In Prob. 15.201, the speed of point B is known to be constant. For the position shown, determine (a) the angular acceleration of the guide arm, (b) the acceleration of point C.
15.201 Several rods are brazed together to form the robotic guide arm shown that is attached to a ball-and-socket joint at O. Rod OA slides in a straight inclined slot, while rod OB slides in a slot parallel to the z axis. Knowing that at the instant shown vB = (9 in./s)k, determine (a) the angular velocity of the guide arm, (b) the velocity of point A, (c) the velocity of point C.
Fig. P15.201
(a)
The angular acceleration of the guide arm.
Answer to Problem 15.202P
The angular acceleration of the guide arm is
Explanation of Solution
Given information:
The velocity
Calculation:
Sketch the Free Body Diagram of the robotic guide arm as shown in Figure 1.
Refer to Figure 1.
The velocity at D along the x axis is
The velocity at A along the x axis is
Calculate the position vector
The position of A.
The position of B.
The position of C.
Consider the angular velocity
Consider the velocity at A
Calculate the angular velocity of each plane using the relation of
Substitute
Resolving i, j and k components.
For i component.
For k component.
Calculate the angular velocity of each plane using the relation of
Substitute
Substitute
Resolving i, j and k components.
For i component.
For j component.
For k component.
Calculate the velocity at A
Substitute
Calculate the angular velocity along y direction
Substitute
Calculate the angular velocity
Substitute
Calculate the velocity of point A
Substitute
Calculate the velocity of point C
Substitute
Consider the angular acceleration
Consider the acceleration at A
Consider the acceleration at B
Calculate the angular acceleration at B of each plane using the relation of
Substitute
Resolving i, j and k components.
For i component.
For j component.
For k component.
Calculate the angular acceleration at A of each plane using the relation of
Substitute
Substitute 0 for
Resolving i, j and k components.
For i component.
For j component.
For k component.
Refer to Figure 1.
The acceleration at A along the x axis
Substitute
Calculate the angular acceleration along y axis
Substitute
Calculate the angular acceleration
Substitute
Therefore, the angular acceleration of the guide arm is
(b)
Find the acceleration of point C.
Answer to Problem 15.202P
The acceleration of point C is
Explanation of Solution
Given information:
The velocity at B is
Calculation:
Refer to part (a).
The angular velocity of the guide arm is
The angular acceleration of the guide arm is
The velocity at C is
Calculate the acceleration at C
Substitute
Therefore, the acceleration of point C is
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