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## Bungee Jump: Simulate Multibody Systems

This is a simple model of a bungee jumper consisting of a mass attached to a platform by a spring and damper.

## Automatic 3D Animation

Multibody systems have visualizers to show what a real-world system would look like.

To simulate the model and view a 3D animation of it, follow the steps below:

- Click the button in the top-right corner.
- When the build is finished, click the Simulate button .
- Click the Animate button .
- Use your mouse or trackpad to drag the animation to a good angle and zoom in with your scroll wheel or by using the trackpad. Then click the Play button to play the animation.

## Instant Plotting

Explore the how the force on the cord varies with time by simulating and plotting the variable ElasticCord.f .

The variable will automatically be plotted when the model is simulated.

## Change Parameters

Changing parameters for the simulation can be done rapidly in Simulation Center. Switch to the Parameters tab and enter a new value for the parameter you would like to vary.

Simulate again to see the effects of your changes.

## Terms and Conditions of Use

This domain example is an informational resource made freely available by Wolfram Research.

## Forced Vibrations of Damped Single Degree of Freedom Systems: Damped Spring Mass System

We have so far considered harmonic forcing functions acting on *undamped* systems. We will now extend our analysis to include systems which include viscous damping. We will still limit our analysis to harmonic forcing functions of the form

Figure 5.1: Damped spring–mass system subjected to harmonic forcing function

Consider a damped spring-mass system subjected to a harmonic forcing function as shown in Figure 5.1(a). The FBD/MAD for this system is shown in Figure 5.1(b) where is once again the displacement from the static equilibrium position. Applying Newton’s Laws we obtain

(5.1)

Once again, the response will be composed of a *homogeneous* solution (the transient response) and a *particular* solution (the steady state response) as

(5.2)

We have previously found the homogeneous solution. For example, in the underdamped situation the homogeneous solution is given in equation (3.11) as

(5.3)

where and are arbitrary constants. To find the particular solution to equation (5.1), we will

assume a solution of the form

(5.4)

Substituting these into the equation of motion gives

(5.5)

Using the identities

or, collecting the and terms,

Comparing the left and right hand sides leads to two equations

(5.6a)

(5.6b)

From (5.6b) we see that

(5.7)

Now (5.6a) (5.6b) gives

(5.8a)

while (5.6a) – (5.6b) gives

(5.8b)

Squaring each of (5.8a) and (5.8b) and adding the results leads to

(5.9)

Therefore, the particular solution (5.4) for this problem is

(5.10)

where is given by equation (5.7). The total solution (for the underdamped case) is

In many cases we are often primarily interested in the long term steady state response of a system. Since the transient response will eventually damp out as we have seen, we will often ignore the transient part and consider the solution to be simply given by the steady state response as

(5.11)

Note that this can be rewritten as

(5.12)

However, as we have already discussed,

As a result, (5.12) becomes

(5.13)

Here we can see that the amplitude of the response is given by

(5.14)

which represents the dynamic magnification factor in the damped situation. Similarly, since

equation (5.7) can be written as

(5.15)

Equations (5.14) and (5.15) are illustrated in Figures 5.2(a) and (b) respectively.

Figure 5.2: Response of a damped SDOF system: (a) Dynamic Magnification Factor (DMF), (b)Phase Angle

(a) In the limiting case where , these results are the same as those obtained in the undamped case.