Bracket — General Periodic Dynamic Analysis

The steady-state response of a system subjected to a non-harmonic periodic excitation can be computed using two different approaches:

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The most straightforward is a time-dependent analysis in which a large number of cycles is computed so that any traces of startup transients fade away. For structures with a low damping, a fairly high number of cycles may be needed so the computational cost can be high.

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An alternative, far more efficient approach, is to represent the forcing function by a Fourier series. It is then possible to compute a sequence of frequency domain solutions, one for each term in the series. In order to get the results back into the time domain, the results for each harmonic has to be superimposed over the time of a full period.

In this example, you learn how to compute the Fourier series coefficients of the periodic forcing function, and how to use them to perform a frequency response analysis. The results from the frequency response analysis is used as input to an inverse FFT analysis to get the steady-state time-varying response for a complete period. In order to validate the results from this approach, a time-dependent analysis is performed.

In both cases, a full or a modal solution scheme can be used. Compared to full time- dependent or frequency-domain methods, modal methods offer advantages with a reduced problem size if a limited number of eigenmode is excited. By representing the dynamics of the system by a few significant eigenmodes, the modal method reduces the size of problem. In this example, modal solution methods are used in both cases.

It is recommended that you review the Introduction to the Structural Mechanics Module, which includes background information and discusses the bracket_basic.mph model relevant to this example.

Model Definition

This model is an extension of the model example described in the section “The Fundamentals: A Static Linear Analysis” in the Introduction to the Structural Mechanics Module.

The geometry is shown in Figure 1.

Figure 1: Bracket geometry

The time varying periodic load is a force applied in the X direction at the bracket holes. It consists of a triangular pulse with an amplitude of 750 N amplitude and zero mean, varying with 40 Hz frequency as shown in Figure 2.

Figure 2: The load history

Rayleigh damping is chosen since it is applicable both in frequency and time domains. The relative damping is set to 0.03 at the frequencies 100 Hz and 300 Hz.

An eigenfrequency analysis of this structure is performed in the tutorial example Bracket — Eigenfrequency Analysis. It shows that the first resonance frequency is about 115 Hz. The fundamental frequency of the load, having the frequency 40 Hz, will thus mainly excite the first mode of the bracket.

Fourier Series

A periodic function F(t) with period T0 (and corresponding angular frequency ω0) can be decomposed into a discrete Fourier series of the form

(1)

where

(2)

The periodic load in this model is a triangular function which is an odd function with zero mean. The Fourier series coefficients can be shown to be

(3)

Here A is the amplitude of the triangular function. For even values of n, the coefficients Fbn are zero.

Alternatively, the trigonometric functions and Fourier coefficients can be expressed on complex form:

(4)

where Fn are complex valued. This is the notation used in COMSOL Multiphysics. The relation between the Fourier series coefficients in the two formulations are

(5)

Results and Discussion

Since the Fourier coefficients for the given load can be determined analytically, the accuracy of the computed values can be investigated. In Figure 3, the Fourier series coefficients are plotted against the frequency.

Figure 3: Numerical and analytical values of Fourier series coefficients.

As can be seen, the computed values match the values given by Equation 3 very well.

Another way of investigating the accuracy of the computed Fourier series coefficients is to compare the real and imaginary values. Since, for this loading function, all coefficients are purely imaginary, the real parts should be small. This comparison is shown in Figure 4. It can be seen that the error increases with the order of the term, but the accuracy is still good up to 2000–3000 Hz. The accuracy is improved by computing a higher number of terms, that is the setting in the study step.

Figure 4: Real and imaginary parts of the Fourier series coefficients.

In Figure 5 and Figure 6, the von Mises stress at a certain time in the period is plotted for the two different solution methods. The results are very similar. There are two different possible sources of inaccuracy:

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The accuracy in the computation of Fourier series coefficients.

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Errors in the time integration, in particular the number of periods used for letting startup transients fade away. In this case, 21 periods are computed.

In most cases, the time-stepping algorithm will be the larger source of error.

Figure 5: Equivalent stress at t=0.022 using the general periodic approach.

Figure 6: Equivalent stress at t=0.022 using a time-dependent modal approach.

In Figure 7, the contributions to the displacement from each harmonic is plotted. This is the amplitude for each frequency in the frequency sweep. It can be seen that it is mainly the Fourier terms at 40 Hz and 120 Hz that contribute. Thus, it is only some of the few terms with the very best accuracy that are dominating.

Figure 7: Amplitude contribution from each harmonic to the displacement in the X-direction at the tip of the bracket.

In Figure 8, the displacement at the tip of the bracket is shown for one period. The two different solution methods are compared, and the results are almost indistinguishable.

In Figure 9, a similar graph for the σxx stress component at a point in the highly stressed fillet is shown.

In both graphs the effect of the two dominating frequencies 40 Hz and 120 Hz is clearly visible. The frequency content of the excitation is filtered by the dynamics properties of the structure, giving a response that bears little similarity with the excitation.