Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 847

Designating the total reciprocating mass which moves with the piston as Mrec and the

rotating mass moving with the rank as Mrot, the unbalanced inertial forces Pz (along the direc-

tion of the piston) and Px (along a perpendicular direction) may be written as

Pz = (Mrec + Mrot)Rω^2 cos ωt + Mrec

R

L

(^22) ω
cos 2ωt ...(Eq. 20.78)
and Px = MrotRω^2 sin ωt ...(Eq. 20.79)
w
C R R 1
x
z
L
L 2
L 1
P
(M ) 2
(M ) 3
(M ) (^1) wt
O
Fig. 20.28 Simple crank-mechanism
Here ω is the angular velocity and R is the radius of the crank.
For the simple crank-mechanism shown, the reciprocating and rotating masses are given
by the following equations:
Mrec = M 2 + M 3
L
L
F 1
HG
I
KJ ...(Eq. 20.80)
and Mrot = M 1
R
R
M
L
L
(^1) + 3 F 2
HG
I
KJ ...(Eq. 20.81)
Here M 1 = mass of crank,
M 2 = mass of reciprocating parts, i.e., piston, piston rod, and crank-head,
M 3 = mass of connecting rod,
L = length of connecting rod,
L 1 = distance between the CG of the connecting rod and the crank pin C,
L 2 = distance between the CG of the connecting rod and the wrist pin P,
and R 1 = distance between the CG of the crank shaft and the centre of rotation.
(These equations are based on a simplifying assumption regarding the distribution of
the mass of crank shaft).
The first term for Pz in Eq. 20.78 involving cos ωt (first harmonic) is called the “primary”
inertial force and the second involving cos 2ωt (second harmonic) is called the “secondary”