ME3001 A4 Introductory Problems in Vibration

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ME3001 Dynamics of Machinery Monsoon 2012 Assignment 4 Introductory problems in Vibration Submit on 25-10-12 1. An unknown mass of m kg attached to the end of unknown spring k has a natural frequency of 100cpm.When a ½ kg mass is added to m, the natural frequency is lowered to 75 cpm. Determine the unknown mass m and the spring constant k N/m. [Thomson, ans: k= 70.461N/m] 2. Determine the natural frequency of the system shown in Fig.1. The point A (where the ends of the springs are attached to the cylinder) is at a distance a vertically above the centre. The cylinder rolls without slip. [Ghosh and Mallik, ans: [ 4k/(3m)] 1/2 (1+a/r) ] Fig.1 3. Shames 19.30 4. (Energy method) Shames 19.46 5. (Damped forced vibration) A spring–mass is excited by a force F 0 sinωt. At resonance the amplitude is measured to be 0.6 cm. At 0.8 resonant frequency, the amplitude is measured to be 0.45 cm. Determine the damping factor ξ for the system - this is a method for determination of ξ. [ref. Thomson, ans: c=39.81Ns/m] Exercise 4 1. (Springs in series and parallel) Shames 19.2 2. Shames 19.6 3. Shames 19.10 4. Shames 19.12 5. (Equations of motion-Free Vibration, work out by Newton’s law, Energy method) Shames 19.10 6. Shames 19.48 1

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Transcript of ME3001 A4 Introductory Problems in Vibration

Page 1: ME3001 A4 Introductory Problems in Vibration

ME3001 Dynamics of Machinery Monsoon 2012

Assignment 4 Introductory problems in Vibration Submit on 25-10-12

1. An unknown mass of m kg attached to the end of unknown spring k has a natural frequency of 100cpm.When a ½ kg mass is added to m, the natural frequency is lowered to 75 cpm. Determine the unknown mass m and the spring constant k N/m. [Thomson, ans: k= 70.461N/m]

2. Determine the natural frequency of the system shown in Fig.1. The point A (where the ends of the springs are attached to the cylinder) is at a distance a vertically above the centre. The cylinder rolls without slip. [Ghosh and Mallik, ans: [ 4k/(3m)]1/2(1+a/r) ]

Fig.1

3. Shames 19.304. (Energy method) Shames 19.46 5. (Damped forced vibration) A spring–mass is excited by a force F0sinωt. At resonance the amplitude

is measured to be 0.6 cm. At 0.8 resonant frequency, the amplitude is measured to be 0.45 cm. Determine the damping factor ξ for the system - this is a method for determination of ξ. [ref. Thomson, ans: c=39.81Ns/m]

Exercise 4

1. (Springs in series and parallel) Shames 19.22. Shames 19.63. Shames 19.104. Shames 19.125. (Equations of motion-Free Vibration, work out by Newton’s law, Energy method) Shames 19.10 6. Shames 19.487. Shames 19.548. Shames 19.589. Shames 19.10410. Two cylindrical tanks of cross-sectional areas A1 and A2 are connected by a pipe of length l and cross

sectional area Ap as shown in Fig.2. If h is the equilibrium height of the liquid surface, determine the natural frequency of oscillation of the fluid when disturbed. [Ghosh, ans: {g(1+ A1/A2)/[h{1+ A1/A2

+lA1/(hAp)}]}1/2 ]

11. What will be the natural frequency of rocking motion of the rectangular block shown in Fig.3? What happens when R<H/2? Assume no slip condition. [Ghosh, ans: g(R-H/2)(L2/12+H2/3)]1/2 ]

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Fig.2 Fig.3Fig.2

12. The string shown in Fig.4 is in tension T which is assumed to remain constant for small displacements. Find the natural frequency of the vibration of the vertical vibration of the string.

[W.W.Seto, ans: ωn=√ TLma(L−a)

rad/s.]

13. (Undamped forced vibration) Shames 19.56, 14. (Free-damped vibration) Shames19.76,15. (Free-damped vibration) A weight attached to a spring of stiffness 600N/m has a viscous damping

device. When the weight is displaced and released, the period of vibration is 2 s, and the ratio of consecutive amplitudes is 4 to 1. Determine the amplitude and phase when a force F=2cos3t acts on the system. [Thomson, ans: X=7.91 mm, φ=72.11o]

16. A 13.6 kg block is supported by spring arrangements shown in Fig.5. If the block is moved from equilibrium 44 mm vertically downward and released, determine (a) the period and the frequency of the resulting motion, the maximum velocity and acceleration of the block. [Beer and Johnston, ans: (a) 0.361 s; 2.77 Hz (b) 0.765 m/s; 13.31 m/s2]

Fig.4 Fig.5

17. (Logarithmic decrement) For the system shown in Fig.6, (a) obtain the coefficient of critical damping cc (b) if c = cc/2, find the damped natural frequency and (c) determine the logarithmic decrement.

18. Set up the equation of motion of the system shown in Fig.7 and find out the steady state solution.[Ghosh, ans: F0 cos ωt/(1-ω2m/k)]

Fig.6 Fig.7

19. (Damped forced vibration) Show that for the damped spring-mass system, the peak amplitude

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occurs at a frequency ratio given by the expression( ωωn

)p

=√¿¿. [Thomson]

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