Dynamic Behavior of a Spring Loaded Rotating Slider

This model simulates the dynamic behavior of a spring loaded rotating slider. The motion of the slider is analyzed under various forces such as the centrifugal force, spring force and damping force.

This is modeled using the Multibody Dynamics interface present in COMSOL Multiphysics and the results of the analysis are compared with the analytical results.

Model Definition

The slider geometry used in this model is shown in Figure 1. The geometry consists of two parts, a slider and a base. The base is rotating around its center of rotation with a constant angular velocity. The slider is connected to the base such that it is free to translate along the base axis. A prismatic joint is used to connect the slider with the base. An elastic spring and a viscous damper are also attached on the prismatic joint to control the motion of the slider.

Figure 1: Model geometry.

The centrifugal force acting on the slider due to the base rotation moves the slider radially outward along the base axis. Due to the attached spring between the slider and the base, the slider oscillates about a mean position. The oscillations gradually decay due to the damper attached between the two components.

The computed results are compared with the analytical solution, which is obtained by solving an ODE for the equivalent system. In the system the motion of the slider is considered and therefore the external force acting on the system is equal to the centrifugal force acting on the slider.

The equations for the equivalent system can be written as

where m is the point mass, c is the damping coefficient, k is the spring constant, u is the displacement of the slider, F is the centrifugal force, r is the radial distance of the slider, and w is the angular velocity of the base.

Results and Discussion

Figure 2 shows the time history of the displacement of the slider in the radial direction. The computed displacement is in excellent agreement with its analytical counterpart. The time history of the displacement shows an oscillatory motion of the slider with respect to its base. The damping effect is also evident, as the amplitude of the oscillation decays over time.

Figure 3 displays the time variation of the velocity of the slider with respect to its base in the radial direction. The plot shows that the computed value of the radial velocity is also in excellent agreement with the analytical value.

Figure 4 shows the polar plot for the time variation of the radial position of the slider. It shows that the slider never crosses its initial position during one full revolution of its base.

Figure 2: Comparison of radial displacement of the slider with the analytical solution.

Figure 3: Comparison of radial velocity of the slider with the analytical solution.

Figure 4: Polar plot for the radial position of the slider.

Figure 5 shows the polar plot for the time variation of the velocity of the slider with respect to its base in the radial direction. From the plot, it is evident that in one full revolution of the base, the slider completes two and a half oscillations (which is equal to the number of lobes in the velocity plot). The magnitude of the oscillation is also decaying which can be seen from the decreasing size of the successive lobes.

Figure 6 shows the polar plot for the time variation of the kinetic energy of the slider due to its relative motion with respect to the base. In this plot, there are five lobes, and they also indicate that the slider undergoes two and a half cycles of oscillation, as the kinetic energy completes one cycle in half a cycle of oscillation.

Figure 5: Polar plot for the radial velocity of slider.

Figure 6: Polar plot for the kinetic energy of slider due to its radial motion.

Notes About the COMSOL Implementation

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In this model, the slider and the base are modeled as flexible elements using the node. If the stresses and the deformation in the components are not of interest, they can also be modeled as rigid elements using the node.

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A node can establish a direct connection between nodes. For flexible elements, however, nodes are needed to define the connection boundaries.

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Initial rigid body translation and rotation of a system can be defined at the physics node in the section and can be inherited, if needed, by the feature node for rigid as well as flexible elements.

Multibody_Dynamics_Module/Verification_Examples/spring_loaded_slider

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
a	0.05[m]	0.05 m	Side length of slider
p	0.5[m]	0.5 m	Initial position of slider
omega	1[rad/s]	1 rad/s	Angular velocity
k	2.5[N/m]	2.5 N/m	Spring constant
c	0.1[N*s/m]	0.1 N·s/m	Damping coefficient
rho	2700[kg/m^3]	2700 kg/m³	Density
m	rho*a^3	0.3375 kg	Mass of slider

Geometry 1

Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type a.

Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type a.

Locate the section. In the text field, type p-a/2.

In the text field, type a.

Form Union (fin)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Clear the check box.

In the toolbar, click

Add Material

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Click in the window toolbar.

In the toolbar, click

to close the window.

Multibody Dynamics (mbd)

In the window, under click .

In the window for , locate the section.

In the d text field, type a.

To model steady state rotation, define consistent initial values in the physics node. By default, these will be inherited by the node.

Click to expand the section. Specify the Xc vector as


0	x
a/2	y

In the text field, type omega.

Attachment 1

In the toolbar, click

and choose .

Select Boundary 3 only.

Attachment 2

In the toolbar, click

and choose .

Select Boundary 6 only.

Prismatic Joint 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

The default values for the joint properties apply, so no further settings are needed.

Spring and Damper 1

In the toolbar, click

and choose .

In the window for , locate the section.

In the ku text field, type k.

In the cu text field, type c.

Prismatic Joint 1

Constrain the relative motion between the slider and the base for a certain time duration.

In the window, click .

Prescribed Motion 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

In the iup text field, type (t>=0.1).

Use the node to rotate the system.

Rigid Connector 1

In the toolbar, click

and choose .

Select Boundary 1 only.

In the window for , locate the section.

Select the check box.

Locate the section. From the list, choose .

In the text field, type omega*t.

Model Builder

Use a node to compute the analytical solution.

Click the

button in the toolbar.

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

Click .

Global Equations 1

In the toolbar, click

and choose .

In the window for , locate the section.

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)	Description
ua	muatt+cuat+kua-(m(p+ua)omega^2)(t>=0.1)	0	0	Analytical displacement