Theory of Machine
The velocity of piston in a reciprocating steam engine is given by (where ω = Angular velocity of crank, r = Radius of crank pin circle, θ = Angle turned by crank from inner dead center, and n = Ratio of length of connecting rod to the radius of crank)

ωr [cos θ + (cos 2θ/n)]
ω²r [sin θ + (sin 2θ/n)]
ω²r [cos θ + (cos 2θ/n)]
ωr [sin θ + (sin 2θ/n)]

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Theory of Machine
The static balancing is satisfactory for low speed rotors but with increasing speeds, dynamic balancing becomes necessary. This is because, the

Unbalanced forces are not dangerous at higher speeds
Unbalanced couples are caused only at higher speeds
Effects of unbalances are proportional to the square of the speed
Effects of unbalances are directly proportional to the speed

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