Loaded Spring — Using Global Equations to Satisfy Constraints

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Loaded Spring — Using Global Equations to Satisfy Constraints

In this tutorial model, which demonstrates a more generally applicable method, a structural mechanics model of a spring is augmented by a global equation that solves for the load required to achieve a desired total extension of the spring.

Figure 1: A 4.5-turn steel spring is fixed at one end, and has a load applied at the other. The load is a variable which is solved for to achieve a total displacement.

Model Definition

Figure 1 shows the modeled 4.5-turn steel spring. One end of the spring is fixed rigidly, and the other end has a distributed load applied to it, acting in the axial direction of the spring. Rather than an input to the model, this load is a variable being solved for; it is implicitly specified via a global equation in such a way as to give a total spring extension of 2 cm. The extension of the spring is computed by using an average operator on the moving end of the spring. The average operator evaluates the average z-displacement over the boundary at which the load is applied.

The global equation adds one additional degree of freedom to the model, the unknown load. Not all available equations solvers are suited for such problems, but the direct solver used as default for structural mechanics can handle it. Because the structure has a uniform cross section, use a swept mesh.

Results and Discussion

Figure 2 shows the deformed shape of the spring. The average displacement of the end of the spring is 2 cm, as specified by the global equation. The force required to get this displacement is 471 N. Although this problem uses a linear elastic material model, this approach would work equally well if the material model was nonlinear or if geometric nonlinearity was taken into account.

Global equations do have certain restrictions upon their usage. The global equation must be continuous and differentiable with respect to all of the unknowns, and it must not overconstrain, nor underconstrain, the problem. Each global equation should add one constraint and one degree of freedom to the model. Under these conditions, the global equations can be used in a variety of ways beyond what is shown here.

Figure 2: The deformed shape of the spring.

COMSOL_Multiphysics/Structural_Mechanics/loaded_spring

Modeling Instructions

From the menu, choose .

In the window, click

.

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In the window, click

.

2

In the tree, select .

3

4

5

In the tree, select .

6

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
dh	2[cm]	0.02 m	Prescribed extension

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In the window, under click .

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In the window for , locate the section.

3

From the list, choose .

Create a helix for the spring (Figure 1).

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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 4.5.

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Locate the section. In the text field, type 180.

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.

Next, add an operator that you will later use to average the z-directional displacement field on the end of the spring.

Average 1 (aveop1)

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In the toolbar, click

and choose .

Choose wireframe rendering to get a better view on some boundaries where you will assign boundary conditions.

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button in the toolbar.

3

In the window for , locate the section.

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From the list, choose .

5

Select Boundary 6 only.

Solid Mechanics (solid)

Next, set up the physics. Add a global equation to compute the appropriate load for the prescribed extension. As an advanced feature, the entry is not available by default in the context menu.

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button in the toolbar.

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In the dialog box, in the tree, select the check box for the node .

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Global Equations 1

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In the window, under right-click and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	f(u,ut,utt,t) (1)	Initial value (u_0) (1)	Initial value (u_t0) (1/s)
Force	aveop1(w)-dh	0	0

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Locate the section. Click

.

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In the dialog box, type force in the text field.

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In the tree, select .

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In the window for , locate the section.

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.

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In the dialog box, type displacement in the text field.

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In the tree, select .

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Boundary Load 1

1

In the toolbar, click

and choose .

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Select Boundary 6 only.

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In the window for , locate the section.

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From the list, choose .

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Specify the Ftot vector as


0	x
0	y
Force	z

Fixed Constraint 1

1

In the toolbar, click

and choose .

2

Select Boundary 1 only.

Assign material properties. Use for all domains.

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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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Click in the window toolbar.

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In the toolbar, click

to close the window.

Use swept mesh to generate a uniform mesh over the spring domain. Start by specifying the mesh on one end face of the spring.

1

In the toolbar, click

and choose .

2

Select Boundary 6 only.

1

In the window, click .

2

In the window for , locate the section.

3

From the list, choose .

In the toolbar, click

.

1

Right-click and choose .

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In the window for , locate the section.

3

In the text field, type 200.

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.

In the toolbar, click

.

The default plot shows the von Mises stress on the surface of the spring. Compare the plot with Figure 2.

Evaluate the force required to get the displacement specified in the global equations.

Global Evaluation 1

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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 table, enter the following settings:


Expression	Unit	Description
Force	N	State variable Force

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Finish the result analysis by evaluating the average displacement of the end of the spring.

Global Evaluation 2

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In the toolbar, click

.

2

In the window for , locate the section.

3

In the table, enter the following settings:


Expression	Unit	Description
aveop1(w)	dm	Average 1

4