m2k1 m2k2 s 2 k1k2 0 in general the characteristic

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Unformatted text preview: gree-of-freedom system in the region around a resonance. This allows the single degree-of-freedom techniques previously developed, to be used to estimate the modal parameters for multi degree-of-freedom systems. Examining the principle characteristic of a lightly damped, single degree-of-freedom system at resonance: the magnitude peaks, the phase passes through −90° , the imaginary part peaks, and the real part passes through zero. These characteristics, in addition to the shape of the frequency response in the vicinity of the resonance, can be used to estimate the natural frequency and damping. P hase [deg] 180 90 0 -90 2 10 1.5 Im aginary part [m /N] Real part [m /N] 1 0.5 0 -0.5 -1 -1.5 1.5 10 1 10 M agnitude [m /N] 0 10 -1 1 -2 0.5 0 -0.5 -1 10 10 -3 10 -4 -1.5 0 1 2 3 4 5 6 Frequency [Hz] 7 8 9 10 0 1 2 3 4 5 6 Frequenc y [Hz ] 7 8 9 10 Specifically, recall the definition of resonance (excitation at the undamped natural frequency of the system). At resonance, the frequency response for a single degree-offreedom system reduces to H (ω ) = 1 . This is purely imaginary, implying that the jΩc response is 90° behind the forcing function. Therefore, the imaginary part of the frequency response is maximum and the real part is zero. Also, from section eight, the half-power method may be used to estimate the damping as ζ = ∆f . 2 fc Lecture Notes -63- 06/16/06 12:25 PM Mechanical Vibrations I Examining a multi degree-of-freedom frequency response, many of the same characteristics can be observed, in particular, the change of phase through resonance, the peak in the magnitude and imaginary parts, and the zero crossing in the real part. From these characteristics, in addition to the shape of the frequency response in the vicinity of the resonance, the natural frequency and damping can be estimated. But additionally, for multi degree-of-freedom systems, the mode shape can also be estimated. H 1 80 11 .. H 21 .. H 31 H 0.3 0.2 11 .. H 21 .. H 31 90 P has e [deg] 0.1 Imag [m/N] 0 2 4 6 8 10 12 0 0 -0 . 1 -0 . 2 -90 -1 80 0 2 4 6 8 10 12 10 0 0.3 H 11 21 31 H 0.2 0.1 Real [m/N] 0 -0 . 1 -0 . 2 H H 11 21 31 10 M agnitude [m /N] -1 H H 10 -2 10 -3 10 -4 0 2 4 6 F re qu e nc y [ H z ] 8 10 12 0 2 4 6 F re q u e n c y [ H z ] 8 10 12 While the mathematical form of a multi degree-of-freedom frequency response function is more complicated than for a single degree-of-freedom, it is still possible in many situations to reasonably apply single degree-of-freedom theory. Notice that there are several differences between a single and multi degree-of-freedom frequency response function. One important difference is that there is now more than one frequency response function for the system. Additionally, notice that there are now multiple peaks and the phase is no longer restricted to the range 0°, −180° . As a result, the phase at resonance can be ±90° . (It must still lose phase through resonance, though.) The reason that there are multiple frequency re...
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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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