The static deflection of an automobile on its springs is 80 mm. Find the critical speed when the automobile is travelling on a road which can be approximated by a sine wave of amplitude 60 mm and a wave length 12 m (Fig.17.56). Assume the damping to be 0.05. Also calculate the amplitude of

Question

The static deflection of an automobile on its springs is 80 mm. Find the critical speed when the automobile is travelling on a road which can be approximated by a sine wave of amplitude 60 mm and a wave length 12 m (Fig.17.56). Assume the damping to be 0.05. Also calculate the amplitude of vibration at 60 km/h.

Accepted Answer

[latex] \omega_{n}=\sqrt{\frac{g}{\delta_{s t}}}=\sqrt{\frac{9.81}{80 imes 10^{-3}}}=11.07 rad / s [/latex].

For dynamic magnification to be maximum,

[latex] \beta=\sqrt{1-2 \zeta^{2}} [/latex].

[latex] =\sqrt{1-2 imes(0.05)^{2}}=0.9975 [/latex].

[latex] ext { Excitation frequency, } \omega=\beta \omega_{n} [/latex].

[latex] =0.9975 imes 11.07=11.04 rad / s [/latex].

[latex] ext { Critical velocity of vehicle, } v=f \lambda=\frac{\omega \lambda}{2 \pi} [/latex].

[latex] \frac{11.04 imes 12}{2 \lambda}=21.09 m / s [/latex].

[latex] =75.92 km / h [/latex].

Excitation frequency, at 60 km/h,

[latex] \omega=\frac{2 \pi imes 60 imes 10^{3}}{3600 imes 12}=8.827 rad / s [/latex].

[latex] \beta=\frac{\omega}{\omega_{n}}=\frac{8.827}{11.07}=0.7883 [/latex].

[latex] ext { Amplitude of vehicle, } X=\frac{y \sqrt{1+(2 \zeta \beta)^{2}}}{\sqrt{\left(1-\beta^{2} ight)^{2}+(2 \zeta \beta)^{2}}} [/latex].

[latex] =\frac{60 \sqrt{1+(2 imes 0.05 imes 0.7883)^{2}}}{\sqrt{\left\{1-(0.07883)^{2} ight\}^{2}+(2 imes 0.05 imes 0.7883)^{2}}}=0.7883 [/latex].

[latex] =155.65 mm [/latex].

Question 17.32: The static deflection of an automobile on its springs is 80 ...

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