Dynamic Modeling of Reciprocating Compressors...

Dynamic Modeling of Reciprocating Compressors with Vertical Axis

P R. G. Kurka1, Karen L. G. Paulino

2, Jaime H. Izuka

1

1Faculty of Mechanical Engineering, University of Campinas (UNICAMP)

P.O. Box 6122, 13083-970 Campinas, SP, Brazil

Tel.: +55 19 3521 3175; fax: +55 19 3289 3722.

email: [email protected]


2 Department of Science and Earth Studies

Federal University in São Paulo - UNIFESP

R. Prof. Artur Riedel, 275, 09972-270, Diadema, SP, Brazil


Abstract The paper presents the dynamic model of a reciprocating compressor with vertical axis. The model

consists of a rotor, crankshaft, connecting rod and piston, all supported by visco-elastic bearings. The

model incorporates the gyroscopic interactions due to the movement of the bearings of rotating

components. The Newton-Euler method is used in the analysis, establishing the necessary differential

equations that describe the movement of the system. Analysis of the flexible bearing model shows that

oscillations of the rotating internal components of the compressor increase the loads of the bearings, as

compared to the support loads of non-flexible bearing models.

1 Introduction

Reciprocating compressors use a crankshaft mechanism to convert the rotor axis rotation into piston

translation. The compressor moving parts are supported by hydrodynamic bearings, whose lubrication is

maintained by means of a helicoidal/centrifugal pump. Compressor lifetime and performance is highly

affected by the lubrication of its moving parts. An inefficient or inadequate lubrication of the moving parts

may lead to equipment failure or waste of energy [1] and [2]. The hydrodynamic supports cause a

secondary displacement of the rotor components in the radial direction of the bearings. The numerical

simulation of such an operational behavior, through a dynamics analytical model that includes the

secondary radial movements, yields a realistic estimation of the forces supported by the bearings. Accurate

estimation of bearing loads is an important aspect to the design and performance of reciprocating

machines and mechanisms. The o objective of the present work is, therefore, to obtain a detailed

numerical model of the dynamics of a single cylinder, reciprocating refrigeration compressor [3] with

flexible bearings.

The performance of refrigerating compressors, due to the dynamics of its driving system, connecting rod

and piston, together with bearing lubrication effects, has been researched, by Prata [4], Cho [5] and Kim

[6], and Goodwin [7], among others. The works show that optimization of the bearing forces improve the

performance, lifetime and reliability of the system. An efficient dynamic model allows for better results in

the solution of the coolant fluid compression/expansion effects, better parts design, and contribute to the

knowledge database for research on NVH requirements and compressor lubrication performance [8] and

[9].

1573

The necessity of a method to predict the entire compressor movement is highlighted by Dufour et al. [10].

A dynamic slider/crank model is proposed to study the transient and steady state behavior of a single

cylinder compressor. A numerical analysis for a reciprocating compression mechanism is performed by

Kim and Han [6], which considers the coupled dynamic behavior of the piston and crankshaft. Prata et al.

[4] present a numerical investigation for a piston inside the cylinder in a small hermetic compressor, with

the radial clearance between piston and cylinder filled with lubricant oil. The effects of some design

parameters and operating conditions are examined; with regards to the stability of the piston, oil leakage

and friction losses. Emphasis is placed on the investigation of the influence of the pin location, radial

clearance and oil viscosity on the piston dynamics. The piston dynamics is also investigated by Cho and

Moon [5]. In this research the crankshaft and connecting rod is included in the model to represent the

transformation of the rotating motion to a reciprocating translation of the piston.

The sensitivities to dimensional tolerances in reciprocating engine dynamics is researched by Sinha, et al.

[11], for an internal combustion engine. Such an engine model is governed by a nonlinear differential

equation with a discontinuous forcing function, due to the nature of the pressure-volume (P-V) diagram.

The differential equation of motion comes from the kinematic equations of the slider/crank mechanism. A

single-cylinder engine model with flexible crank support is proposed by Goudas et al. [12]. Transient and

steady state dynamic response of a class of slider/crank mechanisms with compliant supporting is

investigated. The attention is focused on investigating the influence of the system parameters on its

dynamics.

All the above mentioned works use a simplified dynamic model of the rotor-connecting rod-piston model,

which does not account for the gyroscopic effects that take place due to the orbital movement of the

rotating components in their bearing cases. The model proposed in the present work introduces such a

behavior, considering as a first approach, the existence of linear elastic bearings. The formulation and

simulation results found in such a model can be easily exported to more realistic bearing behavior,

including the coupled hydrodynamic effects of non-linear stiffness and damping.

The dynamic efficiency of the compressor is affected by operational misalignments. The alignment of

rotor supports, for example, has a great influence on the behavior of rotating machinery [13]. The

dynamical models must be able to quantify the increase of bearing loads as a result of misalignments. A

gap is observed in the literature, with respect to such a dynamic study, since the models available address

only the more simplified aligned or pinned behavior of the bearings. Current models, which are used with

commercial software, are based on constrained or pinned movements of the bearings of axis, connecting

rod and piston. The misalignments introduced due to bearing flexible movements cannot be verified in

such models. In the present modeling proposal, the rotational movement of compressor parts are

described, based on Euler’s angles, comprised of precession, nutation and spin ([14] and [15]). Three

local reference frames are defined, in addition to the inertial coordinates system, whose origins are placed

at the center of mass of the compressor parts (axis, connecting rod, piston).

The bearing reactions are modeled as elastic and viscous forces, which represent the hydrodynamic loads

produced by the relative displacement between the bearings and their casing. External loads come from

the driven torque and the cooling gas pressure on the piston [16]. All parts are considered rigid and the

distance between the bearings is not long enough to produce relevant elastic bending of the elements. The

basic analysis consists on establishing the balance of forces in the piston due to the cylinder pressure

variation, to determine the connecting rod reaction force and consequently the loads in the eccentric

(crankshaft) and compressor axis. The dynamic behavior of the model is based on a numerical simulation

through a fourth order Runge-Kutta method, using the Matlab environment. The computational code

simulates the compressor dynamics under constant operational speeds, allowing for the study of the

influence of different constitutive models of loads that act on the bearings.

2 Description of the model

The reciprocating compressor main parts are the compressor housing, axis, cylinder, connecting rod,

piston and discharge valves, shown in Figure 1. The housing protects the driving elements, moving parts

1574 PROCEEDINGS OF ISMA2010 INCLUDING USD2010

of the compressor and functions also as a lubricant reservoir, since the oil is stored in its lower region.

The lubricating system consists of an oil pump, oil filter and oil cooler.

Figure 1: Representation of reciprocating compressor [17].

The driving element is the axis, which is attached to an electric rotor. The driven elements are the

connecting rod, which is positioned eccentrically to the axis and converts the rotating movement into a

reciprocating translation; and the piston, which is connected to the rod through a pin, and whose

reciprocating movement compresses the cooling gas into the cylinder.

The axis is supported by hydrodynamic bearings: the primary and secondary radial bearings and a vertical

(axial) weight supporting bearing, all attached to the compressor block. The axial bearing is not

considered in the present model, since the most important secondary movements take place in radial

directions. Another hydrodynamic bearing is located at the axis eccentric connection point. Such a

bearing is responsible to the eccentric attachment of the axis to the connecting rod. An articulated pin

attaches the connecting rod to the piston. The small dimensions of the articulated pin connection, when

compared to the dimensions of the remaining components bearings, allow the use of a non flexible force

transmitting element, rather than a hydrodynamic bearing in the analysis. The main piston movement

takes place in the axial direction, but small displacements also occur in the radial directions. For this

reason, two flexible sliding bearings are defined at the piston top and skirt regions, which represent the

actuating points of the forces that cause radial displacements.

2.1 Components

Four reference systems are created for the analysis: an inertial and three local ones, each one of them

located in the center of mass of the compressor components. The motion of the rotor is described by five

movements: two absolute displacements of its center of mass, xr, yr, and three Euler rotations: precession

r, nutation r and spin r. Four movements are considered in describing the motion of the connecting

rod: two absolute displacements of its center of mass, xc, yc, and two Euler rotations: nutation c and spin

c. Four movements are considered in describing the motion of the piston: two absolute displacements of

its center of mass, yp and zp and two Euler rotations: nutation p and spin p , all shown in Figure 2.

DYNAMICS OF ROTATING MACHINERY 1575

Figure 2: Inertial System (XYZ), Local System and their rigid body motion and bearing position (1,

2, 3, 4, 5 and 6)

Rotation matrices are used to convert the local coordinates into the inertial frame. The complete rotation

matrix of the rotor is a product of its three individual rotations (r, r and r), Tr = T rTrTr. The

structures of the three rotations matrices are given in equations (1), below.

100

0)cos()sin(

0)(sin)cos(

rr

rr

r

T,

)cos(0)(sin

010

)(sin-0)cos(

rr

rr

r

T,

100

0)cos()(sin-

0)(sin)cos(

rr

rr

r

T (1)

The rotation matrix of the connecting rod is a product of its two individual rotation (c and c), Tc = Tc

Tc . The structures of the three rotation matrices are given in equations (2), below.

)cos(0)(sin

010

)(sin-0)cos(

cc

cc

cT,

100

0)cos()(sin-

0)(sin)cos(

cc

cc

cT (2)


For the Piston, the complete rotation matrix is given by the product of its two individual rotations

(p and p) : Tp = Tp Tp. The rotation matrices are shown in equations (3) below.

)cos(0)(sin

010

)(sin-0)cos(

T

pp

pp

p

,

100

0)cos()(sin-

0)(sin)cos(

T pp

pp

p

(3)

2.2 Angular velocity

The angular speed of the compressor’s moving parts can be calculated through differentiation of their

Euler rotations [14] and are defined as:

dt

d0

0

TT

0dt

d0

T

dt

d0

0

r

rrr

r

r

r (4)

0dt

d0

T

dt

d0

0c

c

c

c (5)

0dt

d0

T

dt

d0

0p

p

p

p (6)

2.3 Coupling of elements

Couplings are the constraints given by the operational and geometry characteristics of the different

components of the compressor. The first restriction imposes that the connecting rod is pinned to the piston.

Such a condition is expressed in equation (7) below.

pc (7)

The relative displacement between the centerlines of the rotor’s eccentric’s bearing and the connecting

rod’s bearing, Figure 3, is expressed by 3, that is,

c3gcr3gr3 RRRRΔ (8)

The crankshaft mechanism imposes that the position of the center of mass of the connecting rod, Rgc ,

relates to the position of the center of mass of the piston, Rgp, that is,

c4p4gpgc RRRR (9)

Such a restriction is depicted in Figure 3.


Figure 3: Geometric restrictions of the reciprocating compressor: the variable 3 (eccentricity

displacement) and mass center position of the connecting rod (Rgc).

Two additional geometric constant parameters are the piston’s height, z , with regards to the origin of the

inertial system and the piston’s offset value, given by y.

2.4 Constitutive Equations (Elastic and Viscous Force)

The elastic and viscous forces are proportional to the negative value of the bearing displacement and

velocity, respectively. They are defined trough a stiffness matrix K and damping matrix Ci and are

represented as:

i

i

i

i

k00

0k0

00k

K , i = 1,2,….,6 (10)

i

i

i

i

c00

0c0

00c

C , i = 1,2,….,6 (11)

The values assigned to the stiffness and damping constant are obtained by trial and error, in such a way

that the orbits of all bearings do not exceed the eccentricity limit [19].

2.5 Combined Forces

The constitutive equation for the viscous and elastic bearing forces represented by forces F1, to F3 and F5

to F6 , which index describes the bearing position defined at Figure 2, can be written as :

)()( r1r1gr1r1r1gr11 rRRCrRRKF (12)

)()( r2gr2r2r2gr22 RRCrRRKF (13)


33333 ΔCΔKF (14)

gpldp5gp5p5p5gp55 )()y( RCRRCΔrRRKF (15)

gpldp6gp6p6p6gp66 )()z( RCRRCΔrRRKF (16)

Numerical simulations at the end of the paper show the influence on the choice of different damping and

stiffness coefficients for the bearings.

2.6 Equations of motion and their solution

The dynamic equations of motion are obtained from the Newton-Euler equations. Application of such

equations to the rotor yields,

321 FFFRM grr

τFTrFTrFTrωIωωI 3rr32rr21rr1rrrrr (17)

The same equations applied to the connecting rod yields,

c43gcc PFFRM

c_visc4cc43cc3ccccc τFTrFTrωIωωI (18)

Newton-Euler equations applied to the piston yields

pext654gpp PFFFFRM

6pp65pp54pp4ppppp FTrFTrFTrωIωωI (19)

A fourth order Runge Kutta algorithm is used, by means of the MatLab ODE45 toolbox function, to solve

the simultaneous system of differential equations presented in equations (17) to (19).

3 Defining the operational conditions for simulation

3.1 Operating torque

The torque required for maintaining a constant axis rotation is firstly calculated. The bearings are assumed

to be infinitely rigid so that no radial displacement is verified. The operating torque is calculated as a

function of equally spaced angles of rotation of the rotor, along a complete revolution. The driven torque

profile calculated in the pinned system operational condition is used as an input to the flexible system

analysis.

3.2 Flexible System bearing forces and displacements

Actuating loads and displacements of the bearings are calculated using the driving torque profile

computed in the pinned system analysis. The constitutive equations of the bearing displacements provide

the necessary relationship between loads and radial displacements. The constitutive hypothesis for the

bearing force reaction mechanisms are verified in this analysis.


3.3 Gas pressure condition

The gas pressure variation due to the compressor reciprocating movement is extracted from Couto [16]

and is presented in Figure 4 (a)(b). Such a pressure curve is used in the pinned system analysis to calculate

the driving torque which is necessary to maintain a constant angular velocity of the rotor axis.

Figure 4: (a) (b)Gas pressure at compressor cylinder. (c) Driven torque for reciprocating

compressor when submitted to compression loads.

4 Simulation results

Results from numerical simulations are compared, concerning the pinned and flexible systems. This

comparison verifies the relevance of using a complete dynamic modeling in the investigation of the

bearing forces. Different dynamic forces and bearing load conditions are analyzed from the model. The

inertia forces arriving from the movement of the compressor parts, during operation, is analyzed first. The

bearing forces are calculated in the sequence, considering the gas compression load.

The compression load, according to the gas pressure curve described in Figure 4 (a)(b), requires the

calculation of an operational torque to maintain the condition of constant rotor velocity. The torque is

calculated with the hypothesis of pinned bearings and the result is shown in Figure 4 (c).

4.1 Pinned and conservative bearing reaction forces

The bearing loads for the pinned and flexible bearing conditions are shown in Figure 5 and Figure 6. The

bearing stiffness are chosen in such a way as to preserve the maximum allowed eccentricity physical


relationship. For this simulation the adopted stiffness are: k1 = 5,0 x 107 N/m; k2 = 4,0 x 10

7 N/m; k3 = 2,5 x

107 N/m; k5 = k6 = 3,0 x 10

7 N/m.

Figure 5: Bearing load at pinned system .

Figure 6: Bearing load at flexible system.

The RMS value of the bearing force is computed, and it is verified that the elastic bearing reaction forces

are higher than those necessary to maintain the pinned dynamics. Figure 7 shows the bearing orbit in the

flexible system.


Figure 7: Bearing orbit at flexible system.

4.2 Bearing force with dissipative force

This simulation considers the calculation of bearing forces that are proportional to its radial velocity

components. A change in the bearing reaction force is verified with the use of such a dissipative model.

The reaction from the dissipative bearing models shows the same regular distribution pattern and modulus

as the reaction forces of the pinned bearing model, as observed in Figure 8.

Figure 8: Bearing load at flexible system.

The viscous components of the bearing reaction forces are one order of magnitude smaller than the elastic

force components. Figure 9 shows the bearing forces of viscous nature.


Figure 9: Bearing dissipative load at flexible system.

A more regular bearing orbit for the parametric viscous damping model is shown in Figure 10. This orbit

behavior indicated that the dissipative forces work as a regularization/stabilization control of the bearing

displacement, reducing the additional load generated by its decentralized movement.

Figure 10: Bearing orbit at flexible system.

A summary with the RMS load results for each bearing model are shown in Table 1.


Inertial Load Compressive Load Compressive and Dissipative

Load

Pinned System Flexible

System

Pinned System Flexible

System

Geometric

Dissipation

Parametric

Dissipation

1 160,9 215,6 392,1 528,0 524,1 395,4

2 78,0 164,6 154,6 331,7 329,9 159,5

3 83,6 103,7 239,2 289,9 285,3 240,0

4 4,9 8,6 27,9 40,8 39,3 28,5

5 4,6 6,7 25,6 47,9 45,1 26,1

1,2,3,4,5 – bearing position (1-primary, 2-secondary, 3-eccentric, 4-piston bottom, 5-piston top)

Table 1 – Bearing Loads RMS Value (Newton).

4.3 Orbit comparison with experimental results

The bearing model represented by elastic stiffness and parametric dissipation show the most stable orbit

when compared to the simulations of other bearing conditions. The orbit results for the rotor’s primary

and secondary bearing, using such a parametric dissipation model are plotted in Figure 11. An

experimental orbit measurement for the same bearings of the real compressor, available in [16], is shown

in Figure 12.

Figure 11: Bearing orbit: Simulation results for Primary bearing (red) and Secondary bearing

(blue).

The comparison shows the same stable behavior of the orbits in both simulation and experimental results.

The difference in orbit shapes are due to the arbitrary stiffness and damping values used in the numerical

model, as well as to the model itself, which is a simplification of the true hydrodynamic nature of bearing

forces that take place in the real mechanism. Nevertheless, the simplified stiffness-parametric damped

bearing model allows for the determination of a more realistic behavior of the dynamics that take place on

the mechanism components of the compressor.


Figure 12: Bearing orbit: Experimental result from Couto [16].

5 Conclusion

Analysis of the results of simulation using the complete dynamic model of the compressor with oscillation

of the rotating components leads to the following conclusions.

1 – Modeling of undamped oscillations of the axis-crankshaft-piston system produce distinct loads in the

compressor bearing, when compared to the bearing loads obtained with the pinned model.

2- The presence of dissipative forces in the bearings is responsible for the attenuation of secondary

movement in directions other than those necessary to perform the compression cycle. Dissipative forces

are one order of magnitude lower than the elastic forces that take place at the flexible bearings, and are

responsible to keep the bearing force to levels that are similar to those found in the pinned bearing system

model.

3- It is believed that the use of a full hydrodynamic bearing, modeled by Reynolds equation and coupled

with the system dynamics, would bring, for the present system, results that are similar to those calculated

with the pinned model.

4- The proposed model quantifies in a proper way the components displacement in directions that differ

from those necessary to execute the compression cycle. It is believed that such a model has a potential for

more precise calculations of the bearing dissipated energy and maintenance of a more stable compression

cycle.

5- The model can also be successfully employed in the analysis of other compressors and reciprocating

engines whose inertia, velocity or radial gap tolerances may lead to orbital movements that influences in a

more critical way the total load supported by the bearings.

Acknowledgements

The authors wish to thank Brazil´s research support foundation, CNPq for sponsoring this work.


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