Reference no: EM132094173

Learning Outcomes

LO1. Design mathematical models that analyse and evaluate mechanical systems

LO2. Explain and apply control theory and control system approaches to mechanical systems

LO3. Explain the role of engineering assumptions in building mathematical models of mechanical systems

LO4. Relate theory to problems of introducing, operating and maintaining mechanical systems in the industrial context

LO5. Identify and evaluate engineering uncertainty and the limitations of mathematical models

LO6. Work collaboratively in a team to produce high quality outputs

LO7. Create professional documentation using mechanical systems terminology, symbols and diagrams

Question 1: A mass m₁, hangs from a spring k N/m and is in static equilibrium. A second mass m₂, drops through a height h and sticks to m, without rebound, as shown in Fig. P2.5. Determine the subsequent motion.

                                                            FIGURE P2.5.                           FIGURE P2.7.

Question 2. A connecting rod weighing 21.35 N oscillates 53 times in 1 min when suspended as shown in Fig. P2.8. Determine its moment of inertia about its center of gravity, which is located 0.2.54 m from the point of support.

FIGURE P2.8.

Question 3. A wheel and axle assembly of moment of inertia I is inclined from the vertical by an angle a, as shown in Fig. P2.10. Determine the frequency of oscillation due to a small unbalance weight w lb at a distance a in. from the axle.

FIGURE P2.10, FIGURE P2.11.

Question 4. A uniform bar of length L is suspended in the horizontal position by two vertical strings of equal length attached to the ends. If the period of oscillation in the plane of the bar and strings is ti and the period of oscillation about a vertical line through the center of gravity of the bar is r2, show that the radius of gyration of the bar about the center of gravity is given by the expression

k = (t₂/t₁)L/2

Question 5. Determine the effective mass at point n and its natural frequency for the system shown in Fig. P2.24.

Fig. P2.24.

Question 6. Determine the effective rotational stiffness of the shaft in Fig. P2.29 and calculate its natural period.

                                                         FIGURE P2.30.                             FIGURE P2.29.

Question 7. A machine part of mass 1.95 kg vibrates in a viscous medium. Determine the damping coefficient when a harmonic exciting force of 24.46 N results in a resonant amplitude of 1.27 cm with a period of 0.20 s.

Question 8. Show that for the dampled spring-mass system, the peak amplitude occurs at a frequency ratio given by the expression

(ω/ω_n)_p = √(1-2ζ²)

Question 9. A counterrotating eccentric mass exciter shown in Fig. P3.13 is used to determine the vibrational characteristics of a structure of mass 181,4 kg. At a speed of 900 rpm, a stro¬boscope shows the eccentric masses to be at the top at the instant the structure is moving upward through its static equilibrium position, and the corresponding amplitude is 21.6 rnm, If the unbalance of each wheel of the exciter is 0.0921 kg - rn, determine (a) the natural frequency of the structure, (b) the damping factor of the structure, (c) the amplitude at 1200 rpm, and (d) the angular position of the eccentrics at the instant the structure is moving upward through its equilibrium position.

Fig. P3.13

Question 10. Figure P3.20 represents a simplified diagram of a spring-supported vehicle traveling over a rough road. Determine the equation for the amplitude of W as a function of the speed, and determine the most unfavorable speed.

Figure P3.20

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