ASSACHUSETTS INSTITUTE OF TECHNOLOGY Departent of echanical Engineering 2.010: Systes odeling and Dynaics III Final Eaination Review Probles Fall 2000 Good Luck And have a great winter break! page 1
Proble #1 For each of the 3 systes depicted in the Bode plots below, estiate the effective daping ratio (and percentage overshoot) of the closed-loop syste which would result is you ipleented a unity-feedback proportional (G=1) controller. HINTS: Recall the estiator Prof. Hogan presented in class: ζ (The daping ratio, zeta, is roughly.01 ties the phase argin, φ, in degrees.) 100 Look-up Table for ζ vs. % overshoot. (5% corresponds to ζ.707). ζ %-overshoot 0.1 70% 0.2 50% 0.3 37% 0.4 25% 0.5 16% 0.6 9.5% 0.707 5% 0.7 1.5% page 2
Proble #2 A choo-choo train engine, of ass, is linked via a coupling to a single car, of ass L, which is heavily laden with coal. The sketch below depicts this siple syste, where the link is odeled as a spring and daper: = 1e4 kg L = 9e4 kg k = 2e4 N/ b = 0.5e4 N-sec/ F = 1e4 N Since there are two asses, writing state equations for the position and velocity of each (ass) will yield four state variables. i. Derive the 44 A atri and the 41 B vector to coplete the relationship below (where the input is F, the force at the engine ass, ). & & && && = & & [ A] + [ B] F ii. anipulate the algebraic relationships represented by your state space equations in part (i) to derive the transfer function fro input force, F (s) to output velocity (s & ). & ( s) F( s) iii. =? Find and poles and zeroes and plot the on the Real vs. Iag ais ( pzap ). FOR ATLAB: Since this is not an ea, you can use atlab to evaluate the following (to gain further insight on syste behavior): iv. Show the step response for the ass for 30 seconds after a step input, F. Copare with the step response for ass L.(i.e. find the TF fro F to this other ass first ) Refer to hoework proble 1.3(d) to eplain qualitatively what is happening. (You have one ode where the whole syste oves along together (w=0) and a second ode where the two asses oscillate in opposition to one another. ake an accurate, labeled sketch of the location of the poles and zeroes of the transfer function θ (s)/i(s) on a cople plane. iv. Use rlocus to produce a root locus. Does the syste becoe unstable? (i.e. the TF in part (ii) represents an OLTF(s) for the syste if controlled by a proportional (siply a gain, G) controller. For what positive G values is this syste stable?) Also produce and analyze a Bode diagra (with atlab) for this OLTF(s). Do you obtain the sae answer (for stable G values)? page 3
Proble #3 A high-pass filter allows high frequency signals (changes in input) to pass through (but ignors slow, low frequency inforation). A low pass filter, by coparison, easures low frequency inforation (i.e. the response to steady-state input) while rejecting higher frequency signals. Each is characterized by a frequency known as the break-point frequency. i. Derive the transfer function fro V in to V for each of the two RC circuits below. Which one (if either) is a high frequency filter and which is a low frequency filter? Why? What arguent can you use? What are the poles and zeroes (if any) for each TF? [Hint: Equate the current through the resistor with the current through the capacitor ] ii. Obtain a transfer function fro V in to V for the circuit below by eploying a siilar balance of current. What are the poles and zeroes here? iii. A lag (or ore properly lag-lead ) copensator derives its nae fro the fact that the resulting open-loop syste phase will lag that syste alone, due to a pole at a lower (relative) frequency [and will then regain phase due to the presence of a zero at a (relatively) higher frequency. A lead (or ore properly lead-lag ) copensator, by contrast, consists of a zero at a (relatively) lower frequency, which ake the phase of the response lead that of the uncopensated open-loop syste and a (relatively) higher frequency pole. Given C a = (a) (b) (c) C b, what type of copensator is created in each of the following cases: R a = Rb R a << Rb R a >> Rb page 4
Proble #4 An astronaut in space is attepting to aneuver a spinning satellite. The satellite spins at a rate 2 Ω and has an inertia I a = a. While the astronaut works, the satellite is teporarily fied (still spinning freely) at the end of a long, stiff rod of radius R. He anipulates the satellite by pushing on the rod at a distance (R/2) fro the satellite, as shown in the figure below: i. Calculate the direction and agnitude of the force (if any) the astronaut ust be applying if we observe the satellite travelling in a sooth (constant angular velocity) arc in the direction shown. [Hint: This is essentially a restateent of the Quick Quiz proble fro Lecture 23.] ii. The astronaut wishes to ove the satellite in the z direction (down, toward the botto on the page). In what direction should he push? What will happen if he stops pushing? (i.e. Does he need to continue pushing for the satellite to continue oving, once set in otion?) page 5