Development of multiple linear regression-based models for fatigue life evaluation of automotive coil springs
Fatigue analysis of automotive coil springs is crucial because these components are constantly exposed to dynamic loads as the vehicle travels across different terrains. Coil springs are also among the critical components in an automotive chassis because they affect passenger comfort and vehicle aerodynamics. The spring durability and vehicle ride quality are dependent on the design of the coil spring-damper pairs in order to handle different road profiles. Many studies have been carried out to examine the vertical vibrations of the vehicle excited by realistic road profiles. Reza Kashyzadeh et al. studied the effects of road surface roughness on the vibrations of the vehicle suspension system. The road profiles were regarded as random road excitations in order to estimate the fatigue life values of the suspension system components such as the wheel hub, Pitman arm, suspension arm, and joints. Gonzaacute;lez et al. used vehicle acceleration measurements instead of random road profiles to estimate the road surface roughness, where a transfer function was used to establish the relationship between the power spectral densities of the road surface and the vehicle acceleration measurements.
Harsh road profiles do not only affect the fatigue life of automotive coil springs but also the vehicle ride quality. Spring design plays a vital role in determining the vehicle response towards road excitations, which will affect the vehicle ride quality. The vehicle ride quality is defined in terms of the vehicle body acceleration. Seifi et al. used random road excitations as the inputs for simulations of a ground vehicle in order to improve ride comfort by minimizing the vehicle body acceleration. Based on the findings of existing studies, both the fatigue life of automotive coil springs and vertical vibrations of the vehicle are determined by a common source: road excitations. This is further supported by the durability transfer concept, which posits that damage or severity of usage at various points of interest on a vehicle can be predicted by measuring the vehicle acceleration at nominal points. Since vehicle ride quality is influenced by ground excitations, it is possible to correlate the fatigue life of automotive coil spring with vehicle ride quality. Pawar Prasant and Sarafassessed the vehicle ride quality and durability of an automotive coil spring based on the road profile measurement method. The results showed that it is possible to develop models that can simultaneously predict the fatigue life of the automotive coil spring and vehicle ride characteristics. Kong et al. proposed a linear model to express the relationship between spring durability and vehicle ride quality based on power regression. However, it shall be noted that the model was limited to a predefined set of parameters for the vehicle suspension system.
The objective of this study is to develop MLR-based spring durability models in order to predict the fatigue life of automotive coil springs. It is hypothesized that the fatigue life of the automotive coil spring is correlated with the vertical vibrations of the vehicle and natural frequencies of the vehicle suspension system because both of these parameters are related to road excitations. To the best of the authorsrsquo; knowledge, there are no detailed models currently available that are capable of predicting the spring durability and vehicle ride quality. Even though many studies have been carried out to analyze spring durability and vehicle ride quality, none of these studies are focused on modelling the relationship between spring durability, vehicle ride quality, and natural frequencies of the vehicle suspension system. In this study, MLR-based models are established to predict the fatigue life of automotive coil springs.
In this study, 1300 cc sedan was chosen as the case study vehicle. This car model has a front MacPherson strut suspension system, which is commonly used in most vehicle models. Both the accelerometer and strain gauge were attached to the measurement points using a strong adhesive to ensure that the sensors were stable during data collection. The used strain gauge had the specifications like 2 mm gauge length, gauge factor of 2.07 plusmn; 1.0% and strain gauge resistance of 120 Omega; while the accelerometer was a piezoelectric type with sensitivity of 1.02 mV/(m/ ), measurement range of plusmn;4900 m/ and frequency range of 0.5–10 kHz. The accelerometer and strain gauge were connected to the data acquisition system, which recorded the signals detected by the sensors and these signals were transmitted to the laptop computer. The experimental setup like accelerometer, strain gauges and data acquisition were calibrated using a calibration certificate. With the certified calibration of experimental setup, the errors of the measurement were minimized. Subsequently, the strain-time and acceleration-time histories were displayed in real-time in the laptop computer.
MLR was selected to model the relationship between the natural frequencies of the vehicle suspension system, vertical vibrations of the vehicle, and fatigue life of the automotive coil spring. Linear curve fitting was carried out as a preliminary step in MLR analysis in order to ensure that the relationship between the dependent and independent variables is linear. In this study, the fatigue life values were first linearized by logarithmic transformation. In the MLR-based spring durability models, the fatigue life of the automotive coil spring is the dependent variable whereas the natural frequencies of the vehicle suspension system and vertical vibrations of the vehicle are the independent variables. It was assumed that there is a linear relationship between the dependent and independent variables, which is an important assumption in MLR analysis.
The population r
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基于多元线性回归的汽车线圈弹簧疲劳寿命评估模型的建立
汽车线圈弹簧的疲劳分析是至关重要的，因为当汽车在不同的地形上行驶时，这些部件不断地暴露在动态载荷下。线圈弹簧也是汽车底盘的关键部件之一，因为它们影响乘客的舒适度和车辆的空气动力学。弹簧的耐久性和车辆的行驶质量取决于线圈弹簧阻尼器的设计，以处理不同的道路剖面。许多研究已经进行，以检查垂直振动的车辆激励现实的道路轮廓。Reza Kashyzadeh等人研究了路面粗糙度对车辆悬架系统振动的影响。将路面剖面作为随机路面激励，对轮毂、Pitman臂、悬架臂、关节等悬架系统部件进行疲劳寿命评估。Gonzalez等人用车辆加速度测量代替随机路面轮廓来估计路面粗糙度，其中传递函数用来建立路面功率谱密度与车辆加速度测量之间的关系。
不利的路面条件不仅影响汽车线圈弹簧的疲劳寿命，而且影响车辆的行驶质量。弹簧设计在确定车辆对路面激励的响应中起着至关重要的作用，影响着车辆的行驶质量。根据车身加速度来定义车辆的行驶质量。Seifi等人使用随机道路激励作为模拟地面车辆的输入，通过最小化车身加速度来提高乘坐舒适性。基于现有研究的结果，汽车线圈弹簧的疲劳寿命和车辆的垂直振动都是由一个共同的来源决定的:道路激励。耐久性转移的概念进一步支持了这一观点，该概念假定通过测量车辆在标称点处的加速度，可以预测车辆在不同测量点上的损伤或使用的严重程度。由于车辆的行驶质量受地面激励的影响，因此有可能将汽车线圈弹簧的疲劳寿命与车辆的行驶质量联系起来。帕瓦尔·帕桑特(Pawar Prasant)和萨拉夫·巴布布基于路面轮廓测量方法，对汽车线圈弹簧的行驶质量和耐久性进行了评估。结果表明，建立能够同时预测汽车线圈弹簧疲劳寿命和车辆行驶特性的模型是可能的。Kong等人提出了一种基于功率回归的线性模型来表达弹簧耐久性与车辆行驶质量之间的关系。但是，需要注意的是，该模型仅限于为车辆悬架系统预定义的一组参数。
为了预测汽车线圈弹簧的疲劳寿命，建立了基于多元线性回归的弹簧耐久性模型。假设汽车线圈弹簧的疲劳寿命与车辆的垂直振动频率和车辆悬架系统的固有频率有关，因为这两个参数都与路面受到的冲击有关。据作者所知，目前还没有建立能够预测弹簧耐久性和车辆行驶质量的详细模型。尽管已经有许多研究对弹簧耐久性和车辆行驶质量进行了分析，但这些研究都没有集中对弹簧耐久性、车辆的垂直振动频率和车辆悬架系统固有频率之间的关系进行建模。在这项研究中，我们建立了多元线性回归模型来预测汽车线圈弹簧的疲劳寿命。
在这项研究中,排量为1300 cc的轿车被选为案例研究对象。因为该车型采用麦克弗森支柱式前悬架系统，这是最常用的车型。加速度计和应变片都用强力胶附着在测量点上，以确保传感器在数据采集过程中是稳定的。使用的应变仪计量长度等于2 毫米,应变计的灵敏系数为2.07 plusmn; 1.0%，加速度计的电阻为120 Omega;，传感器的灵敏度为1.02 mV / (m / ),测量范围是plusmn;4900 m /, 频率范围为 0.5 -10 kHz。加速度计和应变计连接到数据采集系统，数据采集系统记录传感器检测到的信号，并将这些信号传输到笔记本电脑。利用检定证书对加速度计、应变计、数据采集等实验装置进行校准。通过对实验装置的校准，将测量误差降到最低。然后，在笔记本电脑上实时显示应变时间和加速时间。
选择多元线性回归的方法针对汽车悬架系统的固有频率、汽车的垂直振动的频率和汽车线圈弹簧的疲劳寿命之间的关系进行建模。在多元线性回归分析中，首先进行线性曲线拟合，以保证因变量和自变量之间的关系是线性的。在这项研究中，首先将疲劳寿命的数值通过对数变换线性化。在基于多元线性回归的弹簧耐久性模型中，汽车弹簧的疲劳寿命是因变量，而悬架系统的固有频率和汽车的垂直振动的频率是自变量。假设因变量与各个自变量之间存在线性关系，这是多元线性回归分析中的一个重要假设。
总体回归线性回归模型定义如下:
或者
其中为因变量(汽车线圈弹簧的疲劳寿命)，为自变量。在这项研究中，有两个自变量:车辆悬架系统的固有频率和车辆的垂直振动频率。表示截距，是常数。表示因变量与自变量线性关系的斜率，即变异系数。是期望值为0的随机误差。
选用三种疲劳寿命模型进行疲劳分析，因为部分循环载荷较大，会导致汽车弹簧产生较大的塑性变形，降低疲劳寿命。这种行为被归类为低循环疲劳。然而，应变寿命模型也适用于高周疲劳分析。应变寿命模型已广泛应用于叶片弹簧和阻尼弹簧塔等汽车零部件的疲劳分析。这是因为应变寿命模型与局部循环应力水平相关，局部循环应力水平足够大，出现明显的循环塑性应变，导致疲劳寿命小于10个循环。许多地面车辆部件的设计经常受到过载的影响，足以导致局部产生。在研究中通过应变时程的长短来评估汽车零部件的耐久性。结果表明，有限元弹性模型能够合理准确地估计应力和应变，特别是当载荷低于屈服应力时。然而，当施加的载荷大于屈服应力时，实际应力和应变均大于对应于弹性极限的实际值。
本研究考虑了Coffin-Manson、Morrow和Smith-Watson-Topper模型。在以前的研究表明，当平均应力为0时，Coffin-Manson模型更佳。然而，当主要载荷被压缩时，平均应力被修正后更适用于Morrow模型。当主要载荷被拉伸时，Smith-Watson-Topper模型更精确。
将基于Coffin-Manson、Morrow和Smith-Watson-Topper模型的多元线性回归的弹簧耐久性模型预测的疲劳寿命值与实验微应变时间内获得的历史值进行比较，来验证预测模型的有效性。一般情况下，决定系数()值较高，说明该因变量的变异性很大程度上可以用回归模型解释，说明该模型与实验数据具有较好的拟合性。
其中，Coffin-Manson 模型定义为:
①
Morrow 模型定义为:
②
Smith-Watson-Topper 模型定义为:
③
在等式 ①, ②, ②中,时塑性应变，是疲劳延性系数， 是疲劳强度系数， 疲劳强度指数，是疲劳韧性指数， 是平均应力， 是最大应力 。
采用拉伸滞后能损伤模型和应变-寿命模型对汽车线圈弹簧的疲劳寿命进行了预测。通过分析得到了50组汽车加权加速度值和汽车线圈弹簧疲劳寿命值的数据集。（见表A1）
表A1
|
钢筋直径 |
预测弹簧硬度(N/mm) |
计算的弹簧硬度 (N/mm) |
由有限元计算弹簧的硬度(N/mm) |
不同比例(%) |
|
11.0 |
14.0 |
14.3 |
13.8 |
1.4 |
|
11.3 |
16.0 |
16.1 |
15.7 |
1.9 |
|
11.6 |
18.0 |
18.0 |
17.6 |
2.2 |
|
12.0 |
20.0 |
20.3 |
19.7 |
1.5 |
|
12.2 |
22.0 |
22.3 |
21.6 |
1.8 |
|
12.4 |
24.0 |
24.0 |
23.9 |
0.4 |
|
12.6 |
26.0 |
25.7 |
25.5 |
1.9 |
|
12.8 |
28.0 |
27.5 |
27.0 |
3.5 |
|
13.0 |
30.0 |
29.5 |
28．4 |
5.3 |
|
13.3 |
32.0 |
32.4 |
30.8 |
3.8 |
利用汽车线圈弹簧的疲劳寿命值、车辆的加权加速度值和车辆悬架系统的固有频率等数据集，建立了基于多元线性回归的弹簧耐久性模型。
得到了基于Coffin-Manson 模型的弹簧耐久性模型为:
④
得到了基于Morrow 模型的弹簧耐久性模型为:
⑤
最后，得到了基于Smith-Watson-Topper 模型的弹簧耐久性模型为:
⑥
在方程式④、⑤、⑥中， , 和 分别代表基于Coffin-Manson、Morrow、Smith-Watson-Topper模型的弹簧耐久性模型预测的疲劳寿命值，为车辆悬架系统的固有频率，为车辆的加权加速度(垂直振动频率)。为了分析数据趋势，绘制出回归模型的图像（如图A2所示）。z轴表示疲劳寿命值(因变量)，y轴和x轴分别表示加速度和固有频率(自变量)。图像显示了采样数据生成的回归模型，可以看出，随着加权加速度增大，汽车线圈弹簧的疲劳寿命显著降低，这是不可取的。另外，汽车线圈弹簧的疲劳寿命和加权加速度也随着汽车悬架系统固有频率的增加而降低，这与预期一致。在此基础上，对每个基于多元线性回归的弹簧耐久性模型进行了拟合优度评估。
用决定系数()评估基于Coffin-Manson、Morrow、Smith-Watson-Topper模型的弹簧耐久性模型的拟合优度。决定系数()越高，因变量变异性所显示的百分比越高。基于Coffin-Manson 模型的多元线性回归模型的值为83%，均方差(MSE)为0.5282。基于Morrow 模型的多元线性回归模型的值和MSE略高，分别为88%和0.5855。基于Smith-
Watson-Topper 模型的多元线性回归模型的值与基于Coffin-Manson 模型的多元线性回归模型的值相同(为83%);但是，在该模型下MSE最高(为0.7056)。总体上，值大于80%，说明该模型对汽车线圈弹簧疲劳寿命的预测是可靠的。按照Sivak和Ostertagova的分类，值大于0.90的回归模型被认为“非常好”，而值大于0.80的回归模型被认为“好”。但是需要注意的是，值并没有硬性的规则，因为在一些领域(尤其是生物科学和社会科学领域)，值为0.60的回归
图A3
模型被认为是足够好的。分析表明，所建立的多元线性回归模型均具有较好的预测弹簧疲劳寿命的能力。
在只有一个解释变量的线性回归分析中，可以简单地用因变量与自变量绘制一张二维图表示回归模型。回归模型是曲线上所有数据点的最优拟合直线，被用于评价拟合优度。多元线性回归分析涉及多个解释变量，因此，验证线性、正态性和同方差是很重要的，这样才能确保对基于多元线性回归的弹簧耐久性模型的研究能够得到有效的结论。从图像(图A2)可以看出，疲劳寿命(因变量)与车辆加权加速度和车辆悬架系统固有频率(自变量)呈线性变化，证明了基于多元线性回归的弹簧耐久性模型的线性假设是合理的。
为基于Coffin-Manson、Morrow和Smith-Watson-Topper 方程的多元线性回归的弹簧耐久性模型绘制了标准化残差的正常预测概率(P-P)图（如图A3所示）。可以看出，数据点位于正态线附近(即图中的对角线P-P)，意味着因变量的分布都是正态的。这就证明了基于多元线性回归的弹簧耐久性模型是正态分布的假设是正确的。
通过P-P图检验标准化残差的正态性，采用方差分析(ANOVA)确定了汽车弹簧疲劳寿命(因变量)与悬架系统加权加速度和固有频率(自变量)的关系。用F检验看回归模型与数据是否吻合。基于方差分析结果
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