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
2π. √(q/I)
1/2π
(1/2π). √(q/I)

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Theory of Machine
The inversion of a mechanism is

Turning its upside down
Changing of a higher pair to a lower pair
Obtained by reversing the input and output motion
Obtained by fixing different links in a kinematic chain

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

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