Question 5.5: Find the response of the single degree of freedom system sho...
Find the response of the single degree of freedom system shown in Fig. 5.9 to the rectangular impulsive force shown in Fig. 5.8, where m = 10 kg, k = 9,000 N/m, c = 18N · s/m, and F_0 = 10,000 N. The force is assumed to act at time t = 0 and the impact interval is assumed to be 0.005 s.
The linear impulse I is given by
I = \int_ {t_{1}}^{t_{2}}F(t) dt = \int_0^{0.005} 10,000 dt = 10,000(0.005) = 50 N · s
The natural frequency of the system ω is given by
ω = \sqrt{\frac{k}{m}} =\sqrt{\frac{9000}{10}}= 30 rad/s
The critical damping coefficient C_c is
C_c = 2mω = 2(10)(30) = 600
The damping factor ξ is
ξ = \frac{c}{C_c} = \frac{18}{600} = 0.03
The damped natural frequency ω_d is
ω_d = ω\sqrt{1 − ξ^2} = 30\sqrt{1 − (0.03)^2} = 29.986 rad/s
The system response to the impulsive force is then given by
x(t) = \frac{I}{mω_d} e^{−ξωt} \sin ω_dt\\[0.5 cm] = \frac{50} {(10)(29.986)}e^{-(0.03)(30)t} \sin 29.986t\\[0.5 cm] = 0.1667e^{−0.9t} \sin 29.986t