Concept explainers
(a)
The time interval between a maximum positive displacement and the following maximum negative displacement is
Explanation of Solution
Given information:
The system is underdamped, the time period between two successive points is
The figure shows the underdamped system.
Figure-(1)
Write the expression amplitude of vibrations of an underdamped system..
Here, the amplitude of an underdamped system is
Differentiate Equation (I) with respect to time.
For maximum positive and negative displacement,
The maximum negative displacement occurs after a phase difference of
Calculation:
Substitute
Here, the time interval for maximum positive displacement is
Substitute
Here, the time interval for maximum negative displacement is
Substract Equation (V) and (VI).
Substitute
Conclusion:
The time interval between a maximum positive displacement and the following maximum negative displacement is
(b)
The time interval between two successive zero displacement is
Explanation of Solution
Calculation:
Substitute
Substitute
Here, the time interval for first zero displacement is
Substitute
Here, the time interval for second zero displacement is
Substract Equation (X) from (IX).
Substitute
Conclusion:
The time interval between two successive zero displacement is
(c)
The time interval between maximum positive displacement and zero displacement is greater than
Explanation of Solution
Calculation:
Substract Equation (V) from (IX).
The values of
Substitute
Neglect the time interval difference to be negative.
Substitute
Conclusion:
The time interval between maximum positive displacement and zero displacement is greater than
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Chapter 19 Solutions
Vector Mechanics For Engineers
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