Flat disk Hollow disk Solid sphere. Flat plate Hollow plate Solid block. n y n O O O

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1 NAME ME161 Midterm. Icertify that Iupheld the Stanford Honor code during this exam Tuesday October 29, 22 1:15-2:45 p.m. ffl Print your name and sign the honor code statement ffl You may use your course notes, homework, books, etc. ffl Write your answers on this handout ffl Where space is provided, show your work to get full credit ffl If necessary, attach extra pages for scratch work Problem Value Score Total 1 315

2 idterm.1 (33 pts.) True/False, Multiple Choice, Definitions, etc. (a) (4 pts.) Each object below has a uniform density and an equal mass. Answer the following questions about I zz, the moment of inertia of each object about the line that passes through its mass center and is perpendicular to the plane of the paper. Flat disk Hollow disk Solid sphere Flat plate Hollow plate Solid block ffl (1 pt.) Consider the first row of objects. The flat disk/hollow disk/solid sphere (circle one) has the largest value of I zz, whereas the flat disk/hollow disk/solid sphere (circle one) has the smallest value of I zz. ffl (1 pt.) Consider the second row ofobjects. The flat plate/hollow plate/solid block has the largest value of I zz, whereas the and (fill in the blanks) have equal values of I zz. ffl (2 pts.) Consider all the objects in both rows. The has the largest value of I zz, whereas the has the smallest value of I zz. (b) (3 pts.) The figure below shows six objects, each with a uniform density. For each object, consider I nxny, the product of inertia of the object for lines that pass through point O and are parallel to n x and n y. Below each object, mark whether the product of inertia is negative, zero, or positive. n y n O n x z O O O O O (c) (2 pts.) For both 2D and 3D analysis, the angular velocity of a rigid body B in a reference frame N can always be written in terms of the time-derivative ofanangle and a unit vector k as N! B = ± _ k as long as and k are carefully chosen. True/False. 316

3 (d) (5 pts.) Put the following functions into amplitude/phase form by filling in the blanks. (e) -3Λsin(2t) + 4Λcos(2t) = Λsin(2t + ) -3Λsin(2t + ß ) + 4Λcos(2t) = Λcos(2t + ) 3 (5 pts.) A system is governed by the differential equation ÿ + (k d 4) _y + 4 y = Draw a root locus versus k d by filling in the table and completing the associated graph. Clearly mark the direction of increasing k d. Do not worry if your root locus does not completely fit on the graph. k d Pole location p 1 Pole location p 2 2 Imaginary Axis 4 6 Real Axis 8 1 (f) (g) (1 pt.) The range of values of k d that stabilize y(t) is (1 pt.) The value of k d that results in a purely oscillatory solution for y(t) is (h) (1 pt.) y(t) goes most quickly to zero when k d = (i) (j) (k) (1 pt.) The range of k d that result in some oscillatory behavior of y(t) is (1 pt.) The range of k d that result in an overdamped solution for y(t) is (3 pts.) Circle the phrase that best describes the associated behavior. ffl A pole in the right half plane is stable/neutrally stable/unstable ffl As a pole moves to the left, it is more stable/unstable/oscillatory/damped/fast ffl As a pole moves away from the origin (in any direction), it is more stable/unstable/oscillatory/d 317

4 (l) (6 pts.) Each graph below shows y(t) versus t for a specific value of k d [with y()=1 and _y()=]. Below each graph, fill in the appropriate value of k d, i.e., k d =, 2, 4, 6, 8, or e e+9-2e+9-3e+9-4e+9-5e+9-6e+9-7e+9-8e e e Figure 22.32: k d = Figure 22.33: k d = Figure 22.34: k d = Figure 22.35: k d = Figure 22.36: k d = Figure 22.37: k d = Figure 22.38: Time response of y for various values of k d 318

5 Midterm.2 (24 pts.) Equations of motion for an aircraft The figure below shows an aircraft B flying over Earth N (considered to be a Newtonian reference frame). The aircraft's orientation and speed depends on engine thrust, aerodynamics, and gravity. To design a robust flight controller, one needs the equations that govern the aircraft's motion. To facilitate the analysis, right-handed sets of unit vectors n x ; n y ; n z and b x ; b y ; b z are fixed in N and B, respectively, withn x horizontally right, n y vertically upward, n z =b z,andb x in the aircraft's forward direction. The following table of symbols is available to assist in the analysis. Quantity Symbol Type Local gravitational constant g constant Mass of B m constant Moment of inertia of B about B c (B's mass center) for b z I zz constant Measure of the aerodynamic forces on B in b x direction Fx aero specified Measure of the aerodynamic forces on B in b y direction Fy aero specified Measure of the aerodynamic torque on B in b z direction T aero specified Thrust from jet engine in the b x direction F thrust specified Angle that b x makes with the local horizontal dependent variable Measure of B's velocity inb x direction v x dependent variable Measure of B's velocity inb y direction v y dependent variable Time t independent variable B B c θ N (a) (3 pts.) Form b R n, the rotation matrix that relates b x ; b y ; b z to n x ; n y ; n z. b R n n x n y n z b x b y b z (b) (2 pts.) Find N! B, the angular velocity ofb in N, and express it in terms of b x ; b y ; b z. N! B = (c) (2 pts.) Write the complete definition of N ff B, the angular acceleration of B in N, and then express it in terms of b x ; b y ; b z. 319

6 N ff B 4 = = (d) (5 pts.) The velocity of B c in N is N v Bc = v x b x + v y b y. Write the complete definition of N a Bc, (the acceleration of B c in N), and express it in terms of b x ; b y ; b z. (e) (f) N a Bc 4 = = (2 pts.) The aerodynamic forces acting on B can be replaced with a force Fx aero b x +Fy aero b y applied at B c together with a couple of torque T aero b z. Considering aerodynamic forces, thrust, and gravitational forces, find F B, the resultant of all forces on B. F B = (7 pts.) Using your knowledge of vector algebra to simplify your work, form a set of scalar equations that suffice to predict the motion of the aircraft, i.e., differential equations for v x (t), v y (t), and (t). Put the equations in standard form. (g) (h) (1 pt.) Classify the previous equations by picking the relevant qualifiers from the list below. Uncoupled Linear Homogeneous Constant-coefficient 1st-order Ordinary Algebraic Coupled Nonlinear Inhomogeneous Variable-coefficient 2nd-order Partial Differential (2 pts.) Given initial values of v x, v y,, and _ ; numerical values for m, g, and I zz ; and the functional form of Fx aero, Fy aero, T aero,andf thrust,howwould you simulate the motion of the aircraft, i.e., how would you solve the equations of motion to find v x (t), v y (t), and (t). 32

7 Midterm.3 (18 pts.) Equations of Motion for Control of an Antenna The figure below shows a small motor A which rotates a pulley B which in turn rotates a second pulley C. The purpose of this mechanism is to control the orientation of an antenna which is rigidly welded to C. The following table of symbols is available to assist in the analysis. Quantity Symbol Type Radius of A and inner wheel of B r constant Relevant moment of inertia of A and its motor about A o I A constant Relevant moment of inertia of B about B o I B constant Relevant moment of inertia of C and antenna about C o I C constant Linear torsional viscous damper for A b A constant Linear torsional viscous damper for B b b constant Linear torsional viscous damper for C b C constant Motor torque on A T m specified Angle between a line fixed in A and line fixed in N A dependent variable Angle between a line fixed in B and line fixed in N B dependent variable Angle between a line fixed in C and line fixed in N C dependent variable Time t independent variable N θ B θ C θ A A o B o T m A B 5r C C o 5r (a) Figure 22.39: Motor with belt-drive controlling antenna orientation (2 pts.) Form an expression for K, the kinetic energy of the entire system in a Newtonian reference frame N, and express it in terms of symbols in the table and their time-derivatives. (b) K = (4 pts.) Ignoring the flexibility of the pulley belts, form an expression forp, the power of the system in N, and express it in terms of T m, b A, b B, b C, _ A, _ B,and _ C. P = 321

8 (c) (5 pts.) Form an equation of motion for this system. Use an inextensible model for the pulley belts, i.e., _ A =5 _ B and _ B =5 _ C, to express your results without A, B, or their derivatives. (d) (1 pt.) Cast the equation of motion into the form m e C + b e _ C + k e C = g(t) and determine m e, b e, k e, g(t) in terms of numbers, k p,andk d. m e b e k e g(t) (e) (2 pts.) The motor and A are very small as compared to the larger pulley C and its attached antenna, i.e., I A < :1ΛI C. In view of this, an engineer tells you to simplify the model of the system and ignore I A. Do you think this is a good idea or not? Explain. (f) (4 pts.) At a certain instant of time (t=), C =, _ C =:2 rad=sec, and the motor is off (T m =). Find the solution for C (t) and express it in terms of m e, b e, k e,andt. C (t) = 322

9 Midterm.4 (25 pts.) Motion Control of an Antenna The equation governing C (henceforth designated as ) in an antenna system similar to the one depicted in Figure is a function of the motor torque T m. The equations of motion and the feedback control law fort m have the form _ = T m T m = -k p + -k d _ (a) (2 pts.) Cast the equation of motion into the form m e + be _ + ke = g(t) and determine m e, b e, k e, g(t) in terms of numbers, k p,andk d. m e b e k e g(t) (b) (8pts.) Find values of k p and k d such that the decay ratio decayratio=:75 and the settling time t settling =:1 secs (c) (5pts.) Taking ()=1: rad and _ ()= rad=sec, create a rough sketch of the first few periods of the response

10 (d) (1pts.) Determine the effect of increasing each of the feedback-control constants on the various quantities listed on the left-hand side of the table below. Only consider control constants that produce an underdamped system. Fill in the table with one of the following: smaller", unchanged", larger', or need more information". Effect on various quantities Effect of increasing k p Effect of increasing k d Decay ratio Damping ratio Natural frequency! n Damped natural frequency! d Period of vibration fi period Maximum overshoot M p Settling time t settling 324