Question 13.3.2: Suppose the main mass in the system in Example 13.3.1 has th......
Suppose the main mass in the system in Example 13.3.1 has the value m_1 = 0.8 slug. Evaluate the sensitivity of the absorber design to variations in the input frequency.
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} = 2π(5) = 10π rad/sec. Thus, the stiffness is
k_1=\omega_{n_1}^2 m_1=(10 \pi)^2 0.8=790 lb / ftand
r_1=\frac{\omega}{\omega_{n_1}}=\frac{\omega}{10 \pi}
r_2=\frac{\omega}{\omega_{n_2}}=\frac{\omega}{8 \pi}
Substituting these values into (13.3.5) with F = mϵω², we obtain
T_1(j \omega)=\frac{1}{k_1} \frac{1 – r_2^2}{\left(1 + k_2 / k_1 – r_1^2\right)\left(1 – r_2^2\right) – k_2 / k_1} (13.3.5)
\frac{X_1}{m \epsilon}=\left|\frac{\omega^2\left(1 – \omega^2 / 64 \pi^2\right)}{\left(1.061 – \omega^2 / 100 \pi^2\right)\left(1 – \omega^2 / 64 \pi^2\right) – 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.
