The fig 1.12 represents various main dimentions of circular arc cam.
Fig 1.12
Expression for determining the displacement, velocity and acceleration of the follower when flat face of the follower has contact on the circular flank
Fig 1.13
Let,
r1 = OB = Least base circle radius
r2 = Nose circle radius
R = QD = Flank circle radius
d = Distance between the centre of cam and nose circle
α = Angle of ascent
φ = Angle of contact on circular flank
Displacement:
X=BC=OC−OB=DE−r1$X=BC=OC-OB=DE-{r}_{1}$
=(QD−QE)−r1$=(QD-QE)-{r}_{1}$ =(R−OQcosθ)−r1$(R-OQcos\theta )-{r}_{1}$= R−(R−r1)cosθ−r1$R-(R-{r}_{1})cos\theta -{r}_{1}$
Thus,
\(X = (R - r_1)(1 - cos \theta)\)
Velocity:
\(V = \frac {dx}{dt} = \frac {dx}{d\theta}\times \frac {d\theta}{dt}\)= \(\omega(R - r_1)(sin \theta)\)= \(V = \omega (R - r_1)sin \theta\)
From this equation, it is evident that, at the beginning of the ascent, the velocity is zero (when &theta = 0) and it increases with θ. It will be maximum when the follower is just shift from circular flank to circular nose.
Vmax=ω(R−r1)sinϕ${V}_{max}=\omega (R-{r}_{1})sin\varphi $
Acceleration:
a=dvdt=dvdθ×dθdt$a=\frac{dv}{dt}=\frac{dv}{d\theta}\times \frac{d\theta}{dt}$
V=dxdt=dxdθ×dθdtandhencea=ω2(R−r1)cosθ$V=\frac{dx}{dt}=\frac{dx}{d\theta}\times \frac{d\theta}{dt}\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}hence\phantom{\rule{1em}{0ex}}a={\omega}^{2}(R-{r}_{1})cos\theta $
It is obvious from the above equation that, at the beginning of the ascent when θ = 0, acceleration is maximum and it goes on decreasing and is maximum when θ=ϕ$\theta =\varphi $
amax=ω2(R−r1)${a}_{max}={\omega}^{2}(R-{r}_{1})$ and ,
amin=ω2(R−r1)cosϕ${a}_{min}={\omega}^{2}(R-{r}_{1})cos\varphi $
Expression for determining the displacement, velocity and acceleration of the follower when flat face of the follower has contact on the nose
Fig 1.14
Let,
r1=${r}_{1}=$ OB =least base circle
r2${r}_{2}$= nose circle radius
R=QD=flank circle radius
d= distance between centers of cam and nose circle
α$\alpha $= angle of ascent
ϕ$\varphi $=angle of contact on cicular flank.
Displacement
x=BC=OC−OB=DE−r1$x=BC=OC-OB=DE-{r}_{1}$
= DP+PE−r1$DP+PE-{r}_{1}$=r2+OPcos(α−θ)−r1$={r}_{2}+OPcos(\alpha -\theta )-{r}_{1}$, thus
x=r2+dcos(α−θ)−r1$x={r}_{2}+dcos(\alpha -\theta )-{r}_{1}$
Velocity
v=dxdt=dxdθ.dθdt$v=\frac{dx}{dt}=\frac{dx}{d\theta}.\frac{d\theta}{dt}$,
therefore
the veclocity,
v=ωdSin(α−θ)$v=\omega dSin(\alpha -\theta )$,
the velocity is minimum when α=θ$\alpha =\theta $, this happens when the follower is at the apex of the circular nose and it is maximum when (α−θ)$(\alpha -\theta )$ is maximum and it is so when the contact chnaes from circluar flank to circlular nose i.e when α−θ=ϕ$\alpha -\theta =\varphi $
Acceleration
a=dvdt=dvdθ.dθdt$a=\frac{dv}{dt}=\frac{dv}{d\theta}.\frac{d\theta}{dt}$ =−ω2dcos(α−θ)$-{\omega}^{2}dcos(\alpha -\theta )$,
negative sin indicates retardation. It is maximum when α−θ=0,$\alpha -\theta =0,$, i.e when the follower is at the apex of the nose and minimum when (α−θ)$(\alpha -\theta )$ is maximum, ie when the follower changes the contact from circular flank to circular radius.
amax=−ω2d${a}_{max}=-{\omega}^{2}d$ and amin=−ω2dcosϕ${a}_{min}=-{\omega}^{2}dcos\varphi $