Bracket — Frequency-Response Analysis

PDF

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 115 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

Figure 2 shows the root mean square of the displacement at the tip of the left arm of the bracket for both the pure harmonic load case and the combined harmonic and static load cases.

Figure 2: 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 115 Hz for the unloaded structure in both bracket arms and the frequency shift for the loaded structure. These results are in agreement with the values predicted by solving with the eigenfrequency solver.

You can also verify that the deformation remains small even around the resonance frequency.

Figure 3 shows the phase of the x-displacement at the tip of the bracket right arm.

Figure 3: Phase of x-displacement at the tip of the bracket right arm.

Note the smooth transition around the resonance frequency which results from the damping using a 5% loss factor. The prestressed load case solution shows a double phase change corresponding to the two vibration modes.

Notes About the COMSOL Implementation

For a structural mechanics physics interface in COMSOL Multiphysics, there are four predefined study types available for frequency response analysis: , , and and .

The modal analysis uses the modal solver to compute the frequency response. This analysis type significantly speeds up the computation when compared to the regular frequency domain analysis if the number of frequencies is large.

Use the prestressed frequency response analysis when a structure is subjected to both static and harmonic loads, and the stiffness induced by the static load case can affect the structural response to the harmonic load.

Structural_Mechanics_Module/Tutorials/bracket_frequency

Modeling Instructions

From the menu, choose .

Browse to the model’s Application Libraries folder and double-click the file bracket_basic.mph.

Solid Mechanics (solid)

Linear Elastic Material 1

In the window, expand the node, then click .

1

In the toolbar, click

and choose .

In the frequency domain you can use loss factor damping, viscous damping, or Rayleigh damping. For this example, use loss factor damping.

2

In the window for , locate the section.

3

From the list, choose .

Structural steel (mat1)

1

In the window, expand the node, then click .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Isotropic structural loss factor	eta_s	0.05	1	Basic

You can now apply an external harmonic load to the bracket arms.

Solid Mechanics (solid)

Boundary Load, Harmonic

1

In the toolbar, click

and choose .

2

In the window for , type Boundary Load, Harmonic in the text field.

3

Select Boundaries 4, 5, 75, and 76 only.

4

Locate the section. Specify the FA vector as


10[kPa]	x
0	y
0	z

To define a harmonic load in the frequency domain modal analysis, you need to mark the load as being a harmonic perturbation.

5

Right-click and choose .

1

In the toolbar, click

to open the window.

2

Go to the window.

3

Find the subsection. In the tree, select .

4

Click in the window toolbar.

5

In the toolbar, click

to close the window.

Step 1: Eigenfrequency

1

In the window for , locate the section.

2

Select the check box.

3

In the associated text field, type 2.

Step 2: Frequency Domain, Modal

The frequency range will be 50 Hz–190 Hz with a refined frequency sweep step between 100 Hz and 130 Hz.

1

In the window, click .

2

In the window for , locate the section.

3

In the text field, type 50 70 90 range(100,1,130) 150 170 190.

4

In the toolbar, click

.

1

button in the toolbar.

The default plot group shows the stress distribution on a deformed geometry for the final frequency. You can change the frequency for the plot evaluation in the list in the settings for the plot group.

Plot the root mean square of the displacement at the tip of the left arm of the bracket.

Displacement, RMS

1

In the toolbar, click

and choose .

2

In the window for , type Displacement, RMS in the text field.

3

Locate the section. Select the check box.

4

In the associated text field, type Frequency (Hz).

1

Right-click and choose .

2

Select Point 1 only.

3

In the window for , locate the section.

4

In the text field, type solid.disp_rms.

5

Click to expand the section. From the list, choose .

6

Find the subsection. From the list, choose .

7

From the list, choose .

8

Click to expand the section. Select the check box.

9

From the list, choose .

10

In the table, enter the following settings:


Legends
free harmonic, right arm

Displacement, RMS

1

In the window, click .

2

In the window for , click to expand the section.

3

From the list, choose .

1

Right-click and choose .

2

Select Point 109 only.

3

In the window for , locate the section.

4

In the text field, type solid.disp_rms.

5

Locate the section. From the list, choose .

6

Find the subsection. From the list, choose .

7

From the list, choose .

8

Locate the section. Select the check box.

9

From the list, choose .

10

In the table, enter the following settings:


Legends
free harmonic, left arm

11

In the toolbar, click

.

Now plot the phase shift with respect to the applied load at a specified point location.

1

In the toolbar, click

.

2

In the window for , locate the section.

3

In the text field, type 0.

4

In the text field, type -50e-3.

5

In the text field, type -50e-3.

6

Select the check box.

Displacement phase, X component

1

In the toolbar, click

.

2

In the window for , type Displacement phase, X component in the text field.

3

Locate the section. From the list, choose .

4

Locate the section. Select the check box.

5

In the associated text field, type Frequency (Hz).

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Click in the upper-right corner of the section. From the menu, choose .

5

Locate the section. From the list, choose .

6

Locate the section. Select the check box.

7

From the list, choose .

8

In the table, enter the following settings:


Legends
free harmonic

9

In the toolbar, click

.

In the solution dataset node, you can change the phase used to display the solution.

You will now consider a static load applied to the bracket and perform a prestressed frequency domain analysis.

1

In the toolbar, click

to open the window.

2

Go to the window.

3

Find the subsection. In the tree, select .

4

Click in the window toolbar.

5

In the toolbar, click

to close the window.

Global Definitions

1

In the window, under click .

2

In the window for , locate the section.

3

In the table, enter the following settings:


Name	Expression	Value	Description
P0	30[MPa]	3E7 Pa	Peak load intensity
YC	-300[mm]	-0.3 m	Y coordinate of hole center

Analytic 1 (an1)

1

In the toolbar, click

and choose .

2

In the window for , type load in the text field.

3

Locate the section. In the text field, type F*cos(atan2(py,abs(px))).

4

In the text field, type F, py, px.

5

Locate the section. In the text field, type Pa, m, m.

6

In the text field, type Pa.

Solid Mechanics (solid)

Boundary Load, Prestress

1

In the toolbar, click

and choose .

Apply a boundary load to the bracket holes.

2

In the window for , type Boundary Load, Prestress in the text field.

3

Locate the section. From the list, choose .

4

Locate the section. From the list, choose .

5

Locate the section. Specify the FA vector as


0	t1
0	t2
load(-P0,Z,Y-YC)(sign(X)(Y-YC)<0)	n

The default boundary system is in the deformed configuration. This would make the load behave as a follower load when used in a geometrically nonlinear context. Change to a fixed coordinate system.

Boundary System 1 (sys1)

1

In the window, under click .

2

In the window for , locate the section.

3

From the list, choose .

Step 2: Frequency Domain Perturbation

1

In the window, under click .

2

In the window for , locate the section.

3

In the text field, type 50 70 90 range(100,2,140) 150 170 190.

4

In the toolbar, click

.

1

button in the toolbar.

You have previously created a point graph plot for the unloaded case. Add a new point graph plot to the same figure but use the dataset of the second load case.

Point Graph 1, Point Graph 2

In the window, under , Ctrl-click to select and .

1

Right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. Find the subsection. From the list, choose .

5

Find the subsection. From the list, choose .

6

Locate the section. In the table, enter the following settings:


Legends
prestressed harmonic, right arm

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

4

Locate the section. Find the subsection. From the list, choose .

5

Find the subsection. From the list, choose .

6

Locate the section. In the table, enter the following settings:


Legends
prestressed harmonic, left arm

7

In the toolbar, click

.

1

In the window, under right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

1

In the window, under right-click and choose .

2

In the window for , locate the section.

3

From the list, choose .

4

In the toolbar, click

.

5

Locate the section. In the table, enter the following settings:


Legends
prestressed harmonic

6

In the toolbar, click

.

Stress (solid), prestressed

1

In the window, under click .

2

In the window for , type Stress (solid), prestressed in the text field.