1 s 2 f ms cs k starting from the equation while the

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Unformatted text preview: iscrete, sampled time data. H (ω ) = −ω 2 [ M ] + jω [C ] + [ K ] −1 By plotting both the Transfer Function and the Fourier Transform (over a limited range), the equivalence of both transforms can be shown. Lecture Notes -3506/16/06 12:25 PM Mechanical Vibrations I One interesting aspect of the difference, however, is that while the Transfer Function requires the plotting of two surfaces, the Frequency Response Function requires only two curves, clearly a much more convenient representation. 1 0.9 1 0.8 M agnitude M agnitude 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.6 0.4 0.2 0 10 5 0 -5 Om ega [rad/sec ] -10 -1 -0.5 S igm a [rad/s ec] 0.5 0 1 0.1 0 -10 -8 -6 -4 -2 0 2 Om ega [rad/s ec ] 4 6 8 10 4 3 4 2 2 1 P has e [rad] 0 P has e [rad] 0 -2 -1 -4 10 5 0 -5 Om ega [rad/sec ] -10 -1 -0.5 S igm a [rad/sec ] 0.5 0 1 -2 -3 -4 -10 -8 -6 -4 -2 0 2 Om ega [rad/s ec ] 4 6 8 10 Note that in addition to displacement per unit force input-output relationships, other steady-state input-output relationships can be developed. Some of these alternative relationships will be explored later as steady-state applications. Lecture Notes -36- 06/16/06 12:25 PM Mechanical Vibrations I #8 - The Frequency Response Function (FRF) In general, for most vibrations problems, the input/output relationship known as the Frequency Response Function (FRF) will be used. This function describes the input/output relationship on a frequency-by-frequency basis. Although not generally referred to as a Frequency Response Function, the effect of a graphic equalizer on the sound from a stereo system can be described by the frequency response function. In this case, the input is the raw audio and the output is the modified audio. By adjusting the amplification or attenuation (and relative phasing) of the various bands, the equalizer can give the resulting audio a variety of sound characteristics, from flat response to enhancing male or female voices to concert hall sound effects. Recall that the Frequency Response Function can be expressed by evaluating the Transfer Function at s = jω . H (ω ) X 1 (ω ) = 2 F −ω m + jω c + k Multi-Degree of Freedom Frequency Response Function Just as the Transfer Function has a multidegree of freedom form, so does the Frequency Response Function. By evaluating the multi-degree of freedom (matrix) Transfer Function at s = jω , the multi-degree of freedom Frequency Response Function results. H (ω ) = −ω 2 [ M ] + jω [C ] + [ K ] −1 By contrast with the Transfer Function, the Frequency Response Function is well suited for vibration applications, particularly experimental applications. While there is no discrete equivalent to the Laplace Transform, there is a discrete Fourier Transform that yields essentially the same information as the continuous integral transform. This is particularly important when working with discrete, sampled time data. Input-Output Model For the frequency response function, the following pictorial model is helpful. In fact, when used...
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This note was uploaded on 09/29/2013 for the course MECHANICAL ME taught by Professor Regalla during the Fall '11 term at Birla Institute of Technology & Science, Pilani - Hyderabad.

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