Concept explainers
A 4-lb uniform rod is supported by a pin at O and a spring at A and is connected to a dashpot at B. Determine (a) the differential equation of motion for small oscillations, (b) the angle that the rod will form with the horizontal 5 s after end B has been pushed 0.9 in. down and released.
Fig. P19.165
(a)
The differential equation of motion for small oscillations.
Answer to Problem 19.165RP
The differential equation of motion for small oscillations is
Explanation of Solution
Given information:
The weight of the uniform rod (W) is 4 lb.
The distance between A to O (a) is 6 inch.
The distance between O to B (b) is 18 in.
The spring constant (k) is 5 lb/ft.
The damping coefficient (c) is
The acceleration due to gravity (g) is
Calculation:
Calculate the mass of the uniform rod (m) using the formula:
Substitute 4 lb for W and
Show the free body diagram of the rod as Figure (1).
For small angle, take
Calculate the deflection at the point A
Substitute 6 in. for a.
Calculate the deflection at the point B
Substitute 18 in. for b.
Calculate the deflection at the point C
Substitute 6 in. for a.
Take the moment about O.
Calculate for the spring force
Substitute
Calculate the damping force
Substitute
Calculate the moment of inertia
Substitute 6 in. for a and 18 in. for b.
The angle
Calculate the acceleration
Substitute 6 in. for a and
Substitute
Consider equilibrium.
Take the moment about O.
Substitute 2 for
Substitute
Therefore, the differential equation of motion for small oscillations is
(b)
The angle that the rod
Answer to Problem 19.165RP
The angle that the rod
Explanation of Solution
Given information:
The weight of the uniform rod (W) is 4 lb.
The distance between A to O (a) is 6 inch.
The distance between O to B (b) is 18 inch.
The spring constant (k) is 5 lb/ft.
The damping coefficient (c) is
The acceleration due to gravity (g) is
Calculation:
Consider the equation (3).
Substitute
Solve the above equation.
Since the computed roots are real and distinct.
Write the expression for the general solution for the differential equation as follows:
Substitute 0 for t.
Differentiate the equation (4) with respect to time ‘t’.
Substitute 0 for t.
The above solution corresponds to a no vibratory motion because the roots
Therefore, the angle that the rod
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