Concept explainers
(a)
Show that in case of heavy damping
(a)
Explanation of Solution
Calculation:
Since
The expression for the differential equation of over damping as follows:
Differentiate the above equation with respect to ‘t’.
Since the body is released with no initial velocity.
Substitute 0 for t,
Substitute 0 for t,
Substitute
Substitute
Substitute
Apply boundary condition.
For
As
Thus the positive answer for the ‘t’ greater than 0 for the equation (4) cannot exist because the exponential (e) is increased to positive power be less than one which is not possible. Hence, the value of x is not becomes zero.
Show the graph of x versus t for the above solution as Figure (1).
(b)
Show that in case of heavy damping
(b)
Explanation of Solution
Calculation:
Since the body is started from O with arbitrary initial velocity.
Substitute 0 for t, 0 for x and
Substitute 0 for t, 0 for x, and
Substitute
Substitute
Substitute
Apply boundary condition.
For
For
Show the graph of x versus t for the above solution as Figure (2).
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Chapter 19 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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