Figure Q3 shows an idealised mathematical model of a mechanical vibrating system which has two degrees of freedom. The system is constrained to move along a straight line on a horizontal plane. Coordinates X1 and X2 refer to the linear displacement measured from the static equilibrium position of the relevant masses. a. Assuming that X1 is bigger than X2, draw the free body diagram for the system, showing applied forces on the two masses and their mass accelerations. (4 marks) b. Derive the equations of motion for each mass using Newton's second law. c. If the values of the masses and the spring stiffness are: • M₁ = 11 kg • M2 = 2 kg • k₁ = 2400 N/m • k₂ = 300 N/m i. (2 marks) Determine the two natural frequencies of the vibrating system using the characteristic equation method. (6 marks) ii. Determine the associated mode shapes of the vibrating system using the frequency equation method. (4 marks) iii. Provide a graphical representation of the mode shapes identifying the location of any nodes. (4 marks) x2 X1 k2 Κι M2 M1 Figure Q3

Figure Q3 shows an idealised mathematical model of a mechanical vibrating system which has two degrees of freedom. The system is constrained to move along a straight line on a horizontal plane. Coordinates X1 and X2 refer to the linear displacement measured from the static equilibrium position of the relevant masses. a. Assuming that X1 is bigger than X2, draw the free body diagram for the system, showing applied forces on the two masses and their mass accelerations. (4 marks) b. Derive the equations of motion for each mass using Newton's second law. c. If the values of the masses and the spring stiffness are: • M₁ = 11 kg • M2 = 2 kg • k₁ = 2400 N/m • k₂ = 300 N/m i. (2 marks) Determine the two natural frequencies of the vibrating system using the characteristic equation method. (6 marks) ii. Determine the associated mode shapes of the vibrating system using the frequency equation method. (4 marks) iii. Provide a graphical representation of the mode shapes identifying the location of any nodes. (4 marks) x2 X1 k2 Κι M2 M1 Figure Q3

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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