Homework 6, due October 15, 2004.

[goodwine]

1. An undamped, unforced oscillator has a response frequency of 0.5 Hz. When viscous damping is added, it has a response frequency of 0.45 Hz. What is the damping constant assming the mass of the oscillator is 1 kg?
2. A mass oscillating with viscous damping makes 5 compete oscillations per second and in 50 cycles its amplitude of motion is reduced to 10% of its initial value. Find the damping ratio and logarithmic decrement.
3. A mass-spring-damper system with mass, m=0.2 kg, has a natural frequency of 200 cycles per minute when vibrating with no damping. When damping is added the unforced frequency of oscillation is 180 cycles per minute.
   1. Determine the damping constant, b and damping ratio.
   2. When damped with this damping constant, what will be the displacement at the end of one cycle if the initial displacement is 2 meters and initial velocity is zero?
4. A sping-mass-damper system is subjected to a harmonic force. If the amplitude is 40 mm at resonance and 20 mm at 0.75 times the resonance frequency, determine the damping ratio. You may refer only to the particular solution to solve this problem.
5. Consider the system illustrated in the following figure
   • 
   where
   • 
   and the profile of the road, h(d), where d is in meters is given by
   • 
   1. Determine the amplitude of motion of the mass and the displacement transmissibility.
   2. Determine the magnitude of the force transmitted to the mass and the force transmissibility.
6. Consider the system illustrated in the following figure
   • 
   where
   • 
   and
   • 
   If the matgnitude of the response is 2 meters, then by using an equivalent viscous damping, determine the magnitude of the applied force.
7. Using the system in the previous problem write a computer program that determines a numerical approximation to the solution of the system subjected to Coulomb friction. Use 0.1 as the value for both the initial displacement and initial velocity. Submit a plot that compares the motion of the system with Coulomb friction to the equivalent viscously damped system. You may choose to simply plot the exact solution of the equivalent viscously damped system or compute an approximate numerical solution for it as well.
   
   Hint: I am not aware of a sgn() function in C. You may find fabs(x) which gives the absolute value of x helpful.
8. Consider the robot arm illustrated in the following figure (the same as we did in class).
   • 
   Write a computer program using 4th order R-K to determine an approximate solution to this system when controlled by PID control (the equations were given in class).
9. Using this program, verify the following "rules of thumb" for PID control using the definitions of ise time, peak time, overshoot, settling time and steady state error from the following figure.
   • 
   1. For proportional control, i.e., kp > 0, kd = 0 and kI=0, the solutions are oscillatory, and increasing kp increases the frequency of oscillation (which decreases the rise time and peak time) but decreases the mean steady state error. The settling time is infinite. Hint pick a starting value of kp=5.
   2. If derivative control is added to the proportional controller, i.e., kp > 0, kd > 0 and kI=0, then
      • for small kd the solutions are decaying oscillations;
      • increasing kd decreases the settling time;
      • increasing kd sufficiently eliminates the oscillatory behavior completely, resulting in an solution which exponentially decays to the final, steady state value;
      • increasing kp decreases the final steady state error;
      • increasing kp decreases the rise time.
      Hint pick a starting value of about kd=0.5.
   3. Adding integral control (PID control)
      • eliminates the steady state error, even for small values of kp,
      • increasing kI generally increases the overshoot and settling time;
      • increasing kp decreases rise time, but may increase overshoot;
      • increasing kd increases damping and stability.
      Hint: pick a starting value of about kI=0.5.

[grc585]

Will predictor-corrector methods be covered on the exam? Will spring/mass/damper systems with a moving base be covered on the exam?

[goodwine]

Will predictor-corrector methods be covered on the exam? Will spring/mass/damper systems with a moving base be covered on the exam?

Yes to the first question.

For the second, as I said in class today, it will not be on the exam if I covered it after last Monday, September 27. I still need to check my notes to determine this. I will be sure to do so before I make up the exam.

[monkeybrains]

In reviewing homework probs, I do not understand HW 1: Prob 7: Equation C.

Specifically, I don't understand why this is equation linear. It does not
appear to fit the general solution of the linear equation; I would thus
argue that it cannot be linear. But it is linear. Can you explain why?

[goodwine]

In reviewing homework probs, I do not understand HW 1: Prob 7: Equation C.

Specifically, I don't understand why this is equation linear. It does not
appear to fit the general solution of the linear equation; I would thus
argue that it cannot be linear. But it is linear. Can you explain why?

Quick answer: for the test it doesn't matter since it is a partial differential equation. The answer is that all the partial derivatives only appear linearly, i.e., there are no derivatives inside a sine or cosine function or powers of the derivatives.

[whiskaaaaas]

I have a general (and possibly stupid) question: Does natural frequency of a system change depending on the presence or non-presence of damping? Or is it constant for a system regardless of damping?

[goodwine]

I have a general (and possibly stupid) question: Does natural frequency of a system change depending on the presence or non-presence of damping? Or is it constant for a system regardless of damping?

It's not a stupid question since it wasn't really emphasized. The natural frequency is just sqrt(k/m) regardless of the presence of damping and is the frequency of the homogeneous solution when there is no damping. If there is damping, there is still a "natural frequency" which is sqrt(k/m), but it isn't the frequency of the homogeneous response. In that case, the "damped natural frequency" which is sqrt(1-zeta^2) times the natural frequency is the frequency of the homogeneous response.

[chalkdust]

Is the displacement in terms of x_dot_0? If not, any hints on how to solve for it?

[goodwine]

Is the displacement in terms of x_dot_0? If not, any hints on how to solve for it?

You may assume that the initial velocity is zero. I updated the homework to reflect this.

[goodwine]

I'm trying to email this to the whole class via IrishLink, but as usual, it's not working. As usual, my web page is working, so I'll put it here until I can get the email through.

I added a couple clarifications and corrected one mistake on Homework 6.

-Problem 3b: assume the initial velocity is zero.

-Problem 5: assume the variable d is in meters.

-Problem 6: assume the magnitude of the response is 2 meters. As far as I know,
you cannot even do this problem without this information.

I modified the homework at 5:10 on Monday to reflect these changes.

[bprenger]

I had a question regarding the terminology in Problem 1. Does the response frequency mean that the frequency (f) is 0.5 or that omega_n is 0.5?

[goodwine]

I had a question regarding the terminology in Problem 1. Does the response frequency mean that the frequency (f) is 0.5 or that omega_n is 0.5?

You do need to be sure to sort out the terminology, but basically, \omega_n=0.5 and when damping is added \omega_d is 0.45. What b will cause that to happen if the mass has the value given in the problem?

Since both cases are unforced, "response frequency" means the roots of the characteristic equation corresponding to the homogeneous solution, so it's \omega_n when undamped and \omega_d when damped.

[Grendel]

I don't understand where v = 100 km/hr comes into play in problem 5. I solved part A using the particular solution that was provided in class:

Code: Select all

[        1 + (2 * zeta * w / wn )^2      ] ^ (1/2)
[  {1-(w/wn)^2}^2 + (2 * zeta * w/wn)^2  ]

After solving for xp(t) , then you can just plug that into eq. 15 on the help sheet. But in doing this, the velocity provided is never used.

[goodwine]

I don't understand where v = 100 km/hr comes into play in problem 5. I solved part A using the particular solution that was provided in class:

Code: Select all

[        1 + (2 * zeta * w / wn )^2      ] ^ (1/2)
[  {1-(w/wn)^2}^2 + (2 * zeta * w/wn)^2  ]

After solving for xp(t) , then you can just plug that into eq. 15 on the help sheet. But in doing this, the velocity provided is never used.

You need to use the velocity to find \omega. How fast this thing drives over the road dictates how fast it's going over the sinusoidal bumps, which dictates \omega.

[whiskaaaaas]

On problem 4 I used the following approach, but to no avail:

(1) x_h + x_p = eq(10) + eq(13)

(2) 1 eq(10) + eq(13) (non-resonant case)
--- = ----------------------- = ------------------------
2 eq(10) + eq(13) (resonance)

*for resonance I used the substitution w = w_n
for non-resonance w= 0.75w_n

(3) i assumed t=0, x_o= 0

(4) all of the trigonometric functions dropped out, and I was left with an expression in terms of k, m, w_n, b

Any hints on what to do next, or revisions to my approach thus far??

[whiskaaaaas]

here is step (2) in my previous post, it looks awful above:

(2)

1
--- =
2
eq(10) + eq(13)
--------------------- =
eq(10) + eq(13)

(non-resonant case)
-----------------------
(resonance)

[goodwine]

On problem 4 I used the following approach, but to no avail:

(1) x_h + x_p = eq(10) + eq(13)

(2) 1 eq(10) + eq(13) (non-resonant case)
--- = ----------------------- = ------------------------
2 eq(10) + eq(13) (resonance)

*for resonance I used the substitution w = w_n
for non-resonance w= 0.75w_n

(3) i assumed t=0, x_o= 0

(4) all of the trigonometric functions dropped out, and I was left with an expression in terms of k, m, w_n, b

Any hints on what to do next, or revisions to my approach thus far??

It is admittedly not clear from the problem, but implicitly you are to only consider the particular solution (otherwise the solution would depend upon the choice of initial conditions, which weren't specified). I midified the wording of the problem to make it more clear.

This is basically just a one-equation problem. You need the equation that expresses the magnitude of the particular solution as a function of zeta and omega/omega_n. This was given in class, along with a couple plots on the day I wasn't there and Alice Nightingale gave the lecture. It's a 1/sqrt(mess) sort of equation, but you know everyting except zeta for the two cases, so you can set the first case equal to two times the second case to find zeta.

[mightyduck]

In regards to number 8... do we only have to write the program? No plots? What are all the values for the Ks, and I, or are these not necessary to know? Do we not actually have to compile this program?

[goodwine]

In regards to number 8... do we only have to write the program? No plots? What are all the values for the Ks, and I, or are these not necessary to know? Do we not actually have to compile this program?

Number 8 is the program and number 9 uses it. I split it up just to try to make the amount of credit per problem correspond somewhat to the work. You have to compile it to do number 9, so you can just submit the code for number 8.

[mightyduck]

What's a good theta_d to use for number 9? I know it's up to us but I'd rather not get an entire problem wrong because I'm a terrible guesser. Thanks!

[goodwine]

What's a good theta_d to use for number 9? I know it's up to us but I'd rather not get an entire problem wrong because I'm a terrible guesser. Thanks!

Really, since it's linear it doesn't matter at all except sometimes guessing something like 1 may cause unexpected cancellations. I would use either 2 or pi.

[smg]

I am confused on how to approach Problem 6. I was wondering if you could give us somewhere to start, or what we should be trying to do in this problem.

[goodwine]

I am confused on how to approach Problem 6. I was wondering if you could give us somewhere to start, or what we should be trying to do in this problem.

This is a "plug and chug" problem. You have an equation for the amplitude of motion for the equivalent system A = (mess). The only thing you don't know in (mess) is F_0.

[goodwine]

Is there a given value of Fo for #7?

Use the value you compute in problem 6.

[NDChevy07]

The equation given in class ends up with a Ki*I on the right side, with the stipulation I'=theta_d - theta. Is how do we impliment that integral into the program, or did I copy the equation down wrong?

[goodwine]

The equation given in class ends up with a Ki*I on the right side, with the stipulation I'=theta_d - theta. Is how do we impliment that integral into the program, or did I copy the equation down wrong?

This is right. You have three states

x1 = theta
x2 = theta_dot
x3 = I

The derivatives of the first two come from the theta_ddot equation and I_dot is simply theta_desired - theta.

One thing was a bit sloppy on my part: don't confuse I = moment of inertia of the arm, which I set equal to 1 early in the derivation and I which stands for "integral" so that I_dot=theta_desired-theta.

[mightyduck]

Is there going to be homework due the Wednesday right after break? I vote for NO!!!!

(thanks!)

[goodwine]

Is there going to be homework due the Wednesday right after break? I vote for NO!!!!

(thanks!)

I'm in the process of tallying the votes:

No: 1
Yes: 0
Undecided: 180

There are 90 people in the class.

I get 91 votes

At this point I plan to vote "no" as well.

[student]

I have omega_n and b, but how do I get omega? In the review session, we were told to convert h(d) to h(t), but I don't really know how to do that.

[goodwine]

I have omega_n and b, but how do I get omega? In the review session, we were told to convert h(d) to h(t), but I don't really know how to do that.

d is the distance down the road and you know the velocity.

[student]

For 5a, I was able to get a reasonable value for T (T= 0.7380) with omega_n=18.xx and omega=(2*pi*v)/6. Why did I have to multiply by velocity, instead of dividing? Isn't time=distance/rate?

For 5b, In my notes, I have the force transmissibility equation with (omega/omega_n)^2, but the equation list you provided us with does not have the square. Which one should I use?

For 6, I know that the applied force has to be greater than the friction, but I got a very large force F_0=(about)4800. Is this correct or even close?

[student]

Also, for the magnitude of the force transmitted to the mass, do I use the amplitude of the bumps (0.05)?

[student]

I don't know how to start #7. What equations do I use? x(t)=A*sin(wt-phi) is the solution for the equivalent viscously damped system, right? What do I use for phi in this case?
I tried to use 2*sin(w*t-phi) in xemacs.

For the coulumb friction I used
x1+=dt*x2;
x2+=(fo*sin(w*t)-k*x1-fabs(x2)*M*m*g)*dt;
where //M=mu; w=omega; wn=omega_n and fo is the force.
It didn't work. Any hints?

[goodwine]

For 5a, I was able to get a reasonable value for T (T= 0.7380) with omega_n=18.xx and omega=(2*pi*v)/6. Why did I have to multiply by velocity, instead of dividing? Isn't time=distance/rate?

You want to get the height as a function of time, i.e., h(t). Since d-vt, if you substitute vt for d and v is a constant, you have h(t) which is exactly the same as y(t).

For 5b, In my notes, I have the force transmissibility equation with (omega/omega_n)^2, but the equation list you provided us with does not have the square. Which one should I use?

I don't have my notes with me. I suspect the quotient is squared. If you can't wait until Monday I'd just check the derivation. It's probably only one step that you would have to work out.

For 6, I know that the applied force has to be greater than the friction, but I got a very large force F_0=(about)4800. Is this correct or even close?

Yes, I think it's that large.

[goodwine]

Also, for the magnitude of the force transmitted to the mass, do I use the amplitude of the bumps (0.05)?

The force transmissibility equation has the magnitude of the base oscaillation in it (I think it was called Y in the notes). That's what the 0.05 is.

[goodwine]

I don't know how to start #7. What equations do I use? x(t)=A*sin(wt-phi) is the solution for the equivalent viscously damped system, right? What do I use for phi in this case?
I tried to use 2*sin(w*t-phi) in xemacs.

Are you trying just to plot the exact solution? An easier way may be to determine an numerical approximation to it (since you are looping through everything for the friction case anyway). All you need for that is b_eq.

For the coulumb friction I used
x1+=dt*x2;
x2+=(fo*sin(w*t)-k*x1-fabs(x2)*M*m*g)*dt;
where //M=mu; w=omega; wn=omega_n and fo is the force.
It didn't work. Any hints?

The trick to this was to use x2/fabs(x2) in place of sgn(x2) since I'm not aware of one in C.

So the x2 equation from

m xdd + sgn(xd) mu m g + k x = F0 sin(w t)

is

x2 += (F)*sin(w*t) - k*x1 - my*m*g*x2/fabs(x2))*dt;

(or something very similar).

[student]

I was just wondering if anyone else was having trouble opening the solutions page.
I have not been able to open it.

[Abigail]

I was just wondering if anyone else was having trouble opening the solutions page.
I have not been able to open it.

It's not saved with the usual file extension .PDF, although that's the type of file it is.

Probably the easiest/most reliable way for you to open it will be to save it first (click with the other mouse button, choose "Save to disk" or something like that). Then go to where you saved the file locally, and rename it to "hw6solns.pdf". Then try opening it...

[goodwine]

I was just wondering if anyone else was having trouble opening the solutions page.
I have not been able to open it.

Sorry about that. It's fixed now.