Concept explainers
In the planetary gear system shown, gears A and B are rigidly connected to each other and rotate as a unit about the inclined shaft. Gears C and D rotate with constant angular velocities of 30 rad/s and 20 rad/s, respectively (both counterclockwise when viewed from the right). Choosing the x axis to the right, the y axis upward, and the z axis pointing out of the plane of the figure, determine (a) the common angular velocity of gears A and B, (b) the angular velocity of shaft FH, which is rigidly attached to the inclined shaft.
(a)
The common angular velocity of the gears
Answer to Problem 15.199P
The common angular velocity of the gears
Explanation of Solution
Given Information:
The angular velocity of the gear
Draw the schematic diagram of the given system.
Figure-(1)
Write the expression for the velocity of point 1.
Here, the angular velocity of the gear
Write the expression for the velocity of point 1 when gear
Here, the angular velocity of the shaft unit carries gear
Write the expression for the velocity of point 2.
Here, the angular velocity of the gear
Write the expression for the velocity of point 2 when gear
Write the expression for the angular velocity of the inclined shaft unit which carries gear
Here, the angular velocity of the shaft in x- direction is
Calculation:
Consider the unit vector along
From Figure-(1) the coordinate of the point 1 at the intersection of the gears
From Figure-(1) the coordinate of the point 2 at the intersection of the gears
Since, the shaft
Substitute
Substitute
Substitute
Compare the terms along the x- direction in Equation (VII).
Compare the terms along the z- direction in Equation (VII).
Since, the shaft
Substitute
Substitute
Substitute
Compare the terms along the z- direction in Equation (X).
Substitute
Substitute
Substitute
Conclusion:
The common angular velocity of the gear
(b)
The angular velocity of the shaft
Answer to Problem 15.199P
The angular velocity of the shaft
Explanation of Solution
Draw the diagram to show the motion of the shaft is
Figure-(2)
Write the expression for the velocity of point
Here, the position vector at point
Write the expression for the velocity at point
Here, the angular velocity of shaft
Calculation:
From Figure-(2) the coordinate of the point
Substitute
Substitute
Substitute
Conclusion:
The angular velocity of the shaft
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Chapter 15 Solutions
Vector Mechanics for Engineers: Dynamics
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