A cam with a translating flat-face follower is desired to have the follower motion y (θ) = 4[2πθ – θ²], 0 ≤ θ ≤ 2π Contact stress considerations dictate that the radius of curvature of the cam profile should not be less than 40 mm anywhere. The minimum permissible base circle radius is _____ mm (round off to one decimal place).

The correct answer is 48.

Let us analyze the motion of the translating flat-face follower step-by-step to find the minimum permissible base circle radius.

For a cam with a translating flat-face follower, the radius of curvature of the cam profile, denoted as $R_c$, at any angle $\theta$ is given by the relation:
$R_c=R_b+y\left(\theta \right)+\frac{d^{2}y}{d{\theta }^2}$
where:
- $R_b$ is the base circle radius of the cam.
- $y\left(\theta \right)$ is the displacement of the follower as a function of cam rotation angle $\theta$.
- $\frac{d^{2}y}{d{\theta }^2}$ represents the acceleration term (with respect to $\theta$).

The given displacement equation of the follower is:
$y\left(\theta \right)=4[2\pi \theta -{\theta }^2]$, for $0\le \theta \le 2\pi$

First, let us calculate the first derivative of the displacement with respect to $\theta$:
$\frac{dy}{d\theta }=4(2\pi -2\theta )=8(\pi -\theta )$

Next, we calculate the second derivative of the displacement with respect to $\theta$:
$\frac{d^{2}y}{d{\theta }^2}=\frac{d}{d\theta }\left[8\right(\pi -\theta \left)\right]=-8$

Now, substituting the expressions of $y\left(\theta \right)$ and its second derivative into the radius of curvature formula, we get:
$R_c\left(\theta \right)=R_b+4[2\pi \theta -{\theta }^2]-8$

To avoid undercutting and keep contact stresses within limits, the minimum radius of curvature $(R_c{)}_{min}$ must not be less than 40 mm:
$(R_c{)}_{min}=R_b+{\left[y\right(\theta )+\frac{d^{2}y}{d{\theta }^2}]}_{min}\ge 40\text{mm}$

Since $\frac{d^{2}y}{d{\theta }^2}=-8$ is constant, the minimum value of the expression occurs where the displacement $y\left(\theta \right)$ is at its minimum.
In the range $0\le \theta \le 2\pi$, the minimum displacement $y_{min}=0$ occurs at the boundaries $\theta =0$ and $\theta =2\pi$.

Substituting these minimum values into our limiting condition:
$40=R_b+[0-8]$
$40=R_b-8$
$R_b=40+8=48\text{mm}$

Thus, the minimum permissible base circle radius is 48 mm.