Dynamic loads of reciprocating compressors with flexible bearings

Mechanism and Machine Theory 52 (2012) 130–143

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Review

Dynamic loads of reciprocating compressors with flexible bearings P.R.G. Kurka a,⁎, J.H. Izuka a, K.L.G. Paulino b a b

Faculdade de Engenharia Mecânica Universidade Estadual de Campinas, Brazil Universidade Federal de São Paulo, Campus Diadema, Brazil

a r t i c l e

i n f o

Article history: Received 3 June 2011 Received in revised form 18 January 2012 Accepted 20 January 2012 Available online 20 February 2012

a b s t r a c t The paper analyzes the visco-elastic bearing loads in the dynamic model of a reciprocating refrigeration compressor. The model incorporates the gyroscopic interactions due to the radial movement of the bearings. The Newton–Euler method is used in the analysis, establishing the necessary differential equations that describe the movement of the system, leading also to the calculation of orbital displacements of the bearings. © 2012 Elsevier Ltd. All rights reserved.

Keywords: Dynamic analysis Refrigeration compressor Visco-elastic bearings

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . Description of the model . . . . . . . . . . . . . 2.1. Components . . . . . . . . . . . . . . . 2.2. Rotation matrices . . . . . . . . . . . . . 2.3. Angular velocity . . . . . . . . . . . . . . 2.4. Coupling of elements . . . . . . . . . . . 2.5. Operational loads and force diagrams . . . . 2.6. Constitutive equations . . . . . . . . . . . 2.7. Equations of motion and their solution . . . 3. Simulation results . . . . . . . . . . . . . . . . 3.1. Physical properties . . . . . . . . . . . . 3.2. Driving torque . . . . . . . . . . . . . . 3.3. Bearings parameters . . . . . . . . . . . . 3.4. Numerical simulation results . . . . . . . . 3.4.1. Orbit comparison with experimental 4. Conclusion . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . results . . . . . . . . . . . .

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1. Introduction Refrigerating compressors [1] are rotor-crankshaft-piston mechanisms with moving parts supported by hydrodynamic bearings. Compressor lifetime and performance is highly affected by the lubrication of its moving parts. An inefficient or inadequate lubrication

⁎ Corresponding author at: Caixa Postal 6122, CEP 13083-970, Campinas, SP, Brazil. Tel.: + 55 19 3521 3175; fax: + 55 19 3289 3722. E-mail address: [email protected] (P.R.G. Kurka). 0094-114X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2012.01.014

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131

Nomenclature Roman letters rik Local position vector Fik Reaction forces vector Rgk Global center of gravity position vector Rik Global position vector Tik Rotation matrix g Gravitational acceleration constant

Greek letters ωk Angular speed τ Motor torque φk Precession angle θk Nutation angle ψk Spin angle μ Oil film viscosity

of the hydrodynamic supports cause an excessive displacement of the compressor components in the radial direction of the bearings, which leads to mechanism failure or waste of energy [2] and [3]. Most of the practical and analytical dynamic models of reciprocating compressors found in the literature, assume the bearings to be pinned or rigidly connected to its housings, with no contribution to the overall forces that arise from the movement of rotating and displacing masses of the mechanism. The study of bearing loads, orbits and lubrication effects is often done, separately from the dynamics of the components of the reciprocating mechanism (see, for example, the works of Prata [4], Cho [5], Kim [6], and Goodwin [7]). A reciprocating compressor model proposed by Levecque [8], is based on the dynamics of the rotor-crankshaft, crankcase and hermetic housing. Flexibilities of the connecting rod and piston hydrodinamic bearings were not considered in this study. The rotor is supported by bearings which are represented by their stiffness and damping properties. The optimization of the kinematics of a reciprocating compressor is studied by Sultan [9]. The author assumes a 2D planar model. A more accurate dynamic model should be able to quantify the increase of bearing loads as a result of their operational misalignments, caused by the gyroscopic effect. In the present modeling proposal, the rotational movements of compressor parts are described, in terms of their precession, nutation and spin movements [10]. This work aims to study the bearing behavior when subjected to dynamic loads from the compressed gas and components inertia. Bearing movements in a reciprocating compressor are typically in frequency ranges that are harmonic with regards to the operational driving cycle. Such frequencies are distinctively higher than those of the compressor's suspension mechanism, and significantly lower than those of characteristic structural vibration of the mechanical components and housing block. Therefore, the reciprocating hermetic compressor suspension is not considered in this work. Studies regarding the vibration reduction of the complete compressor justify the need to input the stiffness and damping effects of the compressor suspension, as explained by Dufour [11] and Levecque [8] 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 housing. The external loads assumed in the analysis come from experimental data of the cooling gas pressure cycle on the piston [12]. All parts are considered rigid with no relevant elastic twisting or bending of its elements. The analysis consists of establishing the balance of dynamical forces in each component of the compressor. Results are obtained from numerical integration of the dynamic model. The analysis simulates the dynamics of the compressor under a constant operational speed, leading to the visco-elastic reaction forces and orbital movements of the bearings.

2. Description of the model The compressor is comprised of its housing suspended block, stator, rotor, axis, connecting rod, piston and cylinder, as described in [13] and shown in Fig. 1.The dynamical model therefore accounts for the behavior of the moving parts and their corresponding supporting bearings, that are, axis and rotor assemblage, connecting rod and piston. The axis (and rotor) assemblage is suspended by a vertical (axial) supporting bearing and two radial hydrodynamic bearings (primary and secondary), all attached to the compressor block. The axial bearing is not considered in the present model, since the most important movements take place in radial directions. Another hydrodynamic bearing is located at the axis eccentric connection point. Such a bearing is responsible for 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, when compared to the dimensions of the other components bearings, allows it to be modeled as a non flexible force transmitting element, rather than a bearing, in the analysis. The piston travels essentially in its axial direction, but small operating movements also occur in the piston's radial direction. For this

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Fig. 1. Parts of the reciprocating compressor.

purpose, two flexible sliding bearings are defined at the piston top and skirt regions, which represent the actuating points of the forces that cause radial movements. 2.1. Components Four reference systems are created for the analysis: one inertial and three local ones, located in the center of mass of each of the compressor's components. The motion of the rotor is described by five movements: two absolute horizontal displacements of its center of mass, xr, yr, and three Euler rotations: φr, θr, ψr. The primary, secondary and eccentric bearing positions are described, with respect to the rotor's reference frame, by the location vectors, r1r, r2r and r3r, respectively. 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 θc, ψc. The eccentric bearing and the articulated pin positions are described, with respect to the connecting rod's reference frame, by the location vectors, r3c and r4c, respectively. Four movements are considered in describing the motion of the piston: two absolute displacements of its center of mass, xp, yp, and two Euler rotations θp, ψp. The articulated pin, skirt and top bearing positions are described, with respect to the piston's reference frame, by the location vectors, r4p, r5p and r6p. All geometric elements of the components of the compressor, as well as the numbered location of bearings and articulated pin, are shown in Fig. 2. 2.2. Rotation matrices Rotation matrices are used to transform the local system of coordinates of individual components into the inertial frame. The rotation matrix of the rotor is defined by the product of its precession (Tφr), nutation (Tθr) and spin (Tψr) matrices, Tr ¼ Tψr Tθr Tφr

ð1Þ

where, 2

Tψr

cosðψr Þ ¼ 4 − sinðψr Þ 0 2

Tθr ¼ 4

cosðθr Þ 0 sinðθr Þ

sinðψr Þ cosðψr Þ 0

3 0 0 5; 1

3 0 − sinðθr Þ 5; 1 0 0 cosðθr Þ

and 2

Tφr

cosðφr Þ ¼ 4 − sinðφr Þ 0

3 sinðφr Þ 0 cosðφr Þ 0 5: 0 1

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133

Fig. 2. Geometric location of compressor components and parts.

The rotation matrix of the connecting rod is defined by the product of its precession (Tφc) and nutation (Tθc) matrices, Tc ¼ Tθc Tφc

ð2Þ

where, 2 Tθc ¼ 4

cosðθc Þ 0 0 1 sinðθc Þ 0

3 − sinðθc Þ 5; 0 cosðθc Þ

and 2

Tφc

cosðφc Þ ¼ 4 − sinðφc Þ 0

sinðφc Þ cosðφc Þ 0

3 0 0 5: 1

The rotation matrix of the piston is defined by the product of its precession (Tφp) and nutation (Tθp) matrices, Tp ¼ Tθp Tφp

ð3Þ

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where,   cos θp 6 Tθp ¼ 4 0   sin θp 2

0 1 0

 3 − sin θp 7 0   5; cos θp

and   cos φp 6   ¼6 4 − sin φp 2

Tφp

0

  sin φp   cos φp 0

0

3

7 7 0 5: 1

2.3. Angular velocity The angular speed of the compressor's moving parts can be calculated by the derivative of their Euler rotations [10]. The rotor angular velocity vector, ωr is therefore calculated through the combination of a precession with velocity φ_ r about axis Zr, followed by a nutation with velocity θ_ r about axis Yr, and a spin with velocity ψ_ r , around axis Zr, all projected in the rotor's local system of coordinates. The same combination of projected rotations can be applied to the connecting rod and piston elements, yielding the respective rotation vectors, ωc and ωp, given by Eqs. (4), (5) and (6) below. 2 3 3 2 3 0 0 0 6 dθr 7 6 0 7 6 0 7 ωr ¼ 4 dψ 5 þ Tψr 4 5 þ Tψr Tθr 4 dϕ 5 dt r r 0 dt dt

ð4Þ

2 3 3 0 0 6 dθc 7 6 0 7 ωc ¼ 4 dψ 5 þ Tψc 4 5 dt c 0 dt

ð5Þ

2

2

2

3 2 3 0 0 6 0 7 6 dθp 7 ωp ¼ 4 dψ 5 þ Tψp 4 5 p dt 0 dt

ð6Þ

2.4. Coupling of elements Couplings are restrictions and constraints given by the operational and geometric characteristic of the different components of the compressor. The first geometric restriction is that the nutation angles of the connecting rod and piston are the same. Such a condition is expressed in Eq. (7). θc ¼ θc

ð7Þ

The relative displacement between the centerlines of the rotor's eccentric bearing and the connecting rod's bearing, Fig. 3, is expressed as Δ3, and depends on the absolute position of the centers of mass of the connecting rod, Rgc, the rotor, Rgr, and absolute vectors R3r and R3c, that is, Δ3 ¼ −Rgr −R3r þ Rgc þ R3c

ð8Þ

where, R3r ¼ TTr r3r R3c ¼ TTr r3c

ð9Þ

The position of the center of mass of the connecting rod, Rgc, is linked to the position of the center of mass of the piston, Rgp, Fig. 3, in the following way, Rgc ¼ Rgp þ R4p −R4c

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135

Fig. 3. Geometric restrictions in the reciprocating compressor.

where, R4c ¼ TTc r4c R4p ¼ TTp r4p

ð11Þ

Two additional geometric constant parameters are the piston's height, Δz, with respect to the origin of the inertial system, and the piston's offset value, given by Δy. The height and offset parameters are shown in Fig. 4. 2.5. Operational loads and force diagrams The gas reactive compressing load that occurs during the compressor's working cycle causes the bearing reaction forces that are represented in the force diagrams of each individual component. A motor driving torque, compatible with the operational compressing load and rotor angular velocity is also considered as an active external load of the dynamic system. The calculation of such external loads are discussed and presented in Section 3.2.

Fig. 4. Piston height and offset parameters.

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Fig. 5. (a) Force diagram of the rotor. (b) Force diagram of the crank. (c) Force diagram of the piston.

The force diagram of the rotor in the X and Y directions is shown in Fig. 5(a). The diagram does not account for the movement in the Z direction, which is considered to be small and stable. The Rotor's primary and secondary bearing acting forces are represented respectively by 2

3 F 1x F 1 ¼ 4 F 1y 5 0

ð12Þ

and 2

3 F 2x 4 F 2 ¼ F 2y 5 0

ð13Þ

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137

The interaction force between the eccentric shaft and connecting rod, applied to the rotor eccentric's bearing is represented as 2

3 F 3x F 3 ¼ 4 F 3y 5 0

ð14Þ

The rotor's external driving torque is represented as 2

3 0 τ¼405 τz

ð15Þ

The forces associated with the connecting rod movement are shown in Fig. 5(b). The force between the connecting rod and piston pin is represented by the vector 2

3 F 4x F 4 ¼ 4 F 4y 5 F 4z

ð16Þ

The connecting rod's weight vector is represented by 2

3 0 Pc ¼ 4 0 5 −mc g

ð17Þ

The forces associated to the piston movements are shown in Fig. 5(c). Force exerted on the piston by the compressed refrigerating gas is represented as 2

F ext

3 P ðφr Þ ¼4 0 5 0

ð18Þ

Piston's skirt and piston's top bearing loads are represented as 2

3 0 F 5 ¼ 4 F 5y 5 F 5z

ð19Þ

and 2

3 0 4 F 6 ¼ F 6y 5 F 6z

ð20Þ

The piston's weight is expressed as 2 Pp ¼ 4

3 0 0 5 −mp g

ð21Þ

2.6. Constitutive equations Elastic forces: the elastic forces are proportional to the negative value of the bearing displacement. They are calculated trough a stiffness matrix which is defined by its kii direct stiffness and kij cross coupling terms, given as: 2

kii Ki ¼ 4 ki1 ki2

ki1 kii ki1

3 ki2 ki1 5; i ¼ 1; 2; 3; 4; 5; 6 kii

ð22Þ

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Viscous force: the viscous forces are proportional to the negative velocities of displacement of the bearings. They are calculated through a viscous damping matrix which is defined by its cii direct damping and cij cross coupling terms, given as: 2

cii Ci ¼ 4 ci1 ci2

c cii ci1

3 ci2 ci1 5; i ¼ 1; 2; 3; 4; 5; 6 cii

ð23Þ

The procedure to calculate stiffness and damping terms is described in Section 3.3. Combined forces: the constitutive equation for the viscous and elastic bearing forces F1, to F3 and F5 to F6 can be finally written as     F 1 ¼ −K1 Rgr þ R1r −r1r −C1 R_ gr þ R_ 1r

ð24Þ

    F 2 ¼ −K2 Rgr þ R2r −r2r −C2 R_ gr þ R_ 2r

ð25Þ

F 3 ¼ −K3 Δ3 −C3 Δ_ 3

ð26Þ

    F 5 ¼ −K5 Rgr þ R5p −r5p −Δy −C5 R_ gp þ R_ 5p

ð27Þ

    F 6 ¼ −K6 Rgr þ R6p −r6p −Δz −C6 R_ gp þ R_ 6p

ð28Þ

2.7. 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, € ¼ −F þ F þ F Mr R gr 1 2 3 Ir ω_ r þ ωr  Ir ωr ¼ r1r  Tr F 1 þ r 2r  Tr F 2 −r 3r  Tr F 3 þ τ

ð29Þ

The same equations applied to the connecting rod yields, € ¼ −F þ F þ P Mc R gc 3 4 c Ic ω_ c þ ωc  Ic ωc ¼ −r3c  Tc F 3 þ r4c  Tc F 4

ð30Þ

Newton–Euler equations applied to the piston yields € ¼ −F þ F þ F þ F þ P Mp R gp 4 5 6 ext p Ip ω_ p þ ωp  Ip ωp ¼ −r4p  Tp F 4 þ r5p  Tp F 5 þ r6p  Tp F 6

ð31Þ

Terms Mr, Mc and Mp above refer to the mass distribution matrices of the rotor, connecting rod and piston components, respectively. Also, Ir, Ic and Ip are the inertia tensors of the rotor, connecting rod and piston components, respectively. 3. Simulation results The present section shows the assumptions made for the compressor's physical properties and operational conditions, as well as the results of the numerical analysis of its bearings loads. The solution proposed in this work is implemented in two parts. Firstly, the bearings are assumed to be pinned, that is, having infinite rigidity. In this case, there is no translation of the shaft and the position of the compressor parts are defined by the angle of the rotor, constituting a coupled single degree of freedom system. Such an assumption is made in order to calculate the motor driving torque curve, based on the steady state condition of constant speed of rotation of the rotor, and an experimentally known compression load at the cylinder. The driving torque and reactions forces at the bearings are calculated with the use of Eqs. (29)–(31). The second part of the analysis uses the driving torque calculated previously. Realistic bearings rigidity and damping coefficient values are used this stage of implementation of the numerical solution. The mobility of the model increases with the assumption of viscoelastic bearings, and the compressor dynamic behavior is represented by the 11 degrees of freedom variables ϕr, θr, ψr, Rgr(x), Rgr(y), ψc, θp, ψp, Rgp(x), Rgp(y) and Rgp(z), described in Section 2.1. The Runge Kutta method is applied in the simultaneous solution of Eqs. (29)–(31), yielding the bearing loads and orbits analysis information. In this work, Matlab's [16] ODE45 is used to integrate the differential equations that describe the dynamics of the system.

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139

3.1. Physical properties Table 1 below shows the physical properties assumed for the compressor components. Such properties are typical of a commercial refrigerating compressor. The position of the rotor's center of gravity is at the origin of the inertial coordinates system. The center of gravity of the piston and connecting rod are defined in Fig. 3. 3.2. Driving torque The torque required for a constant axis rotation is calculated from an experimentally known gas compression cycle. The gas pressure variation due to the compressor reciprocating movement is extracted from Couto [12] and is presented in Fig. 6(a), (b). Such a compression load requires the calculation of an operational torque to maintain the condition of constant rotor velocity, which is shown in Fig. 6(c). 3.3. Bearings parameters The lubricant film of a journal bearing that supports the rotor load acts like a spring while squeeze effect provides it's damping characteristic [14,15]. Stiffness and damping coefficients of stationary bearings are found by the variation of the force with journal displacement and angular velocity, respectively. These coefficients are available in the literature, as a function of the Sommerfeld number [14]. Rao [14] provides dimensionless stiffness and damping coefficients that are used as reference values in the present work. Stationary bearing coefficients are calculated for the conditions of rotor angular velocity of 3261 rpm, oil viscosity of 2.85 mPa.s and length to diameter ratio of approximately 0.5. The stationary bearings stiffness and damping reference coefficients are adjusted to the specific conditions assumed in the present dynamic simulation, that are of maximum bearing clearance of 12 μm for the primary and secondary bearing, 12.5 μm for the eccentric bearing and 4 μm for the piston. The last two columns of Table 2 show the bearing coefficients adopted in the present model. 3.4. Numerical simulation results Results from numerical simulations are obtained for the pinned and flexible systems, considering the experimental compression load and the calculated motor driving torque. Analysis of the pinned model yields the cyclic loads supported by each bearing. Table 1 Physical properties of components. Part Rotor

Mass (kg) mr= 0.945

Tensor of inertia (kg.m/s) 2 Ir ¼ 10−3 4

0:99 0 0 0:99 −0:01 0

Relative position of bearings 3

−0:01 0 5 0:39

2 r1r ¼ 4 2 r 2r ¼ 4 2 r3r ¼ 4

Connecting rod

mc= 0.029

2

3 0:10 0 0 0:80 0 5 Ic ¼ 10−5 4 0 0 0 0:85

2 r3c ¼ 4 2 r 4c ¼ 4

Piston

mp= 0.045

2

0:46 0 Ip ¼ 10−5 4 0 0:39 0 0

3 0 0 5 0:33

2 r4p

¼4 2

r5p ¼ 4 2 r 6p ¼ 4

3 0 0 5 0:0382 3 0 5 0 −0:020 3 0:0105 0 5 0:0636 3 −0:0131 5 0 0:0004 3 0:0253 0 5 0:0004 3 0:0006 0 5 0 3 −0:0042 5 0 0 3 0:0058 0 5 0

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Fig. 6. (a) (b) Gas pressure at compressor cylinder. (c) Calculated driven torque.

Solution of the model with flexible supports yields the loads, as well as orbital movements of each bearing. Numerical simulations are obtained using the Runge Kutta solution scheme with a time step of 10 − 4 s. Reaction forces are presented in Fig. 7, assuming that the bearings are pinned, or have infinite rigidity. Fig. 8 shows the bearing reaction forces under the flexible model assumption. Piston bearing forces are plotted regarding the XY plane in the infinite rigidity model and in the YZ plane in the flexible model. Bearings orbits and piston displacements obtained in the flexible model analysis are shown in Fig. 9. The reaction forces in the dissipative bearing model have the same regular distribution pattern and modulus as those of the pinned bearing model. The RMS load values in the combined XY directions for each bearing model are shown in Table 3. These results confirm that loads from the infinitely rigid model can be successfully used to estimate the parameters of the viscoelastic bearings, as explained in Section 3.3.

Table 2 Bearing parameters. Static bearing parameters (literature)

Dynamic bearing parameters (adjusted to operational clearances)

Bearing

Stiffness (N/m)

Damping (Ns/m)

Stiffness (N/m)

Damping (Ns/m)

Primary

2

108 0:8 3:8 4 −3:1 1:6 0 0

3 0 05 0

2

106 0:4 1:8 0

3 0 05 0

2

107 1:1 5:0 4 −4:1 2:1 0 0

3 0 05 0

2

105 0:3 1:1 0

3 0 05 0

2

108 0:8 4:8 4 −3:1 1:6 0 0

3 0 05 0

2

106 0:4 3:0 0

3 0 05 0

2

107 0:4 2:3 4 −1:5 0:7 0 0

3 0 05 0

2

104 0:8 5:7 0

3 0 05 0

108 0:8 4:8 4 −3:1 1:6 0 0

3 0 05 0

2

106 0:4 3:0 0

3 0 05 0

2

107 0:6 3:6 4 −2:3 1:2 0 0

3 0 05 0

2

105 0:1 0:8 0

3 0 05 0

2

103 3 0 0 6 05 0 6

2

103 3 0 0 6 05 0 6

Secondary

Eccentric

Piston

2

2

0 40 0

107 3 0 0 1 05 0 1

2:4 4 0:4 0

4:0 4 0:4 0

4:0 4 0:4 0

0 40 0

107 3 0 0 0 40 1 05 0 0 1 2

1:5 4 0:3 0

7:6 4 0:8 0

1:0 4 0:1 0

0 40 0

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141

Fig. 7. Bearing loads with pinned model.

3.4.1. Orbit comparison with experimental results The orbit results for the rotor's primary and secondary bearing, using such a parametric dissipation model are plotted in Fig. 10. An experimental orbit measurement for the same bearings in a real compressor with similar characteristics, available in [12], is also shown in Fig. 10. 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 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.

Fig. 8. Bearing load with flexible model.

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Fig. 9. Bearing orbit with flexible model.

Table 3 Bearing loads RMS value. Maximum load (N) Bearing

Pinned system

Flexible system

Primary Secondary Eccentric Piston skirt Piston top

380.3 147.8 233.6 16.4 15.1

380.3 148.1 233.5 16.4 15.1

Fig. 10. Bearing orbit: simulation results for primary bearing (red) and secondary bearing (blue). Experimental result from [12] represented by dot and plus signs.

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4. Conclusion Analysis of the results of simulations using the complete dynamic model of the compressor with oscillation of the rotating components leads to a number of conclusions as given below. The dynamic model was able to represent the inclination of the rotor during the compressor operation. This behavior could be verified by the principal and secondary bearing orbits, which are in opposite directions. Bearing stiffness and damping properties could be defined by the forces calculated in the pinned model. The validity of such an assumption is verified through the comparison of pinned and flexible forces in the bearing positions. The orbit comparison with an experimental result shows the limitations of this model. A full hydrodynamic model of the bearing (Reynolds equation) should yield a more realistic orbit result. 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. Acknowledgment The authors wish to thank Brazil's research support foundation, CNPq for sponsoring this work. References [1] W.F. Stoecker, Industrial Refrigeration Handbook, McGraw-Hill, 1998. [2] S.A. Tassou, I.N. Grace, Fault diagnosis and refrigerant leak detection in vapour compression refrigerant systems, International Journal of Refrigeration 28 (2005) 680–688. [3] J.M. House, K.D. Lee, L.K. Norford, Controls and diagnostics for air distribution systems, Journal of Solar Energy Engineering, Transactions of the ASME 125 (2003) 310–317. [4] A.T. Prata, J.R.S. Fernandes, F. Fagotti, Dynamic analysis of piston secondary motion for small reciprocating compressors, Journal of Tribology 122 (2000) 752–760. [5] J.R. Cho, S.J. Moon, A numerical analysis of the interaction between the piston oil film and the component deformation in a reciprocating compressor, Tribology International 38 (2005) 459–468. [6] T.J. Kim, J.S. Han, Comparison of the dynamic behavior and lubrication characteristics of a reciprocating compressor crankshaft in both finite and short bearing models, Tribology Transactions 47 (2004) 61–69. [7] M.J. Goodwin, J.L. Nikolajsen, Reciprocating machinery bearing analysis: theory and practice, Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 217 (2003) 409–426. [8] N. Levecque, J. Mahfoud, D. Violette, G. Ferraris, R. Dufour, Vibration reduction of a single cylinder reciprocating compressor based on multi-stage balancing, Mechanism and Machine Theory 46 (1) (2011) 1–9. [9] I.A. Sultan, A. Kalim, Improving reciprocating compressor performance using a hybrid two level optimization approach, Engineering Computations 28 (5) (2011) 616–636. [10] A.A. Shabana, Dynamics Multibody Systems, John Wiley & Sons, Illinois, 1989. [11] R. Dufour, J. Der Hagopian, M. Lalanne, Transient and steady state dynamic behavior of single cylinder compressors: prediction and experiments, Journal of Sound and Vibration 181 (1) (1995) 23–41. [12] P.R.C. Couto, Análise de Mancais Radiais Hidrodinâmicos com Aplicação em Compressores Herméticos de Refrigeração. Ph. D. Thesis, Federal University of Santa Catarina, Florianopolis 2006 [in Portuguese]. [13] H.J. Wisbeck, A.L. Manke, A.T. Prata, Bearing System in Dynamic Loading Including Solid Contact and Wear, Paper, International Compressor Engineering Conference, 1532, 2002. [14] J.S. Rao, Rotor Dynamics, John Wiley & Sons, 1983. [15] G.W. Stachowiak, A.W. Batchelor, Engineering Tribology, 2nd ed. Buttterworth Heinemann, 2001. [16] S.R. Otto, J.P. Denier, An Introduction to Programming and Numerical Methods in MATLAB, Springer – Verlag, London, 2005.