An integral pulley of mass moment of inertia J about its axis, shown in Fig.17.55, is restrained in its movement about its own axis by a torsional spring of stiffness kt and a smaller pulley by means of an inextensible string. Determine the natural frequency.

Question

An integral pulley of mass moment of inertia J about its axis, shown in Fig.17.55, is restrained in its movement about its own axis by a torsional spring of stiffness [latex] k_{t} [/latex] and a smaller pulley by means of an inextensible string. Determine the natural frequency.

Accepted Answer

The equation of motion is,

[latex] J \ddot{ heta}+m \ddot{x} r+k y R+k_{t} heta=0 [/latex].

[latex] ext { Where } x=r heta ext { and } y=R heta [/latex].

[latex] J \ddot{ heta}+m r^{2} \ddot{ heta}+\left(k R^{2}+k_{t} ight) heta=0 [/latex].

[latex] \left(J+m r^{2} ight) \ddot{ heta}+\left(k R^{2}+k_{t} ight) heta=0 [/latex].

or [latex] \omega_{n}=\left(\frac{k R^{2}+k_{t}}{J+m r^{2}} ight)^{1 / 2} rad / s [/latex].

Question 17.31: An integral pulley of mass moment of inertia J about its axi...

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