Exam
Dynamics Final (Engr 3340, Fall 2006)
14 December 2006
Instructions
You have 3 hours to work the exam. Work alone. The exam is closed book and closed note with the exception of... a single page of notes (submit these notes with your exam) the provided table of mass moment of inertia values and the provide tables of Laplace transform properties and pairs You may use, but probably won't need, a calculator. There are 6 problems on this exam. The weighting of these problems is shown in table the relative weight of each problem is given in the title. Problem Problem Problem Problem Problem Problem Problem Value 20 20 15 10 15 20
1 2 3 4 5 6
Table 1: Point Distribution
Good Luck!
1
Problem 1: 3D Kinematics
Despite detailed analysis the naval architects miscalculated the acceleration of our radar antenna in figure 1 - resulting in an instrument failure. An engineer now has to climb the mast to fix the problem. Said engineer ascends the vertical mast toward the antenna with a velocity ve as the ship pitches forward with constant angular velocity 2 and rolls with constant angular velocity 1 . The ship is moving forward with a constant velocity vo . Suggest a local coordinate frame to analyze the kinematics. Label this frame clearly in a sketch. At a moment when the engineer is a distance h above point O, find the linear velocity and acceleration of our intrepid engineer with respect to the fixed inertial frame (OXYZ). Are there Coriolis and centripetal terms in this answer? If so, label these terms in your acceleration expression.
Figure 1: A pitching and rolling ship.
2
Problem 2: Four Pins on a Rotating Disk
Four pins slide in four separate slots cut in a circular plate as shown in figure 2. The circular plate rotates with a constant angular velocity (counter clockwise) about the fixed point O. Each pin moves with a constant magnitude velocity (u), with respect to the rotating disk. The direction of the velocity of each pin is shown in the figure. At the instant shown all pins are a distance r form the axis of rotation O. a) Write and expression for the acceleration of pin 1 (P1 ). b) Write and expression for the acceleration of pin 2 (P2 ). c) Write and expression for the acceleration of pin 3 (P3 ). d) Write and expression for the acceleration of pin 4 (P4 ). In each of your answers be specific about the coordinate frame in which you are expressing your answer.
Figure 2: Rotating Disk with Four Pins.
3
Problem 3: Spinning Rod
The slender rod of mass m and length l rotates about the y-axis as the element of a right-circular cone as shown in figure 3. If the angular velocity about the y-axis is : 1. Determine an expression for the angular momentum of the rod, H, with respect to the principle axes of the slender rod for the particular position shown. 2. Determine an expression for the angular momentum of the rod, H, with respect to the x - y - z axes for the particular position shown. 3. Determine an expression the for torque required to rotate the rod as shown (neglect gravity).
Figure 3: Rotating Slender Rod.
Problem 4: Gyroscopic Motion
The jet aircraft (figure 4) at the bottom of an inside vertical loop has a tendency, due to gyroscopic action of the engine rotor, to yaw to the right (as seen by the pilot and as indicated by the dashed orange wingtip movements). Determine the direction of rotation of the engine rotor as depicted in the expanded view, i.e. is the engine rotation p1 or p2 ? Justify your answer with a brief explanation and/or equations.
Figure 4: Jet aircraft at the bottom of an inside vertical loop.
4
Problem 5: Solving ODE's with Laplace
Given each of the following constant coefficient ordinary differential equations Case A: y + 3y = x(t) Case B: y + 33y + 90y = 30x(t) Case C: y + 2y + 100y = 100x(t) Perform the following analysis for each case: 1. Write the Transfer Function - G(s) =
Y (s) X(s) .
2. What are the poles of the system? Sketch the pole locations on the complex plane. 3. Sketch the unit step-response for each system (label the time axis with actual values and label the steady-state value). Hint: the roots of a quadradic equation as2 + bs + c = 0 are: -b b2 - 4ac s= 2a
5
Problem 6: Frequency Response Analysis
The Bode plot in figure 5(a) illustrates the frequency response of a particular second-order system. The corresponding transfer function is G(s) =
2 X(s) Kn = 2 2 Y (s) s + 2n s + n
1. The DC gain of the system is defined as the gain in the system (|G(j)|) for low frequency. What is DC gain of the system in decibels? What is the corresponding ratio of output to input, i.e., what is the ratio |X| for low frequency? |Y | 2. The bandwidth (b ) of the system is defined as the maximum frequency where the am...