Chapter 2 Laplace Equation

Teaching objectives:

1. Understand General Dynamics equations.

2. Use the Laplace equation correctly to establish the moving differential equations of the point system.

Highlights and difficulties of this chapter:

Select generalized coordinates, and express the particle kinetic energy as a function of generalized coordinates and generalized velocity.

Calculate the generalized force or represent the potential function of the conservative system as a generalized coordinate function.

Teaching Process:

Introduction: This chapter combines the DA long principle and the virtual displacement principle to export

The most common equation for solving the dynamic problem of the point system is the basis for analyzing the dynamics.

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1. General dynamics equation

It is a mass system composed of several particles.

The quality of the intermediate point is as follows:

The active force is used, and the constraint is,

Inertial Force = -,

Based on the principle of "da rang ",

Any virtual displacement given to the particle,

The principles of virtual displacement include:

Add the above equations:

Ideal constraints include:

Replace =-with the following values:

------- General dynamics equation

Or

For example, 16.1 known: In the pulley system shown in Figure 16.2, the weight of the heavy weight hanging on the movable pulley is not counted as the weight of the pulley and rope: acceleration of heavy objects

Solution: 1. Study object: overall system = 1

2. analyze the main power ()

3. Analysis of motion, virtual and inertial force,

16.2, where

,

,

Kinematics relationship:

4. Show any virtual displacement to the system,

5. Solving the general dynamics equation:

-(

Place the inertial force and virtual displacement relationships into the above formula,

Include:

Because it is an independent variable

Ii. Laplace Equation

1. Equation derivation,

Transform the general dynamic equation

()

Set

Substitution ()

Right side = (B)

Two classic relationships:

(1), (2)

Proof of formula (1:

Because they are independent from each other

Proof of formula (2:

So

Replace formula (1) and (2) into formula (c), including:

Left =

=

= (D)

Replace formula (B) (d) into formula (a) and move the item:

In the complete system, they are independent of each other and can be obtained: -------------- the second type of equations of the Laplace.

The nature of the equation. The second-order differential equations can be used to solve the motion and the main force, and cannot be used as binding force.

2. Laplace equation of conservative systems

Potential Function: the generalized force is substituted into the Laplace equation,

Yes

Because the above formula is changed:

Command, calledLath Function, Which can be:

------------- The Laplace equation of conservative systems

3. Application of the Laplace Equation

For example 16.2, it is known that the quality of the triangular prism is smooth in contact with the horizontal plane, the quality of the homogeneous cylinder is, the half longitude is, and is placed on the inclined surface of the triangular prism. There is no relative slide between the cylindrical and triangular prism, regardless of the scroll friction, set (Figure 16.3)

The acceleration of the Three-prism and the center of the cylindrical,

Solution: 1. Study object: whole,

Generalized coordinates,

2. analyze the main power and calculate the advertising Power:

Ling,

Ling,

3. Analysis of motion, calculation of kinetic energy, and translation of triangular prism,

The kinetic energy of the prism:

Wheel kinetic energy:

Total system kinetic energy

4. Calculate the partial derivative and substitute it into the Laplace equation.

,

,

,

,,

5. Solving

And form simultaneous solutions.

, So

For example 16.3, it is known that the end of a homogeneous rod with a mass of a length is connected to a spring with a steel factor and is restricted to the vertical direction. The rod can also swing through the horizontal axis, as shown in 16.4: moving Differential Equations of bar,

Solution: 1. Study object: whole,

Generalized Coordinates (Fig. 16.4)

2. analyze motion and calculate kinetic energy

Pole for plane motion,

3. analyze the main power, calculate potential energy, and write down the Laplace Function

Set the position of the equilibrium point to the coordinate origin, and set the equilibrium position to the zero potential energy point of elasticity and gravity:

Here, after the above formula is entered, you can:

Lath Function

4. Calculate the partial derivative and substitute it into the Laplace equation.

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