Theory of Machine
The natural frequency of free torsional vibrations of a shaft is equal to (where q = Torsional stiffness of the shaft, and I = Mass moment of inertia of the disc attached at the end of a shaft)

2π qI
1/2π
2π. √(q/I)
(1/2π). √(q/I)

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Theory of Machine
In the below figure, PC is the connecting rod and OC is the crank making an angle θ with the line of stroke PO and rotates with uniform angular velocity at ω rad/s. The Klien's acceleration diagram for determining the acceleration of the piston P is shown by quadrilateral CQNO, if N coincides with O, then

Acceleration of the piston P is maximum and its velocity is zero
Acceleration of the piston P is zero and its velocity is maximum
Acceleration and velocity of the piston P is maximum
Acceleration and velocity of the piston P is zero

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Theory of Machine
The primary unbalanced force due to inertia of reciprocating parts in a reciprocating engine is given by (where m = Mass of reciprocating parts, ω = Angular speed of crank, r = Radius of crank, θ = Angle of inclination of crank with the line of stroke, and n = Ratio of the length of connecting rod to radius of crank)

m.ω².r (sin 2θ/n)
m.ω².r cosθ
m.ω².r sinθ
m.ω².r (cos 2θ/n)

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