BALANCING OF RECIPROCATING MACHINES

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 𝑚𝑐 mounted on the crank at a

radial distance 𝑟𝑐 from main bearing 𝑂1 and at an angular position equal to

𝜙 + 180°. This mass will create a constant-magnitude centrifugal force at

𝑂1 that rotates with speed ω. 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 𝑚𝑐𝑟𝑐,

the maximum magnitude of the shaking force can be reduced considerably.

2

(37)

3


Correction amounts typically used range from 𝑚𝑐𝑟𝑐 = 𝑚𝑟/2 to 𝑚𝑐𝑟𝑐 =

2𝑚𝑟/3. For example, consider the case of 𝑚𝑐𝑟𝑐 = 0.6𝑚𝑟 for a mechanism

with a ratio of crank length to connecting-rod length given by 𝑟

𝑙= 0.25. For

this, the shaking force is

The magnitude of this force in terms of crank angle 𝜙 is

Figure 5

4 Figure 5 (c)


5


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 𝜙. The maximum magnitude of the shaking force is 𝑭𝑠 𝑚𝑎𝑥 =0.66𝑚𝑟𝜔2 and occurs when 𝜙 equals 100° and 260°. Superimposed on

Figure 5c is the initial shaking-force variation (dashed line) without the

counterweight. The maximum shaking force is 𝑭𝑠 𝑚𝑎𝑥 = 1.25𝑚𝑟𝜔2 at

𝜙 = 0°. 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 𝑃1 , 𝑃2 and 𝑃3 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 𝜔 is constant.

7


Assumptions: continued…….

The cylinder orientations are defined by angles 𝜃𝑛, 𝑛 = 1, 2, 3, … . 𝑁 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 𝜓𝑛, 𝑛 = 2, 3, 4, ……𝑁 which do not vary with time (i.e., each

crank is rigidly attached to the same crankshaft).

Figure 6

8


Each slider-crank mechanism will generate a shaking force with a line of

action along that particular cylinder’s centerline (i.e., at angle 𝜃𝑛 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 𝜓1 = 0 from the previous definition of angle 𝜓𝑛. The resultant

shaking force will be

(38)

(39)

(40)

9


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


Canceling 𝑚𝑟𝜔2 , which is nonzero, and factoring further yields the

following relations

(44)

(45)

and

11


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 𝜃𝑛

and the angular crank spacing 𝜓𝑛, 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


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 𝑠𝑛, 𝑛 =1, 2, 3…… ,𝑁, measured from that cylinder designated as number 1

(therefore, 𝑠1 = 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


(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


(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.


(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 𝜃1 = 𝜃2 =

𝜃3…… = 𝜃𝑛 …… = 𝜃𝑁 =𝜋

2

Suppose further that the cylinders are equally spaced axially with a spacing

s; then, 𝑠𝑛 = 𝑛 − 1 𝑠, 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 180° cranks; that

is, N=2, 𝜓1 = 0,𝜓2 = 𝜋. Substituting into Eqs (53), we obtain


Figure 7

Figure 7 shows a two-cylinder, in-line arrangement with 180° cranks; that

is, N=2, 𝜓1 = 0,𝜓2 = 𝜋. 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


20

with a maximum magnitude of 2𝑚𝑟𝜔2 𝑟

𝑙. 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.


In an opposed engine, all the cylinders lie in the same plane, with half on

each side of the crank axis. Selecting 𝜃1 = ⋯ = 𝜃𝑁 2 = ⋯ =𝜋

2and

𝜃𝑁 2 +1 = ⋯𝜃𝑁 = ⋯ =3𝜋

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 180° cranks, where N = 2, 𝜃1 =𝜋

2, 𝜃2 =

3𝜋

2, 𝜓1 = 0,𝜓2 = 𝜋, 𝑠1 =

0, 𝑠2 = 𝑠. 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


Figure 7: (a) An opposed two-cylinder engine with cranks. (b) Double connecting rods for cylinder 1.

23


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).


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 90° between banks. The

four-throw crankshaft has 90° 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


28


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:


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 𝑀𝑠

refers to the shaking moment:


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 𝑚𝑟𝑠𝜔2√10 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 𝜔 where at any instant the angle of the

moment vector with respect to the y-direction is 𝜔𝑡 − 71.6° . 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 𝑚𝑐𝑟𝑐𝑠𝑐𝜔2 =

𝑚𝑟𝑠𝜔2√10 and the locations are as depicted in Figure 9b, where 𝑚𝑐 is the

mass, 𝑟𝑐 is the radial position, and 𝑠𝑐 is the axial spacing of the

counterweights. Because this engine can be completely balanced in this

fashion, it exhibits smooth-running performance.


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BALANCING OF RECIPROCATING MACHINES