dynamics — Structural Dynamics Analysis

affect.dynamics

Summary

This module contains useful functions and postprocessors concerned with analyzing the behavior of physical structures when subjected to dynamic forces. This module is useful when the applied dynamic forces result in accelerations high enough to excite the structure’s natural frequency.

Dynamic analysis can be used to find dynamic displacements, time history, and modal analysis.

frf

Frequency Response Function

Frequency response is the quantitative measure of the output spectrum of a system or device in response to a stimulus, and is used to characterize the dynamics of the system. It is a measure of magnitude and phase of the output as a function of frequency, in comparison to the input. The frequency response function is a transfer function used to identify the resonant frequencies, damping and mode shapes of a physical structure.

\begin{gather*} \begin{split} & \text{Input Force} \\ & \quad F(ω) \end{split} & \longrightarrow & \begin{split} & \text{Transfer Function}\\ & \qquad H(ω) \end{split} & \longrightarrow & \begin{split} & \text{Displacement Response}\\ & \qquad\quad X(ω) \end{split} \end{gather*}

Here, \(F\) is the input force as a function of the frequency \(\omega\), and \(H\) is the transfer function, while \(X\) is the displacement (or velocity or acceleration) response function. Each function is a complex function, with real and imaginary components, which may also be represented in terms of magnitude and phase, and thus the functions are spectral functions. For sake of computation and simplicity, we consider each to be a Fourier transform.

Thus, in the frequency domain, the structural response X(ω) is usually expressed as the product of the frequency response function H(ω) and the input or applied force F(ω). Usually the response X(ω) may be in terms of displacement, velocity, or acceleration.

\[X(ω) = H(ω)⋅F(ω)\]
\[H(ω) = \frac{X(ω)}{F(ω)}\]

Using a frequency response function, the following can be observed:

  • Resonances - Peaks indicate the presence of the natural frequencies of the structure under test
  • Damping - Damping is proportional to the width of the peaks. The wider the peak, the heavier the damping
  • Mode Shape – The amplitude and phase of multiple FRFs acquired to a common reference on a structure are used to determine the mode shape

Nomenclature:

Various transfer functions are useful for measuring system response and these have common names:

Quantity Name of Frequency Response Function
displacement / force admittance, compliance, receptance
velocity / force mobility
acceleration / force accelerance, inertance
force / displacement dynamic stiffness
force / velocity mechanical impedance
force / acceleration apparent mass, dynamic mass

Example:

Examine the natural frequencies in the computational results of a structural model. Find the peak values of the accelerance frequency response function, where the response is acceleration in the z-direction given an input stimulus of force in the z-direction.

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from affect.exodus import DatabaseFile
from affect.dynamics import frf
from scipy.signal import argrelmax

with DatabaseFile('./SRS-FRF-example/model/1/p1f-out.h') as e:
    times = e.globals.times()
    num_times = times.size
    node_vars = e.nodal.variables
    az = e.nodal.variable_at_times(node_vars['AccZ'], 0, 0, num_times)
    fz = e.nodal.variable_at_times(node_vars['AForceZ'], 1, 0, num_times)

frequency, h_transfer = frf(fz, az, times)
peaks = argrelmax(h_transfer)
for i, j in enumerate(peaks):
    print(i, frequency[j], h_transfer[j])