﻿ 刚弹耦合动力学初值问题拟变分原理及其应用<sup>*</sup>
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1. 哈尔滨工程大学 机电工程学院, 哈尔滨 150001;
2. 黑龙江科技大学 机械工程学院, 哈尔滨 150022;
3. 北京强度环境研究所 可靠性与环境工程技术重点实验室, 北京 100076

Quasi-variational principle and application of initial value problem for rigid-elastic coupling dynamics
ZHOU Ping1,2, LI Haibo3, LIANG Lifu1
1. College of Mechanical Electrical Engineering, Harbin Engineering University, Harbin 150001, China;
2. College of Mechanical Engineering, Heilongjiang University of Science and Technology, Harbin 150022, China;
3. Science and Technology on Reliability and Environment Engineering Laboratory, Beijing Institute of Structure and Environment Engineering, Beijing 100076, China
Received: 2016-11-04; Accepted: 2017-01-13; Published online: 2017-02-27 09:37
Foundation item: National Natural Science Foundation of China (111172046, 10272034)
Corresponding author. LIANG Lifu, E-mail: lianglifu@hrbeu.edu.cn
Abstract: The rigid-elastic coupling dynamics has been widely used in the national defense and civil economic construction, but there are still no mature theoretical research results. In this view, the quasi-variational principle of the initial value problem was established, according to rigid-elastic coupling characters, and the quasi-stationary condition of the quasi-variational principle was derived by the variational method. This condition is the governing equation of rigid-elastic coupling dynamics. Two examples were given to show the application of this condition. One was the analytical solution of the odd order vibration mode of free beam obtained by the governing equation. The other is the analytical solution of the even order vibration mode of the free beam obtained by the variational direct method Ritz method. The results show that the quasi-variational principle of the initial value problem of the rigid-elastic coupling dynamics provides the basis for the establishment of finite element model.
Key words: rigid-elastic coupling dynamics     initial value problem     quasi-variational principle     governing equation     free beam

1 建立拟变分原理

1.1 刚体动力学初值问题拟变分原理

 (1)

1.2 弹性动力学初值问题拟变分原理

 (2)

1.3 刚弹耦合动力学初值问题拟变分原理

 (3)

 (4)

 (5)

 (6)

2 推导控制方程 2.1 两类变量刚弹耦合动力学控制方程

 (7)

 (8)

 (9)
 (10)
 (11)
 (12)

 (13)

 (14)

 (15)
 (16)
 (17)
 (18)

2.2 一类变量刚弹耦合动力学控制方程

 (19)
 (20)
 (21)
 (22)
3 算例

 (23)

3.1 自由梁奇数阶振型

 图 1 无约束梁自由振动的三阶振型 Fig. 1 Third-order vibration mode of unrestrained beam in a free vibration

 (24)

 (25)

 (26)

 (27)

 (28)

 (29)

 (30)

 (31)

 (32)

 (33)
 (34)

 (35)

 (36)

 (37)

3.2 自由梁偶数阶振型

 图 2 无约束梁自由振动的二阶振型 Fig. 2 Second-order vibration mode of unrestrained beam in a free vibration

 (38)

 (39)

 (40)

 (41)

 (42)

 (43)

 (44)

 (45)

 (46)

 (47)
 (48)

 (49)

 (50)

3.3 两点探讨

1) 3.1节和3.2节算例给出的是一般情况。当i取2时，由奇数阶振型可知广义位移为

 (51)

 (52)

2) 考虑刚弹耦合效应的计算结果，根据式(37) 和式(50) 计算可得

 (53)

 (54)

4 结论

1) 结合刚体动力学初值问题拟变分原理和非保守弹性动力学初值问题拟变分原理，经过确定耦合项和相应的初值项，建立刚弹耦合动力学初值问题拟变分原理。为建立有限元计算模型提供了依据。

2) 应用变分方法，借助于Laplace变换中的卷积理论及Green定理，推导出刚弹耦合动力学控制方程。

3) 应用刚弹耦合动力学初值问题拟变分原理解决自由梁的振动问题，得到问题的近似的解析解。

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#### 文章信息

ZHOU Ping, LI Haibo, LIANG Lifu

Quasi-variational principle and application of initial value problem for rigid-elastic coupling dynamics

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(7): 1321-1329
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0849