A simulation related to Problem 8/72

In today's lecture, we talked about how the amplitude of response depends on the ratio of omega/omega_n in base-excited systems (such as Problem 8/72):

• For small omega/omega_n, the response tracks closely with that of the base.
• For omega/omega_n near 1, near-resonance (large response) is expected. This is usually undesirable for obvious reasons.
• For large omega/omega_n, the response is diminished as compared to the base motion. This is usually good.

One way to make omega/omega_n small is to increase the value of the excitation frequency omega. For Problem 8/72, this means driving fast. This is undesirable since, as was pointed out in class, this creates a large transmitted force in the suspension (and possible an undesirable ticket from the police!). Another way to make omega/omega_n large is to decrease the value of omega_n. Note that omega_n can be decreased by INCREASING the apparent mass of the system. This is called "vibration isolation".

Earlier in the system, Ryan (rvanklom) posted an interesting idea in vibration isolation for Formula 1 vehicles called an "inerter" (see the following post). This inerter allows for the apparent mass to increase without a large increase in actual mass (it converts translational motion into rotational motion).

Shown below is a simulation of the response of one quarter of the body of a car without and with an inerter. Carefully compare the two responses. 

• As you can see, the response with the inerter is less than half of that response without the inerter. 
• Also note that the damping ratio has also decreased with the inerter (the transients take longer to die away) although the damping constant c is unchanged. Do you know why this is true? (Look at how the damping ratio is related to the damping constant and the mass of the system.)

Let us know if you have any thoughts on this.