## 不同截面形状罐体液罐车液体质心特性1)

*浙江警察学院交通管理工程系,杭州 310053

## THE CHARACTERISTICS OF THE CENTER OF MASS OF THE LIQUID IN THE TANKER WITH DIFFERENT CROSS SECTION SHAPES1)

HE Lieyun,*,2), LI Guojun, ZHOU Yan*

*Department of Transportation Management and Engineering, Zhejiang Police Collage, Hangzhou 310053, China

Department of Public Foundation, Zhejiang Police College, Hangzhou 310053, China

 基金资助: 1)浙江省软科学资助项目(2020C35G2321127)

Abstract

In order to investigate the motion of the center of mass of liquid in a turning tank truck, tanks with four common section shapes, i.e., circular, oval, multi-section arc and square, are considered in this research. The liquid in the tank is assumed to be quasi-static during the turning. The liquid center of mass is calculated through the integration of the center of mass for each homogeneous section plane. Using MATLAB, the centroid coordinates and centroid trajectory images are obtained. The lateral stability characteristics of tank trucks with different shape of cross-section is analyzed. The research results provide guidelines for the design and driving stability of liquid tank truck.

Keywords： tank truck; the center of mass; quasi static; cross section; filling rate; lateral acceleration

HE Lieyun, LI Guojun, ZHOU Yan. THE CHARACTERISTICS OF THE CENTER OF MASS OF THE LIQUID IN THE TANKER WITH DIFFERENT CROSS SECTION SHAPES1). Mechanics in Engineering, 2022, 44(2): 328-336 DOI:10.6052/1000-0879-21-303

## 1 准静态条件下液体质心计算模型

### 图1

Fig.1   Four common cross-sectional shapes of tanks

### 1.1 液罐车转向运动时液面倾斜率

$k=\frac{{g}\sin {(\theta -\alpha )}-{a}_{\rm n}\cos {(\theta -\alpha )}}{{g}\cos {(\theta -\alpha)}-{a}_{\rm n}\sin {(\theta -\alpha )}}$

### 图2

Fig.2   Incline analysis of steering motion liquid level of elliptical tank truck

《公路工程技术标准》(JTG B01—2014)规定,道路圆曲线超高值较小,在特殊环境时极限值也不超过10%,即$\theta <0.1$ rad,一般情况下车辆悬架变形引起罐体最大侧倾角度$\alpha \approx 0.06$ rad[17],且$\alpha$与$\theta$接近相互抵消的关系,因此当车辆行驶速率为$v$,转向半径为$r$,液罐车转向行驶时液面斜率表达式可简化为

$k=\frac{v^{2}}{gr}$

### 1.2 液体质心计算模型

《公路路线设计规范》(JTG D20—2017)规定,直线与圆曲线连接处、以及半径不同的同向圆曲线相连接处,一般设有回旋曲线和超高渐变段,液罐车按匀速率驶入或驶离圆曲线路段过程时,受到的离心加速度均匀变化。因此车辆在圆曲线路段行驶过程,只要加速度不发生突变且路面是平面的,计算液体质心位置变化就接近准静态模型,自由液面是一条斜线。液体满足准静态变化可以通过简单的试验加以验证,将装有液体的透明罐体放到车辆上,当车辆在曲线路段行驶时,可以清楚观察到罐体内液体的倾斜过程。若液罐车装载为密度均匀的液体,均质平面质心坐标计算公式为

$\left. {\begin{array}{l} x_{\rm c}=\displaystyle\int \dfrac{x{\rm d}A}{A} \\ y_{\rm c}=\displaystyle\int \dfrac{y{\rm d}A}{A} \\ \end{array}} \right\}$

### 图3

Fig.3   Calculation working conditions of centroid of eight-section arc section tank

### 图4

Fig.4   Section model of tank with eight segment arc structure

(1) $A_{\rm II}$区域面积及质心坐标计算

$A_{\mathrm{II}}=2 \int_{y_{\mathrm{I}}}^{y_{\mathrm{II}}}\left[x_{2}+\sqrt{r_{2}^{2}-\left(x-y_{2}\right)^{2}}\right] \mathrm{d} y$
$x_{\rm IIC}=0$
$y_{\mathrm{IIC}}=\frac{2}{A_{\mathrm{II}}} \int_{y_{\mathrm{I}}}^{y_{\mathrm{II}}} y\left[x_{2}+\sqrt{r_{2}^{2}-\left(x-y_{2}\right)^{2}}\right] \mathrm{d} y$

(2)$A_{\rm V}$区域面积及质心坐标计算

\begin{aligned}A_{\mathrm{V}}=& \frac{x_{m}\left(y_{m}-y_{\mathrm{IV}}\right)}{2}+x_{\mathrm{IV}}\left(y_{\mathrm{IV}}-y_{m}\right)+\\& \int_{x_{\mathrm{IV}}}^{x_{\mathrm{III}}}\left[y_{4}+\sqrt{r_{4}^{2}-\left(x-x_{4}\right)^{2}}-y_{m}\right] \mathrm{d} x\end{aligned}
$\begin{array}{l}\int_{0}^{x_{\mathrm{IV}}} x\left(y_{\mathrm{IV}}-y_{m}\right) \mathrm{d} x+ \\\left.\int_{x_{\mathrm{IV}}}^{x_{\mathrm{III}}} x\left[y_{4}+\sqrt{r_{4}^{2}-\left(x-x_{4}\right)^{2}}-y_{m}\right] \mathrm{d} x\right\}\end{array}$
\begin{aligned}y_{\mathrm{VC}} &=\frac{1}{A_{\mathrm{V}}}\left\{\int _ { y _ { m } } ^ { y _ { \mathrm { IV } } } y \left[x_{4}+\sqrt{r_{4}^{2}-\left(y-y_{4}\right)^{2}}-\right.\right.\\&\left.\left.\left(\frac{y-y_{m}}{k}+x_{m}\right)\right] \mathrm{d} y\right\}\end{aligned}

$\left.\begin{array}{rl}x_{\mathrm{AC}} & =\frac{\sum_{i=\mathrm{I}}^{\mathrm{V}} x_{i \mathrm{C} \times} A_{i}}{\sum_{i=\mathrm{I}}^{\mathrm{V}} A_{i}}=\frac{\sum_{i=\mathrm{I}}^{\mathrm{V}} x_{i \mathrm{C} \times} A_{i}}{e \times S} \\y_{\mathrm{AC}}= & \frac{\sum_{i=\mathrm{I}}^{\mathrm{V}} y_{i \mathrm{C} \times} A_{i}}{\sum_{i=\mathrm{I}}^{\mathrm{V}} A_{i}}=\frac{\sum_{i=\mathrm{I}}^{\mathrm{V}} x_{i \mathrm{C} \times} A_{i}}{e \times S}\end{array}\right\}$

(1) 圆形：以国内某厂家生产的CLW9401GYQC型液化气体运输半挂车为例,该车的圆形截面罐体内直径为2525 mm,筒体长度为10 692.0 mm。

(2) 椭圆形：以国内某厂家生产的HRT5250GYY型运油车为例,该车的椭圆形截面罐体内长半轴为1190 mm,短轴为755 mm,筒体长度为6320 mm。

(3) 多段弧形：以国内某厂家生产的5070GJYH型加油车为例,该车的八段圆弧组合形截面罐体外形尺寸如下。宽度为1800 mm,高度为1250 mm,圆弧半径$r_1=500$ mm,$r_3=1950$ mm,$r_4=500$ mm,$r_5=2950$ mm。

(4) 方形截面罐体并不常见,少部分洒水车罐体采用。方形截面罐体液罐车罐体长度为10 000 mm,宽度为2400 mm,高度为1600 mm。

## 2 四种罐体截面液罐车质心坐标与充装率关系

### 图5

Fig.5   Filling rate-liquid centroid coordinate diagram}

### 2.2 充装率对液体质心位置影响分析

(1) 随着充装率逐渐减小,液罐车转向行驶时,液体质心偏移幅度逐渐增大;

(2) 在同一充装率时,液罐车转向运动时质心纵坐标及横坐标同时增大,但横坐标增大量明显大于纵坐标,说明液体的横向晃动量大于纵向晃动量;

(3) 对图5(a)~图5(d)液体质心坐标连线图像采用一次线性函数"$y=ax+b$"进行拟合时,结果如表1所示。

Table 1  Mass coordinate of liquid center line

## 3 四种罐体截面液罐车质心轨迹与侧向加速度

### 图6

Fig.6   Lateral acceleration-center of mass trajectory diagram}

### 3.2 侧向加速度对液体质心位置影响分析

(1) 随着侧向加速度增大,液体质心偏移平衡位置距离越大,且质心沿横向偏移的距离明显比纵向偏移大,说明液罐车的侧翻阈值主要是受质心横向偏移影响;

(2) 液体质心轨迹图像均为光滑凹曲线,说明伴随着侧向加速度增大,液体质心变化是连续的,即使方形罐体也不存在突变点;

### 图7

Fig.7   Distance between liquid center of mass and tank geometric center-lateral acceleration diagram

(1) 圆形截面罐体内液体质心与罐体几何中心间距为常量,其余三种截面形状罐体内液体质心与罐体几何中心间距则随着侧向加速度增大而增大。

(2) 当侧向加速度$a$小于0.15$g$后,椭圆形和多段弧形两种截面,$r$与$a$是非线性关系,且$r$增加相对缓慢;当侧向加速度$a$大于0.15$g$后,$r$与$a$接近线性关系;而方形截面罐体$r$与$a$始终为非线性关系,且$a$在0.15$g$~0.35$g$间时,$r$的变化较为明显。

## 4 结论

(1) 四种罐形液罐车在侧向加速度作用下,罐内液体质心发生偏移,且横向偏移量均明显大于纵向偏移量。

(2) 当侧向加速度相同时,随着充装率减小,四种罐形液罐车的液体质心偏移量均增大。圆形和椭圆形截面罐体内液体质心坐标连线满足一次线性函数,其余两种罐形液罐车的液体质心坐标连线线性拟合程度较差。

(3) 当充装率相同时,随着侧向加速度增大,四种罐形的液体质心偏移量均增大,液体质心轨迹为一段光滑曲线,且圆形截面罐体液罐车的液体质心轨迹是一段圆弧。

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