âComputational Civil Engineering - "Intersections" International Journal

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“Computational Civil Engineering 2005”, International Symposium 2313 g/NINERTANCE2 g/Na1 g/Na/√100 105 110 115 120125? 1? 0? 2130FREQUENCY (Hz)Fig. 4. Third resonance plotted on enlarged linear scalesIn the calculation of the damping by this method, it is assumed that each peakrepresents motion in only one mode, i.e. the contribution from the off-resonantmodes is negligible. For resonances that are well separated, however, sufficientlyaccurate estimates of damping can be achieved, provided η < 0, 1 . If the offresonantvibration is not negligible, but however, constant, it may appear at firstthat the off-resonant vibration could be extracted from the measured amplitude.Unfortunately, this cannot be done directly, as accelerations are vector quantitiesand phase angles of the responses have to be taken into consideration. Thus theeffect of the off-resonant vibration will be different above and below resonance ofthe mode under consideration, resulting in a non-symmetrical peak [5], [6].To determine the mode shape, the superstructure of bridge is excited at a naturalfrequency, and the ratios of the amplitudes at various points on the structure aredetermined (fig. 5). As the acceleration goes through a phase change of 180 0 as onepasses through a node, the relative phase of the accelerations at various pointsshould also be taken into account.Since the beam has uniform cross-section, the mode shape ideally be symmetrical;however, the asymmetry of the mode shape is evident. This is because more thatone mode is represented in general, at any one frequency, and therefore the modeshape is not a true principal mode. In other words, the phase angle between theforce and the acceleration would not be exactly 90 0 at all points on thesuperstructure.
232 C. C. ComisuFREE - FREE BEAMFIRST MODESECOND MODETHIRD MODEFig. 5. Mode shape for the third natural frequency3.2. Contribution from Off-Resonant ModesThe response of a continuous structure (multiple degree of freedomsystem) can be represented by the superposition of the responses in its individualmodes, considering each mode to responses as a single degree of freedom system.Differences between the resonance characteristics of the response of a singledegree of freedom system and those of a multiple degree of freedom system are,however, manifested by the contribution of the off-resonant modes to the mode thatis excited.These responses can be derived from equation of motion for a singledegree of freedom system with structural (hysteretic) damping [5]:or••γ ⋅ km ⋅ x+ω•j⋅ω⋅tx+k ⋅ x = F ⋅ ewhere( + j ⋅γ )••m ⋅ x+kj⋅ω⋅t( 1 + j ⋅γ) ⋅ x = F ⋅ ek 1 - is called the complex stiffnessγ - is called the structural damping factor.The steady state solution is given by:
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COMPUTATIONAL CIVIL ENGINEERING2005
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40 F.-Zs. Gobesz, D. Turdamany info
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