Chapter 10 Rigid Body Plane Motion 1

1. Rigid Body Plane Motion
   

Teaching Objectives

1. Identify the characteristics of the Plane Motion of the rigid body, master the method of studying the plane motion, and correctly judge the rigid body in the plane motion.

2. Be proficient in the use of various methods-the baseline method, the instantaneous method, and the velocity projection method to find the velocity of any point in a plane image.

3. The base point method is used to calculate the acceleration of any point in a plane image.

Highlights of this Chapter

Based on the decomposition and synthesis of motion, this paper studies the base point method for finding the velocity and acceleration of each point on the surface map, aiming at the speed. The Velocity Projection Method and the instantaneous center method are derived from the base point method.

Difficulties in this Chapter

Correctly understand the meaning of the selected base point and the moving characteristics of the relative base point when the plane motion is decomposed to the point moving and the point rotation with the base point; the concept of a new instantaneous speed.

Teaching Process

1. Concepts of Plane Motion

1. Concepts of Plane Motion

Example 1: the movement of the wheel when the car is traveling along a straight line (Figure 10.1)

The movement of the wheel follows the moving of the body and the rotation of the opposite body.

Example 2. Movement of the connecting rod AB in the crank-Linkage Mechanism

Example 3. Arbitrary board movement on the blackboard

What are the commonalities of the above movements?

Plane Motion definition: the distance from any point in a rigid body to a fixed plane remains unchanged during motion, that is, all points in the body are moving in parallel to a fixed plane.

2. Simplified Mechanical Model

Allows a rigid body to move in parallel to a fixed plane.

Point A represents the movement of a line segment.

Point B represents the movement of a line segment

The Plane Chart S represents the motion of a rigid body.

Conclusion: The Plane Motion of a rigid body can be simplified to that of a plane image in its own plane.

3. Equations of Motion

To determine the position of a plane image S in the coordinate system, you only need to determine the position of a line AB in the coordinate system. The coordinate must be determined.Base point. So the motion equation of Plane Motion:

(10.1)

The motion equation of M is

(10.2)

The AM length and constant in the formula, so as long as the equation (10.1) is determined, the M point motion can be determined.

4. Analysis of decomposition and Decomposition Characteristics of motion

Special Case Analysis: in equation (10.1), if "S" is used for translation, if "S" is used for fixed axis rotation, the plane movement can be regarded as the synthesis of the translation and the rotation of the fixed axis.

Motion decomposition: research object: plane graphics s

Static series: Fixed plane.

Dynamic System: (where a is the "S" point, accompanied by a's translation, is a fictitious coordinate system ).

Implicated motion: The motion changes with point.

Relative Motion: rotate around a point

Therefore, the plane motion rotates with the base point a translation + relative base point.

Decomposition motion characteristics:

Translation: varies with the base point

Rotation: the rotation angle, angular velocity, and angular acceleration of different base points are the same, that is, the rotation is irrelevant to the selection of the base point.

Proof 1:

Proof 2:

Constant

2. Analysis of the angular velocity of a plane image and the speed of each point on the Image

1. base point method (synthesis)

The plane movement rotates with the base point translation + relative base point. The known a-point speed and angle are used to calculate the speed of any point B in the graph.

The speed of point B is: (10.3)

Formula, where formula (10.3) can only obtain two knowledge quantities, the common known amount

Is the direction of the sum.

Formula (10.3) can also be obtained by vector derivation, where it is a constant.

Where, that is

2. Speed Projection Method

Projection (10.3) to the positive vertical projection of the AB Line and the AB line.

(10.4)

(10.5)

Formula (10.4) is calledVelocity Projection TheoremIs the non-deformation attribute of the rigid body. In formula (10.5), the positive direction of the X-ray that has been projected over point B is the positive direction and the direction indicated in section 10.9.

As shown in example 10.111, In the crank-link mechanism, it is known that the crank OA length is R, and the rotation around the O axis is counter-clockwise. When it is obtained, the speed of the slider B and the angular velocity of the connecting rod AB are obtained.

Solution: 1. analyze motion:

Rotate the fixed axis of the OA rod, and the AB rod moves in a plane.

2. Analysis speed

Oa rod:, AB ROD:

There are only two unknown numbers, which can be solved. The speed synthesis diagram 11 contains

Obtain,

While

Another solution: Use the Velocity Projection Method:

AB:

Set direction 10.12

:

(The negative number is the opposite of the Assumption)

(The axis points to positive)

(Negative signs indicate clockwise conversion)

Problem. If this parameter is obtained (point c is the midpoint of the AB rod)

For example, in the plane mechanism shown in Figure 10.2, it is known that, the OA rod rotates at a constant speed around the O axis. In the position shown in the figure, OA and CB follow the horizontal direction, AC along the vertical direction, try to find the angle velocity of this instantaneous (1) rod. (2) Speed of point C on the board.

Solution: 1. analyze motion

OA rod, fixed shaft rotation

ABC for Plane Motion

2. Analysis speed

OA:

ABC:

Speed synthesis diagram:

:

3. Find

From figure 10.13:

Q: If we do not analyze the speed of point B, can we find it?

For example, in Figure 10.3, a plane hinge mechanism is provided. It is known that the Rod's angular velocity is, And the Rod's angular velocity is 10.14. In the illustration, the pole is instantaneous, the pole is straight, the pole is AC and the horizontal, while the pole BC is at a straight angle to the lead ,. Evaluate the speed of the instantaneous point C.

Solution: 1. analyze motion

And the rod rotates on a fixed axis,

The AC and BC poles are used for Plane Motion.

2. Analysis speed

:

:

AC: ()

BC: (B)

Formula (a) and (B:

3. Instantaneous Speed Method

Introduction: If yes

The speed distribution of each point on the graph is 10. As shown in

Instantaneous SpeedThe instantaneous speed center of an instantaneous plane image, which is short for speed Instant Center, is usually represented by "P.

Theorem:In general, the instantaneous speed exists only on each instantaneous plan..

Proof: We have known the velocity of a plane image at any point M and the angular velocity of the plane image,

As shown in figure 10.16 after M, the speed of p on mn is:

In the opposite direction.

At that time,

At that time, there was only one definite value and only a point on the MN straight line that satisfied the condition, so the theorem was proved.

Several methods to find the Instantaneous Heart:

1) known speed direction of two points

A) Renewal

B) instantaneous Translation

C) the desired size (Figure 10.19)

2) It is known that a plane image is rolled along a line or plane, and the contact point is instantaneous (Figure 10.20)