Cork Institute of Technology. Summer 2007 Mechanics of Machines (Time: 3 Hours)


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1 Cork Institute of Technology Bachelor of Engineering (Honours) in Mechanical Engineering Award Instructions Answer FOUR questions. All questions carry equal marks. (NFQ Level 8) Summer 2007 Mechanics of Machines (Time: 3 Hours) Examiners: Mr. P. Clarke Prof. M. Gilchrist Dr. F. S. Murphy Q1. (a) Many vibrationrecording instruments are based on the principle of the seismograph. Describe the principle of operation of this instrument and some of the range of other vibration measuring instruments available. (b) Find the expression for the frequency ratio at which a damped vibrometer gives the maximum ratio of recorded amplitude of vibration to the amplitude of actual vibration. (10marks) (c) If the maximum positive error recorded by the instrument is not to exceed 2 percent, determine the damping ratio necessary. (d) Determine the frequency ratio at which the instrument could be used if an error of ±2% could be tolerated. Q2. (a) For a simply supported statically loaded beam, carrying a uniformly distributed load of w per unit length, there are theoretically an infinite number of frequencies at which the beam displays characteristics of transverse vibration. Sketch the first three modes of vibration. (b) Show that the fundamental frequency of the system described in part (a), is given by the expression:
2 N = π 2 gei wl 4 (10 marks) (c) A uniform steel bar has a length of 1.2m. The density of steel is 7834 kg/m 3 and the modulus of elasticity (E) is 207 GPa. Determine the frequency of the first mode of longitudinal vibration if both ends are clamped. (d) Determine the frequency of the first mode of longitudinal vibration of the system described in part (c), if both ends are free. Q3. (a) Describe how the Holzer method of determining natural frequencies can be used in the interpretation of results obtained from measurements of vibration in practice. (b) A sixcylinder engine and flywheel are coupled to a load of moment of inertia of 2248 kgm 2. The load is coupled to the flywheel by a shaft of stiffness 26.2 x 10 6 Nm/rad. The flywheel has a moment of inertia of 8194 kgm 2 and is coupled to the engine by a shaft of stiffness 65 x 10 6 Nm/rad. Each cylinder can be represented by a small flywheel of moment of inertia 15.5 kgm 2. The six small flywheels, which represent the crank throws and associated masses of the cylinders can be regarded as coupled together by relatively stiff shafts of stiffness 64.7 x 10 6 Nm/rad. By reducing the system to a two mass system or otherwise obtain a first approximation for the first mode of torsional vibration. (10 marks) (c) Using the obtained first approximation produce the Holzer table and comment on the result. (10 marks) Q4. During the operation of a machine, vibration isolators are used to protect the foundations from the large forces that develop. (a) Show that the transmissibility ratio in such a case is of the form: T = 1+ (2ζr) 2 (1 r 2 ) 2 + (2ζr) 2 where ζ is the damping ratio and r is the frequency ratio. (8 marks) 2
3 (b) A 400 kg turbine operates at speeds between 1000 and 2000 rev/min. The turbine has a rotating unbalance of 0.25 kgm. Determine the required stiffness of an undamped isolator such that the force transmitted to the foundations is 1000 N. (8 marks) (c) Determine the required stiffness of the isolator, if the damping ratio is 0.1, the maximum force transmitted to the foundations remaining the same. (4 marks) (d) With a damping ratio of 0.1, a maximum force transmitted to the foundations of 1000 N and an upper operating speed of 2500 rev/min, examine the vibration isolator for suitability. Q5. Figure (Question 5a) shows schematically a three storey building with each floor represented by lumped masses. The first floor mass is 3m, the second floor mass is 2m and the top floor mass is m. The value of m is 800x10 3 kg. Between the ground floor and first floor the stiffness is 3k. Between first floor and second floor the stiffness is 2k. Between the second floor and top floor the stiffness is k. The value of k is 400x10 6 N/m as shown by the values given in the diagram. Figure (Question 5a) 3
4 (a) Determine the natural frequencies of the building. (b) Determine the corresponding normal mode configurations of the building. Figure (Question 5b) shows two identical discs of centroidal mass moment of inertia I attached to a steel shaft that is fixed at the right hand end. Each section of the shaft has a diameter, d = 32 mm and a length, l = 600 mm. The mass moment of inertia of the disc, I = 1.36 kgm 2 and the modulus of rigidity of the shaft material = 83 GPa. Figure (Question 5b) (c) Determine the eigenvalues of the system. (d) Determine the natural frequencies of the system. (e) Determine the eigenvectors of the system. 4
5 Q6. An 18 kg wheel shown in Figure (Question 6) is attached to a balancing machine and made to spin at a rate of 750 rev/min. It is found that the forces exerted by the wheel on the machine are equivalent to a forcecouple system consisting of a force F = (160 N)j applied at C and a couple M c =  (14.7 Nm)k, where the unit vectors form a right handed triad which rotates with the wheel. (a) Determine the distance from the axis of rotation to the mass centre of the wheel. (b) Determine the products of inertia I xy and I zx. (10 marks) (c) If only two corrective masses are to be used to statically and dynamically balance the wheel, determine their values. (d) Determine which of the points A, B, D or E should the balance masses be placed. Figure (Question 6) 5
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