Bracket — Frequency-Response Analysis

A frequency response analysis solves for the linear steady-state response of a structure when subjected to harmonic loads. The problem is solved in the frequency domain and you can set a range of frequencies at which to compute the structural response.

In this example you learn how to perform a frequency response analysis of a structure under harmonic loads, but also how to perform a frequency response analysis of a prestressed structure.

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

You study two load cases. In the first case, a harmonic load in the X-direction, with a an amplitude of 10 kPa, is applied to the boundaries of the bracket holes. The second load case consists of a combination of a static preload and the same harmonic perturbation.

An eigenfrequency analysis of this structure is performed in the tutorial Bracket — Eigenfrequency Analysis. It shows that the first resonance frequency is about 114 Hz. For the prestressed case, the eigenfrequency solution shows that the first resonance frequency is about 107 Hz when the arm is under a compressive load, and about 128 Hz when the arm is under a tensile load. In order to capture the resonance peaks properly, you can refine the frequency stepping around these values.

Results and Discussion

The default plot in a frequency-domain analysis shows the variable <phys>.mises_peak. This is a special variable that, in each point, contains the maximum von Mises stress over the whole cycle. The standard von Mises stress variable, <phys>.mises, contains the stress at the current phase angle. This may be far from the peak stress, if there are significant phase shifts. In Figure 2, the stress at the last computed frequency, 200 Hz is shown. More interesting is to look at the results at 114 Hz at which the first natural frequency is located. This is shown in Figure 3. Here, the peak value is 110 MPa, to be compared with 3 MPa in the previous case.

Figure 2: von Mises stress at 200 Hz.

Figure 3: von Mises stress at 114 Hz.

Figure 4 shows the root mean square of the displacement at the tip of the arms of the bracket for both the pure harmonic load case and the combined harmonic and static load cases.

Figure 4: Root mean square of the displacement at the tip of the left (red) and right (blue) arms for both pure harmonic loaded case (solid) and a combined static and harmonic loaded case (dashed).

The curves show resonance peaks around 114 Hz for the unloaded structure in both bracket arms and a frequency shift for the loaded structure. These results are in agreement with the values predicted by solving with the eigenfrequency solver. The curves for the left and right arms coincide as long as there is no prestress.

You can also verify that the deformation remains small even around the resonance frequency. Thus, the linearity assumption within the frequency-domain studies is fulfilled.

Figure 5 shows the phase of the x-displacement at the tips of both arms.

Figure 5: Phase of x-displacement at the tip of the bracket right arm.

Note the smooth transition where the displacement is in phase with the load at lower frequencies and in counterphase for higher frequencies. This is an effect of the damping using a 5% loss factor. The prestressed load case solution shows interesting properties where the phase flips at different frequencies in each arm. This can be interpreted so that the two arms move synchronously for low an d high frequencies, but against each other for intermediate frequencies.

In Figure 6 and Figure 7, the perturbation of the von Mises stress is shown at 107 Hz and 128 Hz. This result is the linearized deviation from the constant stress caused by the static preload, and thus the values are both positive and negative. Each arm dominates the response close to its own eigenfrequency.