Basic Mechanical Vibrations by A.J Pretlove (Auth.)

By A.J Pretlove (Auth.)

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The only snag is that these coordinates do not have a simple physical interpretation and may be difficult to find. 1 will now be found. 13) where a is an unknown constant. 1. Note that mode shape information is contained within the principal coordinates. 1 it is known that in the fundamental mode of vibration X1/X2 = 1 and hence y2 = 0; similarly, in the second mode of vibration Xi/X2 = — 2 and hence yr = 0. Thus, at a natural frequency of vibration only the principal coordinate corresponding to that frequency has a non-zero value.

Fourier series analysis 37 The points also correspond approximately to frequencies at which the phase lags are 45° and 135°. 5 Fourier series analysis A steady and continuous forcing of vibration may not necessarily be sinusoidal. It may be repetitive however, such as a square wave or a saw-tooth wave. If this is so then the forcing term can be expressed as a convergent series sum of sinusoids using the method of Fourier analysis. Because the system is linear the response may also be obtained as a sum of the responses to the individual sinusoids into which the force waveform has been decomposed.

16) so that X is the amplitude of the motion and φ is the phase lag of the motion with respect to the applied force. 17) is the classic resonance curve. 3. It is plotted as the Dynamic Magnification Factor (DMF) against the frequency ratio (ω/ωη). 17) the quantity F0/k is the deflection of the mass under the action of a static force of magnitude F0. Under the action of a dynamic force F0 the deflection amplitude is X. 19) The peak D M F occurs at a frequency of ω η ν /ΐ — 2ζ2 and this is called the resonance frequency.

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