Determine the undamped and damped natural frequencies of the system shown in Fig.17.19. k1 = 2 kN/m, k2 = 3 kN/m, c1 = 100 N.s/m, c2 = 200 N.s/m, and m = 15 kg.

Question

Determine the undamped and damped natural frequencies of the system shown in Fig.17.19. [latex] k_{1}=2 kN / m , k_{2}=3 kN / m , c_{1}=100 N \cdot s / m , c_{2}=200 N \cdot s / m , ext { and } m=15 kg [/latex].

Accepted Answer

Equivalent stiffness, [latex] k_{e}=\frac{k_{1} k_{2}}{k_{1}+k_{2}} [/latex]. for springs in series.

[latex] =\frac{2 imes 3}{2+3}=1.2 kN / m [/latex].

Equivalent damping coefficient, [latex] c_{e}=\frac{c_{1} c_{2}}{c_{1}+c_{2}} [/latex]. for dampers in series.

[latex] =\frac{100 imes 200}{100+200}=\frac{200}{3} N \cdot s / m [/latex].

Undamped natural frequency,

[latex] \omega_{n}=\sqrt{\frac{k_{e}}{m}} [/latex].

[latex] =\sqrt{\frac{1.2 imes 10^{3}}{15}}=8.94 rad / s [/latex].

Critical damping coefficient, [latex] c_{c}=2 m \omega_{n}=2 imes 15 imes 8.94=268.328 N \cdot s / m [/latex].

Damping ratio, [latex] \zeta=\frac{c_{e}}{c_{c}}=\frac{200}{3 imes 268.328}=0.24845 [/latex].

Damped natural frequency, [latex] \omega_{d}=\omega_{n} \sqrt{1-\zeta^{2}} [/latex].

[latex] =8.94 \sqrt{1-(0.24845)^{2}} [/latex].

[latex] =8.66 rad / s [/latex].

Question 17.7: Determine the undamped and damped natural frequencies of the...

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