balancing of reciprocating machines
TRANSCRIPT
DEPARTMENT OF MECHANICAL ENGINEERING
AMU ALIGARH
BALANCING OF RECIPROCATING MACHINES
MEC3120: Dynamics of Machinery
1
Balancing of Single Cylinder Machines
Generally, partial balancing of forces in single-cylinder engines and
compressors is to add a rotating counterbalance to the crank. This
counterweight supplements that described in the preceding section, which
is used to counteract the rotating unbalance due to the crank mass and the
rotating part of the connecting-rod mass. Figure 5a shows the mechanism
of Figure 4a with a counterweight of mass ๐๐ mounted on the crank at a
radial distance ๐๐ from main bearing ๐1 and at an angular position equal to
๐ + 180ยฐ. This mass will create a constant-magnitude centrifugal force at
๐1 that rotates with speed ฯ. The total shaking force will then be the vector
sum of the centrifugal force and the force of Eq. (36), (previous part MD-
balancing II.pdf) as shown in Figure 5b. In terms of x and y unit vectors,
It is clear from Eq. (37) that the this counterweight cannot eliminate the
shaking force completely. This happens because this introduces a nonzero y
component force, and also, the x component is reduced, but will not be
identically equal to zero. However, by properly sizing the correction ๐๐๐๐,
the maximum magnitude of the shaking force can be reduced considerably.
2
(37)
3
Balancing of Single Cylinder Machines
Correction amounts typically used range from ๐๐๐๐ = ๐๐/2 to ๐๐๐๐ =
2๐๐/3. For example, consider the case of ๐๐๐๐ = 0.6๐๐ for a mechanism
with a ratio of crank length to connecting-rod length given by ๐
๐= 0.25. For
this, the shaking force is
The magnitude of this force in terms of crank angle ๐ is
Figure 5
4 Figure 5 (c)
Balancing of Single Cylinder Machines
5
Balancing of Single Cylinder Machines
Figure 5c shows a polar plot of the shaking force, where each point on the
curve defines the magnitude and direction of the force for a corresponding
value of ๐. The maximum magnitude of the shaking force is ๐ญ๐ ๐๐๐ฅ =0.66๐๐๐2 and occurs when ๐ equals 100ยฐ and 260ยฐ. Superimposed on
Figure 5c is the initial shaking-force variation (dashed line) without the
counterweight. The maximum shaking force is ๐ญ๐ ๐๐๐ฅ = 1.25๐๐๐2 at
๐ = 0ยฐ. Thus, a 47-percent reduction in magnitude has been achieved
through the addition of a rotating counterweight. The optimum size of the
counterweight would be that which produces equal shaking force
magnitudes at points ๐1 , ๐2 and ๐3 on the polar-force plot of Figure 5c.
Examination of the figure shows that the correction used in this example is
close to optimum, and therefore, little improvement beyond the 47-percent
reduction could be obtained.
6
Balancing of Multi-Cylinder Machines
Many applications of the slider-crank mechanism in engines, pumps, and
compressors involve the use of multiple mechanisms, which are designed
to provide smoother flow of fluid or transmission of power than can be
accomplished in a single-cylinder device.
These multi-cylinder systems facilitate one of the more effective means of
reducing the consequences of shaking forces. By a proper arrangement of
the individual mechanisms, the shaking forces will partially, and perhaps
totally, cancel one another.
We will first develop general shaking-force-balancing relationships for
multi-cylinder machines and then examine some specific configurations.
Figure 6 shows an N cylinder machine with general arrangement of
cylinder and piston assembly. (Only three cylinders are shown)
Assumptions:
All the slider-crank mechanisms have the same crank length r, connecting-
rod length l and reciprocating mass m.
Crank angular velocity ๐ is constant.
7
Balancing of Multi-Cylinder Machines
Assumptions: continuedโฆโฆ.
The cylinder orientations are defined by angles ๐๐, ๐ = 1, 2, 3, โฆ . ๐ which
are fixed angular positions with respect to the y-axis.
The angular crank throw spacings with respect to crank 1 are represented
by angles ๐๐, ๐ = 2, 3, 4, โฆโฆ๐ which do not vary with time (i.e., each
crank is rigidly attached to the same crankshaft).
Figure 6
8
Balancing of Multi-Cylinder Machines
Each slider-crank mechanism will generate a shaking force with a line of
action along that particular cylinderโs centerline (i.e., at angle ๐๐ with
respect to the y-axis). From Eq. (36), (previous part MD-balancing II.pdf),
the expression for the individual shaking forces is
From Figure 6, substituting the angle relationships, Eq. (38) can be
expresses as
where ๐1 = 0 from the previous definition of angle ๐๐. The resultant
shaking force will be
(38)
(39)
(40)
9
Balancing of Multi-Cylinder Machines
In order for the forces to be completely balanced in the arrangement, the y
and z components of Eq. (40) must be identically zero; that is,
Substituting Eq. (39), we see that the conditions of Eq. (41) become
(41)
(42)
(43)
10
Balancing of Multi-Cylinder Machines
Canceling ๐๐๐2 , which is nonzero, and factoring further yields the
following relations
(44)
(45)
and
11
Balancing of Multi-Cylinder Machines
The only way that these expressions can be identically zero is if the
individual coefficients of the time-dependent sine and cosine functions are
all zero. This yields the following eight necessary conditions for complete
balance of the shaking forces:
(46) (47)
The first four conditions (Eqs. 46) account for the primary parts of the
shaking forces, and if these are all satisfied, then the primary shaking forces
are balanced.
The other four conditions (Eqs. 47) represent the secondary parts, and if
those conditions are satisfied, then the secondary shaking forces are
balanced.
Note that the eight conditions are in terms of the cylinder orientations ๐๐
and the angular crank spacing ๐๐, and it follows that some arrangements of
these parameters may balance the forces while other arrangements will not.
Further, some arrangements may result in only primary force balancing or
only secondary force balancing.
Of these two possibilities, primary balancing is preferred, because it
represents cancellation of the larger parts of the shaking forces.
12
Balancing of Multi-Cylinder Machines
13
In most multi-cylinder machines, the slider-crank mechanisms must be
spaced axially along the crankshaft in order to avoid interference during
their operation.
This axial spacing is represented in Figure 6 by distances ๐ ๐, ๐ =1, 2, 3โฆโฆ ,๐, measured from that cylinder designated as number 1
(therefore, ๐ 1 = 0 ).
Since the individual shaking forces will not, in general, lie in a single
transverse plane, they will produce a net shaking moment, as well as a net
shaking force, that will tend to cause an end-over-end rotational vibration
of the crankshaft.
A set of conditions for balancing shaking moment can be established by
imposing the requirement that the sum of shaking-force moments about any
arbitrary axial location must be zero. Taking moments about the axial
location of cylinder 1 yields
Balancing of Multi-Cylinder Machines
(48)
Eq. 48 can be re-expresses as
In order for this equation to be satisfied, the individual j and k components
of the second factor in the cross product must be identically zero; that is
14
Balancing of Multi-Cylinder Machines
(49)
and (50)
15 15
These equations are similar to Eqs. (41)1 and (41)2 and lead to the
following similar set of conditions for balancing shaking moments.
The first four conditions (Eqs. 51) guarantee primary shaking-moment
balance, while the other four conditions (Eqs 52) yield secondary shaking-
moment balance. Taken together, the eight equations account for the axial
configuration of the cylinders, as well as for their angular orientation and
the angular crank spacing.
Sixteen equations (46)1-4 , (47)1-4 , (51)1-4 and (52)1-4 can be used to
investigate the balancing of any piston engine or compressor.
Balancing of Multi-Cylinder Machines
(51) (52)
16
Consider an engine, all of whose cylinders lie in a single plane and on one
side of the crank axis. Suppose that these locations are given by ๐1 = ๐2 =
๐3โฆโฆ = ๐๐ โฆโฆ = ๐๐ =๐
2
Suppose further that the cylinders are equally spaced axially with a spacing
s; then, ๐ ๐ = ๐ โ 1 ๐ , where the cylinders are numbered consecutively
from one end of the crankshaft to the other. Substituting this, Eqs. (46),
(47), (51) and (52) reduce to the following conditions:
Balancing of In-Line Engines
(53)
17
Figure 7 shows a two-cylinder, in-line arrangement with 180ยฐ cranks; that
is, N=2, ๐1 = 0,๐2 = ๐. Substituting into Eqs (53), we obtain
Balancing of In-Line Engines
Figure 7
Figure 7 shows a two-cylinder, in-line arrangement with 180ยฐ cranks; that
is, N=2, ๐1 = 0,๐2 = ๐. Substituting into Eqs (53), we obtain
18
Balancing of In-Line Engines (53)
19
Thus, the primary parts of the shaking forces are always equal and
opposite; therefore, they cancel, but because they are offset axially, they
form a nonzero couple. This is shown in Figure 7. On the other hand, the
secondary parts of the shaking forces are always equal with the same sense,
and they therefore combine to produce a net force and also cause a net
moment. From Eq. (40), the net shaking force is
Balancing of In-Line Engines
20
with a maximum magnitude of 2๐๐๐2 ๐
๐. Although this shaking force is
nonzero, it nevertheless represents a significant improvement in
comparison to a single-cylinder engine with respect to typical ratios.
However, as noted, a shaking couple has been introduced.
Balancing of In-Line Engines
In an opposed engine, all the cylinders lie in the same plane, with half on
each side of the crank axis. Selecting ๐1 = โฏ = ๐๐ 2 = โฏ =๐
2and
๐๐ 2 +1 = โฏ๐๐ = โฏ =3๐
2, we note that half of Eqs. (46), (47), (51) and
(52) are automatically satisfied; these are Eqs. (46)1, (46)2, (47)1, (47)2,
(51)1, (51)2, (52)1, and (52)2. This is because there will be no y-direction
forces or z-direction moments in the general force and moment equations.
As an example, consider the two-cylinder opposed engine of Figure 8(a),
with 180ยฐ cranks, where N = 2, ๐1 =๐
2, ๐2 =
3๐
2, ๐1 = 0,๐2 = ๐, ๐ 1 =
0, ๐ 2 = ๐ . Substituting into Eqs. (46)3, (46)4, (47)3, (47)4, (51)3, (51)4,
(52)3, and (52)4 yield
21
Balancing of In-Line Engines (Opposed Engines)
22
Balancing of In-Line Engines (Opposed Engines)
Figure 7: (a) An opposed two-cylinder engine with cranks. (b) Double connecting rods for cylinder 1.
23
Balancing of In-Line Engines (Opposed Engines)
24
The net shaking force is zero, because both parts of the individual shaking
forces cancel. This is an improvement over the two-cylinder, in-line engine
of Figure 7, but there will be a significant shaking couple (both primary and
secondary) due to the staggering of the crank throws. Clearly, the smaller
the spacing s, the better will be the design from the point of view of
balancing. One method of reducing s to zero and thereby eliminating the
shaking couple is to use double connecting rods for one of the cylinders, as
shown in Figure 8(b).
Balancing of In-Line Engines (Opposed Engines)
25
Figure 9(a) shows A V-8 engine with cranks. This arrangement can be
completely balanced with the addition of rotating counterweights on the
crankshaft. (b) Location of the counterweights
Due to its compact form, the V engine is common in automotive and other
applications. Consider, for example, the V-8 engine of Figure 9a, consisting
of two banks of four cylinders with an angle of 90ยฐ between banks. The
four-throw crankshaft has 90ยฐ cranks, with an axial spacing s between
cranks. The following quantities are determined from the figure:
Balancing of V Engines
26
The force-balance conditions, as evaluated from Eqs. (46) and (47), are
27
Balancing of V Engines
28
Balancing of V Engines
Thus, the engine is completely force balanced. In fact, this configuration
is force balanced for any angle between the cylinder banks, because each
bank of four cylinders is force balanced independently.
29
Next, we examining the shaking-moment conditions which leads to the
following results:
Balancing of V Engines
30 30
There is a primary shaking couple, but no secondary shaking couple; hence,
the engine arrangement, by itself, does not yield a complete force and
moment balance. However, the shaking couple has a special nature that
facilitates total balancing by means of a relatively straightforward
modification. To understand that nature, we consider Eq. (48), where ๐๐
refers to the shaking moment:
Balancing of V Engines
31
In this equation, the secondary parts of the shaking forces have been
disregarded, since they will cancel. Rearranging terms and substituting the
results obtained earlier, we have
The magnitude of this moment is ๐๐๐ ๐2โ10 which is constant for all
values of time t, and the direction of the moment is perpendicular to the
crank axis and rotates with speed ๐ where at any instant the angle of the
moment vector with respect to the y-direction is ๐๐ก โ 71.6ยฐ . This is
exactly the same as the rotating, unbalanced dynamic couple discussed
earlier. Thus, the net effect of this engine arrangement is what appears to be
rotating dynamic unbalance. Therefore, the shaking couple can be balanced
by a set of rotating counterweights that produce an equal, but opposite,
rotating couple. The magnitude of this couple is given by ๐๐๐๐๐ ๐๐2 =
๐๐๐ ๐2โ10 and the locations are as depicted in Figure 9b, where ๐๐ is the
mass, ๐๐ is the radial position, and ๐ ๐ is the axial spacing of the
counterweights. Because this engine can be completely balanced in this
fashion, it exhibits smooth-running performance.
Balancing of V Engines