Would a bungee jumper represent a damped system
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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.
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.
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
Once again, the response will be composed of a homogeneous solution (the transient response) and a particular solution (the steady state response) as
We have previously found the homogeneous solution. For example, in the underdamped situation the homogeneous solution is given in equation (3.11) as
where and are arbitrary constants. To find the particular solution to equation (5.1), we will
assume a solution of the form
Substituting these into the equation of motion gives
Using the identities
or, collecting the and terms,
Comparing the left and right hand sides leads to two equations
From (5.6b) we see that
Now (5.6a) (5.6b) gives
while (5.6a) – (5.6b) gives
Squaring each of (5.8a) and (5.8b) and adding the results leads to
Therefore, the particular solution (5.4) for this problem is
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
Note that this can be rewritten as
However, as we have already discussed,
As a result, (5.12) becomes
Here we can see that the amplitude of the response is given by
which represents the dynamic magnification factor in the damped situation. Similarly, since
equation (5.7) can be written as
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.