纯电动汽车高速齿轮传动全工况抑振修形方法

杜进辅，胡亮; Du Jinfu; Hu Liang

引 en

纯电动汽车高速齿轮传动全工况抑振修形方法^*

杜进辅
机构：
西安理工大学机械与精密仪器工程学院,西安 710048
×
，胡亮
机构：
西安理工大学机械与精密仪器工程学院,西安 710048
×

西安理工大学机械与精密仪器工程学院,西安 710048；

Vibration Suppression Modification Method of Battery Electric Vehicle High-speed Gear Transmission under Full Working Conditions^*

Du Jinfu
Affiliation：
School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048 , China
×
，Hu Liang
Affiliation：
School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048 , China
×

School of Mechanical and Precision Instrument Engineering, Xi’an University of Technology, Xi’an 710048 , China；

通讯作者:

杜进辅,E-mail:dujinfu@xaut.edu.cn

中图分类号:TH132.4

文献标识码:A

文章编号:1672-6553-2023-21(10)-094-009

DOI:10.6052/1672-6553-2023-110

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出版信息

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目录contents

摘要 Abstract
关键词 Keywords
1 系统动力学分析模型
2 激励计算
2.1 时变啮合刚度
2.2 啮合冲击
2.3 齿面摩擦
3 全工况最优齿面修形
3.1 齿廓和齿向抛物线修形
3.2 齿面修形参数优化
3.3 工况点选取
4 算例分析
5 结论
参考文献

摘要

为改善纯电动汽车齿轮传动在整个工况范围内的工作性能,提出一种计及负载扭矩和转速工况影响的齿面修形方法.考虑时变啮合刚度、啮合冲击、齿面摩擦激励,建立了系统动力学分析模型.结合轮齿几何接触分析和承载接触分析,采用遗传算法优化获得全工况最优的齿廓、齿向抛物线修形系数,并对比分析了不同修形方案的抑振效果.结果表明：计及工况影响的齿面修形在全工况下的振动加速度均方根都保持在较低水平,说明这一齿面修形策略具有更好的全局抑振效果.

Abstract

To improve the performance of battery electric vehicle gear transmission over the entire range of operating conditions, a tooth surface modification method considering the influence of load torque and speed is proposed. The system dynamics analysis model is established considering time-varying meshing stiffness, meshing impact and tooth surface friction excitation. Combined with Tooth Contact Analysis (TCA) and Loaded Tooth Contact Analysis (LTCA), the optimal tooth profile and axial parabolic modification coefficient are obtained by genetic algorithm optimization for full working conditions. The vibration suppression effects of different modification schemes are compared and analyzed. Simulation results show that the root mean square of vibration acceleration of the tooth surface modification, that considering the influence of working conditions, remains at a low level under full working conditions, indicating that this strategy of tooth surface modification has better global vibration suppression effect.

关键词

齿面修形；全工况；啮合刚度；啮合冲击；齿面摩擦

Keywords

Tooth surface modification ； full working conditions ； meshing stiffness ； meshing impact ； tooth surface friction

引言
节能减排大趋势下，新能源汽车发展迅猛，其中，纯电动汽车因其在智能化和舒适性方面的优势备受关注.失去了发动机噪声的掩蔽，纯电动汽车齿轮减速器的振动噪声问题凸显.另一方面，乘用车电机高速化趋势下，与其直连的齿轮传动系统输入转速升高进一步加剧振动，从而影响整机可靠性和稳定性.因此，纯电动汽车高速齿轮传动的抑振降噪至关重要.
齿面修形是齿轮传动系统抑振降噪的重要手段，国内外学者围绕这一问题开展了大量卓有成效的研究.文献^[1，2]设计了多种修形方式，并分析了不同修形方式对齿面啮合性能的影响.文献^[3，4]针对准双曲面齿轮提出了Ease-off拓扑修形方法.文献^[5-8]基于轮齿几何接触分析（Tooth Contact Analysis，TCA）和轮齿承载接触分析（Loaded Tooth Contact Analysis，LTCA），开展齿面修形研究，并通过优化算法获得最佳修形参数.文献^[9]提出了一种基于传动误差峰峰值最小的拓扑修形方法，以改善齿轮传动系统的振动噪声问题.文献^[10]建立了含不同齿顶修缘量的斜齿轮啮合模型，文献^[11]考虑人字齿轮的轴向窜动，分析了修形对刚度和振动的影响.文献^[12]采用圆弧曲线对直齿轮进行齿廓修形.文献^[13]研究了主动修形对齿轮抑振的有效性.文献^[14]通过齿面修形设计了具有高阶传动误差的弧齿锥齿轮传动.文献^[15]系统阐述了齿轮传动的低噪声设计理论与方法.文献^[16]研究了转速、齿侧间隙、啮合误差和负载等参数对行星齿轮动力学特性的影响.
上述工作为齿轮传动基于齿面修形的抑振降噪奠定了理论基础，但更关注齿轮传动在特定工况（即负载和扭矩不变）下的修形效果，修形齿面在其他工况下的适用性需进一步明确.为此，本文以纯电动汽车齿轮传动为研究对象，基于TCA、LTCA技术开展激励模拟和齿面修形研究，采用遗传算法获得全工况（整个转速和扭矩范围内）最优的齿廓、齿向抛物线修形系数，并对比不同修形方案在各种工况下的抑振效果，以探究纯电动汽车齿轮传动在全工况下的最优修形策略，并进一步推广应用于其他齿轮传动.
1 系统动力学分析模型
以图1（a）所示某乘用纯电动汽车单挡两级斜齿轮传动系统为研究对象，建立如图1（b）所示弯扭轴摆耦合分析模型，齿轮传动基本参数如表1所示.
图1 减速器模型（a:三维模型 b:系统级动力学模型）
Fig.1 Reducer model (a: 3D model b: System-level dynamic model)
表1 减速器齿轮副基本参数
Table1 Basic parameters of reducer gear pair
根据牛顿第二定律，由图1（b）可得两级斜齿轮传动系统的动力学方程组为：

\{\begin{matrix} {\ddot{x}}_{1} m_{1} + {\dot{x}}_{1} c_{1 x} + x_{1} k_{1 x} = F_{12 x} - F_{f 1} c o s φ_{1} \\ {\ddot{y}}_{1} m_{1} + {\dot{y}}_{1} c_{1 y} + y_{1} k_{1 y} = - F_{12 y} - F_{f 1} s i n φ_{1} \\ {\ddot{z}}_{1} m_{1} + {\dot{z}}_{1} c_{1 z} + z_{1} k_{1 z} = F_{12 z} \\ {\ddot{θ}}_{1 x} I_{1} + {\dot{θ}}_{1 x} c_{1 θ x} + θ_{1 x} k_{1 θ x} = F_{12 z} s i n φ_{1} r_{b 1} \\ {\ddot{θ}}_{1 y} I_{1} + {\dot{θ}}_{1 y} c_{1 θ y} + θ_{1 y} k_{1 θ y} = - F_{12 z} c o s φ_{1} r_{b 1} \\ {\ddot{θ}}_{1 z} I_{1} = - F_{12 y} r_{b 1} - F_{s t 1} r_{b 1} - F_{f 1} s_{1} + T_{1} \end{matrix}

(1)

\{\begin{matrix} {\ddot{x}}_{2} m_{2} + {\dot{x}}_{2} c_{2 x} + x_{2} k_{2 x} = - F_{12 x} + F_{f 1} c o s φ_{1} \\ {\ddot{y}}_{2} m_{2} + {\dot{y}}_{2} c_{2 y} + y_{2} k_{2 y} = F_{12 y} + F_{f 1} s i n φ_{1} \\ {\ddot{z}}_{2} m_{2} + {\dot{z}}_{2} c_{2 z} + z_{2} k_{2 z} = - F_{12 z} \\ {\ddot{θ}}_{2 x} I_{2} + {\dot{θ}}_{2 x} c_{2 θ x} + θ_{2 x} k_{2 θ x} = F_{12 z} s i n φ_{1} r_{b 2} \\ {\ddot{θ}}_{2 y} I_{2} + {\dot{θ}}_{2 y} c_{2 θ y} + θ_{2 y} k_{2 θ y} = - F_{12 z} c o s φ_{1} r_{b 2} \\ {\ddot{θ}}_{2 z} I_{2} + c_{θ} ({\dot{θ}}_{2 z} - {\dot{θ}}_{3 z}) + k_{θ} (θ_{2 z} - θ_{3 z}) = F_{12 y} r_{b 2} + F_{s t 1} r_{b 2} + F_{f 1} s_{2} \end{matrix}

(2)

\{\begin{matrix} {\ddot{x}}_{3} m_{3} + {\dot{x}}_{3} c_{2 x} + x_{3} k_{2 x} = F_{23 x} - F_{f 2} c o s φ_{2} \\ {\ddot{y}}_{3} m_{3} + {\dot{y}}_{3} c_{2 y} + y_{3} k_{2 y} = F_{23 y} - F_{f 2} s i n φ_{2} \\ {\ddot{z}}_{3} m_{3} + {\dot{z}}_{3} c_{2 z} + z_{3} k_{2 z} = F_{23 z} \\ {\ddot{θ}}_{3 x} I_{3} + {\dot{θ}}_{3 x} c_{3 θ x} + θ_{3 x} k_{3 θ x} = F_{23 z} s i n φ_{2} r_{b 3} \\ {\ddot{θ}}_{3 y} I_{3} + {\dot{θ}}_{3 y} c_{3 θ y} + θ_{3 y} k_{3 θ y} = - F_{23 z} c o s φ_{2} r_{b 3} \\ {\ddot{θ}}_{3 z} I_{3} - c_{θ} ({\dot{θ}}_{3 z} - {\dot{θ}}_{2 z}) - k_{θ} (θ_{3 z} - θ_{2 z}) = - F_{23 y} r_{b 3} - F_{s t 2} r_{b 3} - F_{f 2} s_{3} \end{matrix}

(3)

\{\begin{matrix} {\ddot{x}}_{4} m_{4} + {\dot{x}}_{4} c_{3 x} + x_{4} k_{3 x} = - F_{23 x} + F_{f 2} c o s φ_{2} \\ {\ddot{y}}_{4} m_{4} + {\dot{y}}_{4} c_{3 y} + y_{4} k_{3 y} = - F_{23 y} + F_{f 2} s i n φ_{2} \\ {\ddot{z}}_{4} m_{4} + {\dot{z}}_{4} c_{3 z} + z_{4} k_{3 z} = - F_{23 z} \\ {\ddot{θ}}_{4 x} I_{4} + {\dot{θ}}_{4 x} c_{4 θ x} + θ_{4 x} k_{4 θ x} = F_{23 z} s i n φ_{2} r_{b 4} \\ {\ddot{θ}}_{4 y} I_{4} + {\dot{θ}}_{4 y} c_{4 θ y} + θ_{4 y} k_{4 θ y} = - F_{23 z} c o s φ_{2} r_{b 4} \\ {\ddot{θ}}_{4 z} I_{4} = F_{23 y} r_{b 4} + F_{s t 2} r_{b 4} + F_{f 2} s_{4} - T_{2} \end{matrix}

(4)

式中：x_i，y_i，z_i为第i（i=1，2，3，4）个齿轮沿x，y，z方向的振动位移；θ_ix，θ_iy，θ_iz为第i（i=1，2，3，4）个齿轮饶X，Y轴的摆动位移和绕Z轴的扭转位移；F₁₂_x，F₁₂_y，F₁₂_z，F₂₃_x，F₂₃_y，F₂₃_z分别表示输入、输出级沿x，y，z三个方向的动态啮合力；F_st₁，F_st₂为输入、输出级啮合冲击力；F_f₁，F_f₂为输入、输出级齿对的齿间摩擦力；m_i（i=1，2，3，4）和I_i（i=1，2，3，4）分别表示各轮质量和转动惯量；T₁，T₂为电机输入扭矩和负载扭矩；s_i（i=1，2，3，4）为各轮摩擦力臂；r_bi（i=1，2，3，4）为各轮基圆半径；φ₁，φ₂为输入、输出级齿对啮合平面与Y轴正向的夹角；k_jx，k_jy，k_jz（j=1，2，3）为各轴在x，y，z方向的等效支撑刚度；c_jx，c_jy，c_jz（j=1，2，3）为各轴在x，y，z方向的等效支撑阻尼；k_iθx，k_iθy，c_iθx，c_iθy为第i（i=1，2，3，4）个齿轮x，y方向摆动的刚度系数、阻尼系数；k_θ，c_θ为中间轴的扭转刚度和扭转阻尼.
2 激励计算
齿轮传动系统的内外部激励因素众多，本文主要考虑时变啮合刚度、啮合冲击和齿面摩擦激励.
2.1 时变啮合刚度
本文基于LTCA计算齿轮副的时变啮合刚度^[17].通过LTCA获得齿轮副在修形和误差条件下的承载传动误差，则其啮合刚度K_t可用下式表示：

K_{t} = \frac{P}{Z}

(5)

式中，P表示力或力矩，Z表示线位移或角位移变形即承载传动误差.
根据文献^[17]中时变啮合刚度计算方法，计算获得表1中输入级齿对在输入扭矩100N·m下的时变啮合刚度曲线，如图2所示.
图2 时变啮合刚度曲线
Fig.2 Time-varying meshing stiffness curve
2.2 啮合冲击
齿轮传动中由于制造安装误差及轮齿承载变形的存在，导致轮齿在啮合过程中啮合点偏离理论啮合线，从而出现线外啮合冲击等不平稳啮合现象并产生强烈的振动及噪声.考虑啮入冲击对齿轮传动的影响高于啮出冲击^[18]，本文仅考虑啮入冲击，啮入冲击形成原理如图3所示.
图3 啮入冲击原理图
Fig.3 Principle diagram of meshing impact
图中，O₁，O₂分别为主从动轮圆心，P为节点，D为提前啮入起始点，E为正常啮入点为正常啮入点的反转点，δ为啮合齿对综合变形量（由LTCA求得），根据啮合原理和几何关系可得啮入初始点D的准确位置.在ΔO₁DO₂和ΔPDO₂中：

r_{O 1 D} = \sqrt{a_{1}^{2} + r_{a 2}^{2} - 2 a_{1} r_{a 2} c o s ∠ P O_{2} D}

(6)

∠ P O_{2} D = φ_{k} + Δ φ_{2} + γ_{2}

(7)

γ_{2} = \frac{π}{2} - α_{t} - ∠ P E O_{2}

(8)

∠ P E O_{2} = a r c s i n \frac{r_{2} s i n (\frac{π}{2} + α_{t})}{r_{a 2}}

(9)

φ_{k} = \frac{δ}{r_{a 2}}

(10)

r_{O_{1} E} = \sqrt{a_{1}^{2} + r_{a 2}^{2} - 2 a_{1}^{2} r_{a 2} c o s γ_{2}}

(11)

∠ P O_{1} D - Δ φ_{1} - ∠ P O_{1} E = ∠ D O_{1} E

(12)

式中：r_ai，r_i，r_bi（i=1，2）分别为主从动轮齿顶圆、节圆、基圆半径；a₁为两齿轮啮合中心距；α_t为分度圆端面压力角；Δφ₁，Δφ₂分别为主从动轮的反转角.联立式（6）~式（12）求得主动轮反转角Δφ₁，从而得到冲击时间Δt为：

Δ t = \frac{Δ φ_{1}}{ω_{1}}

(13)

式中：ω₁为主动轮角速度.
根据冲击动力学理论，啮入冲击力幅值表达式为^[19]：

F_{S} = v_{s} \sqrt{\frac{J_{1} J_{2}}{(J_{1} r_{b 2}^{'}_{b 2}^{2} + J_{2} r_{b 1}^{2}) q_{s}}}

(14)

式中：v_s为线外啮入点冲击速度；J₁，J₂为主从动轮瞬时转动惯量；r′_b₂为从动轮瞬时基圆半径，q_s为线外啮入点的单齿对柔度.
图4 啮入冲击曲线
Fig.4 Impact curve of meshing
由于啮入冲击的时间很短，因此通常将啮入冲击力处理为锯齿波函数.根据上文所述得到如图4所示啮入冲击力曲线，图中T为啮合周期，T₁为输入扭矩，n₁为主动轮转速.
2.3 齿面摩擦
齿面摩擦与齿面的几何形貌、表面粗糙度、相对滑动速度等因素有关，本文采用Buckingham半经验公式^[20]计算摩擦因数：

μ = 0.05 e^{- 0.125 v} + 0.002 \sqrt{v}

(15)

式中，v为齿面相对滑动速度.齿轮啮合如图5所示，其中N₁N₂为理论啮合线，B₁B₂为实际啮合线.当两齿廓在啮合线上某点K相啮合时：

v = ω_{1} s_{1} - ω_{2} s_{2}

(16)

其中：摩擦力臂s₁，s₂可依据如下公式计算获得：

s_{1} = (r_{b 1} + r_{b 2}) t a n α_{t} - \sqrt{r_{a 2}^{2} - r_{b 2}^{2}} + ω_{1} r_{b 1} t

(17)

s_{2} = \sqrt{r_{a 2}^{2} - r_{b 2}^{2}} - ω_{1} r_{b 1} t

(18)

图5 齿轮啮合示意图
Fig.5 Gear meshing diagram
在齿轮啮合过程中，当啮合点由节圆一侧移动到另一侧时，摩擦力将改变方向，引入方向函数sign（v）后齿面摩擦力计算公式为：

F_{f} = s i g n (v) μ F

(19)

3 全工况最优齿面修形
3.1 齿廓和齿向抛物线修形
齿廓和齿向修形是重要的齿面修形手段，齿廓修形可以减小齿对的啮合冲击，齿向修形则使齿向载荷分布更加合理.
本文基于轮齿接触分析，通过对齿条刀具的齿廓修形来实现斜齿轮齿廓方向的修形^[8].展成小齿轮的齿条刀具的法面齿廓见图8.坐标系S_a为建立在刀具法面齿廓并与刀具共同运动的坐标系，S_b为建立在刀具法向齿面并与刀具共同运动的坐标系，S_c为建立在齿条刀具横向齿面沿齿宽和齿距线中点的坐标系，S_d为建立在小齿轮上的随动坐标系，S_e为建立在小齿轮上的固定坐标系.d₀为二分之一的法向齿宽，a为当顶点在O_a处的抛物线齿廓修形系数，α_n为法向压力角.在坐标系S_a中齿条刀具法向齿廓的坐标表示为：

(20)

图6 齿条刀具法面齿廓
Fig.6 Normal tooth profile of rack cutter
根据图6~图8通过坐标变换可得小齿轮齿面方程

r_{d} (x_{a}, d_{1}, φ_{3}) = M_{d e} M_{e c} M_{c b} M_{b a} r_{a} (x_{a})

(21)

$\begin{matrix} M_{b a} = [\begin{matrix} - c o s α_{n} & s i n α_{n} & 0 & 0 \\ - s i n α_{n} & - c o s α_{n} & 0 & - d_{0} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ M_{c b} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c o s β & s i n β & d_{1} s i n β \\ 0 & - s i n β & c o s β & d_{1} c o s β \\ 0 & 0 & 0 & 1 \end{matrix}] \\ M_{e c} = [\begin{matrix} 1 & 0 & 0 & - r_{1} \\ 0 & 1 & 0 & - d_{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ M_{d e} = [\begin{matrix} c o s φ_{3} & - s i n φ_{3} & 0 & 0 \\ s i n φ_{3} & c o s φ_{3} & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \end{matrix}$
式中：M_de，M_ec，M_cb，M_ba为坐标变换矩阵.其中β为螺旋角，φ₃为齿轮加工过程中小齿轮的转角，d₁为坐标S_b系与坐标系S_c沿O_bz_b轴方向的距离，r₁为小齿轮的分度圆半径，d₂=φ₃r₁.
根据齿轮啮合原理：

f (x_{a}, d_{1}, φ_{3}) = {\vec{n}}_{c} \cdot {\vec{v}}_{c}^{(c, d)} = 0

(22)

图7 齿条投影
Fig.7 Rack projection
图8 齿轮齿条啮合
Fig.8 Pinion and rack meshing
齿向修形可直接在小齿轮齿面方程中实现^[8].齿向修形采用了沿螺旋线方向的抛物线修形，齿向修形示意图见图9.坐标系S_f为建立在小齿轮齿廓的辅助坐标系，S_g为建立在小齿轮齿廓上沿任意螺旋角方向的坐标系.在坐标系S_g中：

y_{g} = b z_{g}^{2}

(23)

式中：b为齿向修形系数.
图9 齿向修形示意图
Fig.9 Diagram of axial modification
在坐标系S_f中：

z_{f} = z_{g} c o s β_{g} + b z_{g}^{2} s i n β_{g}

(24)

y_{f} = - z_{g} s i n β_{g} + b z_{g}^{2} c o s β_{g}

(25)

相比于未修形齿面，修形后齿面坐标变化值为：

Δ y = b z_{g}^{2} c o s β_{g}

(26)

根据公式（24）和图9：

z_{g} = \frac{- c o s β_{g} + \sqrt{{c o s}^{2} β_{g} + 4 b z_{f} s i n β_{g}}}{2 b s i n β_{g}}

(27)

将式（27）代入式（26）可得：

Δ y = b {(\frac{- c o s β_{g} + \sqrt{{c o s}^{2} β_{g} + 4 b z_{f} s i n β_{g}}}{2 b s i n β_{g}})}^{2} c o s β_{g}

(28)

其中： $β_{g} = a r g t a n (r_{g} t a n β / r_{1}) ， r_{g} = \sqrt{x_{d}^{2} + y_{d}^{2}} ， z_{f} = z_{d}$
根据公式（28），可获得齿廓修形和齿向修形后的小齿轮齿面坐标：

r_{d} (x_{a}, d_{1}, φ_{3}) = M_{d e} M_{e c} M_{c b} M_{b a} r_{a} (x_{a}) + [\begin{matrix} 0 \\ Δ y \\ 0 \\ 0 \end{matrix}]

(29)

3.2 齿面修形参数优化
遗传算法具有良好的全局搜索能力，能够自适应地控制搜索过程以求得最优解，故本文选用遗传算法结合齿轮的TCA和LTCA对齿廓及齿向抛物线修形系数进行优化.
齿轮副啮合线上的相对振动加速度是引起齿轮传动系统振动噪声的重要原因.本文提出考虑工况对修形效果影响，以整个工作范围内各工况下的振动加速度均方根的均值最小为目标进行优化.在遗传算法中目标函数表达式为：

m i n (f) = \frac{\sum_{i = 1}^{N} X (i)}{N}

(30)

其中：

X = \sqrt{\frac{\sum_{j = 1}^{M} y^{2} (j)}{M}}

(31)

式中：X（i）为某一工况下的振动加速度均方根，N为整个工作范围内选取的工况数目，Y为啮合线方向的振动位移，M为一个啮合周期内选取的啮合点数目.
齿廓及齿向修形系数优化流程如图10所示：
3.3 工况点选取
纯电动汽车动力传动系统中，扭矩和转速工况的选取要考虑电机低速恒扭和高速恒功率输出特性.本文中，电机额定功率为100kW，额定扭矩为180N·m，根据图11该电机外特性曲线，电机额定转速以下减速器仅输入转速发生变化而输入扭矩保持不变；额定转速以上根据T=9550P/n确定输入扭矩与转速的关系，模拟工作过程中减速器扭矩转速的变化.
图10 修形系数优化流程
Fig.10 Modification coefficient optimization process
图11 电机外特性曲线
Fig.10 External characteristic curve of motor
4 算例分析
采用前文修形方法，对输入级齿对主动轮进行修形，结合遗传算法获得特定工况及全工况下最优的修形齿面（其中考虑全工况的修形选取n=3000rpm，T=180N·m; n=5000rpm，T=180N·m; n=7000rpm，T=136.4N·m; n=9000rpm，T=106.1N·m; n=12000rpm，T=79.6N·m等离散工况点），优化获得的修形系数如表2所示：
表2 遗传算法优化结果
Table2 Genetic algorithm optimization results
图12 3000rpm时输入级啮合线方向振动加速度
Fig.12 Vibration acceleration in the direction of the input stage mesh line at 3000rpm
图13 3000rpm时输出级啮合线方向振动加速度
Fig.13 Vibration acceleration in the direction of the output stage mesh line at 3000rpm
对不同修形方案下的齿轮系统进行时域分析，获得各方案下输入、输出级齿对在不同工况下啮合线方向振动加速度见图12~图17.图12、图14和图16中方案1~3修形齿轮系统分别在转速3000rpm、5000rpm、7000rpm时输入级齿对振动加速度表现最优即振动加速度最小.对比图12、14、16不难看出方案4修形后的齿轮系统输入级齿对加速度虽并非表现最优，但在各转速下始终保持在较低水平.对比图13、图15、图17各方案下输出级齿对的振动加速度，可见方案4所获齿轮系统输出级齿对振动加速度在各转速下保持在前二的水平.由此可见考虑全工况效果的修形在不同工况下均能较好的抑振降噪.
图14 5000rpm时输入级啮合线方向振动加速度
Fig.14 Vibration acceleration in the direction of the input stage mesh line at 5000rpm
图15 5000rpm时输出级啮合线方向振动加速度
Fig.15 Vibration acceleration in the direction of the output stage mesh line at 5000rpm
图16 7000rpm时输入级啮合线方向振动加速度
Fig.16 Vibration acceleration in the direction of the input stage mesh line at 7000rpm
图17 7000rpm时输出级啮合线方向振动加速度
Fig.17 Vibration acceleration in the direction of the output stage mesh line at 7000rpm
图18为各修形方案下输入、输出级齿对啮合线方向振动随转速的变化图.在图18（a）中方案1及方案2分别在转速大于6000rpm、7500rpm时，输入级啮合线方向振动加速度大于未修形齿轮系统，因此仅考虑特定工况下的修形在其他工况下对系统的抑振甚至会产生负面效果.对比图18中各方案曲线可以看出全工况下最优修形齿面即方案4，输入、输出级齿对在整个工作范围中减振效果最优.因仅对输入级齿对主动轮进行修形，所以在输入级减振效果更为明显.对比各方案在不同转速下输入、输出级齿对啮合线方向加速度均方根的大小，可以看出方案4并非在所有工况下减振效果都为最优，但其在全工况下输入、输出级齿对加速度均方根的平均值最小.
图18 不同修形方案下啮合线方向振动比较（a:输入级；b:输出级）
Fig.18 Vibration comparison of meshing line direction under different modification schemes (a : input stage; b : output stage)
5 结论
本文考虑时变啮合刚度、啮合冲击及齿面摩擦激励，构建了纯电动汽车齿轮传动系统的动力学分析模型，探讨了全工况下最优的抑振修形方法.选取转速3000~12000rpm、负载79.6~180N·m范围内8个工况点代表全工况，以所有工况点处啮合线方向振动加速度均方根的平均值来评价修形效果，输入级结果为77.27m/s²，而最好的单工况修形策略3则为98.39m/s²，输出级结果为110.13m/s²，而最好的单工况修形策略2则为115.2m/s².表明只针对特定转速和扭矩的修形策略1~3，最终的修形效果均只保证了其特定工况下最优，而考虑工况变化影响的修形方法虽无法保证所有工况点处均为最优，但却具有更好的全局抑振效果.
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基本信息

中图分类号: TH132.4
文献标识码: A
DOI: 10.6052/1672-6553-2023-110
文章编号: 1672-6553-2023-21(10)-094-009

基金信息

国家自然科学基金(51705419)；
中国博士后科学基金(2018M633540)；
陕西自然科学基础研究计划(2022JM271)；
陕西省博士后科研项目(2018BSHEDZZ10)；
National Natural Science Foundation of China(51705419)；
China Postdoctoral Science Foundation (2018M633540)；
Natural Science Basic Research Program of Shaanxi Province (2022JM271)；
Postdoctoral Research Project of Shaanxi Province (2018BSHEDZZ10)；

引用信息

稿件历史

收稿日期: 2023-05-12

参考文献

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纯电动汽车高速齿轮传动全工况抑振修形方法*

Vibration Suppression Modification Method of Battery Electric Vehicle High-speed Gear Transmission under Full Working Conditions*

摘要

Abstract

关键词

Keywords

1 系统动力学分析模型

(1)

(2)

(3)

(4)

2 激励计算

2.1 时变啮合刚度

(5)

2.2 啮合冲击

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

2.3 齿面摩擦

(15)

(16)

(17)

(18)

(19)

3 全工况最优齿面修形

3.1 齿廓和齿向抛物线修形

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

3.2 齿面修形参数优化

(30)

(31)

3.3 工况点选取

4 算例分析

5 结论

参考文献

基本信息

基金信息

引用信息

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纯电动汽车高速齿轮传动全工况抑振修形方法^*

Vibration Suppression Modification Method of Battery Electric Vehicle High-speed Gear Transmission under Full Working Conditions^*