Theory of Machine
A single degree of freedom system is given by, m × (d²x/dt²) + c × (dx/dt) + sx = F.cosω.t with usual notations. It represents

Free vibration with damping
Forced vibration without damping
Forced vibration with damping
Free vibration without damping

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

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