Theory of Machine
The equation of motion for a single degree of freedom system with viscous damping is 4(dx²/dt²) + 9(dx/dt) + 16x. The damping ratio of the system is

44090
30195
9/128
44082

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Theory of Machine
In a steam engine, the earlier cut-off with a simple slide valve may be obtained by increasing the steam lap and the angle of advance of the eccentric but keeping constant the travel and lead of the valve, this method will

Cause withdrawing or throttling of steam
Reduce maximum opening of port to steam
Reduce length of effective stroke of piston
All of the listed here

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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 sinθ
m.ω².r cosθ
m.ω².r (cos 2θ/n)

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