PAGE 2 ME2115 AY-18/19: QUESTION 3 (a) A uniform slender rod is bent into the shape as shown in Figure 3(a). It is 4 m long and has a mass of 4 kg. Show that its mass moment of inertia about A is given by πΌπ΄ = 1.833 πππ2 (b) This object is assembled into a mechanism as shown in Figure 3(b). It is hinged at point A, which is frictionless. The linear spring attached at the mid-point of AB has a spring constant k = 100 N/m, and an unstretched length of 0.5 m. Suppose it is released from rest at the position shown in the figure, determine the magnitude of velocity at point D when it is rotated by 90o counter-clockwise (i.e. DAB becomes horizontal) Take g = 9.81 m/s2 in your calculations. (c) Suppose you can add a point mass at either C or D to increase the velocity of D in part (b), which one will you choose. Explain your answer without doing the detail calculations. Figure 3(a) Figure 3(b) PAGE 3 ME2115 AY-18/19: QUESTION 4 The same object given in Figure 3(a) is deployed in another mechanism, as shown in Figure 4. It is hinged at A, and the spring constant is k, and damping coefficient of the damper is c. The system is in static equilibrium in Figure 4. (a) Suppose the system is being set into small oscillation. Draw a free-body diagram (FBD) of the object, when it is rotated slightly about its equilibrium position. Include all the essential information in the FBD to make it self-explanatory. And write the corresponding equation of motion, expressing your answer in terms of m, k, c, g, πΌπ΄ . Note: You can ignore the weight of the rod and initial spring force at static equilibrium state because they counter-balanced each other during oscillation. (b) Given that k = 100 000 N/m and c = 100 N.s/m, and using the mass information in QUESTION 3, determine the damping ratio οΊ, and the damped natural frequency ππ of this system. (c) Suppose a vertical sinusoidal force is applied at point C, as given by the expression π(π‘) = 30π ππ(100π‘) Determine the steady-state amplitude of vibration of the system at point D. Figure 4 PAGE 4 ME2115 AY19-20: QUESTION 3 A 5 kg rod BC is attached by pins to two uniform disks as shown in the figure. The mass of the large disk is 5 kg and that of the smaller disk is 1 kg. A spring with spring constant of 50 N/m is attached to the pivot joint at C, which is un-stretched at this given position. Knowing that the system is released from rest in the position shown, and the two disks roll without slipping, determine the velocity of the rod after the large disk has rotated through 90°. Figure 3 PAGE 5 ME2115 AY19-20: QUESTION 4 Figure 4 (a) shows a double-pulley which has an inner spool of radius 10 cm, and an outer spool of radius 30 cm. This double-pulley has a mass of 10 kg, and its radius of gyration is 15 cm. It is free to rotate about a hinge at O. A 1 kg block is attached to the inner spool, while a spring with spring constant of 100 N/m is attached to the outer spool, as shown in the figure. (a) Using the energy method, determine the natural angular frequency of this system for small amplitude of oscillation. Note that you do not need to include the elastic potential energy of the spring due to its initial elongation, and the gravitational potential energy of the mass because they cancel each other absolutely during the vibration motion. (b) Now, a damper is attached as shown in the Fig. 4(b), its damped vibration response is recorded as shown in the time-history plot. Using the logarithmic decrement, determine the damping ratio of this system, and its damped oscillation frequency. (b) (a) Figure 4 PAGE 6 ME2115 AY20/21: QUESTION 1 A slender rod BC is attached to an inextensible cable AB (vertically) and a collar mass at point C as shown in the Fig 1(a). It is held stationary at this position where the spring that is supporting the collar mass is not deformed at this state. It is then released from rest and the collar mass is observed to be sliding along the vertical rod until the system reaches the position shown in Fig. 1(b), where the rod becomes horizontal. Fig. 1 (a) Fig. 1 (b) Given the following information: Length of cable AB Length of rod BC Angular position, π1 Angular position, π2 0.80 m 0.70 m 42o 73o Mass of collar Mass of rod BC Spring constant, k 1 kg 3 kg 80 N/m (a) Assuming zero friction in the entire system, determine the speed of the collar mass at the final position in Fig. 1 (b). Use g = 9.81 m/s2 in your calculation. (b) Suppose it is required that the rod to reach a momentarily stop when it is at the horizontal position in Fig 1(b). Determine the spring constant that should be used to realize this requirement. PAGE 7 ME2115 AY20/21: QUESTION 2 A rigid body of mass 1.0 kg is pivoted at point O and its centre of mass G is located at a distant of d = 6.0 cm from O when it is at the static equilibrium position as shown in Fig 2(a). Fig. 2(a) (a) Now when it is set into small amplitude oscillation, the period is observed to be 1.10 seconds. Determine the radius of gyration of this rigid-body about its center of mass G. Now, a spring with spring constant of 180 N/m is attached to the rigid body, which is undeformed at the static equilibrium state shown in Fig 2(b). Fig. 2(b) (b) Suppose the rigid body undergoes a small angular rotation ο± in the counter-clockwise direction. Sketch the corresponding FBD, and hence determine the natural oscillating frequency for this new configuration. PAGE 8 ME2115 AY20/21: QUESTION 3 A given spring-mass-damper system is set into free oscillation and its time-history response is plotted in Fig. 3. Fig. 3 (a) It is recorded that the amplitudes π₯1 = 1.0 ππ and π₯4 = 0.1 ππ. The period of oscillation is also measured to be 2.2 seconds. Suppose the mass of the system is 10 kg. determine the damping coefficient used in the system. (b) Suppose the mass is doubled, what will be the corresponding damping ratio for the system? Now the ceiling at which the system is mounted onto is subjected to harmonic motion with an amplitude of 1.0 cm and forcing frequency of 1.2 Hz. (d) For this given based excitation, determine the steady-state response amplitude of the mass? (e) What is the maximum damping force exerted by the damper on the mass? Hints: Forces on the mass depend on the relative motion between the mass and the base excitation. PAGE 9 ME2115 AY21/22-S1: QUESTION 3 Figure 3 shows a mechanism that consists of two identical slender rods and a circular disk connected by pin joints. One end of the rod AB is fixed to the wall at A, and the disk is placed on a rough surface. A spring is connected at the mid-span of the rods. The information of the system is given in the table in Figure 3. Figure 3 This mechanism is held at stationary when π = 30o . At this initial state, the spring is undeformed. It is then released from rest, and the disk is observed to be rolling without slipping on the ground. Determine the linear velocity of the disk when π = 90o . (25 marks) Hints: (i) Use the velocity equation to establish the relations between the angular velocities of the components. (ii) Look for their instantaneous centre of rotations at the final state. PAGE 10 ME2115 AY21/22-S1: QUESTION 4 (a) A uniform disk with a hole at the top part is pivoted at point O as shown in Figure 4(a). Given that the radius of gyration for the centroidal mass moment of inertia of this mass is ο« = 1.4505L. Show that this mass will oscillate freely under gravity at a frequency given by π πΏ ππ = 0.1778√ (10 πππππ ) Hints: The centroid of the area and centre of mass for the body is identical since is uniform. Use point O as the reference to determine its centroid. (b) Now, a spring of stiffness constant k = 10 N/m and a damper with damping constant c = 2 N/m·s is attached to the two sides of the mass, as shown in Figure 4(b). Suppose the length and its mass are given by L = 0.5 m and M = 1.0 kg, respectively. Determine the damped oscillating frequency of this system. Note: Sketch the FBD in your solution, and remember to include the weight. Figure 4(a) Figure 4(b) (15 marks)