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ME 302 Dynamics of Machinery Lecture Notes

© 2011

1

CHAPTER 3

DYNAMICS OF RECIPROCATING ENGINES

3-1 INTRODUCTION AND KINEMATICS

This chapter studies the dynamics of a slider crank mechanisms in an analytical way. This is an example for the analytical approach of solution instead of the graphical accelerations and force analyses. The gas equations and models for combustion is not a concern of this chapter.

Fig: Indicator diagram showing the pressure versus crank rotation.

A 1

4

B

2 3

Pθ φ ω

l r

x

x

y

G3 G2

I2,m2 I3,m3

m4

intake compression power exhaust

Crank angle

Pr es

su re

ME 302 Dynamics of Machinery Lecture Notes

© 2011

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Fig:Slider crank mechanism

Loop closure equation can be:

φθ sinsin lr = (1) φθ coscos lrx += (2)

from trigonometric identities

⇒=+ 1cossin 22 φφ φφ 2cos1sin −= (3) from first equation

θφ sinsin l r

= (4)

Substitute this equation into 3 and then resulting equation into (2) 2

sin1cos

−+= θθ

l rlrx (5)

Dynamic analysis of reciprocating engines was done in late 1800's and by that time extensive

calculations had to be avoided. So there are many approximations in the analysis to simplify the arithmetic. In equation (5) square root term can be replaced by simplest expression. Taylor series expansion of square root term, first two term included is as follows:

θθ 22 22

sin 2

1sin1 l

r

l r

−=

− (6)

Squaring is also an arithmetically difficult process:

2 2cos1

sin 2 θθ −= (7)

Substituting equation 6 and 7 into 5

− −+=

2 2cos1

2 1cos 2

2 θθ l

rlrx

++−= t

l r

tr l

rlx ωω 2cos 4

cos 4

2

(8)

ME 302 Dynamics of Machinery Lecture Notes

© 2011

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Equation (8) defines the displacement of the slider. Velocity and acceleration expressions are by successive differentiation of this equation with respect to time. If we assume that the angular velocity of the crank is constant then velocity and acceleration of the slider become:

−−=

−−= t

l r

trt l

r trx ωωωω

ω ωω 2sin

2 sin2sin

2 sin& (9)

−−=

−−= t

l r

trt l

r trx ωωωω

ω ωωω 2coscos2coscos 2&& (10)

In dynamic force analysis, we put inertia and external forces on top of existing mechanism and then solve statically. Under the action of external and inertia forces, too many forces exist on...

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