Concept explainers
An automobile wheel-and-tire assembly of total weight 47 lb is attached to a mounting plate of negligible weight that is suspended from a steel wire. The torsional spring constant of the wire is known to be K = 0.40 lb·in/rad. The wheel is rotated through 90° about the vertical and then released. Knowing that the period of oscillation is observed to be 30 s, determine the centroidal mass moment of inertia and the centroidal radius of gyration of the wheel-and-tire assembly.
Fig. P19.159
The centroidal mass moment of inertia
Answer to Problem 19.159RP
The centroidal mass moment of inertia
Explanation of Solution
Given information:
The total weight of the automobile (W) is 47 lb.
The total spring constant of wire (K) is
The wheel rotates at an angle
The time period of oscillation
The acceleration due to gravity (g) is
Calculation:
Calculate the frequency of oscillation (f) using the formula:
Substitute 30 s for
Calculate the natural circular frequency of oscillation
Substitute 0.03333 Hz for f.
When an angular displacement of
Take moment for external forces as follows:
Here,
Restoring couple acts on the system due to the angular displacement of
Take moment for effective forces as follows;
Here,
Equate the moment for external and effective forces in the system using the relation:
The expression for the general differential equation of motion as follows:
Find the expression for the natural circular frequency of vibration:
Compare the differential equations (1) and (2).
Substitute
The expression for the centroidal mass moment of inertia as follows:
Here, m is the mass of the wheel-and-tire assembly and
Calculate the mass of the wheel-and-tire assembly (m) using the formula:
Substitute 47 lb for W and
Substitute
Therefore, the centroidal mass moment of inertia
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Chapter 19 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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