Nonlinear Harmonic Response

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Nonlinear Harmonic Response

This tutorial shows how to evaluate the harmonic response of a structure with a moderately nonlinear behavior. To solve such a nonlinear problem accurately, it is necessary to use a time-domain analysis. Solving in time domain can, however, be time consuming as it requires several periods until a steady-state solution is reached. A way to speed up such an analysis is to use a linearized frequency response analysis to provide good initial conditions. This approach is described in this tutorial example.

Model Definition

The model consists of a circular membrane, supported along its outer edge.

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Membrane radius, R = 0.25 m

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Membrane thickness, h = 0.2 mm

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Young’s modulus, E = 200 GPa

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Poisson’s ratio, ν = 0.33

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Mass density, ρ = 7850 kg/m3

The outer edge of the membrane is supported in the transverse direction. Two points have constraints in the in-plane direction in order to avoid rigid body motions.

The membrane is pretensioned by in the radial direction with σi = 100 MPa, giving a membrane force T0 = 20 kN/m.

A harmonically varying pressure with a 50 Hz frequency and a magnitude of F0 = 10 kPa is applied on the membrrane.

Results and Discussion

Figure 1 shows the vertical displacement at the disk center. The peak displacement is about 6.8 mm, which is significantly less than for the linearized solution (about 8.6 mm). The steady state is reached within a few periods, thanks to the initial condition using the linearized frequency-domain solution. This way, the analysis time is significantly reduced.

Figure 1: Vertical displacement of the disk center versus time near steady state.

For comparison, Figure 2 shows the same vertical displacement at the disk center, but this time computed using the default initial condition. While the same peak displacement value is obtained, it requires more than 10 periods to reach the steady state.

Figure 2: Vertical displacement of the disk center versus time (computed from zero displacement/velocity).

In Figure 3 the displacement at t = 0.075 s, corresponding to the peak deformation of the membrane, is shown.

Figure 3: Vertical displacement from time domain solution.

For comparison, Figure 4 shows the displacement from the frequency domain solution.

Figure 4: Vertical displacement from frequency domain solution.

Notes About the COMSOL Implementation

To save computational time to reach a steady-state solution using a periodic time-domain analysis, the initial conditions must be carefully selected. For a structural dynamics problem you need to set both the displacement and velocity fields at the initial time. The solution from a linearized harmonic response is a good choice and easy to obtain.

In COMSOL Multiphysics, you can use the withsol() operator to evaluate expression from a different solution. The first argument of the operator has to be the tag of the solution to use and the second argument the expression to evaluate.

Table 1 shows the expressions used as the initial condition for the displacement field.

Table 1: Initial conditions for the displacement field.
withsol('sol1',real(u))	X
withsol('sol1',real(u))	Y
withsol('sol1',real(u))	Z

Similarly, Table 2 shows the initial structural velocity field.

Table 2: Initial conditions for the structural velocity field.
withsol('sol1',real(mbrn.u_tX))	X
withsol('sol1',real(mbrn.u_tY))	Y
withsol('sol1',real(mbrn.u_tZ))	Z

Structural_Mechanics_Module/Tutorials/nonlinear_harmonic

Modeling Instructions

From the menu, choose .

In the window, click

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In the window, click

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In the tree, select .

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4

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In the tree, select .

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Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
R	250[mm]	0.25 m	Radius
th	0.2[mm]	2E-4 m	Thickness
T0	100[MPa]*th	20000 N/m	Pretension force
F0	10[kPa]	10000 Pa	Excitation load magnitude
freq0	50[Hz]	50 Hz	Excitation frequency

Cylindrical System 2 (sys2)

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In the window, expand the node.

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Right-click and choose .

Work Plane 1 (wp1)

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In the toolbar, click

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In the window, click .

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In the window for , click

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Work Plane 1 (wp1)>Circle 1 (c1)

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In the toolbar, click

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In the window for , locate the section.

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In the text field, type R.

Work Plane 1 (wp1)>Point 1 (pt1)

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In the toolbar, click

.

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In the window, right-click and choose .

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button in the toolbar.

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In the toolbar, click

to open the window.

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Go to the window.

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In the tree, select .

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Right-click and choose .

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In the toolbar, click

to close the window.

Membrane (mbrn)

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In the window, right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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In the f1 text field, type 170.

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In the ζ1 text field, type 2e-2.

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In the f2 text field, type 400.

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In the ζ2 text field, type 4e-2.

Thickness and Offset 1

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In the window, under click .

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In the window for , locate the section.

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In the d0 text field, type th.

Prescribed Displacement 1

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Locate the section. Select the check box.

Prescribed Displacement 2

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In the toolbar, click

and choose .

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Select Point 3 only.

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In the window for , locate the section.

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Select the check box.

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Select the check box.

Prescribed Displacement 3

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In the toolbar, click

and choose .

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Select Point 1 only.

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In the window for , locate the section.

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From the list, choose .

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Locate the section. Select the check box.

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In the toolbar, click

and choose .

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In the window for , locate the section.

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From the list, choose .

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Locate the section. From the list, choose .

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Locate the section. Specify the FL vector as


T0	r
0	phi
0	a

Add a spring with an arbitrary, small stiffness in order to suppress the out-of-plane singularity of the unstressed membrane.

Spring Foundation 1

1

In the toolbar, click

and choose .

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Select Boundary 1 only.

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In the window for , locate the section.

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From the list, choose .

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In the kA table, enter the following settings:


0	0	0
0	0	0
0	0	10

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In the toolbar, click

and choose .

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Select Boundary 1 only.

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In the window for , locate the section.

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From the list, choose .

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In the p text field, type F0.

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Right-click and choose .

In a frequency domain analysis, the zero phase corresponds to cos(time). Change the excitation load phase so that it it represents sin(time) since such a load will be used in the time-domain analysis.

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In the toolbar, click

and choose .

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In the window for , locate the section.

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In the φ text field, type -pi/2.

Step 2: Frequency Domain Perturbation

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In the window, under click .

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In the window for , locate the section.

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In the text field, type freq0.

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Locate the section. Select the check box.

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In the tree, select .

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Right-click and choose .

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In the window, click .

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In the window for , locate the section.

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Clear the check box.

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In the toolbar, click

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Add Predefined Plot

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Go to the window.

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In the tree, select .

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Click in the window toolbar.

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In the toolbar, click

.

Displacement - Frequency Domain

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In the window, under click .

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In the window for , type Displacement - Frequency Domain in the text field.

Study 1/Solution 1 (sol1)

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type -90.

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type w.

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From the list, choose .

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In the toolbar, click

.

Point Evaluation 1

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In the toolbar, click

.

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Select Point 3 only.

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In the window for , locate the section.

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In the table, enter the following expression and unit:


Expression	Unit	Description
w	mm	Displacement field, Z-component

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From the list, choose .

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Membrane (mbrn)

The next step is to set initial conditions using the solution computed by the frequency response analysis.

Initial Values 2

1

In the toolbar, click

and choose .

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Select Boundary 1 only.

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In the window for , locate the section.

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Specify the u vector as


withsol('sol1',real(u))	X
withsol('sol1',real(v))	Y
withsol('sol1',real(w))	Z

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Specify the Structural velocity field as:


withsol(’sol1’,real(mbrn.u_tX))	X
withsol(’sol1’,real(mbrn.u_tY))	Y
withsol(’sol1’,real(mbrn.u_tZ))	Z

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In the toolbar, click

and choose .

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Select Boundary 1 only.

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In the window for , locate the section.

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From the list, choose .

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In the p text field, type sin(2*pi*t*freq0)*F0.

Point Probe 1 (point1)

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In the toolbar, click

and choose .

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In the window for , locate the section.

3

.

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Select Point 3 only.

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Locate the section. In the text field, type w.

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From the list, choose .

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In the toolbar, click

to open the window.

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Go to the window.

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Find the subsection. In the tree, select .

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Right-click and choose .

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In the toolbar, click

to close the window.

Step 1: Time Dependent

With good initial conditions you only need to compute for a few periods to reach steady state.

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In the window for , locate the section.

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In the text field, type 0 range(3,1/20,4)/freq0.

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Select the check box.

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Locate the section. Select the check box.

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In the tree, select .

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Right-click and choose .

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In the tree, select .

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Right-click and choose .

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In the window, click .

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In the window for , locate the section.

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Clear the check box.

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In the toolbar, click

.

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Add Predefined Plot

1

Go to the window.

2

In the tree, select .

3

Click in the window toolbar.

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In the toolbar, click

.

Displacement - Time Domain

1

In the window, under click .

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In the window for , type Displacement - Time Domain in the text field.

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Locate the section. From the list, choose .

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type w.

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From the list, choose .

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In the toolbar, click

.

1

In the window, under click .

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

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

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In the text area, type Vertical displacement at disk center.

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Locate the section.

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Select the check box. In the associated text field, type Displacement (mm).

Probe Table Graph 1

1

In the window, expand the node, then click .

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In the window for , locate the section.

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From the list, choose .

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Click to expand the section. Clear the check box.

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In the toolbar, click

.

The modeling part is now finished. Follow the steps below if you want to run the model later with modified parameter values.

Step 1: Stationary

1

In the window, under click .

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In the window for , locate the section.

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Select the check box.

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In the tree, select , , and .

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Right-click and choose .

Step 2: Frequency Domain Perturbation

1

In the window, click .

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In the window for , locate the section.

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In the tree, select .

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Right-click and choose .

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In the tree, select .

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Right-click and choose .