|
|
|
|
|
|||
 |
 |
 |
 |
 |
 |
|
|
|
||||||
| Answers are provided for you
to check your work, though these answers are not guaranteed to be
correct. You receive credit only for the work you show that follows
from the basic principles, knowns, givens, and assumptions you cite. If
you get a different answer than the answers provided here, be sure to
talk to your professor.
Complete solutions are posted in the library after the due date. |
| HW1 | ||
| problem | hint | answer |
| 1.1 | keq=13k/7 | |
| 1.2 | Draw equivalent spring schematics, i.e. in a) you have 6 in parallel, in b) you have 5 in series. | case a: 6K, case b: K/5, case c: 2K/3 |
| 1.3 | Model the beam alone as a mass-spring system, then add the additional spring. Don't use more significant figures than are warranted. | keq= 1098 lb/in meq= 0.031 slugs |
| top | ||
| HW2 | ||
| problem | hint | answer |
| 2.1 | The area moment of inertia of the beam is given by I = (bh^3)/12. There is more than one correct answer. One can explore the solution space by selecting a range of h and plotting b(h) . Note that the stress requirement is an inequality. Be sure show your design meets the specifications. | |
| 2.2 | No Hint | D1=0.01885 N-m-s/rad |
| 2.3 | Draw FBD=KD of load disk only. Note that the fluid between the disks acts like a damper with both ends of the damper rotating. | |
| 3.1 | Draw FBD and KD for each of the masses. Assume that x3>x2>x1 | |
| 3.2 | The input is an angular displacement. | |
| 3.3 | Draw FBD and KD of the disk and block separately. Your answer should be a set of first order ODEs and an output equation. | |
| 4.1 | To reduce your equations to just three you need to eliminate the contact force between disks 2 and 3. Solve for F in the equation obtained from looking at disk 2 and substitute into the equation found from looking at disk 3. To eliminate q2 use kinematics to relate q2 and q3. | |
| 4.2 |
To save time in writing the transfer functions just define the denominator polynomial to be Ds = (something) and then write the transfer functions as q1/T2 = (Cs + k2)/Ds, etc. For part (e) don't forget the output equations y=Cx+Du, where the output variables are y1 and y2, given by y1 = q1 and y2 = q2. The answer given uses the state the state variables: {x1 x2 x3 x4} = {q1 q2 q1dot q2dot} |
|
| top | ||
| HW3 | ||
| problem | hint | answer |
| 5.1 | You may want to define a coordinate, x3, on the right side of K1. You can then use dependent motion to relate x3 to x1. | |
| 5.2 | For part b) after
you solve for the highest derivative put the equation in the Laplace
domain and divide the equation by s to get rid of the derivative of the
input. You will have an integrator in one of the feedback loops.
The start of the diagram is shown below.
|
not provided |
| 6.1 | No hint. | |
| 6.2 | Place the ground right below vi. v0 is the voltage across C2 plus (or minus) the voltage across R2. Be careful with the direction of current in each branch. | |
| 6.3 | No hint. | |
| 7.1 | Use 'evalf' or 'vpa' in Maple to get rid of large rationals in answer. | |
| 7.2 | Comment on the results of the simulations. | |
| top | ||
| HW4 | ||
| problem | hint | answer |
| 8.1 | In part (c), steady-state gain is the magnitude of the transfer function when transients have died out, that is, when the d/dt terms are zero (this is the same as s=0). | |
| 8.2 | There is more than one correct answer to part (c) and the coefficients in the transfer function, part (d), depend on your answer to part (c). Be sure to show that your values meet specs. | |
| 9.1 | ||
| 9.2 | Answer not given, but you should have about 9 equations and 9 unknowns. Assume the force F is large enough so that the tape is always in tension. | not given |
| top | ||
| HW5 | ||
| problem | hint | answer |
| 10.1 | ||
| 10.2 | ||
| 12.1 | The model is first-order in velocity. | |
| 12.2 | Write the EOM in the angular displacement, theta. The basic DE is nonlinear. What assumption can you make to linearize it? In Matlab, multiply vectors element by element using the '.*' syntax. The plot given for part b) is for the problem assigned last year. Note that some of the initial conditions have changed for this year. | |
| 13.1 | Model the tail section as the spring
damper model below and set up the differential equation: The position y(t) is a known input and function of time, t. |
|
| 13.2 | Use logarithmic decrement. Also, note carefully the units of time on the figure. | |
| top | ||
| HW6 | ||
| problem | hint | answer |
| 14.1 | Make sure the maximum deflection is not exceeded when you selected a spring. You will find that this will be a limiting factor in your design. Be sure to confirm that the spring you select meets all the specifications. | not provided |
| 15.1 | Like eating an elephant--try to solve in small pieces. Each state derivative is the input to an integrator. Use gains and sums to make the integrator input obey its governing equation. | not provided |
| 15.2 | The answer is provided on the problem statement. | |
| top | ||
| HW7 | ||
| problem | hint | answer |
| 17.1 | You should find that the current through the capacitor (Cs), in parallel with the current source, is zero. | |
| 17.2 | "Negligible thermal capacitance" implies that negligible energy is stored in the thermocouple bead. The thermocouple in the gas stream radiates to the wall. Don't forget to use units of Rankine. | |
| 18.1 | Recall that the exact solution for a 1st-order ODE with a step input is known. The solution with the heat sink attached assumes the total area is 0.0055 m^2 (it might be better to use 0.005 m^2, but this is an OK assumption without more detailed information). The heat sink is made of aluminum, so you will need to look up the properties of aluminum. | |
| 18.2 | (a) Treat the
two layers of ceramic as separate systems. (b) System inputs are Tr and Ta. (c) There is more than one correct answer. (That's why we use a design-space approach.) Limit your solutions to dimensions less than a meter. |
|
| 19.1 | The answer to part (a) assumed that the heat transfer only occurs out the top of the chip. Part (d): Recall the known response of a first-order system to a step input. | |
| 19.2 | The time should be between 5.8 and 6.8 minutes (this is the upper and lower range) | |
| top | ||
| HW8 | ||
| problem | hint | answer |
| 20.1 | Measure the amplitudes and set the Mag. Factor equal to the Amplitudes ratio. Find the phase shift and express as an angle to find time constant. | |
| 20.2 | Notice that undamped frequency is given. Measure plots to find phase shift and amplitude ratios. From these you can find damping ratio and static gain. | |
| 20.3 | For small angles x, sin(x) = x and cos(x) = 1 (approximately). | |
| 21.1 | Use superposition. | |
| 21.2 | ||
| top | ||
| HW9 | ||
| problem | hint | answer |
| 24.1 | Draw vertical lines at the two different input frequencies and then read off the Bode magnitude and phase angle. Your answer may be slightly different from those shown depending on how you read the figures. | |
| 24.2 | Recall that the SS output magnitude is the product of the TF magnitude and the input magnitude. | |
| 24.3 | Don't forget to factor and put in the form found in the table. | Not provided |
| 25.1 | Don't forget to factor and put in the form found in the table. | Not provided |
| 25.2 | Don't forget the gain. From the straight line approximation you should be able to determine the break points. | |
| top | ||
| HW10 | ||
| problem | hint | answer |
| 26.1 | Don't forget to sketch the straight line approximation on top of the exact. | |
| 26.2 | Don't forget to sketch the straight line approximation on top of the exact. | |
| 26.3 | Everybody's answer will be slightly different. For a first cut as to the form of the transfer function determine the minimum order for the numerator and denominator to get the shape provided. | Not provided |
| 27.1 | You can follow the wires backwards to determine enough equations. Or label the integrator outputs as states, write the state-space description from the block diagram and convert to a TF. OR follow the patterns in the standard forms handout. | |
| 27.2 | Don't forget to label the roots for kp = 0.001, kp = 0.04 and for the critically damped case. You can do this problem each case at a time (find the roots using Maple or Matlab or your calculator and then write down the real and imaginary part of each root and then plot it) or by writing a Matlab program to do this. Either method is fine. | |
| 28.1 | b) Match powers of s between the desired closed-loop char. eqn. and the denominator of the closed-loop TF. You should get 2 eqns for the unknown gains. | |
