Concept explainers
The 9-kg uniform rod AB is attached to springs at A and B, each of constant 850 N/m, which can act in both tension and compression. If end A of the rod is depressed slightly and released, determine (a) the frequency of vibration, (b) the amplitude of the angular motion of the rod, knowing that the maximum velocity of point A is 1.1 mm/s.
Fig. P19.37
(a)
The frequency
Answer to Problem 19.37P
The frequency
Explanation of Solution
Given information:
The mass (m) of the uniform rod AB is 9 kg.
The spring constant of spring (k) A and B is
The length (l) is 600mm.
Calculation:
Show the free body diagram of the rod AB as in Figure (1):
Write the expression for the force at A
Here, k is the spring constant and
Write the expression for the force at B
Here,
The external forces in the system are spring force and force due to the mass of the rod. The effective force in the system is
Taking moment about C,
Substitute
Take moment about C in the system for effective forces.
Equate the moment about C in the system for external and effective forces
At the equilibrium position
Substitute 0 for
Substitute equation (3) in equation (2),
Write the expression for moment of inertia
Write the expression for the tangential acceleration
Calculate the natural circular frequency
Substitute
Substitute 9 kg for m and 600 mm for l.
Divided the above equation by
Divide the above equation by 0.09333.
Substitute
Compare the differential Equation (5) with the general differential equation of motion
Calculate the frequency
Substitute
Therefore, the frequency
(b)
The amplitude
Answer to Problem 19.37P
The amplitude
Explanation of Solution
Given information:
The mass (m) of the uniform rod AB is 9 kg.
The spring constant of spring (k) A and B is
The length (l) is 600mm.
The maximum velocity
Calculation:
Write the expression for angular deflection:
At time (t) is zero the deflection
Substitute 0 for t and 0 for
Differentiate the equation (6).
Substitute 0 for t and 0 for
Write the express the maximum velocity
Substitute
Calculate the angular amplitude
Substitute
Therefore, the amplitude
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Chapter 19 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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