1. This is a CLOSED BOOK, CLOSED NOTES exam. A list of equations for your reference is provided below.

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1 _ Instructions 1. This is a CLOSED BOOK, CLOSED NOTES exam. A list of equations for your reference is provided below. 2. Begin each problem on the printed sheet and continue on the lined paper provided. 3. Work on one side of each sheet with only one problem on each sheet. 4. On each problem, take time to ensure that you have shown your procedure and reasoning in a clear and orderly manner. 5. When handing in the test, RESTAPLE EACH PROBLEM SEPARATELY and staple this cover sheet to Problem 1. Make sure that all sheets for each problem are in the correct sequential order with your name and lab section number on each sheet. Provided Equations a a b c = = sinα sin β sinγ + b 2abcosγ = c φ Q = 180 φ R M.A. = R 4 2 sinγ sin β s + l p + q For an angular input at link i: ω = θ ωi ɺR = R ω i I1 iii θ = ± I I 1 i R = ± I I 1i i For a linear input at link i: ω = θ Ri ɺ ɺR = R Ri ɺ θm θk ρn = ± θ θ ρ RING n ωsun ωarm ρ = ω ω ρ k ARM m R θm θk = ± ρ RING SUN m n Problem 1 Problem 2 Problem 3 Problem 4 Total Score

2 Problem 1. (30 points). Part I. (15 Points). For the mechanism shown in Figure 1(a): (i) Determine the mobility of the mechanism using the Kutzbach criterion. Clearly number each link and label the lower pairs (1) and higher pairs (2) on the given figure. (ii) Draw and label vectors that are suitable for a complete kinematic analysis of the mechanism. (iii) Identify a suitable input, or inputs, for the mechanism and write the vector loop equation(s) for the mechanism. Clearly list the known variables, the unknown variables, and any constraints. If you identified constraints then write the constraint equation(s). Figure 1(a). A planar mechanism. 2

3 Problem 1. (continued) Part II. (15 Points). For the mechanism shown in Figure 1(b): (i) Determine the mobility of the mechanism using the Kutzbach criterion. Clearly number each link and label the lower pairs (1) and higher pairs (2) on the given figure. (ii) Draw and label vectors that are suitable for a complete kinematic analysis of the mechanism. (iii) Identify a suitable input, or inputs, for the mechanism and write the vector loop equation(s) for the mechanism. Clearly list the known variables, the unknown variables, and any constraints. If you identified constraints then write the constraint equation(s). Figure 1(b). A planar mechanism. 3

4 Problem 2. (25 Points). For the planar mechanism in the position shown in Figure 2, the velocity of the input link 2 is V2 = Link 4 is a wheel with a radius ρ4 = 6 cm. The length of the coupler link AB = 20 cm, and the dimensions AD = DE = 10 cm. (i) Draw appropriate vectors on Figure 2 and write the vector loop equation(s). Write any constraint equation(s) that are required for a complete kinematic analysis of the mechanism. (ii) Solve for the first-order kinematic coefficients of the mechanism. (iii) Determine the magnitude and direction of the angular velocity of the wheel (link 4). (iv) Determine the magnitude and direction of the velocity of the coupler point E. Figure 2. A planar mechanism. 4

5 Problem 3. (25 Points). For the planar mechanism in the position shown in Figure 3 (which is drawn full scale): (i) List the primary instant centers and the secondary instant centers. (ii) Using the given Kennedy circle, show the locations of all the instant centers on Figure 3. (iii) Determine the first-order kinematic coefficients of links 2, 3, and 4 given that link 5 is the input. Specify both the numerical values and the signs. (iv) Determine the first-order kinematic coefficients of links 3, 4, and 5 given that link 2 is the input. Specify both the numerical values and the signs. Kennedy Circle Figure 3. A planar mechanism (drawn full scale). 5

6 Problem 4. (20 Points). Part I. (10 Points). The gear train shown in Figure 4(a) is composed of two sets of epicyclic gears, each containing: (i) a sun gear; (ii) a planetary gear; (iii) a ring gear; and (iv) an arm. An unknown input angular velocity ω in is applied to the ring gear of the first set while the arm of the second set acts as the output of the gear train. The output of gear Set #1 drives the input of gear Set #2. Note: planetary gears in the two gear sets are not used as input or output. The radii of the sun and ring gears are as shown in Figure 4(a). The output angular velocity of the gear train is ω out = 20 rad / s, counterclockwise. For gear Set #1, determine: (i) the magnitude and direction of the input angular velocity, (ii) the magnitude and direction of the output angular velocity, ω INT. ω in ; and Figure 4(a). A gear train mechanism. Part II. (10 Points). For the planar mechanism in the position shown in Figure 4(b), the velocity of the input link is V2 = 15 cm/s to the right and the velocity of point P is VP = 24 cm/s at 45. If the input velocity is changed to V2 = 8 cm/s to the left then, for the given position, determine the magnitude and direction of the velocity of point P. Figure 4(b). A planar mechanism. 6