$为基函数向量, A为响应系数向量
$$ {\boldsymbol{\varPhi }}({x_i}) = {\left[ \begin{gathered} 1 \\ \cos {\lambda _1}{x_i} \\ \vdots \\ \cos {\lambda _n}{x_i} \\ {\xi _1}({x_i}) \\ {\xi _2}({x_i}) \\ {\xi _3}({x_i}) \\ {\xi _4}({x_i}) \\ \end{gathered} \right]^{\text{T}}} \text{, }\quad{\boldsymbol{A}} = \left[ \begin{gathered} {a_1} \\ {a_2} \\ \vdots \\ {a_n} \\ {b_1} \\ {b_2} \\ {b_3} \\ {b_4} \\ \end{gathered} \right] $$ (C6) 将式(C5)代入(C2)中, 并取i = 1, L = L1, 可得梁1的拉格朗日方程
$$ \begin{split} & L = \frac{1}{2}{{\mathrm{e}}^{{\text{2j}}\omega {\text{t}}}}{{\boldsymbol{A}}^{\text{T}}}[{E_1}{I_1}\int_0^{{L_1}} {{\boldsymbol{\varPhi }}_{xx}^{\text{T}}} ({x_1}){{\boldsymbol{\varPhi }}_{xx}}({x_1}){\text{d}}{x_{\text{1}}} + {k_{\text{L}}}{{\boldsymbol{\varPhi }}^{\text{T}}}(0){\boldsymbol{\varPhi }}(0) + \\ &\qquad {k_{\text{I}}}{{\boldsymbol{\varPhi }}^{\text{T}}}({L_1}){\boldsymbol{\varPhi }}({L_1}) + {K_{\text{L}}}{\boldsymbol{\varPhi }}_x^{\text{T}}(0){{\boldsymbol{\varPhi }}_x}(0) - \\ &\qquad {\rho _1}{S_1}{\omega ^2}\int_0^{{L_1}} {{{\boldsymbol{\varPhi }}^{\text{T}}}({x_1})} {\boldsymbol{\varPhi }}({x_1}){\text{d}}{x_{\text{1}}}]{\boldsymbol{A}}\end{split} $$ (C7) 由于在端部非线性单元耦合弹性梁系统中, 梁1的左端无旋转约束弹簧, 所以此处KR取值为0, 并将kR替换为kI.
结合瑞利-里兹步骤, 可以得到标准矩阵特征值问题, 具体表达式为
$$ ({\boldsymbol{K}} - {\omega ^2}{\boldsymbol{M}}){\boldsymbol{A}} = 0 $$ (C8) 求解式(C8)的特征值问题, 可以得到梁1的模态参数即响应系数向量A, 将所对应的特征向量代入横向振动位移边界光滑傅里叶级数形式(C3), 即可得到梁1的模态振型函数. 对于梁2, 采用与梁1同样的推导过程. 取i = 2, L = L2, 将式(C7)中约束弹簧替换为对应于端部非线性单元耦合弹性梁系统中梁2的边界约束弹簧. 得到梁2的拉格朗日方程
$$ \begin{split} & L = \frac{1}{2}{{\mathrm{e}}^{{\text{2j}}\omega t}}{{\boldsymbol{A}}^{\text{T}}}[{E_2}{I_2}\int_0^{{L_2}} {{\boldsymbol{\varPhi }}_{xx}^{\text{T}}} ({x_2}){{\boldsymbol{\varPhi }}_{xx}}({x_2}){\text{d}}{x_{\text{2}}} + {k_{\text{R}}}{{\boldsymbol{\varPhi }}^{\text{T}}}({L_2}){{\boldsymbol{\varPhi }}_2}({L_2}) + \\ &\qquad {K_{\text{R}}}{{\boldsymbol{\varPhi }}^{\text{T}}}({L_2}){\boldsymbol{\varPhi }}({L_2}) - {\rho _2}{S_2}{\omega ^2}\int_0^{{L_2}} {{{\boldsymbol{\varPhi }}^{\text{T}}}({x_2})} {\boldsymbol{\varPhi }}({x_2}){\text{d}}{x_{\text{2}}}]{\boldsymbol{A}}\end{split} $$ (C9) 与梁1同样地分析推导过程即可得到梁2的模态振型函数. 值得注意的是, 本文根据狄拉克函数将边界控制方程含端部非线性单元参数的复杂项和质量项代入弹性梁系统的振动控制方程.
附录D
R1m, R2n, Cmi和dnj的具体表达式为
$$\begin{split} &{R_{{\text{1}}m}}{\text{ = }}\int_0^{{L_{\text{1}}}} \left\{ {E_{\text{1}}}{I_{\text{1}}}\sum\limits_{i{\text{ = 1}}}^{{{N}_{\text{1}}}} {\frac{{{{\text{d}}^{\text{4}}}{\varphi _{1i}}({x_{\text{1}}})}}{{{\text{d}}{x_{\text{1}}}^{\text{4}}}}} {q_{1i}}(t) + {C_{{\text{B1}}}}\sum\limits_{i = 1}^{{{N}_{\text{1}}}} {\frac{{{\text{d}}{q_{1i}}(t)}}{{{\text{dt}}}}{\varphi _{1i}}({x_{\text{1}}})} +\right. \delta ({x_{\text{1}}} - {L_{\text{1}}})\left\{ {C_{\text{E}}}\left[\sum\limits_{i = 1}^{{{N}_{\text{1}}}} {\frac{{{\text{d}}{q_{1i}}(t)}}{{{\text{d}}t}}{\varphi _{1i}}({x_{\text{1}}})} - \frac{{{\text{d}}{u_{\text{2}}}({\text{0}},t)}}{{{\text{d}}t}}\right] +\right.\\ &\qquad {k_{\text{E}}}\left[\sum\limits_{i = 1}^{{{N}_{\text{1}}}} {{\varphi _{1i}}({x_{\text{1}}}){q_{1i}}(t)} - {u_{\text{2}}}({\text{0}},t)\right] +\left.\left.k{n_{\text{E}}}{\left[\sum\limits_{i = 1}^{{{N}_{\text{1}}}} {{\varphi _{1i}}({x_{\text{1}}}){q_{1i}}(t)} - {u_{\text{2}}}({\text{0}},t)\right]^3} \right\} \right\} {\psi _{{\text{1}}m}}({x_1}){\text{d}}{{{x}}_1}\end{split} $$ (D1) $$\begin{split} &{R_{{\text{2}}n}} = \int_0^{{L_2}} \left\{{E_2}{I_2}\sum\limits_{i = 1}^{{{N}_{\text{2}}}} {\frac{{{{\text{d}}^4}{\varphi _{{\text{2}}i}}({x_{\text{2}}})}}{{{\text{d}}{x_2}^4}}{q_{{\text{2}}i}}(t)} + {C_{{\text{B2}}}}\sum\limits_{i = 1}^{{{N}_{\text{2}}}} {\frac{{{\text{d}}{q_{{\text{2}}i}}(t)}}{{{\text{d}}t}}} {\varphi _{{\text{2}}i}}({x_{\text{2}}}) + \right. \delta ({x_{\text{2}}} - {x_{\text{S}}}){k_{\text{S}}}\sum\limits_{i = 1}^{{{N}_{\text{2}}}} {{\varphi _{{\text{2}}i}}({x_{\text{2}}}){q_{{\text{2}}i}}(t)} + \delta ({x_{\text{2}}} - {L_{\text{2}}}){F_{\text{0}}}{\text{sin}}(\omega t) + \delta ({x_{\text{2}}})\cdot\\ &\qquad \left\{ {C_{\text{E}}}\left[\sum\limits_{i = 1}^{{{N}_{\text{2}}}} {\frac{{{\text{d}}{q_{{\text{2}}i}}(t)}}{{{\text{d}}t}}{\varphi _{{\text{2}}i}}({x_{\text{2}}})} - \frac{{{\text{d}}{u_{\text{1}}}({L_{\text{1}}},t)}}{{{\text{d}}t}}\right] +\right. {k_{\text{E}}}\left[\sum\limits_{i = 1}^{{{N}_{\text{2}}}} {{\varphi _{{\text{2}}i}}({x_{\text{2}}}){q_{{\text{2}}i}}(t)} - {u_{\text{1}}}({L_{\text{1}}},t)\right] + \left.\left.k{n_{\text{E}}}{\left[\sum\limits_{i = 1}^{{{N}_{\text{2}}}} {{\varphi _{{\text{2}}i}}({x_{\text{2}}}){q_{{\text{2}}i}}(t)} - {u_{\text{1}}}({L_{\text{1}}},t)\right]^{\text{3}}} \right\} \right\} {\psi _{{\text{2}}n}}({x_{\text{2}}}){\text{d}}{x_{\text{2}}}\end{split} $$ (D2) $$ {C_{mi}} = \int_0^{{L_{\text{1}}}} {\sum\limits_{i = {\text{1}}}^{{{{N}}_{\text{1}}}} {[{\rho _{\text{1}}}{S_{\text{1}}} + \delta ({x_{\text{1}}}){m_{\text{A}}} + \delta ({x_{\text{1}}} - {L_{\text{1}}}){m_{\text{L}}}]{\varphi _{{\text{1}}i}}({x_{\text{1}}}){\psi _{{\text{1}}m}}({x_{\text{1}}}){\text{d}}{x_{\text{1}}}} } $$ (D3) $$ {{\text{d}}_{nj}} = \int_0^{{L_{\text{2}}}} {\sum\limits_{i{\text{ = 1}}}^{{{{N}}_{\text{2}}}} {[{\rho _{\text{2}}}{S_{\text{2}}} + \delta ({x_{\text{2}}}){m_{\text{R}}}]{\varphi _{{\text{2}}i}}({x_{\text{2}}}){\psi _{{\text{2}}n}}({x_{\text{2}}}){\text{d}}{x_{\text{2}}}} } $$ (D4) -
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图 1 通过端部非线性单元耦合弹性梁系统物理模型
Figure 1. Physical model of elastic beam system coupled by end nonlinear element
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图 2 不同截断数下梁1和梁2的振动响应曲线
Figure 2. Vibration energy response curves of beam 1 and beam 2 under different truncation numbers
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图 3 在伽辽金法、拉格朗日法和谐波平衡法下得到的梁1和梁2振动响应曲线对比
Figure 3. Comparison of vibration response curves of beam 1 and beam 2 obtained by Galerkin method, Lagrange method, and harmonic balance method
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图 4 梁1和梁2在不同端部耦合非线性刚度下的振动响应曲线
Figure 4. Vibration response curves of beam 1 and beam 2 under different end-coupled nonlinear stiffnesses
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图 5 梁1和梁2在不同端部耦合黏性阻尼下的振动响应曲线
Figure 5. Vibration response curves of beam 1 and beam 2 with different end-coupled viscous damping
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图 6 端部耦合非线性恢复力在不同端部耦合非线性刚度和端部耦合黏性阻尼下的响应曲线
Figure 6. Response curves of end-coupled nonlinear restoring force under different end-coupled nonlinear stiffness and end-coupled viscous damping
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图 7 梁1和梁2在单频激励下随端部耦合非线性刚度变化的振动响应曲线
Figure 7. Vibration response curves of beam 1 and beam 2 with end-coupled nonlinear stiffness variation under single frequency excitation
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图 8 梁1和梁2在单频激励下随端部耦合黏性阻尼变化的振动响应曲线
Figure 8. Vibration response curves of beam 1 and beam 2 with end-coupled viscous damping variation under single frequency excitation
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图 9 梁1和梁2在准周期振动状态下的动能响应和动能响应峰值
Figure 9. Kinetic energy response and peak kinetic energy response of beam 1 and beam 2 under quasi-periodic vibration state
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