Question 11.2: In Fig. 11.10, the mass m1 and spring k1 represent a single-......
In Fig. 11.10, the mass m_{1} and spring k_{1} represent a single-DOF system with low inherent damping. Its response z_{1}, due to the applied force, F_{0}, is found to be excessive. The obvious solution of adding a damper in parallel with k_{1} is found not to be feasible in this case, and it is proposed to add a damped vibration absorber consisting of m_{2}, k_{2} \mathrm{and} c, where the mass of m_{2} is one-tenth that of m_{1}.
(a) Define the properties of the vibration absorber non-dimensionally, and plot the non-dimensional magnitude of the displacement, \left|\underline{z}_{1} \right| / z_{s}, \mathrm{versus} \Omega_{1} = \omega / \omega_{1}. Discuss the improvement compared with the original system.
(b) If the mass of m_{1} is 100 kg, and the natural frequency of the original single-DOF system is 10 Hz, define the optimum properties of the absorber dimensionally.
Part (a):
The value of the mass ratio, \mu = m_{2}/m_{1} = 0.1.
From Eq. (11.16) the optimum value of R is
R_{\mathrm{opt}} = \frac{1}{1+\mu } = \frac{1}{1 +0.1} = 0.909 (A)From Eq. (11.17), with \mu = 0.1, the optimum value of the damping parameter \gamma = c/2m_{2}\omega _{1} \mathrm{is} \\ \gamma_{\mathrm{opt}} = \sqrt{\frac{3\mu }{8(1+\mu )^{3}} } = 0.168 (B)
Substituting the above values of \mu , R_{\mathrm{opt}} \mathrm{and} \gamma_{\mathrm{opt}} into Eq. (11.15), the plot shown as a solid line in Fig. 11.12 was obtained using a standard spreadsheet. The maximum magnification of the system, with the optimized damper, is about 4.6.
An assessment of the improvement effected by adding the damper can be made by plotting the magnification of the original single-DOF system, without the added damper, for comparison. This is given by Eq. (4.12), which with the present notation is
\frac{\left|z\right| }{z_{s}} = \frac{1}{\sqrt{(1 – \Omega^{2})^{2} + (2\gamma \Omega )^{2}} } (4.12) \\ \frac{\left|z_{1}\right| }{z_{s}} = \frac{1}{\sqrt{(1 – \Omega^{2}_{1})^{2} + (2\underline{\gamma }\Omega _{1} )^{2}} } (C)where \underline{\gamma } is the damping coefficient of the original single-DOF system before adding the auxiliary damper. Taking \underline{\gamma } = 0.02, the original magnification curve is shown as a dashed line in Fig. 11.12. In comparing the two curves in Fig. 11.12 it should be remembered that the damping in the original single-DOF system was neglected when deriving the response for the system with the added damper, giving a small but conservative error. Ignoring this, the maximum magnification at the worst excitation frequency in each case is seen to be reduced by a factor of at least 5.
Part (b):
To define the properties of the vibration absorber dimensionally, we have
Thus the auxiliary damper has
Mass = m_{2} = 10 kg;
Spring stiffness = k_{2} = 32 620 N/m;
Damper constant = c = 211 N/m/s.
