Question 11.3: (a) Find the optimum damping for a springless vibration abso......
(a) Find the optimum damping for a springless vibration absorber where the absorber mass is one-half of the mass of the main system. Use Eq. (11.15) to plot the nondimensional response of the optimized system, and compare it with non-optimized cases.
(b) Compare the optimized response above with that for an absorber with the same mass ratio, but having an optimized spring as well as a damper.
Part (a):
The mass ratio \mu = m_{2} / m_{1} = 0.5 in this case. The optimum value of the damping parameter, \gamma _{\mathrm{opt}}, is given by Eq. (11.18) as:
The response curve is plotted from Eq. (11.15) in Fig. 11.13, using \gamma _{\mathrm{opt}} = 0.365, \mathrm{with} R = 0 \mathrm{and} \mu = 0.5. Curves for the non-optimized values of the damping parameter, \gamma = 0.2 \mathrm{and} \gamma = 0.8, are seen to give increased maximum response.
Part (b):
For an absorber with the same mass ratio, \mu = 0.5, except that a spring is used, and both spring and damper are optimized, the optimum value of R is given by Eq. (11.16):
and the optimum damping is given by Eq. (11.17) with \mu = 0.5: \\ \gamma _{\mathrm{opt}} = \sqrt{\frac{3\mu }{8(1+\mu )^{3}} } = 0.236 (C)
The resulting response curve is plotted from Eq. (11.15) in Fig. 11.14, together with the optimized response with a springless absorber. A system with a tuned spring is seen to give considerably lower (i.e. better) maximum response than a springless design.
