Question 13.3.2: Sensitivity Analysis of Absorber Design Suppose the main mas...

The absorber values from the previous example are k_{2} = 48  lb/ft and m_{2} = 0.076  slug. The natural frequency of the machine with its supports, in radians per second, is ω_{n_{1}} = 2 π(5) = 10 π  rad/sec. Thus, the stiffness is
k_{1} = ω^{2}_{n_{1}} m_{1} = (10 π)^{2} 0.8 = 790  lb/ft

and
r_{1} = \frac{ω}{ω_{n_{1}}} = \frac{ω}{10 π}
r_{2} = \frac{ω}{ω_{n_{2}}} = \frac{ω}{8 π}
Substituting these values into (13.3.5) with F = m ε ω², we obtain

T_{1}(jω) = \frac{1}{k_{1}} \frac{1  −  r^{2}_{2}}{(1  +  k_{2}/k_{1}  −  r^{2}_{1})(1  −  r^{2}_{2})  −  k_{2}/k_{1}}                (13.3.5)

\frac{X_{1}}{m ε} = \left| \frac{ω^{2}(1  −  ω^{2}/64 π^{2})}{(1.061  −  ω^{2}/100 π^{2})(1  −  ω^{2}/64 π^{2})  −  0.061} \right| \frac{1}{790}
Figure 13.3.5 shows a plot of X_{1}/mε versus ω. The plot shows that resonance occurs when the input frequency ω is near one of the two natural frequencies, which can be found from the roots of the denominator of X_{1}/mε . These frequencies are 23.54 and 33.53 rad/sec. The plot indicates the sensitivity of the design to changes in ω. From this plot, we can see that the motion amplitude of the main mass will be large if the input frequency is less than approximately 95% of its design value of 8π = 25.13 rad/sec.