Balancing of reciprocating masses

[monty37]

my book gets this expression for inertia force of the reciprocating masses of the engine

F=(R/g)ω.ω.r(cosθ+(cos2θ)/n) ,but there is no derivation given as to how he gets the acceleration part,is it using the acceleration diagram?and why should n be involved,
n supposed to be the l/r ratio.

 

[CFDFEAGURU]

What book and what type of engine? An inline, a V, a radial a ...

Thanks
Matt

 

[monty37]

the book is theory of machines by abdulla shariff
well this expression is a general expression for a common reciprocating engine..
later using the same expression..other formulae are being derived for v and radial engines.

 

[ank_gl]

The formula is derived by writing piston displacement as a function of crank angle & then differentiating it twice.

And you should get a good book.

 

[CFDFEAGURU]

A good book for this subject is.

https://www.amazon.com/dp/007329098X/?tag=pfamazon01-20

There are chapters and numerous sections about balancing.

Thanks
Matt

 

[xxChrisxx]

The formula is derived by writing piston displacement as a function of crank angle & then differentiating it twice.

And you should get a good book.

+1

This was the way I did it in my thesis.

 

[monty37]

is thomas bevan a good book ?
in another book,an acceleration diagram has been drawn and this formula is
shown..but tell me why consider n(l/r ratio)?

 

[xxChrisxx]

because that alters the acceleration characterisitcs of the piston. The longer the rod the closer it gets to sinusoidal motion.

That equation should explain perfectly why you were having trouble asserting that pistions had SHM.

Where Z = R/L

I lumped my terms differently, and its negative because I set the convention that inertia forces acts oppoiste gas pressure forces (ie it resists pistion motion). As you can see as 1/Z tends to zero, (ie as the conrod gets longer and approaches infinite) you get closer to SHM. The higher the 1/Z Ratio the flatter the acceleration curve at BDC.

 

[monty37]

i get it now..does the same force act during every stroke of the piston?

 

[xxChrisxx]

Bah. EDIT for the post above, the labels on the graph are the wrong way round.

When Z tends to zero you get closer to SHM.

Silly mistake on my part.Well it depends what you mean by is every stroke the same? For a certain crankshaft angular velocity (RPM), the acceleration characterisitcs and therefore the force are the same.

The cylinder pressures vary throughout the cycle which alters the resultant force on the piston and rod.

 

[ank_gl]

i get it now..does the same force act during every stroke of the piston?

Do you mean suction, compression... by every stroke? If so, then no, same force doesn't act. the graph posted above is acceleration of the piston when crankshaft is rotating at constant angular velocity. Engine is a different animal, motion of piston causes the rotation of cs. You can get a plot of pressure inside a cylinder during a complete cycle. Try heywood.

 

[xxChrisxx]

Force acting down Cylinder Axis due to Gas Pressure

Resultant Force over all 4 strokes

Note that the above was a low to medium speed engine speed. At high speeds the inertia force dominates the resultant force.

Also stictly speaking each cycle undergoes minor variaction, due to slightly alternig GPF. For the typical analysis cases this can be conveniently ignored.