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.

Student Generated Solutions

The following are solutions for textbook problems written by students in the course:

• Problem No. 2/216 - submitted by Derek
• Problem No. 3/7 - submitted by RyanA
• Problem No. 3/24 - submitted by RyanA
• Problem No. 3/50 - submitted by Derek
• Problem No. 3/54 - submitted by Imtiaz Ahmed
• Problem No. 3/56 - submitted by Imtiaz Ahmed
• Problem No. 5/59 - submitted by Steve-O
• Problem No. 5/66 - submitted by Leonard Pinto
• Problem No. 5/81 - submitted by Leonard Pinto
• Problem No. 6/118 - submittted by Bradle Harner
• Problem No. 6/129 - submitted by Anonymous
• Problem No. 6/148 - submitted by Anonymous
• Problem No. 8/1 - submitted by Robert Shingleton
• Problem No. 8/3 - submitted by Robert Shingleton
• Problem No. 8/14 - submitted by Robert Shingleton
• Problem No. 8/19 - submitted by Dale Szul
• Problem No. 8/26 - submitted by Madhi Alhasan
• Problem No. 8/32 - submitted by Warren
• Problem No. 8/45 - submitted by Madhi Alhasan
• Problem No. 8/50 - submitted by Warren
• Problem No. 8/65 - submitted by Warren
• Problem No. 8/66 - submitted by Warren
• Problem No. 8/74 - submitted by ayu_djaputra

[Disclaimer: I have not yet checked through the details for all these solutions. Let us know if you find any errors. CMK]

Homework Problem 8/72

I'm having some trouble understanding somethings with problem 8/72. First, it says the mass of the trailer is 500 kg, and that each 75 kg added to the load during the loading caused the trailer to sag 3 mm on its springs. Is the 500 kg the mass of the trailer itself (i.e. without any 75 kg additions to the load), and how many 75 kg additions were added to it? And how do each of those 3 mm amounts that the trailer sags from each addition affect the problem?
Also, how do we find omega for the trailer? Are we supposed to divide the speed of the trailer (25 km/hr) by the "wavelength" between two bumps on the road (1.2 m)? I would assume omega is necessary in solving the problem, I don't see how it wouldn't be if the contour of the road is a sinusoidal function.

Homework Problem 8/67

I worked through this problem but I had the coefficient of y as 2k/m, not 4k,m. For my FBD I had two forces of k(y-y_b), going up. This gave me the right result for the term in front of y_b, but not for the term in front of y. Ideas?

Homework Problem 8/61

I am having trouble understanding exactly what 8/61 is asking exactly. I solved for "A" which is the amplitude. I then set this equal to less than 2b and solved for omega. I tried to compare this to omega_n but still wasn't completely understanding the question.

Homework Problem No. 8/46

I am having troubles setting up 8/46. My main issues deal with the effect of the arm with the dashpot on the system.

Homework Problem 8/130 - more discussion

There have been several good questions raised in the post below. Maybe I can jump in at this point to say a few things about these questions and some other important points.

• The EOM that I get for this system is [by summing moments about the no-slip contact point C -- see the lecture example on page 8 of the notes]: (3*m*r/2)*x_dot_dot + k*r*x = 0. Therefore the natural frequency, omega_n, is given by: omega_n = sqrt((2*k)/(3*m)). This natural frequency is NOT equal to sqrt(k/m)!!
• The actual IC's that you use to get a particular x_max in the response is somewhat arbitrary. That is, you can start the system out with an infinite number of sets of IC's x(0) and x_dot(0) and get the same x_max. [Can you look at the animations above to see what I used for IC's in this simulation?] For your work, I recommended using x(0) as some non-zero number and x_dot(0) = 0, as Phil B explains in a comment in the following post.
• The amplitude of oscillation for the orange wheel above is roughly twice that of the blue wheel. However, the two wheels take EXACTLY the same amount of time to make one complete cycle of oscillation -- do you know why this is true? Does this agree with your intuition?
• A friction force is required to prevent slipping as the disk rolls. Watch the friction force (FF) in the above animation as the system moves. The friction force for the orange wheel is larger than the friction force for the blue wheel -- why is this necessary? At what point in one cycle of oscillation is the friction force a maximum? Why does it occur that that time?

Homework Problem 8/130

How do you find the max amplitude if you don't know the displacement. for the equation x(t)= C cos(wn t) + S sin(wn t)?

Homework Problem VIB-3

Right now I am stuck on VIB-3. I am unable to determine how to relate the force of the spring and the force of the block on the disk in terms of theta. I feel like the point of no slip should be at a 45 degree angle above the horizontal from point O on the disk. I don't know if this is a correct assumption or if this will even help me out. Any help would be great!