Velocity, Acceleration and Cams

Once a mechanism's input speed is known, the speeds and accelerations of all its links follow from geometry. GATE tests three tools: relative velocity, instantaneous centres and the slider-crank formulas, plus the basics of cams.

Relative velocity

For two points A and B on the same rigid link rotating at ω: v_BA = ω × AB, perpendicular to AB. Velocity polygons are drawn from this rule, link by link.

Instantaneous centres (I-centres)

• A mechanism with n links has n(n − 1)/2 instantaneous centres.
• Kennedy's theorem: the three I-centres of any three links lie on a straight line.
• The velocity of a point = ω × (its distance from the link's I-centre relative to the frame).

Acceleration

• A point on a link rotating about a fixed point has centripetal acceleration ω²r and tangential acceleration αr.
• Coriolis acceleration appears when a slider moves along a rotating link: 2vω, perpendicular to the link. It is needed in quick-return and slotted-lever mechanisms.

Slider-crank mechanism

For crank radius r, connecting rod length l, n = l/r and crank angle θ from inner dead centre, at crank speed ω:

• Piston velocity v ≈ ωr (sin θ + sin 2θ/(2n))
• Piston acceleration a ≈ ω²r (cos θ + cos 2θ/n)
• Angular velocity of the rod ≈ ω cos θ / n

Cams

A cam turns rotary motion into a set follower motion.

• Terms: base circle, pitch curve, pressure angle (angle between the follower's path and the normal to the cam at contact), lift (stroke), dwell.
• Follower motions:
• Uniform velocity: infinite acceleration at the ends (jerk), so it is used only with modifications.
• Simple harmonic motion: max velocity = (πh/2)(ω/β); max acceleration = (π²h/2)(ω/β)², where β is the cam angle for the rise.
• Uniform acceleration and retardation (parabolic).
• Cycloidal: zero acceleration at the start and end, best for high speeds.

Worked example

A crank 100 mm long turns at 300 rpm; the rod is 400 mm (n = 4). Find the piston velocity at θ = 90°.

• ω = 2π × 300/60 = 31.42 rad/s.
• v = 31.42 × 0.1 × (sin 90° + sin 180°/8) = 3.14 m/s.

Practice