﻿ 新型三轴离心机系统构型及数学建模
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Configuration and mathematical modeling for advanced three-axis centrifuge system
CHANG Le, LIU Zhenghua , WEN Nuan, WU Sentang
School of Automation Science and Electrical Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Abstract:The traditional centrifuge system usually uses a high speed, high precision single-axis rotator system to achieve one-way precise simulation of unidirectional centrifugal acceleration in a broad range. Nowadays, in order to realize immense three-axis normal overload of an aviation apparatus in three dimensions, an advanced three-axis centrifuge system was proposed. The high precision three-axis rotator was composed of a foundation and three rotational axes. It succeeded in simulating normal acceleration overload by regulating speed of outer axis, and precise position control of inner and middle axis. Based on the configuration above, a mathematical model of a three-axis centrifuge system was built; related kinematics simulation and theoretic calculation were analyzed. Given the three directions overload of x,y,z, offset angle of internal axis, middle axis and rotation speed of outer axis was solved inversely by means of the force-balance equation. In the end, through the analysis of dynamics simulations, correctness and effectiveness of the new centrifuge system configuration were validated further.
Key words: centrifuge system     high-speed aircraft     overload simulation     mathematical modeling     kinematic analysis

1 过载模拟系统结构

 图 1 过载模拟系统机械本体三维图Fig. 1 Three-dimensional structure of overload simulation system
2 数学建模 2.1 过载的定义

N为作用在对象上的除重力G以外的所有外力的合力,定义过载为n=N/G,方向与N一致[11].

1) 离心机过载.

2) 机动过载.

2.2 坐标系的建立及绕旋转矩阵的定义

 图 2 过载模拟系统坐标系示意图Fig. 2 Coordinate system schematic diagram of overload simulation system

2.3 数学模型

3 运动学分析 3.1 离心机以恒定的角速率ω做匀速转动

1) 在x3轴方向上过载由向心力在该轴上的一个分量产生,即G31=rω2.其中r为质量块绕回转中心匀速转动的半径在x3轴上的投影长度.由于回转中心在外框坐标系内的坐标为(R,0,0),而质量块坐标为

2) 外框转动时,近似认为在y3轴方向上加载为0,即G32=0.

3) 离心机做匀速转动,不产生切向加速度,但z3轴方向上的过载由向心力Gc在该轴上的一个分量产生,即G33=Gcsin θ00为外框转角.其中,由加载的矢量合成,可得 .对于sin θ0的计算,有

 图 3 转角计算图Fig. 3 Calculating diagram of θ0

θ1 = arctan(G12/G11)    θ1∈[－π,π]

3.2 离心机以加速度α做匀加速转动

4 仿真验证

 图 5 恒定加载时模拟系统各变量曲线Fig. 5 Variable curves of simulation system at constant loading

 图 6 恒定加载时切向加速度产生的误差曲线Fig. 6 Error curves caused by tangential acceleration at constant loading

 图 8 加载变化时模拟系统各变量曲线Fig. 8 Variable curves of simulation system at variational loading

 图 9 加载变化时切向加速度产生的误差曲线Fig. 9 Error curves caused by tangential acceleration at variational loading

1) 过载大小恒定且为最大过载,即x1向过载为45g0;y1向过载为40g0;z1向过载为15g0.

2) x1,y1,z1 3方向的过载为如图 7图 9所示的正弦曲线.

5 结 论

1) 进行过载模拟时,所需要的持续性过载由法向过载提供,但在离心机外框角速度改变时,会出现瞬时的切向过载,该切向加速度会对过载需求[G11 G12 G13]带来影响,但从图中可以看出误差在1.5g0之内.

2) 文中提出的建模和解算方法是假设过载模拟系统是刚性体,并且轴系传动有很好的刚度,但过载模拟系统在实际运行时特别是在离心机高速转动时，结构上一定会产生形变[12],这也会对最终结果带来影响.

3) 从仿真结果可以看出,当给定的过载大小和频率变化较快,解算出的内框、中框的角度值和外框的转速变化也会复杂,这就对离心机的数字伺服控制系统提出了很高的要求,要求其能满足对内框、中框角度及外框转速的高动态、精确跟踪控制,控制中的滞后、非线性和外界未知干扰[13]也会对结果带来误差,因此必须通过设计先进的自适应控制器[14, 15]使系统拥有很强的干扰抑制和稳定特性.

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#### 文章信息

CHANG Le, LIU Zhenghua, WEN Nuan, WU Sentang

Configuration and mathematical modeling for advanced three-axis centrifuge system

Journal of Beijing University of Aeronautics and Astronsutics, 2015, 41(2): 283-288.
http://dx.doi.org/10.13700/j.bh.1001-5965.2014.0148