DESIGN AND APPLICATION OF NONLINEAR ENERGY SINK IN VIBRATION CONTROL OF A HALF-VEHICLE SYSTEM
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摘要: 为提高乘车的舒适性, 提出一种针对半车系统的新型悬架系统控制方案. 该方案的核心是分布式布置非线性能量汇, 即将多个非线性能量汇分别安装于前车身、后车身和前后车轮. 基于牛顿第二定律建立了耦合非线性能量汇的半车系统的动力学方程. 利用谐波平衡法求解系统的近似解析解, 并使用四阶龙格库塔法验证了近似解析解的准确性. 随后, 为说明所提方案的有效性, 比较了原半车系统以及仅在车身前、后安装非线性能量汇和应用所提控制方案时系统的幅频响应. 进一步, 分析了所提控制方案中非线性能量汇质量、非线性弹簧刚度和阻尼对减振效果的影响. 结果表明, 在不改变附加非线性能量汇质量的条件下, 所提方案通过分布式布置非线性能量汇实现了车身和车轮的振动控制, 提升了乘车的舒适性和安全性. 对于车轮, 增大非线性能量汇质量有利于提高减振效率, 而非线性刚度和阻尼的增大可能存在减振效率的恶化; 对于车身, 增大非线性能量汇质量或减小阻尼同样有利于提高减振效果, 而非线性刚度的增大可能存在减振效率的恶化. 所提方案为车辆悬架设计提供了有用的指导, 同时也为非线性能量汇的工程应用提供了理论基础.Abstract: To enhance ride comfort, this study proposes a novel suspension control scheme for a half-vehicle system. The core innovation of this scheme lies in the distributed arrangement of nonlinear energy sinks (NESs), which involves strategically installing multiple NESs at the front body, rear body, and front and rear wheels. The dynamic equations of the half-vehicle system, coupled with NESs, are derived based on the Newton's second law. The approximate analytical solution of the system is obtained using the harmonic balance method, and the accuracy is rigorously verified through numerical simulations employing the fourth-order Runge-Kutta method. To thoroughly evaluate the effectiveness of the proposed scheme, the amplitude-frequency responses of three configurations are carefully analyzed: the original half-vehicle system, the system with NESs installed only at the front and rear of the vehicle body, and the system utilizing the proposed distributed control scheme. The results clearly demonstrate that the proposed approach significantly enhances vibration control performance compared to the other configurations. Additionally, the study systematically investigates the influence of key parameters, including NES mass, nonlinear spring stiffness, and damping, on the system's vibration reduction performance. The findings indicate that the distributed arrangement of NESs effectively controls vibrations of both the vehicle body and wheels without increasing the total mass of the NES. This configuration significantly improves ride comfort and safety. For wheel vibrations, increasing the NES mass enhances vibration reduction efficiency, whereas excessive increases in nonlinear stiffness or damping may lead to performance deterioration. For body vibrations, increasing the NES mass or reducing damping similarly improves vibration control, but excessive nonlinear stiffness may result in reduced performance. The proposed scheme not only offers an effective solution for vibration control but also provides valuable guidance for the design and optimization of vehicle suspension systems. Furthermore, it offers a solid theoretical foundation for the engineering applications of NES.
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引 言
近年来, 汽车行业发展迅速, 人们的生活水平有了很大提升. 汽车成为人们非常普遍的出行方式, 乘坐的舒适性和驾驶的安全性成为关键因素. 然而, 行驶过程中的振动问题是存在的, 这降低了乘坐的舒适性和驾驶的安全性[1]. 研究表明, 车身和轮胎的振动都会影响车辆的乘坐舒适性、噪音、粗糙度甚至安全性[2]. 按照车辆振动传递路径, 其减振设计主要是通过调节以下部件, 即轮胎[3]、悬架[4]和座椅[5]. 通过调节车辆悬架进行减振是较为普遍的方式.
悬架系统通常包括3个主要的机械部件, 分别是连接车轮和车身的悬臂架, 将运动转化为势能的弹簧和旨在耗散动能的减振器[6]. 根据车辆悬架控制方式的不同, 其可分为主动悬架[7-8]、半主动悬架[9-10]和被动悬架[11-12]. 被动悬架因其结构简单、成本低和可靠性高等优点, 广泛应用于车辆悬架中, 许多学者对于被动悬架做了研究工作. 桑志国等[13]提出了一种双气室油气悬挂系统, 与传统的悬架相比效果更好. 动态减振器在被动悬架中应用广泛. Brötz等[14]将一种流体动力减振器应用于车辆悬架中, 能够提高乘坐舒适性并减少车轮负载波动. Xu等[15]提出一种基于两端质量的可变转动惯量被动车辆悬架减振器, 提高了乘坐舒适性并减少了悬架偏转. 上述研究为悬架系统减振器的设计提供了有用指导. 然而线性减振器减振频带较窄, 仅能在主系统固有频率附近减振. 因此, 基于非线性减振器设计的被动悬架受到了越来越多的关注.
非线性能量汇[16-18](nonlinear energy sink, NES)是一种被动非线性减振器. NES由惯性元件、非线性立方刚度元件和阻尼元件组成, 可实现主系统能量的靶向传递[19]. NES可在很宽的频率范围内耗散能量, 并基本不改变主系统的固有频率[20]. 近年来NES被广泛应用于航空航天[21]、土木工程[22]和机械工程[23-24]中. Zhang等[25]将NES用于整艘航天器的减振设计, 研究发现NES在高频范围内随着非线性能量汇黏性阻尼的增加而表现良好. Chen等[26]提出了一种集成了负刚度的新型NES, 其能够在系统受到地震激励时保护主要结构. Tehrani等[27]将NES和TMD应用于转子系统的减振设计, 并讨论了单、双吸振器在不同间隙下的减振性能. Zeng等[28]提出了一种结合了准零刚度隔振器、非线性能量吸收器和惯容器的悬架系统振动控制方案. 该方案可以获得更好的乘坐舒适性和操控稳定性. Wang等[29]将惰性非线性能量汇应用于半车系统, 研究其在垂直和俯仰方向上的动态性能. Wang等[30]将NES应用于1/4悬架系统中, 通过研究系统与NES间的能量传递解释了NES在悬架系统中的减振机理.
用于模拟的车辆模型主要分为3类, 即1/4车辆模型[31]、半车模型[32]和整车模型[33]. 1/4车辆模型仅包含两自由度, 忽略了车辆前后部件间的联系, 使得分析结果可能与实际差别较大; 而整车模型建模复杂, 分析较为困难. 因此本文选择具有4自由度的半车模型进行分析. 相对于1/4车辆模型, 半车模型可以更全面地反映车辆的动态响应, 并考虑了前后轮之间的相互作用.
目前多数研究重点关注车身的振动, 而忽略了轮胎的振动. 然而轮胎的振动同样会对车辆的乘坐舒适性、噪音、粗糙度甚至安全性产生负面影响[2]. 本文的新颖之处在于提出了一种用于半车系统的新型被动悬架控制方案, 以提升乘车的舒适性和安全性. 采用牛顿第二定律建立了耦合NES的半车系统的动力学方程, 使用谐波平衡法对所得方程进行求解. 通过龙格库塔法验证上述结果的正确性. 分析了NES质量、非线性刚度和阻尼对减振效率的影响, 所得结论可为车辆悬架系统的设计与参数选择提供参考.
1. 半车系统模型
图1给出了耦合4个NES的半车系统示意图. 考虑前轮位移z1、后轮位移z2、车身位移z3和车身俯仰角θ 四个自由度. 本文分析了多种NES的布置方式, 包括将4个NES分别安装于前、后车身和前、后车轮 (方案1), 仅在前、后车身各安装一个NES (方案2)和不安装NES (方案3). 其中, 方案1为所提控制方案, 方案2为一般NES控制方案, 方案3即为原半车系统. 这里分别给出方案1和方案2的车辆动力学方程.
1.1 方案1动力学方程
根据牛顿第二定律, 可得车身的动力学方程为
$$ \begin{split} & {m_3}{{\ddot z}_3} + {k_1}({z_f} - {z_1}) + {c_1}({{\dot z}_f} - {{\dot z}_1}) + {k_2}({z_r} - {z_2}) + \\ &\qquad {c_2}({{\dot z}_r} - {{\dot z}_2}) + {k_{bf}}{({z_f} - {z_{bf}})^3} + {c_{bf}}({{\dot z}_f} - {{\dot z}_{bf}}) + \\ &\qquad {k_{br}}{({z_r} - {z_{br}})^3} + {c_{br}}({{\dot z}_r} - {{\dot z}_{br}}) = 0 \end{split} $$ (1) $$ \begin{split} & I\ddot \theta - a[{k_1}({z_f} - {z_1}) + {c_1}({{\dot z}_f} - {{\dot z}_1}) + {k_{bf}}{({z_f} - {z_{bf}})^3} + \\ &\qquad {c_{bf}}({{\dot z}_f} - {{\dot z}_{bf}})] + b[{k_2}({z_r} - {z_2}) + {c_2}({{\dot z}_r} - {{\dot z}_2}) + \\ &\qquad {k_{br}}{({z_r} - {z_{br}})^3} + {c_{br}}({{\dot z}_r} - {{\dot z}_{br}})] = 0 \end{split} $$ (2) 前轮的动力学方程为
$$ \begin{split} & {m_1}{{\ddot z}_1} + {k_{1d}}({z_1} - {x_E}) + {k_1}({z_1} - {z_f}) + {c_1}({{\dot z}_1} - {{\dot z}_f}) + \\ &\qquad {k_{tf}}{({z_1} - {z_{tf}})^3} + {c_{tf}}({{\dot z}_1} - {{\dot z}_{tf}}) = 0 \end{split} $$ (3) 后轮的动力学方程为
$$ \begin{split} & {m_2}{{\ddot z}_2} + {k_{2d}}({z_2} - {x_F}) + {k_2}({z_2} - {z_r}) + \\ &\qquad {c_2}({{\dot z}_2} - {{\dot z}_r}) + {k_{tr}}{({z_2} - {z_{tr}})^3} + {c_{tr}}({{\dot z}_2} - {{\dot z}_{tr}}) = 0 \end{split} $$ (4) 4个NES的动力学方程为
$$ {m_{tf}}{\ddot z_{tf}} + {k_{tf}}{({z_{tf}} - {z_1})^3} + {c_{tf}}({\dot z_{tf}} - {\dot z_1}) = 0 $$ (5) $$ {m_{tr}}{\ddot z_{tr}} + {k_{tr}}{({z_{tr}} - {z_2})^3} + {c_{tr}}({\dot z_{tr}} - {\dot z_2}) = 0 $$ (6) $$ {m_{bf}}{\ddot z_{bf}} + {k_{bf}}{({z_{bf}} - {z_f})^3} + {c_{bf}}({\dot z_{bf}} - {\dot z_f}) = 0 $$ (7) $$ {m_{br}}{\ddot z_{br}} + {k_{br}}{({z_{br}} - {z_r})^3} + {c_{br}}({\dot z_{br}} - {\dot z_r}) = 0 $$ (8) 其中, m3为车身质量, m1和m2分别为前轮和后轮质量. I为车身俯仰转动惯量, a和b分别为前后轴距. k1, k2, c1和c2分别为前后悬架刚度和阻尼. k1d和k2d分别为简化的前后轮与地面间的连接刚度. mbf和mbr分别为安装在车身前后NES的质量, kbf和kbr分别为安装在车身前后NES的非线性刚度, cbf和cbr分别为安装在车身前后NES的阻尼. mtf和mtr分别为安装在前后轮NES的质量, ktf和ktr分别为安装在前后轮NES的非线性刚度, ctf和ctr分别为安装在前后轮NES的阻尼. zbf和zbr分别表示安装在车身前后NES的位移. ztf和ztr分别表示安装在前轮和后轮NES的位移. xE = A1cos(ω1t)和xF = A1cos(ω1t)分别为前轮和后轮受到的位移激励. zf和zr分别是车身前部和车身后部的位移, 表示为
$$ {z_f} = {z_3} - a \sin \theta $$ (9) $$ {z_r} = {z_3} + b \sin \theta $$ (10) 1.2 方案2动力学方程
由1.1节类似的过程, 可得到方案2中车身的动力学方程为
$$\begin{split} & {m_3}{{\ddot z}_3} + {k_1}({z_f} - {z_1}) + {c_1}({{\dot z}_f} - {{\dot z}_1}) + {k_2}({z_r} - {z_2}) + \\ &\qquad {c_2}({{\dot z}_r} - {{\dot z}_2}) + {k_{bf}}{({z_f} - {z_{bf}})^3} + {c_{bf}}({{\dot z}_f} - {{\dot z}_{bf}}) + \\ &\qquad {k_{br}}{({z_r} - {z_{br}})^3} + {c_{br}}({{\dot z}_r} - {{\dot z}_{br}}) = 0\end{split} $$ (11) $$ \begin{split} & I\ddot \theta - a[{k_1}({z_f} - {z_1}) + {c_1}({{\dot z}_f} - {{\dot z}_1}) + {k_{bf}}{({z_f} - {z_{bf}})^3} + \\ &\qquad {c_{bf}}({{\dot z}_f} - {{\dot z}_{bf}})] + b[{k_2}({z_r} - {z_2}) + {c_2}({{\dot z}_r} - {{\dot z}_2}) + \\ &\qquad {k_{br}}{({z_r} - {z_{br}})^3} + {c_{br}}({{\dot z}_r} - {{\dot z}_{br}})] = 0 \end{split} $$ (12) 前轮的动力学方程为
$$ {m_1}{{\ddot z}_1} + {k_{1d}}({z_1} - {x_E}) + {k_1}({z_1} - {z_f}) + {c_1}({{\dot z}_1} - {{\dot z}_f}) = 0 $$ (13) 后轮的动力学方程为
$$ {m_2}{{\ddot z}_2} + {k_{2d}}({z_2} - {x_F}) + {k_2}({z_2} - {z_r}) + {c_2}({{\dot z}_2} - {{\dot z}_r}) = 0 $$ (14) 位于前和后车身的两个NES的动力学方程为
$$ {m_{bf}}{\ddot z_{bf}} + {k_{bf}}{({z_{bf}} - {z_f})^3} + {c_{bf}}({\dot z_{bf}} - {\dot z_f}) = 0 $$ (15) $$ {m_{br}}{\ddot z_{br}} + {k_{br}}{({z_{br}} - {z_r})^3} + {c_{br}}({\dot z_{br}} - {\dot z_r}) = 0 $$ (16) 相较于方案1, 2未在前和后车轮安装NES. 为便于进行比较, 设置方案1和2中附加的NES总质量相同. 两个方案中安装于车身的NES非线性刚度和阻尼也分别取值相同.
2. 近似解析分析与数值验证
采用谐波平衡法求解系统动力学方程. 假设方程的解为[34]
$$ {z_p}(t) = {a_{p,0}} + \sum\limits_{i = 1}^n {{a_{p,i}}\cos (i\omega t} ) + \sum\limits_{i = 1}^n {{b_{p,i}}\sin (i\omega t)} $$ (17) 其中, ap,i和bp,i是谐波系数; p描述自由度, 取值范围为1, 2, ···, 8; i描述谐波阶数I, 取值范围为1, 2, ···, n; n是谐波平衡法的阶数. 为便于表达, 车身俯仰角θ及位于前轮、后轮、前车身和后车身的NES位移分别用z4, z5, ···, z8表示. 由式(17)可得, 相应的速度和加速度方程为
$$ \frac{{{\mathrm{d}}{z_p}}}{{{\mathrm{d}}t}} = \sum\limits_{i = 1}^n {i\omega {b_{p,i}}} \cos (i\omega t) - \sum\limits_{i = 1}^n {i\omega {a_{p,i}}} \sin (i\omega t) $$ (18) $$ \begin{split} & \frac{{{{\mathrm{d}}^2}{z_p}}}{{{\mathrm{d}}{t^2}}} = - \sum\limits_{i = 1}^n {{{(i\omega )}^2}{a_{p,i}}} \cos (i\omega t) - \\ &\qquad \sum\limits_{i = 1}^n {{{(i\omega )}^2}{b_{p,i}}} \sin (i\omega t)\end{split}$$ (19) 将式(17) ~ 式(19)代入式(1) ~ 式(10)中, 平衡常数项、cos(iωt)和sin(iωt)的系数. 可以得到如下方程组
$$ \begin{split} & {F_{zj,0}}({\boldsymbol{\varLambda}} ,\omega ) + {F_{zj,c1}}({\boldsymbol{\varLambda}} ,\omega )\cos (\omega t) + \\ &\qquad {F_{zj,s1}}({\boldsymbol{\varLambda}} ,\omega )\sin (\omega t) + {F_{zj,c2}}({\boldsymbol{\varLambda}} ,\omega )\cos (2\omega t) + \\ &\qquad {F_{zj,s2}}({\boldsymbol{\varLambda}} ,\omega )\sin (2\omega t) +\cdots + {F_{zj,cn}}({\boldsymbol{\varLambda}} ,\omega )\cos (n\omega t) + \\ &\qquad {F_{zj,sn}}({\boldsymbol{\varLambda}} ,\omega )\sin (n\omega t) = 0,\quad j = 1,2,\cdots, 8\\[-1pt] \end{split} $$ (20) 其中, Λ = [ap,0, ap,1, bp,1,···, ap,n, bp,n]; F(·)表示各谐波分量的系数; 下标cn和sn分别表示偶谐波和奇谐波阶数.
取所有一次谐波系数为0, 可以得到(16n + 8)个非线性方程
$$ \begin{split} & {F_{zj0}}({\boldsymbol{\varLambda}} ,\omega ) = 0,{F_{zjc1}}({\boldsymbol{\varLambda}} ,\omega ) = 0,{F_{zjs1}}({\boldsymbol{\varLambda}} ,\omega ) = 0 , \\ &\cdots ,{F_{zjcn}}({\boldsymbol{\varLambda}} ,\omega ) = 0,{F_{zjsn}}({\boldsymbol{\varLambda}} ,\omega ) = 0,\quad j = 1,2,\cdots, 8 \end{split} $$ (21) $$ {\boldsymbol{F}}({\boldsymbol{\varLambda}} ,\omega ) = {\boldsymbol{0}} $$ (22) $$\begin{split} & {\boldsymbol{F}} = [{F_{z1,0}},{F_{z1,c1}},{F_{z1,s1}},\cdots ,{F_{z1,cn}},{F_{z1,sn}} , \\ & \cdots ,{F_{z8,0}},{F_{z8,,c1}},{F_{z8,s1}},\cdots ,{F_{z8,cn}},{F_{z8,sn}}{]^\text{T}} \end{split} $$ (23) 方程组(22)可写为
$$ {\boldsymbol{F}}({\boldsymbol{Q}}) = {\boldsymbol{0}} $$ (24) 其中, Q = {Q1, Q2, ···, Q16n + 9}, Q = (Λ, ω).
方程(24)的雅可比矩阵为
$$ {\mathrm{D}}{\boldsymbol{F}}({\boldsymbol{Q}}) = {\left( {\begin{array}{*{20}{c}} {\dfrac{{\partial {F_1}}}{{\partial {Q_1}}}}& \cdots &{\dfrac{{\partial {F_1}}}{{\partial {Q_{16n + 9}}}}} \\ \vdots & \ddots & \vdots \\ {\dfrac{{\partial {F_{16n + 8}}}}{{\partial {Q_1}}}}& \cdots &{\dfrac{{\partial {F_{16n + 8}}}}{{\partial {Q_{16n + 9}}}}} \end{array}} \right)_{(16n + 8) \times (16n + 9)}} $$ (25) (16n + 9)维向量构造如下
$$ {\boldsymbol{\psi}} ({\boldsymbol{Q}}) = {[{\psi _1}{\text{ }}{\psi _2}{\text{ }} \cdots {\text{ }}{\psi _{16n + 9}}]^{\text{T}}} $$ (26) $$ \begin{split} & {\psi _i} = {( - 1)^{i + 1}}\det \left[\frac{{\partial F}}{{\partial {Q_1}}} \cdots \frac{{\partial F}}{{\partial {Q_{i - 1}}}}\frac{{\partial F}}{{\partial {Q_{i + 1}}}} \cdots \frac{{\partial F}}{{\partial {Q_{16n + 9}}}}\right]\\ &\qquad (i = 1,{\text{ }}2,{\text{ }}\cdots ,{\text{ }}16n + 9) \end{split} $$ (27) 由于雅可比矩阵的每一行与其对应的代数余子式向量的点积恒为0, 可以得到
$$ \left\{ \frac{{\partial {F_i}}}{{\partial {Q_1}}}\frac{{\partial {F_i}}}{{\partial {Q_2}}} \cdots \frac{{\partial {F_i}}}{{\partial {Q_{16n + 9}}}}\right\} \cdot {\boldsymbol{\psi}} = 0{\text{ }}(i = 1,{\text{ }}2,{\text{ }}\cdots ,{\text{ }}16n + 8) $$ (28) 其中, $\boldsymbol{\psi}({\boldsymbol{Q}}) $是解曲线的切向量, 其单位切向量表示
$$ \tau ({\boldsymbol{Q}}) = \frac{{\boldsymbol{\psi}} }{{||{\boldsymbol{\psi}} ||}} $$ (29) 其中, $\|\boldsymbol{\psi}\| $表示向量$\boldsymbol{\psi} $的模.
为了避免用改进的欧拉法计算系统幅频响应时出现拐点, 在局部伪弧长法的基础上引入了弧长s. 该非线性系统包含(16n + 8)个方程, 需要添加一个约束方程使(16n + 9)个未知量可解. 补充方程为
$$ \tau \cdot \Delta {\boldsymbol{Q}} - \Delta s = 0 $$ (30) 其中, 符号Δ表示变量的微小变化. 然后, 求解方程式(24)的问题变换为求解以下方程
$$ \left.\begin{aligned} & {{\boldsymbol{F}}({\boldsymbol{Q}}) = {\boldsymbol{0}}} \\ & {\tau \cdot \Delta {\boldsymbol{Q}} - \Delta s = 0} \end{aligned}\right\} $$ (31) 方程(31)被转换为柯西问题
$$ \frac{{{\text{d}}{\boldsymbol{Q}}}}{{{\text{d}}s}} = \tau ({{\boldsymbol{Q}}_0}),{\text{ }}{\boldsymbol{Q}}(0) = {{\boldsymbol{Q}}_0} $$ (32) 方程(32)可以用改进的欧拉法来求解. 可以得到预测解. 通过控制解的精度, 用牛顿迭代法校正预测解.
$$ \left.\begin{aligned} & {Q_v^0 = {Q_v}} \\ & {Q_v^l = Q_v^{l - 1} - \left[ {\begin{array}{*{20}{c}} {{\mathrm{D}}{\boldsymbol{F}}({\boldsymbol{Q}})} \\ {{\boldsymbol{\psi}} ({Q_k})} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{\boldsymbol{F}}({\boldsymbol{Q}}_v^{l - 1})} \\ 0 \end{array}} \right]} \end{aligned}\right\} $$ (33) 通过上述伪弧长延伸过程, 可以求出代数方程的解.
定义NES减振效率表达式如下
$$ \eta =\frac{{{A_{\mathrm{N}}} - {A_{{\mathrm{NES}}}}}}{{{A_{\mathrm{N}}}}} \times 100\% $$ (34) 其中, AN和ANES分别为原半车系统的幅值与耦合NES系统的幅值.
选取半车系统的结构参数如表1所示. 表2列出了方案1中各NES的参数值. 采用4阶龙格库塔法求解方程式(1) ~ 式(10), 可以得到所提控制方案(方案1)所对应的系统在路面位移激励下的数值解. 谐波阶数n = 3. 假设系统受到的道路激励幅值A1和A2均为0.03 m. 图2(a) ~ 图2(c)分别展示了前、后轮和车身的幅频响应曲线. 图中实线和圆标点分别为通过谐波平衡法和4阶龙格库塔法得到的结果. 从图2中结果可以看出, 谐波平衡法的结果和4阶龙格库塔法的结果吻合较好, 说明了谐波平衡法结果的准确性.
表 1 半车系统结构参数Table 1. Structural parameters of the half-vehicle systemParameter Notation Value vehicle body mass m3/kg 690 front wheel mass m1/kg 40 rear wheel mass m2/kg 45 pitch moment of inertia I/(kg·m) 1222 front axle distance a/m 1.3 rear axle distance b/m 1.5 front suspension stiffness k1/(N·m−1) 2.0 × 104 rear suspension stiffness k2/(N·m−1) 2.2 × 104 front suspension damping c1/(N·s·m−1) 1000 rear suspension damping c2/(N·s·m−1) 1000 front wheel stiffness k1d/(N·m−1) 2.0 × 105 rear wheel stiffness k2d/(N·m−1) 2.0 × 105 表 2 NES结构参数(方案1)Table 2. Structural parameters of the NES (scheme 1)Parameter Notation Value mass of the front wheel NES mtf/kg 20 mass of the rear wheel NES mtr/kg 20 mass of the front vehicle body NES mbf/kg 50 mass of the rear vehicle body NES mbr/kg 50 stiffness of the front wheel NES ktf/(N·m−3) 2.1 × 106 stiffness of the rear wheel NES ktr/(N·m−3) 2.1 × 106 stiffness of the front vehicle body NES kbf/(N·m−3) 1.3 × 105 stiffness of the rear vehicle body NES kbr/(N·m−3) 1.3 × 105 damping of the front wheel NES ctf/(N·s·m−1) 1000 damping of the rear wheel NES ctr/(N·s·m−1) 1000 damping of the front vehicle body NES cbf/(N·s·m−1) 300 damping of the rear vehicle body NES cbr/(N·s·m−1) 300 ![]()
图 2 路面简谐激励下谐波平衡法与龙格库塔法结果对比Figure 2. Comparison of results between the harmonic balance method and the Runge-Kutta method under harmonic road excitation讨论不同NES安装方案下的减振性能. 为便于比较, 设置方案2中附加的NES总质量与方案1相同. 表3列出了方案2中各NES的参数值. 以下算例均是基于表1的半车系统参数. 图3(a) ~ 图3(c)分别给出了方案1 (4个NES分别安装在前后车身和前后车轮)、方案2 (仅在车身前后各安装1个NES)和方案3 (不安装NES)的幅频响应结果. 从图中方案3的结果(蓝色实线)可以发现, 前、后车轮在第二阶共振峰处具有较大的幅值, 而车身在第一阶共振峰处具有较大的幅值. 因此, 本文重点关注车轮的第二阶共振峰和车身的第一阶共振峰的减振. 如图3(a)和图3(b)所示, 应用方案2 (黑色虚线)并不能显著降低前轮和后轮的共振幅值. 而方案1 (红色虚线)在不增加附加NES质量的条件下, 通过在车身和车轮上分布式安装NES使得前轮和后轮的第二阶共振幅值分别降低了31.73%和31.64%. 图3(c)结果进一步表明, 方案1会略微降低车身的减振效率, 但仍可有效降低车身的振动幅值. 通过方案1实现了同时控制车身与车轮的振动.
表 3 NES结构参数(方案2)Table 3. Structural parameters of the NES (scheme 2)Parameter Notation Value mass of the front vehicle body NES mbf/kg 70 mass of the rear vehicle body NES mbr/kg 70 stiffness of the front vehicle body NES kbf/(N·m−3) 1.3 × 105 stiffness of the rear vehicle body NES kbr/(N·m−3) 1.3 × 105 damping of the front vehicle body NES cbf/(N·s·m−1) 300 damping of the rear vehicle body NES cbr/(N·s·m−1) 300 ![]()
图 3 方案1, 2和3得到的幅频响应曲线Figure 3. Amplitude-frequency response curves obtained from schemes 1, 2, and 33. 参数分析
本节将方案1应用于半车系统减振. 同样关注车轮的第二阶共振峰和车身的第一阶共振峰. 首先分析附加于车轮的NES质量对减振效率的影响. 除附加于车轮的NES质量外, 其余参数如表2所示. 图4(a)和图4(b)分别为不同车轮NES质量下, 前轮和后轮的幅频响应曲线. 从图4中结果可以看出, 随着mtf的增大, 前轮的第二阶共振峰值减小. 与此同时, 随着mtr的增大, 后轮的第二阶共振峰值也减小. 结果表明, 增大附加于车轮的NES质量有利于提高方案1的减振效率. 换句话说, 随着附加于车轮的NES质量的增大, 车辆的乘坐舒适性和安全性得到改善.
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图 4 (a)不同mtf下前轮幅频响应曲线和(b)不同mtr下后轮幅频响应曲线Figure 4. (a) Amplitude-frequency response curves of the front wheel under different mtf conditions and (b) amplitude-frequency response curves of the rear wheel under different mtr conditions其次分析附加于车身的NES质量对减振效率的影响. 除附加于车身的NES质量外, 其余参数如表2所示. 图5为不同车身NES质量下的车身幅频响应曲线. 从图5中结果可以看出, 随着mbf和mbr的增大, 车身的第一阶共振峰逐渐减小. 结果表明, 增大附加于车身的NES质量有利于提高方案1的减振效率, 从而提高车辆的乘坐舒适性.
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图 5 不同车身NES质量下车身幅频响应曲线Figure 5. Amplitude-frequency response curves of the vehicle body under different NESs masses接下来分析附加于车轮的NES非线性刚度对减振效率的影响. 除附加于车轮的NES非线性刚度外, 其余参数如表2所示. 图6(a)和图6(b)分别为不同车轮NES刚度下, 前轮和后轮的幅频响应曲线. 从图6中结果可以看出, 随着ktf的增大, 前轮的第二阶共振峰先减小, 后增大. 当ktf增大到一定值时, 前轮的第二阶共振峰会出现向左偏移的现象, 同时减振效率也会显著降低. 随着ktr的增大, 后轮表现出与前轮相同的现象. 因此, 当车轮NES的非线性刚度取一个合理的值时, 能够显著提高减振效率, 改善车辆的舒适性和安全性.
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图 6 (a)不同ktf下的前轮幅频响应曲线和(b)不同ktr下的后轮幅频响应曲线Figure 6. (a) Amplitude-frequency response curves of the front wheel under different ktf conditions and (b) amplitude-frequency response curves of the rear wheel under different ktr conditions然后分析附加于车身的NES非线性刚度对减振效率的影响. 除附加于车身的NES非线性刚度外, 其余参数如表2所示. 图7为不同车身NES非线性刚度下的车身幅频响应曲线. 从图7中可以看出, 随着kbf和kbr的增大, 车身的第一阶共振峰先减小, 后增大. 系统响应受NES非线性刚度影响较小. 结果表明, 优化设计附加于车身的NES非线性刚度, 对其减振性能的提升不明显.
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图 7 不同车身NES非线性刚度下车身幅频响应曲线Figure 7. Amplitude-frequency response curves of the vehicle body under different NESs nonlinear stiffnesses分析附加于车轮的NES阻尼对减振效率的影响. 除附加于车轮的NES阻尼外, 其余参数如表2所示. 图8(a)和图8(b)分别为不同车轮NES阻尼下, 前轮和后轮幅频响应曲线. 从图8中结果可以看出, 随着ctf增大, 前轮的二阶共振峰先减小后增大. 随着ctr增大, 后轮的二阶共振峰也先减小后增大. 结果表明, 附加于车轮的NES阻尼过高或过低都会降低减振效率.
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图 8 (a)不同ctf下的前轮幅频响应曲线和(b)不同ctr下的后轮幅频响应曲线Figure 8. (a) Amplitude-frequency response curves of the front wheel under different ctf conditions and (b) amplitude-frequency response curves of the rear wheel under different ctr conditions最后分析附加于车身的NES阻尼对减振效率的影响. 除附加于车身的NES阻尼外, 其余参数如表2所示. 图9为不同车身NES阻尼下的车身幅频响应曲线. 从图9中可以看出, 随着cbf和cbr的增大, 车身的一阶共振峰逐渐增大. 说明较小的车身NES阻尼可以获得较好的减振效果.
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图 9 不同车身NES阻尼下的车身幅频响应曲线Figure 9. Amplitude-frequency response curves of the vehicle body under different NESs damping4. 结 论
本文将NES应用于车辆悬架设计, 通过将NES合理分布在车辆的不同位置, 以提高乘坐的舒适性和安全性. 利用牛顿第二定律建立了耦合NES的半车系统的动力学方程. 利用谐波平衡法求解系统在道路谐波激励下的位移幅值响应, 并通过4阶龙格库塔法进行数值验证, 证明了谐波平衡法结果的准确性. 分析了NES质量、非线性刚度和阻尼对方案1减振效率的影响. 所得结论如下.
(1) 提出了一种新的悬架系统控制方案. 该方案将多个NES分别安装在车身前部、车身后部、前轮和后轮上, 在不增加NES附加质量的条件下, 可以同时控制车身和车轮的振动, 提升了车辆舒适性和安全性.
(2) 耦合NES的半车系统是一个非线性系统. 利用谐波平衡法得到了该系统的近似解析解, 并且其结果与数值结果吻合良好, 验证了该方法在分析非线性振动系统方面的有效性.
(3) 参数分析结果表明, 对于车轮, 可通过增大NES质量来提高减振效率, 而对于非线性刚度和阻尼, 应避免过度调整, 防止减振效果变差; 对于车身, 增大NES质量或减小阻尼是提高减振效果的有效途径, 然而, 非线性刚度的增大可能会降低减振效率, 应避免过度调整.
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图 2 路面简谐激励下谐波平衡法与龙格库塔法结果对比
Figure 2. Comparison of results between the harmonic balance method and the Runge-Kutta method under harmonic road excitation
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图 3 方案1, 2和3得到的幅频响应曲线
Figure 3. Amplitude-frequency response curves obtained from schemes 1, 2, and 3
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图 4 (a)不同mtf下前轮幅频响应曲线和(b)不同mtr下后轮幅频响应曲线
Figure 4. (a) Amplitude-frequency response curves of the front wheel under different mtf conditions and (b) amplitude-frequency response curves of the rear wheel under different mtr conditions
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图 5 不同车身NES质量下车身幅频响应曲线
Figure 5. Amplitude-frequency response curves of the vehicle body under different NESs masses
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图 6 (a)不同ktf下的前轮幅频响应曲线和(b)不同ktr下的后轮幅频响应曲线
Figure 6. (a) Amplitude-frequency response curves of the front wheel under different ktf conditions and (b) amplitude-frequency response curves of the rear wheel under different ktr conditions
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图 7 不同车身NES非线性刚度下车身幅频响应曲线
Figure 7. Amplitude-frequency response curves of the vehicle body under different NESs nonlinear stiffnesses
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图 8 (a)不同ctf下的前轮幅频响应曲线和(b)不同ctr下的后轮幅频响应曲线
Figure 8. (a) Amplitude-frequency response curves of the front wheel under different ctf conditions and (b) amplitude-frequency response curves of the rear wheel under different ctr conditions
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图 9 不同车身NES阻尼下的车身幅频响应曲线
Figure 9. Amplitude-frequency response curves of the vehicle body under different NESs damping
表 1 半车系统结构参数
Table 1 Structural parameters of the half-vehicle system
Parameter Notation Value vehicle body mass m3/kg 690 front wheel mass m1/kg 40 rear wheel mass m2/kg 45 pitch moment of inertia I/(kg·m) 1222 front axle distance a/m 1.3 rear axle distance b/m 1.5 front suspension stiffness k1/(N·m−1) 2.0 × 104 rear suspension stiffness k2/(N·m−1) 2.2 × 104 front suspension damping c1/(N·s·m−1) 1000 rear suspension damping c2/(N·s·m−1) 1000 front wheel stiffness k1d/(N·m−1) 2.0 × 105 rear wheel stiffness k2d/(N·m−1) 2.0 × 105 
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表 2 NES结构参数(方案1)
Table 2 Structural parameters of the NES (scheme 1)
Parameter Notation Value mass of the front wheel NES mtf/kg 20 mass of the rear wheel NES mtr/kg 20 mass of the front vehicle body NES mbf/kg 50 mass of the rear vehicle body NES mbr/kg 50 stiffness of the front wheel NES ktf/(N·m−3) 2.1 × 106 stiffness of the rear wheel NES ktr/(N·m−3) 2.1 × 106 stiffness of the front vehicle body NES kbf/(N·m−3) 1.3 × 105 stiffness of the rear vehicle body NES kbr/(N·m−3) 1.3 × 105 damping of the front wheel NES ctf/(N·s·m−1) 1000 damping of the rear wheel NES ctr/(N·s·m−1) 1000 damping of the front vehicle body NES cbf/(N·s·m−1) 300 damping of the rear vehicle body NES cbr/(N·s·m−1) 300
下载: 导出CSV
表 3 NES结构参数(方案2)
Table 3 Structural parameters of the NES (scheme 2)
Parameter Notation Value mass of the front vehicle body NES mbf/kg 70 mass of the rear vehicle body NES mbr/kg 70 stiffness of the front vehicle body NES kbf/(N·m−3) 1.3 × 105 stiffness of the rear vehicle body NES kbr/(N·m−3) 1.3 × 105 damping of the front vehicle body NES cbf/(N·s·m−1) 300 damping of the rear vehicle body NES cbr/(N·s·m−1) 300
下载: 导出CSV
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