《理论力学 Theoretical Mechanics——动力学》（双语版）第十六章 虚位移原理 Theorem of virtual displacements

Theoretical Mechanics Chapter 16: Theorem of virtual displacements

Theoretical Mechanics

理论力学 第十原

Dynamic In the first part, statics, we started from the axioms of statics, then obtained the equilibrium conditions of a rigid body by simplification of the system of forces. They are used to investigate equilibrium problems of rigid bodies and of system of rigid bodies. In this chapter, we shall introduce a theorem which is suitable to study equilibrium problems of any system of particle in general. Using the concepts of displacements and work, the theorem defines the equilibrium conditions for any system of particles. It is called the theorem of virtual displacements. It is the most general theorem to study equilibrium problems. In addition, combining it with D'Alembert's principle, we can obtain a general equation of dynamics which can be used to solve the problems of dynamics

3 In the first part, statics, we started from the axioms of statics, then obtained the equilibrium conditions of a rigid body by simplification of the system of forces. They are used to investigate equilibrium problems of rigid bodies and of system of rigid bodies. In this chapter, we shall introduce a theorem which is suitable to study equilibrium problems of any system of particles, in general. Using the concepts of displacements and work, the theorem defines the equilibrium conditions for any system of particles. It is called the theorem of virtual displacements. It is the most general theorem to study equilibrium problems. In addition, combining it with D‘Alembert’s principle, we can obtain a general equation of dynamics which can be used to solve the problems of dynamics

力单 在第一篇静力学中,我们从静力学公理出发,通过力系 的简化,得出刚体的平衡条件,用来研究刚体及刚体系统的 平衡问题。在这一章里,我们将介绍普遍适用于研究任意质 点系的平衡问题的一个原理,它从位移和功的概念出发,得 出任意质点系的平衡条件。该原理叫做虚位移原理。它是研 究平衡问题的最一般的原理,不仅如此,将它与达朗伯原理 相结合,就可得到一个解答动力学问题的动力学普遍方程

4 在第一篇静力学中，我们从静力学公理出发，通过力系 的简化，得出刚体的平衡条件，用来研究刚体及刚体系统的 平衡问题。在这一章里，我们将介绍普遍适用于研究任意质 点系的平衡问题的一个原理，它从位移和功的概念出发，得 出任意质点系的平衡条件。该原理叫做虚位移原理。它是研 究平衡问题的最一般的原理，不仅如此，将它与达朗伯原理 相结合，就可得到一个解答动力学问题的动力学普遍方程

Chapter 16: Theorem of virtual displacements □§16-1 Constraints and their classification D$16-2 Degrees of freedom and generalized coordinates D 816-3 Virtual displacements and virtual work 心§164 Ideal constraints D 8 16-5 Theorem of virtual displacements

5 §16–1 Constraints and their classification §16–2 Degrees of freedom and generalized coordinates §16–3 Virtual displacements and virtual work §16–4 Ideal constraints §16–5 Theorem of virtual displacements Chapter 16: Theorem of virtual displacements

第十六章虚位移原理 §16-1约束及其分类 §16-2自由度广义坐标 §163虚位移和虚功 心§164理想约束 回§16-5虚位移原理

6 §16–1 约束及其分类 §16–2 自由度 广义坐标 §16–3 虚位移和虚功 §16–4 理想约束 §16–5 虚位移原理 第十六章 虚位移原理

Dynamic 816-1 Constraint and their classification 1. Constraints and the equations of constraints All kinds of conditions which limit the motion of a particle or a system of particles are called constraints The equations which express these limiting conditions are called the equations of constraints. For example: A(A,JA B(xB,e) 4(xy) A crankguide A single pendulum in a plane xa+ya x2+y2=.(x2-x1)2+(ya-y)2=12,ya=0 7

7 §16-1 Constraint and their classification 1. Constraints and the equations of constraints All kinds of conditions which limit the motion of a particle or a system of particles are called constraints. The equations which express these limiting conditions are called the equations of constraints. A single pendulum in a plane . 2 2 2 x + y = l A crankguide , 2 2 2 x y r A + A = ( ) ( ) , 0. 2 2 2 xB − xA + yB − yA = l yB = For example:

学 §16-1约束及其分类 约束及约束方程 限制质点或质点系运动的各种条件称为约束 将约束的限制条件以数学方程来表示,则称为约束方程。 例如: A(xA,JA B(xB,yR) 7777 M(x,y) 曲柄连杆机构 平面单摆 xa +ya xty= (xB-x2+(y-y)2=12,yg=0 8

8 §16-1 约束及其分类 一、约束及约束方程 限制质点或质点系运动的各种条件称为约束。 将约束的限制条件以数学方程来表示，则称为约束方程。 平面单摆 2 2 2 x + y = l 例如： 曲柄连杆机构 2 2 2 x y r A + A = ( ) ( ) , 0 2 2 2 xB − xA + yB − yA = l yB =

Dynarnics 2. Classification of constraints By their forms and characters constraints can be classified to into different types as follows (1) Geometrical constraints and constraints of motion Conditions which limit the geometric position in space of a particle or of a system of particles are called geometrical constraints Examples are, the limiting conditions shown in the cases given above Conditions which limit the motion of a particle or of a system of particles are called constraints of motion An example is the pure rolling of wheels along a tangent track

9 By their forms and characters constraints can be classified to into different types as follows 2. Classification of constraints (1) Geometrical constraints and constraints of motion Conditions which limit the geometric position in space of a particle or of a system of particles are called geometrical constraints. Examples are, the limiting conditions shown in the cases given above. Conditions which limit the motion of a particle or of a system of particles are called constraints of motion. An example is, the pure rolling of wheels along a tangent track

力单 约束的分类 根据约束的形式和性质,可将约束划分为不同的类型,通 常按如下分类: 1、几何约束和运动约束 限制质点或质点系在空间几何位置的条件称为几何约束。 如前述的平面单摆和曲柄连杆机构例子中的限制条件都是几 何约束。 当约束对质点或质点系的运动情况进行限制时,这种约 束条件称为运动约束。 例如:车轮沿直线轨道作纯滚动时。 10

10 根据约束的形式和性质，可将约束划分为不同的类型，通 常按如下分类： 二、约束的分类 1、几何约束和运动约束 限制质点或质点系在空间几何位置的条件称为几何约束。 如前述的平面单摆和曲柄连杆机构例子中的限制条件都是几 何约束。 当约束对质点或质点系的运动情况进行限制时，这种约 束条件称为运动约束。 例如：车轮沿直线轨道作纯滚动时