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32.5 Balancing of Reciprocating Machines

Back

A reciprocating mechanism has a slider that moves rectilinearly back and forth along some guideway. A

piston-cylinder device is a good example. Often, reciprocating machines contain rotatory components in

addition to the reciprocating mechanisms. The purpose is to either covert a reciprocating motion to a

rotary motions (as in the case of an automobile engine), or to convert a rotary motion to a reciprocating

motions (as in the opto-slider mechanism of a photocopier). Irrespective of the reciprocating machine

employed, it is important to remove the vibratory excitations that arise in order to realize the standard

design goals of smooth operation, accuracy, low noise, reliability, mechanical integrity, and extended

service life. Naturally, in view of their rotational asymmetry, reciprocating mechanisms with rotary

components are more prone to unbalance than purely rotary components. Removing the “source of

vibration” by proper balancing of the machine would be especially applicable in this situation.

32.5.1 Single-Cylinder Engine

A practical example of a reciprocating machine with integral rotary motion is the internal combustion

(IC) engine of an automobile. A single-cylinder engine is sketched in Figure 32.16. Observe the

nomenclature of the components. The reciprocating motion of the piston is transmitted through the

connecting rod and crank into a rotatory motion of the crankshaft. The crank, as sketched in

Figure 32.16, has a counterbalance mass, the purpose of which is to balance the rotary force (centrifugal).

Piston

Crosshead

Connecting Cylinder

Rod

Crank

Shaft

Counterbalance

Mass

FIGURE 32.16 A single-cylinder reciprocating engine.

32-26 Vibration and Shock Handbook

We will ignore this in our analysis because the goal is to determine the unbalance forces and ways to

balance them.

Clearly, both the connecting rod and the crank have distributed mass and moment of inertia. To

simplify the analysis, we approximate as follows:

1. Represent the crank mass by an equivalent lumped mass at the crank pin (equivalence may be

based on either centrifugal force or kinetic energy).

2. Represent the mass of the connecting rod by two lumped masses, one at the crank pin and the

other at the cross head (piston pin).

The piston itself has a significant mass, which is also lumped at the crosshead. Hence, the equivalent

system has a crank and a connecting rod, both of which are considered massless, with a lumped mass mc

at the crank pin and another lumped mass mp at the piston pin (crosshead).

Furthermore, under normal operation, the crankshaft rotates at a constant angular speed (v). Note

that this steady speed is realized not by natural dynamics of the system, but rather by proper speed

control (a topic which is beyond the scope of the present discussion).

It is a simple matter to balance the lumped mass mc at the crank pin. Simply place a countermass mc at

the same radius in the radially opposite location (or a mass in inverse proportion to the radial distance

form the crankshaft, but remaining in the radially opposite direction). This explains the presence of the

countermass in the crank shown in Figure 32.16. Once complete balancing of the rotating inertia (mc) is

thus achieved, we still need to completely eliminate the effect of the vibration source on the crankshaft. To

achieve this, we must compensate for the forces and moments on the crankshaft that result from:

1. The reciprocating motion of the lumped mass mp

2. Time-varying combustion (gas) pressure in the cylinder

Both types of forces act on the piston in the direction of its reciprocating (rectilinear) motion. Hence,

their influence on the crankshaft can be analyzed in the same way, except that the combustion pressure is

much more difficult to determine.

The above discussion justifies the use of the simplified model shown in Figure 32.17 for analyzing the

balancing of a reciprocating machine. The characteristics of this model are as follows:

1. A light crank OC of radius r rotates at constant angular speed v about O, which is the origin of the

x – y coordinate frame.

2. A light connecting rod CP of length l is connected to the connecting rod at C and to the piston at P

with frictionless pins. Since the rod is light and the joints are frictionless, the force fc supported by

it will act along its length. (Assume that the force fc in the connecting rod is compressive, for the

purpose of the sign convention). Connecting rod makes an angle f with OP (the negative x axis).

3. A lumped mass mp is present at the piston. A force f acts at P in the negative x direction. This may

be interpreted as either the force due to the gas pressure in the cylinder or the inertia force mpa

where a is the acceleration mp in the positive x direction. These two cases of forcing are considered

separately.

4. A lateral force fl acts on the piston by the cylinder wall, in the positive y direction.

Again, note that the lumped mass mc at C is not

included in the model of Figure 32.17 because it is

assumed to be completely balanced by a countermass

in the crank. Furthermore, the lumped mass

mp includes both the mass of the piston and also

part of the inertia of the connecting rod.

There are no external forces at C. Furthermore,

the only external forces at P are f and fl; where f is

interpreted as either the inertia force in mp or the

gas force on the piston. Hence, there should be

fl = f tan f

f ωt

FIGURE 32.17 The model used to analyze balancing of

a reciprocating engine.

Vibration Design and Control 32-27

equal and opposite forces at the crankshaft O, as shown in Figure 32.17, to support the forces acting

at P. Now, let us determine fl:

Equilibrium at P gives

f ¼ fc cos f

fl ¼ fc sin f

Hence,

fl ¼ f tan f ð32:52Þ

This lateral force fl acting at both O and P, albeit in the opposite directions, forms a couple t ¼ xfl or, in

view of Equation 32.52:

t ¼ xf tan f ð32:53Þ

This couple acts as a torque on the crankshaft. It follows that, once the rotating inertia mc at the crank

is completely balanced by a countermass, the load at the crankshaft is due only to the piston load f and it

consists of:

1. A force f in the direction of the piston motion (x)

2. A torque t ¼ xf tan f in the direction of rotation of the crankshaft (z)

As discussed below, the means of removing f at the crankshaft will also remove t to some extent.

Hence, we will discuss only the approach of balancing f :

32.5.2 Balancing the Inertia Load of the Piston

First, consider the inertia force f due to mp: Here,

f ¼ mpa ð32:54Þ

where a is the acceleration x€; with the coordinate x locating the position P of the piston (in other words

OP ¼ x). We notice from Figure 32.17 that

x ¼ r cos vt þ l cos f ð32:55Þ

However,

r sin vt ¼ l sin f ð32:56Þ

Hence,

cos f ¼ 1 2

􀀏 􀀐2

sin2vt

􀀒 􀀓1=2

ð32:57Þ

which can be expanded up to the first term of Taylor series as

cos f ø 1 2

􀀏 􀀐2

sin2vt ð32:58Þ

This approximation is valid because l is usually several times larger than r and, hence, ðr=lÞ2 is much small

than unity. Next, in view of

sin2vt ¼

2 ½1 2 cos 2vt�� ð32:59Þ

we have

cos f ø 1 2

􀀏 􀀐2

½1 2 cos 2vt􀀉 ð32:60Þ

32-28 Vibration and Shock Handbook

Substitute Equation 32.60 into Equation 32.55. We get, approximately

x ¼ r cos vt þ

􀀏 􀀐2

cos 2vt þ l 2

􀀏 􀀐2

ð32:61Þ

Differentiate Equation 32.61 twice with respect to t to get the acceleration

a ¼ x€ ¼ 2rv2 cos vt 2 l

􀀏 􀀐2

v2 cos 2vt ð32:62Þ

Hence, from Equation 32.54, the inertia force at the piston (and its reaction at the crankshaft) is

f ¼ 2mprv2 cos vt 2 mpl

􀀏 􀀐2

v2 cos 2vt ð32:63Þ

It follows that the inertia load of the reciprocating piston exerts a vibratory force on the crankshaft which

has a primary component of frequency v and a smaller secondary component of frequency 2v; where v is

the angular speed of the crank. The primary component has the same form as that created by a rotating

lumped mass at the crank pin. However, unlike the case of a rotating mass, this vibrating force acts only

in the x direction (there is no sin vt component in the y direction) and, hence, cannot be balanced by a

rotating countermass. Similarly, the secondary component cannot be balanced by a countermass rotating

at double the speed. To eliminate f ; we use multiple cylinders whose connecting rods and cranks

are connected to the crankshaft with their rotations properly phased (delayed), thus canceling out the

effects of f :

32.5.3 Multicylinder Engines

A single-cylinder engine generates a primary component and a secondary component of vibration load at

the crankshaft, and they act in the direction of piston motion (x). Because there is no complementary

orthogonal component (y), it is inherently unbalanced and cannot be balanced using a rotating mass. It

can be balanced, however, by using several piston-cylinder units with their cranks properly phased along

the crankshaft. This method of balancing multicylinder reciprocating engines is addressed now.

Consider a single cylinder whose piston inertia generates a force f at the crankshaft in the x direction

given by

f ¼ fp cos vt þ fs cos 2vt ð32:64Þ

Note that the primary and secondary forcing amplitudes fp and fs; respectively, are given by Equation

32.63. Suppose that there is a series of cylinders in parallel, arranged along the crankshaft, and the crank

of cylinder i makes an angle ai with the crank of cylinder 1 in the direction of rotation, as schematically

shown in Figure 32.18(a). Hence, force fi on the crankshaft (in the x direction, shown as vertical in

Figure 32.18) due to cylinder i is

fi ¼ fp cosðvt þ aiÞ þ fs cosð2vt þ 2aiÞ for i ¼ 1; 2; …; with a1 ¼ 0 ð32:65Þ

Not only do the cranks need to be properly phased, but the cylinders should also be properly spaced along

the crankshaft to obtain the necessary balance. Consider two examples.

32.5.3.1 Two-Cylinder Engine

Consider the two-cylinder case, as shown schematically in Figure 32.18(b) where the two cranks are in

radially opposite orientations (i.e., 1808 out of phase). In this case, a2 ¼ p: Hence,

f1 ¼ fp cos vt þ fs cos 2vt ð32:66Þ

f2 ¼ fp cosðvt þ pÞ þ fs cosð2vt þ 2pÞ ¼ 2fp cos vt þ fs cos 2vt ð32:67Þ

It follows that the primary force components cancel out. However, they form a couple z0fp cos vt where

z0 is the spacing of the cylinders. This causes a bending moment on the crankshaft, and it will not vanish

Vibration Design and Control 32-29

unless the two cylinders are located at the same point along the crankshaft. Furthermore, the secondary

components are equal and additive to 2fs cos 2vt: This resultant component acts at the midpoint of the

crankshaft segment between the two cylinders. There is no couple due to the secondary components.

32.5.3.2 Six-Cylinder Engine

Consider the six-cylinder arrangement shown schematically in Figure 32.18(c). Here, the cranks are

arranged such that a2 ¼ a5 ¼ 2p=3; a3 ¼ a4 ¼ 4p=3; and a1 ¼ a6 ¼ 0: Furthermore, the cylinders are

equally spaced, with spacing z0: In this case, we have

f1 ¼ f6 ¼ fp cos vt þ fs cos 2vt ðiÞ

f2 ¼ f5 ¼ fp cosðvt þ 2p=3Þ þ fs cosð2vt þ 4p=3Þ ðiiÞ

f3 ¼ f4 ¼ fp cosðvt þ 4p=3Þ þ fs cosð2vt þ 8p=3Þ ðiiiÞ

Now, we use the fact that

cos u þ cos u þ

􀀏 􀀐

þ cos u þ

􀀏 􀀐

¼ 0 ðivÞ

which may be proved either by straightforward trigonometric expansion or by using geometric

interpretation (i.e., three sides of an equilateral triangle, the sum of whose components in any

Crank

fi f1

(a)

y O

Crank 1

Crank i

( f1)

( f2)

(b)

y O

Crank 1

Crank 2

f1 = fp cos ωt

+ fs cos 2wt

f2 f3 f4 f5 f6

(c)

y O

1,6

2,5

Crank

Shaft

Crank

(0°)

fp cos wt

fs cos 2wt

−fp cos wt

fs cos 2wt

zo zo zo zo zo

1 3,4

(0°)

(120°)

(240°)

(120°)

(0°)

(180°)

FIGURE 32.18 (a) Crank arrangement of a multicylinder engine; (b) two-cylinder engine; (c) six-cylinder engine

(balanced).

32-30 Vibration and Shock Handbook

direction vanishes). The relation iv holds for any u; including u ¼ vt and u ¼ 2vt: Furthermore,

cosð2vt þ 8p=3Þ ¼ cos 2vt þ

􀀏 􀀐

Thus, from Equation i to Equation iii, we can conclude that

f1 þ f2 þ f3 þ f4 þ f5 þ f6 ¼ 0 ð32:68Þ

This means that the lateral forces on the crankshaft that are exerted by the six cylinders will completely

balance. Furthermore, by taking moments about the location of crank 1 of the crankshaft, we have

ðz0 þ 4z0Þ fp cos vt þ

􀀏 􀀐

þ fs cos 2vt þ

􀀒 􀀏 􀀐􀀓

þ ð2z0 þ 3z0Þ fp cos vt þ

􀀏 􀀐

þ fs cos 2vt þ

􀀒 􀀏 􀀐􀀓

þ 5z0½fp cos vt þ fs cos 2vt􀀉

ðvÞ

which also vanishes in view of relation iv. Hence, the set of six forces is in complete equilibrium and, as a

result, there will be neither a reaction force nor a bending moment on the bearings of the crankshaft from

these forces.

In addition, it can be shown that the torques xifi tan fi on the crankshaft due to this set of inertial

forces fi will add to zero, where xi is the distance from the crankshaft to the piston of the ith cylinder and

fi is the angle f of the connecting rod of the ith cylinder. Hence, this six-cylinder configuration is in

complete balance with respect to the inertial load.

Example 32.5

An eight-cylinder in-line engine (with identical cylinders that are placed in parallel along a line) has its

cranks arranged according to the phasing angles 0, 180, 90, 270, 270, 90, 180, and 08 on the crankshaft.

The cranks (cylinders) are equally spaced, with spacing z0: Show that this engine is balanced with respect

to primary and secondary components of reaction forces and bending moments of inertial loading on the

bearings of the crankshaft.

Solution

The sum of the reaction forces on the crankshaft are

2 fp cos vt þ fs cos 2vt þ fp cosðvt þ pÞ þ fs cosð2vt þ 2pÞ þ fp cos vt þ

􀀏 􀀐

þ fsð2vt þ pÞ

􀀒

þfp cos vt þ

􀀏 􀀐

þ fs cosð2vt þ 3pÞ

􀀓

¼ 2

fp cos vt 2 fp cos vt 2 fp sin vt þ fp sin vt

þ fs cos 2vt þ fs cos 2vt 2 fs cos 2vt 2 fs cos 2vt

¼ 0

Hence, both primary forces and secondary forces are balanced. The moment of the reaction forces about

the crank 1 location of the crankshaft is

ðz0 þ 6z0Þ½fp cosðvt þpÞ þ fs cosð2vt þ 2pÞ􀀉 þ ð2z0 þ 5z0Þ fp cos vt þ

􀀏 􀀐

þ fs cosð2vt þpÞ

􀀒 􀀓

þ ð3z0 þ 4z0Þ fp cos vt þ

􀀏 􀀐

þ fs cosð2vt þ 3pÞ

􀀒 􀀓

þ 7z0½fp cos vt þ fs cos 2vt􀀉

¼ 7z0½2fp cos vt þ fs cos 2vt 2 fp sin vt 2 fs cos 2vt þ fp sin vt 2 fs cos 2vt þ fp cos vt þ fs cos 2vt􀀉 ¼ 0

Hence, both primary bending moments and secondary bending moments are balanced. Therefore, the

engine is completely balanced.

The formulas applicable for balancing reciprocating machines are summarized in Box 32.4.

Vibration Design and Control 32-31

Finally, it should be noted that, in the configuration considered above, the cylinders are placed in

parallel along the crankshaft. These are termed in-line engines. Their resulting forces fi act in parallel

along the shaft. In other configurations such as V6 and V8, the cylinders are placed symmetrically around

the shaft. In this case, the cylinders (and their inertial forces, which act on the crankshaft) are not parallel.

Here, a complete force balance may be achieved without having to phase the cranks. Furthermore, the

bending moments of the forces can be reduced by placing the cylinders at nearly the same location along

the crankshaft. Complete balancing of the combustion/pressure forces is also possible by such an

arrangement.

32.5.4 Combustion/Pressure Load

In the balancing approach presented above, the force f on the piston represents the inertia force due to the

equivalent reciprocating mass. Its effect on the crankshaft is an equal reaction force f in the lateral

direction (x) and a torque p ¼ xf tan f about the shaft axis (z). The balancing approach is to use a series

of cylinders so that their reaction forces fi on the crankshaft from an equilibrium set so that no net

reaction or bending moment is transmitted to the bearings of the shaft. The torques ti also can be

balanced by the same approach, which is the case, for example, in the six-cylinder engine.

Box 32.4

BALANCING OF RECIPROCATING MACHINES

Single cylinder engine:

Inertia force at piston (and its reaction on crankshaft)

f ¼ 2mprv2 cos vt 2 mpl

􀀏 􀀐2

v2 cos 2vt ¼ fp cos vt þ fs cos 2vt

where

v ¼ rotating speed of crank

mp ¼ equivalent lumped mass at piston

r ¼ crank radius

l ¼ length of connecting rod

fp ¼ amplitude of the primary unbalance force (frequency v)

fs ¼ amplitude of the secondary unbalance force (frequency 2v)

Multicylinder engine:

Net unbalance reaction force on crankshaft ¼

i¼1 fi

Net unbalance moment on crankshaft ¼

i¼1 zifi

where

fi ¼ fp cosðvt þ aiÞ þ fs cosð2vt þ 2aiÞ

ai ¼ angular position of the crank of ith cylinder, with respect to a body (rotating) reference

(i.e., crank phasing angle)

zi ¼ position of the ith crank along the crankshaft, measured from a reference point on the

shaft

n ¼ number of cylinders (assumed identical)

Note: For a completely balanced engine, both the net unbalance force and the net unbalance

moment should vanish.

32-32 Vibration and Shock Handbook

Another important force that acts along the direction of piston reciprocation is the drive force due to

gas pressure in the cylinder (e.g., created by combustion of the fuel – air mixture of an internal

combustion engine). As above, this force may be analyzed by denoting it as f : However, several important

observations should be made first:

1. The combustion force f is not sinusoidal of frequency v: It is reasonably periodic but the shape is

complex and depends on the firing/fuel-injection cycle and the associated combustion process.

2. The reaction forces fi on the crankshaft, which are generated from cylinders i; should be balanced

to avoid the transmission of reaction forces and bending moments to the shaft bearings (and

hence, to the supporting frame — the vehicle). However, the torques ti in this case are in fact the

drive torques. Obviously, they are the desired output of the engine and should not be balanced,

unlike the inertia torques.

Therefore, although the analysis completed for balancing the inertia forces cannot be directly used

here, we can employ similar approaches to the use of multiple cylinders for reducing the gas-force

reactions. This is a rather difficult problem given the complexity of the combustion process itself. In

practice, much of the leftover effects of the ignition cycle are suppressed by properly designed engine

mounts. Experimental investigations have indicated that in a properly balanced engine unit, much of the

vibration transmitted through the engine mounts is caused by the engine firing cycle (internal

combustion) rather than the reciprocating inertia (sinusoidal components of frequency v and 2v).

Hence, active mounts, where stiffness can be varied according to the frequency of excitation, are being

considered to reduce engine vibrations in the entire range of operating speeds (e.g. 500 to 2500 rpm).

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