balancing reciprocating masses

5/24/2018 Balancing Reciprocating Masses

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Balancing ReciprocatingMassesThe dynamic analysis of any mechanism will lead to the determination of the forces andmoments which are required to sustain the motion. These can be considerable and if notunderstood and designed for could result in catastrophe. To reduce or eliminate these forces,appropriate balancing masses are sometimes added to the mechanism. It is usually not prac-tical to completely eliminate these forces, but they can often be substantially reduced. Tobegin, consider the simple slider crank mechanism in which link rotates at a constant angularvelocity as shown in Figure 7.1.

Figure 7.1: Attempting t o balance a slider crank mechanismWe are interested in only the portion of the forces which are caused by the motion of thesystem. T he sta ti c forces are st il l there but we wil l assume the y are small comparedt o those due t o the iner t ia o f the system.The motion of the crank, link 2, is a constant rotation about a fixed axis, however, themotion of the connecting rod, link 3, is more complex as it involves both translation androtation. To approximately balance this mechanism, the connecting rod is mode le d as a rodin which the distributed mass is lumped as two masses, m and mc respectively, at each endand C In order for the orignal rod and the model to have the same effect dynamically,three conditions must be met:

where ZG s the moment of inertia of the connecting rod about its center of mass, GQ .and are, respectively, the distances from and C t o the center of mass of the connectingrod. That is, the sum of the lumped masses must equal the total mass of the connecting rodequation 7.1)), the center of gravity of the lumped masses must be at G 3 equation 7.2))and the moment of inertia of the lumped masses about G3 must be the same as that of theconnecting rod equation 7.3)). These three conditions cannot be met with only two variables


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:BALANCING RECIPROCATING MASSES 8

mBand mc) , nd as a result only equations 7.1)and 7.2) are usually satisfied. Thereforegiven the mass of a connecting rod ma, and its geometry and c equations 7.1) and 7.2)can be solved for the required lumped masses as

where is the overall length of the connecting rod. The original slider crank mechanism isthen modeled by the equivalent mechanism shown in Figure 7 2

Figure 7.2: Equivalent slider crank mechanism with a lumped mass connect-ing rods the crank AB is rotating at a constant angular velocity the dynamic force on thesupporting bearing a t caused by the two rotating masses mz ocated a distance d from Aand r n ~ocated a distance r from A) is only that due to the normal acceleration component.Therefore the dynamic force mBr mzd)w2can be balanced by simply adding balancemass m*, at a position diametrically opposite to the crank such that

where is the distance from the support bearing to the balance mass as indicated inFigure 7.3.

Mec E 36 Mechanics of Machines


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:B L NCING RECIPROC TING MASSES 87

Figure 7.3: Balancing the crank of a slider crank mechanism

Note that m and can be chosen independently, th e only restriction on their values is th atequation 7.6) must be satisfied.Now the only masses left to deal with are those at the piston. The secal led effective

reciprocating mass, m,, is defined asm mc mp 7.7)

where mp is the mass of the piston.The reciprocating motion of m will cause a force in the direction of the axis of th e

piston. It is this force which we now must determine and a ttempt to balance. To find thefarce due to m,, the acceleration of the piston is required. The piston acceleration, ac isdetermined by twice differentiating the position vector of the piston. Consider the slider crankmechanism shown in Figure 7.4.

Figure 7.4: Finding the acceleration of the piston

Mec E 36 - Mechanics of Machines


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7: BALANCING RECIPROCATING MASSES 88

The position vector of the piston (i.e., point C) is

where is the length of the crank, is the length of the connecting rod and and are theangles s indicated in Figure 7.4. In particular, note that 8 is measured pos it ive counte r-clockwise fr om the piston axi s with the positive direction of t h e p ist on axis beingoutward from the crank shaft at A Note that is related to 8 by the relationship

which leads to(7.10)

This expression can be substituted into equation (7.8) and the result differentiated twice withrespect to time to obtain the exact acceleration of the piston. The result is complicatedand, given the fact t hat we have already approximated the mass distribution of the connectingrod, we will also introduce an approximation for equation (7.10) in order t o simplify things abit. This is done by using the binomial expansion for the expression

which is an infinite series that holds for s2< 1. With the substitution2=sin = f) sin2&

and since r is less than 1 and sin cannot exceed 1 the series in equation (7.11) can be used.Further, since the ratio of the crank length to the connecting rod length found in most enginesand compressors is considerably less than 1 (i.e r/1


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where is the constant angular velocity of the crank.Important Note

Th e angle t is measured from the axis of the pistonpositive outward) to th e c rank in a counterclockwise direction.This should be permanently e tched somewhere in your onboard ROM

The FBD/MAD of the slider crank is shown in Figure 7.5, which neglects the static forcesi.e., the weights) and assumes that the forces due to the normal acceleration of the crankand the portion of the connecting rod m ~ )ave been balanced by adding a mass m* at aneccentricity of e a s discussed pre vi~usly.~c s the torque acting on the crank required tomaintain the constant angular velocity.

Figure 7.5: FBD/MAD of slider crank mechanismThe only mass left to balance is the reciprocating one, m,. Summing moments aboutpoint on both the FBD and MAD gives

and since there is no net component of acceleration in the vertical direction we get

Note that the external vertical forces and moments acting on the system consisting of thecrank, connecting rod and piston) balance.2As a result, the accelerations of these m sses mz,ms,m*) re ot shown on the MAD since theywill cancel out when using Newton s second law. Please be aware, however, tha t thes e forces are in fact there.



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Finally, the third equation of motion in the direction gives

The force R is the force which the c rank pin applies t o t h e cr ank to maintain the motionof the piston. The force that the cra nk appli es t o t h e cr an k pin is, of course, equal andopposite so that

where F is the so-called s haking for ce as it is the force that the machine frame is subjectedto due to the reciprocating motion. When the sign of F in equation 7.17) is positive, i t isdirected outward s along the xis of the piston outwards positive), or in this casein the positive x direction. The first term of F,, which is a function of the crank angle 0 iscalled the primary inertia force while the second term, which is a function of twice thecrank angle 20, is called the s ec on da ry in er ti a force. It is critical to remember that theseare the forces on t h e cra nk p in and tha t they are directed outwards along the axis of thepiston when F is positive. The magnitude of the forces as well as the direction in or outalong the axis of the piston) change with the crank angle 0 but the line of action remainsalong the axis of the piston.It is possible to partially balance the shaking force by using a balance weight on the crank.As a balance weight on the crank is a rotating mass and not a reciprocating one, it is notpossible to completely balance the shaking force in this way. However, the maximum forcethat the machine frame is subjected to can be reduced using this approach.


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