Kinematics of Rigid Body

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3.2 RELATIVE Motion

in Plane

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3.2.1 General Concepts and Relative Equations of Motion

A : A point of object (disc) number 1 that is slipped on

P : A point of object 1 having the same coordinates as B (a point at the bottom of the channel)

If point B were fixed, it would have the same velocity and acceleration as P. However, since it slides in the channel, it has an additional velocity and acceleration with respect to P, which are called relative velocity and relative acceleration.

Let us consider a ball B that slides freely in a channel opened on disk number 1.

(It only makes the movement of object number 1)

Since A and P are two points of the same object, equations 3.4 and 3.5 are valid between them:

Since P and B have the same coordinates:

the angular velocity of the object on which slided

(3.6)

(3.7)

(3.8)

Note: A and B are two points on different rigid bodies, but these bodies are in contact with each other. Now we examine the figure below:

Coriolis acceleration and its effect are explained in detail on the following pages>

Our aim in this section is to write the equations of velocity and acceleration of two points that are not on the same object relative to each other.

3.2 Kinematics of Rigid Body / Relative Motion in Plane

wanted values :

Figure 3.32

at a time t

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3.2.2 What is Coriolis Acceleration and When Does It Appear?

• If there is a point that slides (displaces) on a rotating object; an independent acceleration of this sliding point occurs, which is called the coriolis acceleration (a_cor).

• Coriolis acceleration has the effect of changing the direction of the relative velocity of the sliding object.

Click to see the moving picture

• What is meant by a rotating object is an object that has an angular velocity of ω at the moment of examination, in which case the object is making instantaneous rotation. It is not necessary for this object to be in a constant rotational motion.

• Now we will prove this equation with another example:..>>

3.2 Kinematics of Rigid Body / Relative Motion in Plane

• The sliding point can be a ball or a pin attached to another object.

Figure 3.33

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3.2.3 Derivation of Coriolis Acceleration Equation: : Consider a collar B sliding with velocity u and acceleration a_k on a rod rotating with angular velocity 𝜔 and angular acceleration 𝛼 about point A.

​

(from equ. 3.4) :

tangent

tangent

• Since A and P are points of the same object, the absolute velocity and absolute acceleration of point P are:

(1.12)

(I)

(II)

• P and H : Points belonging to the rod that have the same coordinates as the collar at the B and 𝐵′ positions of the collar.

y

• at a time

absolute acceleration of collar B:

• Our aim now is to find explicit expressions for the question-marked terms in equation (I).

?

?

absolute velocity of collar B:

Figure 3.34

(from equ. 3.3.d)

x

z

3.2 Kinematics of Rigid Body / Relative Motion in Plane

DYNAMICS - Lecture Notes / Mehmet Zor

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3.2 Kinematics of Rigid Body / Relative Motion in Plane

(III)

If we substitute equations II and III into equation I:

The last term is the Coriolis acceleration, which is an independent acceleration:

Accordingly :

According to the right hand rule :

( It is considered equal to the arc length ds opposite the angle 𝑑𝜃.)

unit vectors

Figure 3.35

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3.2.4 Determination of Coriolis Acceleration from Polar Coordinates

In topic 1.2.5, we derived the acceleration components in polar coordinates in the curvilinear motion of the particle. Now let's write the acceleration equations and represent each term with a different symbol:>>

tangential acceleration:

normal acceleration:

angular acceleration:

angular velocity:

Acceleration equations in Polar Coordinates:

Acceleration Components in Circular Motion (see topic 1.2.4.1)

Linear path

;

The collar B sliding on the rod rotating about A undergoes general planar motion (rotation + translation) and therefore must have all of the terms in the acceleration equations 1.29 and 1.30. Now we will separate this motion into two different motions and specify the accelerations for each motion and which terms they correspond to in the acceleration equations in polar coordinates.

Rotation around A (circular motion)

General Planar Motion

rectilinear translation in r direction

(1.25)

(1.26)

(1.29)

(1.30)

If there was only rotational motion

If there was only translational motion

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.36

Figure 3.37.a

Figure 3.38

Figure 3.40

Figure 3.39

Figure 3.37.b

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Click to see the moving Picture.

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Another example of Coriolis acceleration.

Figure 3.41

Figure 3.42

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On the horizontal plane, the angular velocity of the 1m radius disk rotating around the center A is ω= 2rd/s in the counterclockwise direction and its angular acceleration is α = 1rd/s² in the clockwise direction. In the dc channel opened on the disk, ball B moves freely. In the position shown, its velocity relative to the channel is 2m/s in the c direction and its acceleration is 4m/s² in the d direction. Accordingly,

a-) Calculate the absolute velocity and absolute acceleration of ball B for this position.

Question 3.2.1

b-) If there was a circular channel with a radius of 0.8m instead of a linear dc channel, calculate the velocity and acceleration of ball B and its angular velocity relative to the channel for the same position of ball B. (Assume that the ball will rotate counterclockwise in the circular channel with a constant velocity of 2m/s.)

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Solution:..>>

Figure 3.43

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The angular velocity of the disc on which ball B slides.

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Let's calculate the absolute acceleration of ball B:

a-) B is the point belonging to the sliding object and A is the point belonging to the "object being slid on".

Solution:

Equations 3.6 and 3.7 can be used between A and B.

Figure 3.44

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b-) If there was a circular channel with a radius of 0.8m instead of a linear dc channel, calculate the velocity and acceleration of ball B and its angular velocity relative to the channel for the same position of ball B. (Assume that the ball will rotate counterclockwise in the circular channel with a constant velocity of 2m/s.)

Since the relative velocity of ball B in the position in the figure is to the right:

Although the relative velocity is constant, there is normal acceleration due to circular motion in the channel.

Relative

tangential acceleration:

Relative

normal acceleration:

angular velocity of body B with respect to the channel:

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.45

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C and A are on different objects,

C and D are on the same object (number 1),

Let's write the velocity of C from arms 1 and 2. :

Solution:

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Question 3.2.2

C : is a sliding point on the object number 2, to which A belongs.

Since ACD is an isosceles triangle, AC=DC=40cm

It should be in the direction of the channel. It was selected in the direction of AB.

It is in the opposite direction to the selected direction.

;

Figure 3.46

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Let's write the acceleration of C from arms 1 and 2 :

3.2 Kinematics of Rigid Body / Relative Motion in Plane

It should be in the direction of the channel. It was selected in the direction of AB.

Figure 3.47

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ω₁ = -2rd/s,

α₁ = -1rd/s²

V_E =? , a_E = ?

• A and C are two different points belonging to the 3rd arm.

Since they are located on the same object, the absolute velocity

equation (equation 3.4) can be written between A and C:

• Similarly, C and D are points belonging to arm 2:

(D is the fixed point)

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Question 3.2.3 : In the mechanism in the figure, objects 1, 2 and 3 move on the fixed frame 4. In the position shown, the angular velocity and angular acceleration of disk 1 centered at O are clockwise and their magnitudes are ω₁=2rd/s and α₁=1rd/s² , respectively. Ball B attached to the disk can slide freely in the channel on arm 2, and element E can slide freely in the vertical channel opened on fixed frame 4. Accordingly, calculate the velocity and acceleration of sliding element E for the position shown.

To find the velocity and acceleration of element E, it is sufficient to find the velocity and acceleration of point A.

The intensity of the velocity is assumed to be in the positive (+y) direction.

Figure 3.48

• Since the movement of A is limited in the vertical direction, the velocity vector :

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Now we will write the velocity of the common point B from arms 1 and 2.

Since B and O are points belonging to object 1:

(O is the fixed point)

0

B and D are two points belonging to different objects.

D is a point belonging to arm no. 2; B is a sliding point on arm no. 2.

0

magnitude

Then equation 3.6 can be used between them:

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.49

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Since it showed +,

the direction we chose was correct.

Since it comes out negative, it is in the clockwise direction.

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.48

Figure 3.49

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We perform similar calculations for accelerations.

Examine the operations below. Paying attention to the fact that we use equation 3.5 for points on the same object and equation 3.7 for points on different objects (provided that one is a sliding point and the other is a point on the object on which slid on).

3.2 Kinematics of Rigid Body / Relative Motion in Plane

..>>

Figure 3.50

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3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.50

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N

N

Question 3.2.1

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.51

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Final Question

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.52

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3.2 Kinematics of Rigid Body / Relative Motion in Plane

Question 3.2.3 (*)

Figure 3.53

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3.2.5 Schematic Summary of Rigid Body Planar Kinematics

If A and B are on the same object

B is a sliding point on the object where A is located

A and B are on different objects (but these bodies are in contact with each other)

the angular velocity of the object on which slided.

A is a point of the object that is slipped on

Relative Acceleration: Additional acceleration of point B with respect to the channel (or point P)

Relative Velocity: Additional velocity of point B with respect to the channel (or point P)

Coriolis acceleration

3.2 Kinematics of Rigid Body / Relative Motion in Plane

Figure 3.54