Reciprocating Shaking Forces

Reciprocating compressors include linear translational motion as a part of their normal operation. A single cylinder compressor will have an inherent set of shaking forces as part of its operating characteristics. The forces are resolved to reaction forces on the main bearings, where they transmit through the frame and the foundation. The purpose here will be to show the reader the basics of how these forces are generated and a general philosophy on how they can be minimized by balancing or by cylinder

Table 9-1

General Normalized Functions Describing the Dynamic Characteristics of the Crank and Connecting Rod Mechanisms

Table 9-1

General Normalized Functions Describing the Dynamic Characteristics of the Crank and Connecting Rod Mechanisms

Mr^jl

¿r

DISPLACEMENT

R

(l + 4i)~(cos* + !rcos20)

VELOCITY

R il

sin e + jfjsin Z9

ACCELERATION

£

COS 9 + -J- cos 28

PRESSURE TORQUE

Tp pAR

sin d + yi sin 25

INERTIA TORQUE

-3(j-)sin 30 -^(f-) sin 49

EQUIV. INERTIA OF

RECIP. MASS

W

f1 + "i (T)*] * I (T) cos ® "" caos 2®

-(•f)cos3®-i(r) cos 48]

AVERAGE EQUIVALENT DISK

JkI_

r ■ - ^ (f-r J - •

Source: [4] Reprinted with permission of John Wiley and Sons, Inc.

Source: [4] Reprinted with permission of John Wiley and Sons, Inc.

arrangement. One compressor arrangement used is the balance opposed configuration. Cylinders are oriented in the horizontal position and located on each side of the crankshaft. See Figure 3-3. This is made more difficult with a compressor, even if the arrangement is made completely symmetrical, as the volumetric reduction due to compression requires each cylinder bore, and, therefore, piston size, to be different. This is one reason, however, lighter materials such as aluminum are preferred for the larger pistons.

Table 9-1 is a handy little chart to visualize a vertical, single-cylinder compressor and the basic functions. The functions are normalized to keep them in a dimensionless form [4]. With the following set of equations, the x and y components of the inertia forces for a single cylinder can be calculated. For the derivation, the reader is referred to references [4, 5]. Figure 9-4 depicts the generalized stage to aid in the definition of terms

I ,.. - (mH;, + mrec)(r(O2)(cos0)(cos(j))—(mrot)(ra)2)(sin0)(sin(j)) (9.41 IV - (mrec)(r2co2/l)(cos2G)(cos(j)) (9.5)

I yp = (mrot + mrec)(rco2)(cos9)(sin(j)) + (mrot)(rio2)(sinG)(cos^) (9.6) Fy, = (mrot,)(r2co2/ l)(cos 26)(sin<)>) (9.7)

where

Fxp = primary shaking force in the horizontal direction Fxs = secondary shaking force in the horizontal direction Fyp = primary shaking force in the vertical direction Fys = secondary shaking force in the vertical direction oj = angular velocity, rad/sec 8 ~ crankangle, also (0t

<|> - cylinder travel angle relative to horizontal direction t = time, sec mroI = total mass of the rotating parts, per cylinder mrec = total mass of the reciprocating parts, per cylinder r = crankthrow, also stroke/2 I = connecting rod length

To obtain the mass of the reciprocating elements, the individual reciprocating parts (piston, piston rod, crosshead, crosshead pin) and all other hardware such as nuts must be weighed and the weights added. One more item has to be considered. The connecting rod has both translation-al as well as rotational motion. A simple way of obtaining each part is to assume the connecting rod consists of two masses, one at each end connected by a massless center section. Figure 9-5 shows a direct method of obtaining these values by using two scales. The value obtained from the crank end is assigned to the rotating mass, mrot.

At this point, as far as shaking forces go, the gas forces do not make a contribution. If the rod load or bearing loads are to be analyzed, the gas forces must be calculated and added vectorially to the inertia forces to

Figure 9-5. Method of obtaining weights to a two mass connecting rod model

Figure 9-5. Method of obtaining weights to a two mass connecting rod model

Connecting Rod

Connecting Rod get a net bearing force. This is mentioned at this point because it is some times misunderstood.

After the forces are evaluated for each cylinder of a multistage com pressor, all forces must be summed in the x and y direction. For the maximum shaking forces, the value of the crank angle, which contributes the maximum force, should be used. This involves taking the respective sine and cosine functions to their maximum. For example, a vertical cylinder will have the maximum component force at a crank angle of 0° and 180°. At this time, the horizontal components, primary and secondary, are zero.

While on occasion, the forces may cancel, the moments must also be evaluated. If the vectors are assumed to be acting at the centerline of the crankshaft at the line of travel of the piston, and the resulting vectors summed, it can be seen that on a multistage compressor, if the lateral third stage should balance vectorially, a couple still exists, which results in an unbalanced moment. If it is recalled that 9 = cot, then by referring to Equations 9.4 and 9.6, the forces (and resulting moments) are a function of 0. while being proportional to G) is acting at running speed. The secondary forces (and resulting moments) as seen in Equations 9.5 and 9.7 are a function of 2 x 9, or twice co, and thereby act at twice running speed.

The forces and moments discussed so far have included no provisions for balance. There are two methods of balancing the reciprocating compressor. One that has been mentioned is the arrangement of the cylinders. For example, a horizontally opposed design is an attempt to balance by arrangement. The other method can be used alone on a single-cylinder machine, or used together with arrangement on the multistage. Crankshaft counterweights are used, and whenever possible, they are attached to the crankshaft on either side of the connecting rod journal. Other placements are possible. Placing a mass at radius, r, for a given, to, will generate a rotating vector whose direction is a function of the attachment angle. Intuitively, it would be 180° from the crankthrow. Before final locations can be decided, the vectors for the various crank angles must be plotted on a polar plot. The magnitude and location of the weight or weights can then be decided. Refer to Figure 9-6 for counterweight position with respect to the crankshaft.

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