The magnitude of radial force acting on a ball bearing varies in a sinusoidal manner as shown in Fig. 15.14, while the direction remains fixed. Determine the equation for the variation of the force P against the angle of rotation θ.

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Accepted Answer

In order to determine the equation of the sinusoidal curve shown in Fig.15.14, we will use the basic relationships of plane analytical geometry. Figure 15.15 shows the following two co-ordinate systems:
(i) Co-ordinate system (x, y) with origin at O.
(ii) Co-ordinate system [latex] \left(x^{\prime}, y^{\prime} ight) ext { with origin at } O^{\prime} [/latex].
The co-ordinates of the origin [latex] O^{\prime} [/latex] with respect to the origin O are [latex] \left(x_{o}, y_{o} ight) [/latex]. The relationships for transformation of co-ordinates with pure translation are as follows:

[latex] x^{\prime}=x-x_{o} [/latex] (a).

[latex] y^{\prime}=y-y_{o} [/latex] (b).

Refer to the sine curve shown in Fig. 15.16. The equation of this curve with respect to the [latex] \left(x^{\prime}, y^{\prime} ight) [/latex] coordinate system is given by,

[latex] y^{\prime}=\left(\frac{1}{2} P_{\max } ight) \sin x^{\prime} [/latex] (c).

Also, [latex] x_{o}=\frac{\pi}{2} \quad ext { and } \quad y_{o}=\frac{P_{\max }}{2} [/latex] (d)

From expressions (a), (b) and (d),

[latex] x^{\prime}=x-\frac{\pi}{2} \quad ext { and } \quad y^{\prime}=y-\frac{P_{\max }}{2} [/latex] (e).

Substituting expressions (e) in Eq. (c),

[latex] \left(y-\frac{P_{\max }}{2} ight)=\left(\frac{P_{\max }}{2} ight) \sin \left(x-\frac{\pi}{2} ight) [/latex].

[latex] ext { Using the relationship } \sin (- heta)=-\sin heta ext {, } [/latex]

[latex] \left(y-\frac{P_{\max }}{2} ight)=-\left(\frac{P_{\max }}{2} ight) \sin \left(\frac{\pi}{2}-x ight) [/latex].

[latex] =-\left(\frac{P_{\max }}{2} ight) \cos x [/latex].

Therefore,

[latex] y=\frac{P_{\max .}}{2}-\frac{P_{\max .}}{2} \cos x=\frac{P_{\max .}}{2}(1-\cos x) [/latex].

Replacing y by P and x by θ, (Fig.15.14)

[latex] P=\frac{P_{\max }}{2}(1-\cos heta) [/latex].

Question 15.13: The magnitude of radial force acting on a ball bearing varie...

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