1 Time-Domain Analysis Clarence W. de Silva

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The University of British Columbia

1.1 Introduction............................................................................. 1-1

1.2 Undamped Oscillator .............................................................. 1-2

Energy Storage Elements † The Method of Conservation of

Energy † Free Response

1.3 Heavy Springs ........................................................................ 1-12

Kinetic Energy Equivalence

1.4 Oscillations in Fluid Systems................................................ 1-14

1.5 Damped Simple Oscillator.................................................... 1-16

Case 1: Underdamped Motion ðz , 1Þ † Logarithmic

Decrement Method † Case 2: Overdamped Motion

ðz . 1Þ † Case 3: Critically Damped Motion

ðz ¼ 1Þ † Justification for the Trial Solution † Stability and

Speed of Response

1.6 Forced Response .................................................................... 1-27

Impulse-Response Function † Forced Response †

Response to a Support Motion

Summary

This chapter concerns the modeling and analysis of mechanical vibrating systems in the time domain. Useful

procedures of time-domain analysis are developed. Techniques of analyzing both free response and forced

response are given. Examples of the application of Newton’s force– motion approach, energy conservation

approach, and Lagrange’s energy approach are given. Particular emphasis is placed on the analysis of an

undamped oscillator and a damped oscillator. Associated terminology, which is useful in vibration analysis, is

defined.

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