Frequency-Domain Analysis
In a frequency-domain analysis, you study the response to a harmonic steady state excitation for certain frequencies. Such a steady state can prevail once all transient effects have been damped out.
The response must be linear, so that the single frequency harmonic excitation gives a pure harmonic response with the same frequency. The model may, however, contain nonlinearities. The harmonic response is computed around a certain linearization point. In such a case, the frequency-domain analysis can be considered as a very small perturbation around that linearization point.
Harmonic Perturbation
All loads and responses in a frequency-domain analysis are in general complex-valued quantities. If all loads do not have the same phase, you can describe the phase of a certain load in two ways:
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Add a Phase subnode to the load, in which you directly give the phase angle.
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Enter the load as a complex value, for example as
100[N]*(1+0.3*i)/sqrt(1+0.3^2).
Most results from a frequency domain analysis are complex-valued. In many results evaluation nodes, the real value of any result quantity will be shown. Assuming that you want to display for example the displacement in the x direction, u, you have the following options:
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Plot u or real(u). This gives the displacement at current (default zero) phase angle.
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Plot imag(u). This gives the displacement at a phase angle shifted 90 degrees from the current value.
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Plot abs(u). This gives the amplitude of the displacement.
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Plot arg(u). This gives the phase angle of the displacement.
The reference phase, with respect to which the results above are reported can be entered in the settings for the dataset.
Result quantities that are nonlinear in terms of the displacements, such as principal stresses, should be interpreted with great care in frequency domain. They will in general not be harmonic, so the information about amplitude and phase is not reliable.
Some extra variables for postprocessing are created in a frequency-domain analysis. As an example, in a Solid Mechanics interface with the name solid, the following variables are defined:
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solid.disp — norm of displacement (at current phase angle)
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solid.vel — norm of velocity (at current phase angle)
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solid.acc — norm of acceleration (at current phase angle)
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solid.disp_rms — RMS displacement over a cycle
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solid.vel_rms — RMS velocity over a cycle
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solid.acc_rms — RMS acceleration over a cycle
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solid.uAmpX — amplitude of displacement in the X direction
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solid.uAmp_tX — amplitude of velocity in the X direction
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solid.uAmp_ttX — amplitude of acceleration in the X direction
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solid.uPhaseX — phase of X displacement, in radians
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solid.uPhase_tX — phase of X velocity, in radians
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solid.uPhase_ttX — phase of X displacement, in radians
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solid.mises — von Mises equivalent stress at current phase angle.
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solid.mises_peak — maximum von Mises equivalent stress over a cycle.
The components in other coordinate directions are obtained by replacing X by another coordinate name.
Prestressed Analysis
The shift in the natural frequencies in a prestressed structure may have a significant effect on the frequency response. This is particularly important when the frequencies of the load are close to any of the natural frequencies of the structure.
To do a prestressed analysis, Include geometric nonlinearity must be selected in the Frequency Domain study step. This is automatic when you add the Frequency Domain, Prestressed study type.
The prestress loading can include a contact analysis, in which case the subsequent frequency domain analysis provides a linearization around the current contact state.
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Prestressed Structures
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Harmonic Perturbation
Obtaining a Time History
Sometimes you want to study the time history over a period for the results of a frequency domain analysis. You can do that by adding a Frequency to Time FFT study step. The frequency response results are then viewed as terms in a Fourier series, which can be transformed to time domain. It is possible combine the results for several frequencies into a single time history, under the assumption that they are all multiples of the same fundamental frequency.
For examples showing how to obtain a time history from frequency domain results, see
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Viscoelastic Structural Damper: Application Library path Structural_Mechanics_Module/Dynamics_and_Vibration/viscoelastic_damper_frequency.
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Vibration Analysis of a Deep Beam: Application Library path Structural_Mechanics_Module/Verification_Examples/vibrating_deep_beam.