Cycloidal radial cam design: 22-Mec-A2 Kinematics and Dynamics of Machines, December 2013, Q2

Question 2: Cycloidal radial cam design (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Lift $h=20$ mm, $\omega=1000$ rpm $=104.72$ rad/s (constant). Rise angle $\beta_r=120^\circ=2.094$ rad; fall angle $\beta_f=240^\circ=4.189$ rad. Cycloidal (sine-acceleration) programme.

Find. s, v, a, j laws (rise & fall), $a_\mathrm{max}$ and $j_\mathrm{max}$ on the rise, a base-circle radius and profile for a flat-faced follower, and the pressure angles at 60°/240°/300°.

Approach. Write the standard cycloidal laws in the local angle of each segment, differentiate in time via $\dot\theta=\omega$ to get v, a, j, evaluate the closed-form peaks, then size $R_0$ from the flat-faced cusp condition and read off the (identically zero) pressure angle.

Rise laws (local angle $\phi=\theta$, $0\le\theta\le\beta_r$). $$s=h\!\left[\frac{\theta}{\beta_r}-\frac{1}{2\pi}\sin\frac{2\pi\theta}{\beta_r}\right],\quad v=\frac{h\omega}{\beta_r}\!\left[1-\cos\frac{2\pi\theta}{\beta_r}\right],$$ $$a=\frac{2\pi h\omega^{2}}{\beta_r^{2}}\sin\frac{2\pi\theta}{\beta_r},\quad j=\frac{4\pi^{2}h\omega^{3}}{\beta_r^{3}}\cos\frac{2\pi\theta}{\beta_r}.$$
Fall laws (local angle $\psi=\theta-\beta_r$, $0\le\psi\le\beta_f$). $$s=h\!\left[1-\frac{\psi}{\beta_f}+\frac{1}{2\pi}\sin\frac{2\pi\psi}{\beta_f}\right],\quad v=-\frac{h\omega}{\beta_f}\!\left[1-\cos\frac{2\pi\psi}{\beta_f}\right],$$ $$a=-\frac{2\pi h\omega^{2}}{\beta_f^{2}}\sin\frac{2\pi\psi}{\beta_f},\quad j=-\frac{4\pi^{2}h\omega^{3}}{\beta_f^{3}}\cos\frac{2\pi\psi}{\beta_f}.$$

Cycloidal rise programme over [0°,120°]: displacement, velocity, acceleration and jerk. s and v are smooth and zero-ended; a is a full sine (zero at both ends); j is a cosine.

Peak acceleration and jerk on the rise. The sine peaks at $\theta=\beta_r/4$: $$a_\mathrm{max}=\frac{2\pi h\omega^{2}}{\beta_r^{2}}=\frac{2\pi(0.020)(104.72)^2}{2.094^2}=\;$$$a_\mathrm{max} = 314\ \text{m/s}^2$. The cosine peaks at the ends: $j_\mathrm{max}=\dfrac{4\pi^{2}h\omega^{3}}{\beta_r^{3}}=$ $9.87\times10^{4}\ \text{m/s}^3$.
Base circle for the flat-faced follower. The instantaneous radius of curvature is $\rho=R_0+s+\dfrac{d^{2}s}{d\theta^{2}}$; to avoid a cusp (undercut) this must stay positive. The minimum of $\left(s+s^{\prime\prime}\right)$ over the cycle is $-10.7$ mm, so $R_0>10.7$ mm; a practical choice with margin is $R_0=40$ mm.

Cam profile for the flat-faced follower generated from the cycloidal lift on a 40 mm base circle (dashed = base circle).

Pressure angles at 60°, 240°, 300°. For a translating flat-faced follower the contact normal is always parallel to the follower axis, so the pressure angle is identically zero at every cam angle: φ(60°) = φ(240°) = φ(300°) = 0°. (The point of contact shifts sideways by $s^{\prime}(\theta)=ds/d\theta$ from the axis, which sets the required minimum face width, but the force direction — hence the pressure angle — never tilts.)

Quantity	Value
Max rise acceleration	314 m/s² (at θ = 30°)
Max rise jerk	9.87×10⁴ m/s³ (at θ = 0, 120°)
Base-circle radius	R₀ = 40 mm ( > 10.7 mm cusp limit )
Pressure angle (flat follower)	0° at all cam angles

Topic. Cycloidal (sinusoidal-acceleration) cam law and flat-faced-follower geometry.

Key relations. Cycloidal $s=h[\theta/\beta-\sin(2\pi\theta/\beta)/2\pi]$; peaks $a_\mathrm{max}=2\pi h\omega^2/\beta^2$, $j_\mathrm{max}=4\pi^2 h\omega^3/\beta^3$; flat-face cusp condition $\rho=R_0+s+s^{\prime\prime}>0$; pressure angle $\phi\equiv0$ for a flat face.

Why this works. The cycloidal law has zero acceleration at both ends of each segment, so the acceleration curve is continuous through the dwell junctions and the jerk stays finite — the smoothest of the standard full-rise programmes. The flat face means the common normal always coincides with the follower axis, eliminating the pressure angle entirely; the design trade is a larger base circle to prevent the profile from becoming concave (cusp).

Common pitfall. Differentiating with respect to cam angle and forgetting the $\omega$ and $\omega^{2}$ factors when converting to time derivatives; and applying the roller-follower pressure-angle formula to a flat face. Norton, Design of Machinery, ch. 8.

Question 2 of 6: Cycloidal radial cam design

Question 2: Cycloidal radial cam design (20 marks)