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Dynamic behaviors of the crankshafts in single-cylinder and twin-cylinder rotary compressors
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
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Dynamic behaviors of the crankshafts in singlecylinder and twin-cylinder rotary compressors Haifeng Zhang a,b, Jianhua Wu a,*, Fei Xie b, Ang Chen a, Yanzhong Li a a b
School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China Shanghai Hitachi Electrical Appliances Co., Ltd., Shanghai 201206, PR China
article info
abstract
Article history:
Refrigeration rotary compressors are widely used in air-conditioners. A rotary compressor
Received 17 March 2014
has a special rotor-bearing system, since the elastic cantilevered crankshaft is under dy-
Received in revised form
namic transverse forces on different planes. The large deformation of the crankshaft
24 June 2014
would affect the thickness of the oil film, wear the bearings down or even induce the rotor-
Accepted 17 July 2014
to-stator rubs. It is considered as a non-linear fluidestructure interaction problem. To
Available online 1 August 2014
ensure the compressor operating well, the dynamic behaviors of both single-cylinder and twin-cylinder compressors' crankshafts at various speeds are analyzed. The influence of
Keywords:
configuration of balancers on the reliability of rotor system is investigated for the further.
Dynamic behavior analysis
Calculation results suggest that the vulnerable sections of crankshaft vary with the rota-
Crankshaft
tional speed. It is also found that 80% of the dynamic balance is the optimum design
Rotary compressor
condition to reduce the transverse forces on crankshaft and the wear-out of journal bearings. © 2014 Elsevier Ltd and IIR. All rights reserved.
Comportements dynamiques des vilebrequins dans des cylindre unique et a cylindres jumele s compresseurs rotatifs a Mots cles : Analyse du comportement dynamique ; Vilebrequin ; Compresseur rotatif
1.
Introduction
In a rotary compressor used in air-conditioners (shown in Fig. 1), the crankshaft driven by the motor is capable of rotating with a large range of speed. The volume of compression cavity changes periodically with the rotation and the refrigerant gas in cylinder is compressed. However, due to
* Corresponding author. E-mail address: [email protected] (J. Wu). http://dx.doi.org/10.1016/j.ijrefrig.2014.07.014 0140-7007/© 2014 Elsevier Ltd and IIR. All rights reserved.
the special structure of rotary compressor, crankshaft and motor rotor are made as a cantilever. Besides, the crankshaft is under a large dynamic load including gas force and unbalanced mass forces of comprising mechanical parts (eccentric crank, roller, and balancers on motor rotor). As a result, lubrication conditions at bearings would become severe owing to the deformation of crankshaft, as shown in Fig. 2. What is worse, rotor-to-stator rub would occur if the crankshaft bends
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
Nomenclature C e F Fn, Ft h Hc K L m P Pc Pd Ps q R
Radial clearance between crankshaft and bearing(m) Eccentricity (m) Force (N) Normal and tangential force at vane tip(N) Thickness of oil film (m) Height of cylinder (m) Dimensionless stiffness coefficient The length of arms of forces(m) Mass quantities (kg) Pressure (Pa) Pressure in compression chamber(Pa) Discharge pressure(Pa) Suction pressure (Pa) Vector of displacement(m) Radius(m)
more largely. Therefore, to ensure rotary compressor operating well, the dynamic behavior of crankshaft at various speeds should be analyzed. Classic rotor dynamics is a special branch mainly analyzing the dynamic behavior of structures at overcritical high speed with little transverse loads. Ehrich (Ehrich, 1991) studied the bifurcation of a bearing-rotor system and identified a sub-harmonic vibration phenomenon in the rotor's dynamic behavior. Holmes (Holmes et al., 1978) published a paper dealing with a periodic behavior in journal bearings. Brown et al. (1994) developed a simple model of a rigid, hydrodynamically supported journal bearing using short bearing theory. It was shown that the journal behaved chaotically when the rotating unbalance force exceeded the gravity load. Chang-Jian (2010); ChangJian and Chen (2006) discussed about a rotor supported by journal bearings under non-linear suspension and combined with rub-impact effect, turbulent effect and micropolar lubricant into consideration. Kurka et al. (2012) analyzed the visco-elastic bearing loads in the dynamic model of a reciprocating refrigeration compressor. The NewtoneEuler method was used in the analysis, establishing the necessary differential equations that described the movement of the system, leading also to the calculation of orbital displacements of the bearings. In 1990, Hattori and Kawashima (1990) proposed a method to analyze the rotor-journal bearing system in a twin-rotary compressor. The crankshaft was meshed by onedimensional beam elements based on finite element method. The short bearing theory was employed to solve the oil film force. The sub bearing was taken as one short bearing while the main bearing was taken as two. Elastic bending deformation of the crankshaft, the bearing loads profiles, and the pressure distributions of the oil film had been obtained. In 1998, Dufour et al. (1998) also took beam element to mesh the crankshaft; the bearings were modeled by one- node elements with two lateral translations, and the main bearing
Greeks a ε q qj h 4 u
37
Offset angle of roller center (rad) Eccentricity of journal Rotational angle of crankshaft (rad) Circumferential angle of bearing (rad) Viscosity of oil(Pa s) Attitude angle of bearing(rad) Angular velocity (rad s1)
Subscripts bal Balancers c Crank e Eccentricity of crankshaft ec Eccentric mass of roller and crank g Gas j Journal mj Main journal bearing o Oil ro Roller sj Sub journal bearing
was split into two parts. In his work, the oil film forces were calculated by the stiffness and damping characteristics of the three bearings, which were empirical equations involving ve bearing clearance and bearing load. In 2003, Dufour (Se et al., 2003) continued his research on the balancing procedure for variable-speed compressor ignoring the cylinder pressure force. In Xie's study (Xie and et al., 2006),dynamic behaviors of the rotor-journal bearing system for singlerotary compressor at high speed and low speed were analyzed considering both unbalanced mass force and gas force. Wang et al. (2013) applied the finite element model with three-dimensional solid element to study the vibration characteristics of the rotor-journal bearing system in a variable speed rotary compressor. In his work, the dynamic model of the rotor was solved by the finite element software ANSYS. A variable speed rotary compressor is capable of running from 1200 rpm to 7200 rpm, while the rotation speed of a fixed speed rotary compressor is usually between 2850 rpm and 3650 rpm. Both of them are far below the instability of the lubricant. Besides, unlike the classic rotor dynamics model, the width-diameter ratio of a rotary compressor crankshaft is much larger, and its elastic deformation is mainly caused by transverse forces. Moreover, the heightdiameter ratio of the main bearing in rotary compressor is almost 3, so neither the long bearing theory nor the short bearing theory is proper for it. Detailed comparison between is shown in Table 1. Due to the differences, it is necessary, at the design stage, to use a more suitable model to predict the deformation of crankshaft and the distribution of oil film. In this paper, elasticity of the crankshaft is taken into account, 2dimensional finite element method is applied to solve the oil film, and the dynamic behaviors of both single-cylinder and twin-cylinder compressor at various speeds are analyzed. The influence of configuration of balancers on the reliability of rotor system is investigated for the further.
38
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Fig. 1 e Structures of rotary compressor (a) main body of a single-cylinder rotary compressor (b) main body of a twincylinder compressor.
2.
Forces analysis
The transverse loads acting on the crankshaft are gas force (Fg), unbalanced mass forces of eccentric crank and roller (Fec), unbalanced mass forces of balancing weight (Fb1,Fb2), vane contact force (Fn,Ft), and oil film forces of journal bearings (Fmj,Fsj), as shown in Fig. 3.Note that these transverse forces are on different planes, leading to a large elastic deformation of the crankshaft, which would cause the wear-out of journal bearings or even the rub between rotor and stator. To limit the deformation, the main bearing is lengthened. As a result, the oil film forces turn out to be non-linear.
2.1.
Loads acting on the crank part of crankshaft
The cross-section of the cylinder is shown in Fig. 4. The crank part of crankshaft is surrounded by roller. The inner volume of cylinder is divided into suction chamber and compression
chamber by vane and roller. As crankshaft rotates, the volume of compression chamber decreases, and the inside gas pressure increases from Ps to Pc. The gas force Fg acting on the roller can be derived by the following equation (Yanagisawa and Shimizu, 1985). Fg ¼ 2Rro sin
(1)
Gas force Fg, eccentric force Fec, vane contact force Fn and Ft all act on crank part. Magnitude Fen and direction qf of the combined load are shown as follows. Fr and Fq are radial and angular components of crankshaft load given by Equations (2) and (3), respectively (Yanagisawa and et al., 1982). qþa þ Fec Fn cosðq þ aÞ þ Ft sinðq þ aÞ Fr ¼ Fg cos 2 Fq ¼ Fg sin
Fen ¼
Fig. 2 e The transverse loads acting on the crankshaft of a single-cylinder compressor.
qþa Hc ðPc Ps Þ 2
qþa þ Fn sinðq þ aÞ Ft cosðq þ aÞ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F2r þ F2q
qf ¼ tg1
Fq þq Fr
2.2.
Unbalanced mass forces
(2)
(3)
(4)
(5)
The crankshaft rotates with large eccentric force generated by crank part and roller. To reduce this eccentric force, upper and lower balancers are mounted on the motor rotor, as shown in Fig. 5. Fec is the eccentric force of crank and roller, Fb1 and Fb2 are the unbalanced mass forces of upper and lower balancers, respectively. Lec, Lb1, Lb2 are the arms of forces from the lower end of crankshaft. The unbalanced mass forces of crank part, roller and balancers are as follows.
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Table 1 e Differences of rotor-journal bearing systems in present paper and previous study. Differences
Present paper
Literature Classic rotor dynamics
Transverse loads Operating speed Diameter of crankshaft height-diameter ratio Model
Large dynamic transverse force Below 10 K rpm Small Main bearing: 3 Sub bearing：1.5 A special cantilever model
Little Over 10 K rpm
Heavy load bearing Large Large Below 1.0
Jeffcott rotor model
Fig. 3 e Transverse forces acting on the crankshaft in a rotary compressor. (a) Transverse forces on y-z plane. (b) Transverse forces on xey plane.
Fec ¼ ðmro þ mc Þeu
(6)
Fb1 ¼ mb1 eb1 u2
(7)
Fb2 ¼ mb2 eb2 u2
(8)
2
For rigid system, dynamic balance is designed to be 100% simultaneously, take single-cylinder compressor for example. Fb1 ¼ Fec þ Fb2 Fb1 Lb1 ¼ Fec Lec þ Fb2 Lb2
(9) (10)
Hence. mb1 ¼
e Lb1 Lec ðmro þ mc Þ 1 þ eb1 Lb2 Lb1
(11)
mb2 ¼
e Lb1 Lec ðmro þ mc Þ eb2 Lb2 Lb1
(12)
To change the dynamic balance is to change the weight of balancers. For example, 80% of dynamic balance is to reduce 20% of the original mass weight of both upper balancer and lower balancer.
2.3.
Fig. 4 e Cross-section of the cylinder. Gas force, eccentric force and vane contact force are acting on the crank part of the crankshaft.
Film force
The crankshaft of a rotary compressor is supported by main and sub bearings, which are hydrodynamic bearings with small clearance, as shown in Fig. 6. Due to the deflection and deformation of the crankshaft, local friction may occur on the journal bearings in extreme situations. Since the oil film in journal bearings is of ‘finite length, dynamic loads and various thickness on axial direction’, finite element method (FEM) is applied to solve the force of oil film.
40
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Fig. 5 e Unbalanced mass forces of rotary compressor. (a) The unbalanced mass forces of single-cylinder compressor. (b) The unbalanced mass forces of twin-cylinder compressor. The reaction forces of oil film between the crankshaft and the bearings are calculated by the Reynolds equations as follows: (Ito et al., 2006) 3 v h vP 1 v h3 vP vh vh ¼ 6u þ 12 þ vqj vt r2j vqj h vqj l2 vz h vz
(13)
h ¼ C 1 þ ε cos qj
(14)
between domain (P > 0) and (P ¼ 0), G1 and G2 are the boundaries at the ends of bearings. 8 G1 : z ¼ þ1 P¼0 > > > < G2 : z ¼ 1 P¼0 > > > : Gc : undetermined vP; P ¼ 0 vn
(15)
where h is the thickness of oil film between journal and bearing, P is oil pressure, qj is circumferential angle of the bearing, h is viscosity of oil given by discharge pressure and temperature, C is radial clearance between journal and bearing, and ε is eccentricity of journal with respect to bearing center. The bilinear four-node rectangle element is used to discretize the oil film and Reynolds Boundary Condition is written as Equation (15), where Gc is the undetermined boundary
Fig. 7 e The finite element model of crankshaft in singlecylinder compressor. The locations of bearing housings are marked by dashed boxes. Unbalanced mass forces of balancers are carried by node16 and node18.
Fig. 6 e Structure of journal bearing Oj is the crankshaft crank's center, and Ob is the bearing housing's center.
Fig. 8 e The finite element model of crankshaft in twincylinder compressor. The locations of bearing housings are marked by dashed boxes. Unbalanced mass forces of balancers are carried by Node21 and Node23.
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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
Table 2 e Nodes to carry forces in finite element model. Forces
Nodes (single-cylinder)
Nodes (twin-cylinder)
16,18 1,2,3 9,10,11,12,13,14,15 5,6,7
21,23 1,2,3 14,15,16,17,18,19,20 5,6,7,10,11,12
Unbalanced mass force of the balancers Oilfilm force of the sub earing Oilfilm force of the main earing Gas force of the compressed gas
3.
Finite element model
4.
The momentum equation of crankshaft is. € þ ½KfXg ¼ ½F ½M X
(16)
with {X} being the displacement vector, [M] the mass matrix, [K] the stiffness matrix and [F] the outer load. Since the crankshaft runs at a relatively low speed, its inertia force (not including inertia force of balance weight and eccentric mass) is much smaller than outer loads and can be ignored. Then the following equation is built. _ ½Kfqg ¼ Fg ðqÞ þ fFo ðfqg; fqgÞg þ Fbal ðqÞ
Coupling method between solid and fluid
Oil film force is the main nonlinear factor in the rotor-journal bearing system. The nodes, which are loaded by oil film force, are defined as nonlinear nodes. For this system with local non-linearity, its governing equation can be rewritten as follows (Xie and et al., 2006)
K11 K21
K12 K22
q1 q2
¼
Fg1 Fg2
þ
0 Fo
(18)
(17)
_ is vector of velocity, where, {q} is vector of displacement, fqg {q} ¼ {y, z, qy, qz}T, Fg(q) is gas force, Fbal(q) is the unbalanced _ is oil film force, q is the mass force of balancers, Fo ðfqg; fqgÞ rotational angle of crankshaft. The Timoshenko beam element is applied to mesh crankshaft. According to the shape of the cross-section, material and structure, the shafts in single-cylinder compressor and twin-cylinder rotary compressor are meshed as shown in Fig. 7 and Fig. 8, respectively. Take Fig. 7 as example: First, the thinner part must be represented by at least one individual element, like element 3 and 8, since stress may concentrate there. Second, as the rotor and crankshaft are made of different materials, element 16 and 17 should be meshed to define the properties. Finally, the structure of bearings are taken into consideration and element 1e2, 9e14 are meshed. The locations of bearing housings are marked by dashed boxes. The nodes to carry the forces are shown in Table 2.
Table 3 e Working conditions and main parameters of structure. Single-cylinder compressor Inner diameter of cylinder (mm) Height of cylinder (mm) Outer diameter of roller (mm) Displacement (cc) Crankshaft diameter (mm) Main bearing length (mm) Sub bearing length (mm) Clearance (mm) Distance Lb1 (mm) Distance Lb2 (mm) Distance Lec (mm)
Twin-cylinder compressor
60
58
28 46
20.28 47.38
32.6 22 67 25 20 124.1 222.1 39.9
35.6 21 67 25 20 147.3 245.3 Lec1 (mm) Lec2 (mm)
38.31 64.81
Fig. 9 e The value and angle of gas force and oil film forces in single-cylinder compressor. (a) forces in X-direction changing with rotating angle, (b) forces in y-direction changing with rotating angle.
42
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where, {q1} is the displacements of linear nodes, {q2} is the displacement of nonlinear nodes. Fg1 is the angular gas force, Fg2 is the radial gas force, Fo is the oil film force. Hence q1 ¼ K1 11 Fg1 K12 q2
(19)
b 2 ¼ Fg2 þ Fo þ b f Kq
(20)
b b ¼ K22 f ¼ K21 K1 where K 11 Fg1 Eq. (18) is a set of nonlinear algebraic equations, Fg2 and b f _ In the are the functions of time, Fo is the function of {q} andfqg. b is determined only by system studied in this paper, matrix K K21 K1 11 K12 ,
rotational speed. Once a certain initial value is given to {q}, we _ with NewtoneRaphson method, hence are able to solve fqg _ at the {q} at next time step. In this way, the value of {q} and fqg every moment can be calculated. With the initial value being updated after every iterative period, the convergent periodical solution can be obtained after repeating the above procedure
Fig. 10 e The value and angle of gas force and oil film forces in twin-cylinder compressor. (a) forces in x-direction changing with rotating angle, (b) forces in y-direction changing with rotating angle.
Fig. 11 e Vulnerable sections of crankshaft in rotary compressor. Wear-out between journal and bearing usually occurs on four vertical planes, namely MBT (main bearing top), MBB (main bearing bottom), SBB (sub bearing bottom) and SBT (sub bearing top).
Fig. 12 e Journal orbits of the bearings in single-cylinder compressor. (a) Orbits of four vulnerable sections at the speed of 2850 rpm. (b) Orbits of vulnerable sections at the speed of 7200 rpm.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
for several times. The NewtoneRaphson method is presented as follows. 9 8 Dy_ > > > = < _> Dz ¼ ½F ½D Dq > > y > > ; : Dqz
½D ¼
q_x2 ¼ q2 ¼
)
( ¼
In fact,
i q_x2 i q2
)
( þ
i Dq_ x2 i Dq2
) (26)
vfFo g _ g vfqx2
is the derivative of oil film force with respect
to the translational velocity of journal in the Jacobian matrix.
vfFo g vfFg vfq2 g v qx2 _
b qi þ Fg2 þ Fo þ b ½F ¼ K f 2 n
iþ1 q_x2 iþ1 q2
(21)
where
(
43
y_2
z_2
qy2
qz2
oT
T
(22)
5.
Results and discussion
(23)
(24)
(25)
so
Fig. 13 e Journal orbits of the bearings for twin-cylinder compressor. (a) Orbits of four vulnerable sections at the speed of 2850 rpm. (b) Orbits of vulnerable sections at the speed of 7200 rpm.
5.1. The dynamic behavior of crankshaft in rotary compressor The dynamic behaviors of crankshafts in single-cylinder and twin-cylinder rotary compressors are analyzed in this paper. The environmental friendly R410A is adopted as refrigerant, with the suction pressure at 1.095 MPa and discharge pressure at 4.185 MPa. Working conditions and main parameters of structure are listed in Table 3. Dynamic behaviors of crankshaft at2850 rpm and 7200 rpm have been analyzed (dynamic balance are designed to be 100%). The gas force and oil film forces are shown in Figs. 9 and 10. The direction of gas force is changing with rotation angle of the crankshaft. For single-cylinder compressor, although the lengths of the main and sub bearings are not the same, they almost take the load averagely, and bearing force has only one peak from 0 to 360 . For twin-cylinder, oil film force has two peaks in one cycle. This is because that even though the main bearing mainly takes the load from upper cylinder while the sub bearing mainly takes the load from lower cylinder, gas force from the other cylinder with a different phase can still influence the bearing. Due to the elasticity of crankshaft, wear-out between journal and bearing usually occurs on four vertical planes, namely MBT (main bearing top), MBB (main bearing bottom),
Fig. 14 e Journal orbits at MTop (motor top section).
44
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
SBB (sub bearing bottom) and SBT (sub bearing top), as shown in Fig. 11. Besides, the impact between motor rotor and motor stator may occur at the section of MTop, the top rotor motor. Journal orbit plays a key role in the dynamics of rotary compressor. Journal orbits at both ends of bearings are shown in Figs. 12 and 13. For single-cylinder, at low speed of 2850 rpm, the orbits of journal orbit at SBB and MBB are relatively larger, and the wear-out may occur in these sections where oil film is much thinner. At high speed of 7200 rpm, orbits at SBT, SBB and MBB are smaller, while that at MBT is larger due to the eccentric force of balancers. Conversely, 100% of dynamic balance enlarges the deformation of crankshaft. As a result, wear-out could occur at the upper side of the main bearing at high speed. For twin-cylinder compressor, the shaft whirls two circles in one rotational period, which is different from singlecylinder compressor. Eccentricities of orbits at SBB/MBB are large, which means the oil film is thin and wear-out is possible to occur. As speed increases, the orbits at MBB and MBT grow and almost turn into circles.
Fig. 15 e Maximum eccentricities of journal orbits in single-cylinder compressor. (a) Operating speed is 2850 rpm, (b)operating speed is 7200 rpm.
Orbits at MTop are shown in Fig. 14. When rotating at the same speed, the orbit of the twin-cylinder compressor is much smaller than that of the single-cylinder compressor, which suggests that the rotor-bearing system of a twin-cylinder compressor has a better characteristic of stability. The movements of upper and lower crank parts in twin-cylinder are in opposite phases, so their eccentric forces can counteract to each other. As a result, a pair of balancers with less weight is needed and the orbits of bearing top are smaller. Compare (a), (b) in Figs. 12e14, it can be concluded that the dynamic behavior of crankshaft is determined by its structure (single-cylinder or twin-cylinder), rotational speed and the elasticity.
5.2. Influence of configuration of balancers on the reliability of rotor system In rotary compressor, 100% of dynamic balance may not be the optimum solution since bending deformation of crankshaft is
Fig. 16 e Maximum eccentricities of journal orbits for twincylinder compressor. (a) Operating speed is 2850 rpm. (b) Operating speed is 7200 rpm.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
45
detail, including deformation of the crankshaft, journal orbits and etc. 2) For single-cylinder compressor, the main and sub bearings almost take the load averagely and bearing forces only have one peak. However, as to twin-cylinder compressor, bearing forces have two peaks since there are two cylinders, and each bearing mainly takes the load from the closer cylinder. 3) The vulnerable sections of crankshaft vary with the rotational speed. At low speed, the wear may occur at SBB and MBB. At high speed, the wear may occur at MBT and MTop. 4) Dynamic behaviors of the rotor system with different dynamic balances ranging from 60% to 120% had been studied in this paper. 80% of dynamic balance is an optimum solution for rotary compressor because bending deformation of the crankshaft is a more important factor than vibration. Fig. 17 e Orbits of Mtop in single-cylinder and twincylinder compressor with different balancers.
references a more important factor than vibration. Therefore, dynamic behaviors of the crankshaft with different dynamic balances ranging from 60% to 120% are studied in this paper. According to the calculation results, 100% of dynamic balances is the optimum condition for reducing the wear-out of journal bearings at low speed. On the other hand, when the rotor rotates faster, a smaller of dynamic balance could help to decrease the vibration of rotor-journal system. All things considered, 80% of dynamic balancers is the optimum design condition. The maximum eccentricities of the journal orbits are shown in Fig. 15 and 16, respectively. It is found that the trend of wear-out differs. At low speed, maximum eccentricity of journal orbit varies slightly with the increment of dynamic balance. At high speed, maximum eccentricity of journal orbit at MBT increases rapidly with the increment of dynamic balance, and wear-out may occur at upper part of the main bearing. For single-cylinder compressor at high speed, maximum eccentricities of the journal orbits at SBB/MBB decrease with the increment of dynamic balance. For twin-cylinder compressor at high speed, the chance of wearing at SBB is lower, while that at MBB is high. Orbits at motor top section in both single-cylinder and twin-cylinder compressors with different balancers are shown in Fig. 17. When at the same speed, eccentricity of orbit of MTop in single-cylinder compressor is larger than in twincylinder compressor, and become more serious with the increment of dynamic balance.
6.
Conclusions
1) A numerical method to analyze the dynamic behavior of crankshaft in rotary compressor is built up. The elasticity of the crankshaft, the gas force, nonlinear oil film force etc. are considered comprehensively. Based on this method, the dynamic behavior of the crankshaft has been studied in
Brown, R., Addison, P., Chan, A., 1994. Chaos in the unbalance response of journal bearings. Nonlinear Dyn. 5 (4), 421e432. Chang-Jian, C.-W., 2010. Non-linear dynamic analysis of dual flexible rotors supported by long journal bearings. Mech. Mach. Theory 45 (6), 844e866. Chang-Jian, C.-W., Chen, C.-K., 2006. Nonlinear dynamic analysis of a flexible rotor supported by micropolar fluid film journal bearings. Int. J. Eng. Sci. 44 (15e16), 1050e1070. Dufour, R., Charreyron, M., Gerard, M., 1998. Dynamics prediction of refrigerant rotary compressor crankcrankshaft. In: Proceedings of International Compressor Engineering Conference at Purdue. Ehrich, F., 1991. Some observations of chaotic vibration phenomena in high-speed rotordynamics. J. Vib. Acoust. Stress, Reliab. Des. 113 (1), 50e57. Hattori, H., Kawashima, N., 1990. Dynamic analysis of a rotorjournal bearing system for twin rotary compressors. In: Proceedings of International Compressor Engineering Conference at Purdue. Holmes, A., Ettles, C., Mayes, I., 1978. The aperiodic behaviour of a rigid crankshaft in short journal bearings. Int. J. Numer. Methods Eng. 12 (4), 695e702. Ito, Y., Hattori, H., Miura, K., 2006. Numerical analysis on motion of rolling piston in a rotary compressor. In: Proceeding ASIATRIB 2006 KANAZAWA. Kurka, P.R.G., Izuka, J.H., Paulino, K.L.G., 2012. Dynamic loads of reciprocating compressors with flexible bearings. Mech. Mach. Theory 52, 130e143. ve, F., et al., 2003. Balancing of machinery with a flexible Se variable-speed rotor. J. Sound. Vib. 264 (2), 287e302. Wang, Zengli, et al., 2013. Dynamic analysis for the rotor-journal bearing system of a variable speed rotary compressor. Int. J. Refrigeration 36 (7), 1938e1950. Xie, F., et al., 2006. Dynamic analysis of a rotor-journal bearing system of rotary compressor. In: Proceedings of International Compressor Engineering Conference at Purdue. Yanagisawa, T., et al., 1982. Motion analysis of rolling piston in rotary compressor. In: Proceedings of International Compressor Engineering Conference at Purdue. Yanagisawa, T., Shimizu, T., 1985. Leakage losses with a rolling piston type rotary compressor. II. Leakage losses through clearances on rolling piston faces. Int. J. Refrigeration 8 (3), 152e158.
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journal homepage: www.elsevier.com/locate/ijrefrig
Dynamic behaviors of the crankshafts in singlecylinder and twin-cylinder rotary compressors Haifeng Zhang a,b, Jianhua Wu a,*, Fei Xie b, Ang Chen a, Yanzhong Li a a b
School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, PR China Shanghai Hitachi Electrical Appliances Co., Ltd., Shanghai 201206, PR China
article info
abstract
Article history:
Refrigeration rotary compressors are widely used in air-conditioners. A rotary compressor
Received 17 March 2014
has a special rotor-bearing system, since the elastic cantilevered crankshaft is under dy-
Received in revised form
namic transverse forces on different planes. The large deformation of the crankshaft
24 June 2014
would affect the thickness of the oil film, wear the bearings down or even induce the rotor-
Accepted 17 July 2014
to-stator rubs. It is considered as a non-linear fluidestructure interaction problem. To
Available online 1 August 2014
ensure the compressor operating well, the dynamic behaviors of both single-cylinder and twin-cylinder compressors' crankshafts at various speeds are analyzed. The influence of
Keywords:
configuration of balancers on the reliability of rotor system is investigated for the further.
Dynamic behavior analysis
Calculation results suggest that the vulnerable sections of crankshaft vary with the rota-
Crankshaft
tional speed. It is also found that 80% of the dynamic balance is the optimum design
Rotary compressor
condition to reduce the transverse forces on crankshaft and the wear-out of journal bearings. © 2014 Elsevier Ltd and IIR. All rights reserved.
Comportements dynamiques des vilebrequins dans des cylindre unique et a cylindres jumele s compresseurs rotatifs a Mots cles : Analyse du comportement dynamique ; Vilebrequin ; Compresseur rotatif
1.
Introduction
In a rotary compressor used in air-conditioners (shown in Fig. 1), the crankshaft driven by the motor is capable of rotating with a large range of speed. The volume of compression cavity changes periodically with the rotation and the refrigerant gas in cylinder is compressed. However, due to
* Corresponding author. E-mail address: [email protected] (J. Wu). http://dx.doi.org/10.1016/j.ijrefrig.2014.07.014 0140-7007/© 2014 Elsevier Ltd and IIR. All rights reserved.
the special structure of rotary compressor, crankshaft and motor rotor are made as a cantilever. Besides, the crankshaft is under a large dynamic load including gas force and unbalanced mass forces of comprising mechanical parts (eccentric crank, roller, and balancers on motor rotor). As a result, lubrication conditions at bearings would become severe owing to the deformation of crankshaft, as shown in Fig. 2. What is worse, rotor-to-stator rub would occur if the crankshaft bends
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
Nomenclature C e F Fn, Ft h Hc K L m P Pc Pd Ps q R
Radial clearance between crankshaft and bearing(m) Eccentricity (m) Force (N) Normal and tangential force at vane tip(N) Thickness of oil film (m) Height of cylinder (m) Dimensionless stiffness coefficient The length of arms of forces(m) Mass quantities (kg) Pressure (Pa) Pressure in compression chamber(Pa) Discharge pressure(Pa) Suction pressure (Pa) Vector of displacement(m) Radius(m)
more largely. Therefore, to ensure rotary compressor operating well, the dynamic behavior of crankshaft at various speeds should be analyzed. Classic rotor dynamics is a special branch mainly analyzing the dynamic behavior of structures at overcritical high speed with little transverse loads. Ehrich (Ehrich, 1991) studied the bifurcation of a bearing-rotor system and identified a sub-harmonic vibration phenomenon in the rotor's dynamic behavior. Holmes (Holmes et al., 1978) published a paper dealing with a periodic behavior in journal bearings. Brown et al. (1994) developed a simple model of a rigid, hydrodynamically supported journal bearing using short bearing theory. It was shown that the journal behaved chaotically when the rotating unbalance force exceeded the gravity load. Chang-Jian (2010); ChangJian and Chen (2006) discussed about a rotor supported by journal bearings under non-linear suspension and combined with rub-impact effect, turbulent effect and micropolar lubricant into consideration. Kurka et al. (2012) analyzed the visco-elastic bearing loads in the dynamic model of a reciprocating refrigeration compressor. The NewtoneEuler method was used in the analysis, establishing the necessary differential equations that described the movement of the system, leading also to the calculation of orbital displacements of the bearings. In 1990, Hattori and Kawashima (1990) proposed a method to analyze the rotor-journal bearing system in a twin-rotary compressor. The crankshaft was meshed by onedimensional beam elements based on finite element method. The short bearing theory was employed to solve the oil film force. The sub bearing was taken as one short bearing while the main bearing was taken as two. Elastic bending deformation of the crankshaft, the bearing loads profiles, and the pressure distributions of the oil film had been obtained. In 1998, Dufour et al. (1998) also took beam element to mesh the crankshaft; the bearings were modeled by one- node elements with two lateral translations, and the main bearing
Greeks a ε q qj h 4 u
37
Offset angle of roller center (rad) Eccentricity of journal Rotational angle of crankshaft (rad) Circumferential angle of bearing (rad) Viscosity of oil(Pa s) Attitude angle of bearing(rad) Angular velocity (rad s1)
Subscripts bal Balancers c Crank e Eccentricity of crankshaft ec Eccentric mass of roller and crank g Gas j Journal mj Main journal bearing o Oil ro Roller sj Sub journal bearing
was split into two parts. In his work, the oil film forces were calculated by the stiffness and damping characteristics of the three bearings, which were empirical equations involving ve bearing clearance and bearing load. In 2003, Dufour (Se et al., 2003) continued his research on the balancing procedure for variable-speed compressor ignoring the cylinder pressure force. In Xie's study (Xie and et al., 2006),dynamic behaviors of the rotor-journal bearing system for singlerotary compressor at high speed and low speed were analyzed considering both unbalanced mass force and gas force. Wang et al. (2013) applied the finite element model with three-dimensional solid element to study the vibration characteristics of the rotor-journal bearing system in a variable speed rotary compressor. In his work, the dynamic model of the rotor was solved by the finite element software ANSYS. A variable speed rotary compressor is capable of running from 1200 rpm to 7200 rpm, while the rotation speed of a fixed speed rotary compressor is usually between 2850 rpm and 3650 rpm. Both of them are far below the instability of the lubricant. Besides, unlike the classic rotor dynamics model, the width-diameter ratio of a rotary compressor crankshaft is much larger, and its elastic deformation is mainly caused by transverse forces. Moreover, the heightdiameter ratio of the main bearing in rotary compressor is almost 3, so neither the long bearing theory nor the short bearing theory is proper for it. Detailed comparison between is shown in Table 1. Due to the differences, it is necessary, at the design stage, to use a more suitable model to predict the deformation of crankshaft and the distribution of oil film. In this paper, elasticity of the crankshaft is taken into account, 2dimensional finite element method is applied to solve the oil film, and the dynamic behaviors of both single-cylinder and twin-cylinder compressor at various speeds are analyzed. The influence of configuration of balancers on the reliability of rotor system is investigated for the further.
38
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Fig. 1 e Structures of rotary compressor (a) main body of a single-cylinder rotary compressor (b) main body of a twincylinder compressor.
2.
Forces analysis
The transverse loads acting on the crankshaft are gas force (Fg), unbalanced mass forces of eccentric crank and roller (Fec), unbalanced mass forces of balancing weight (Fb1,Fb2), vane contact force (Fn,Ft), and oil film forces of journal bearings (Fmj,Fsj), as shown in Fig. 3.Note that these transverse forces are on different planes, leading to a large elastic deformation of the crankshaft, which would cause the wear-out of journal bearings or even the rub between rotor and stator. To limit the deformation, the main bearing is lengthened. As a result, the oil film forces turn out to be non-linear.
2.1.
Loads acting on the crank part of crankshaft
The cross-section of the cylinder is shown in Fig. 4. The crank part of crankshaft is surrounded by roller. The inner volume of cylinder is divided into suction chamber and compression
chamber by vane and roller. As crankshaft rotates, the volume of compression chamber decreases, and the inside gas pressure increases from Ps to Pc. The gas force Fg acting on the roller can be derived by the following equation (Yanagisawa and Shimizu, 1985). Fg ¼ 2Rro sin
(1)
Gas force Fg, eccentric force Fec, vane contact force Fn and Ft all act on crank part. Magnitude Fen and direction qf of the combined load are shown as follows. Fr and Fq are radial and angular components of crankshaft load given by Equations (2) and (3), respectively (Yanagisawa and et al., 1982). qþa þ Fec Fn cosðq þ aÞ þ Ft sinðq þ aÞ Fr ¼ Fg cos 2 Fq ¼ Fg sin
Fen ¼
Fig. 2 e The transverse loads acting on the crankshaft of a single-cylinder compressor.
qþa Hc ðPc Ps Þ 2
qþa þ Fn sinðq þ aÞ Ft cosðq þ aÞ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F2r þ F2q
qf ¼ tg1
Fq þq Fr
2.2.
Unbalanced mass forces
(2)
(3)
(4)
(5)
The crankshaft rotates with large eccentric force generated by crank part and roller. To reduce this eccentric force, upper and lower balancers are mounted on the motor rotor, as shown in Fig. 5. Fec is the eccentric force of crank and roller, Fb1 and Fb2 are the unbalanced mass forces of upper and lower balancers, respectively. Lec, Lb1, Lb2 are the arms of forces from the lower end of crankshaft. The unbalanced mass forces of crank part, roller and balancers are as follows.
39
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Table 1 e Differences of rotor-journal bearing systems in present paper and previous study. Differences
Present paper
Literature Classic rotor dynamics
Transverse loads Operating speed Diameter of crankshaft height-diameter ratio Model
Large dynamic transverse force Below 10 K rpm Small Main bearing: 3 Sub bearing：1.5 A special cantilever model
Little Over 10 K rpm
Heavy load bearing Large Large Below 1.0
Jeffcott rotor model
Fig. 3 e Transverse forces acting on the crankshaft in a rotary compressor. (a) Transverse forces on y-z plane. (b) Transverse forces on xey plane.
Fec ¼ ðmro þ mc Þeu
(6)
Fb1 ¼ mb1 eb1 u2
(7)
Fb2 ¼ mb2 eb2 u2
(8)
2
For rigid system, dynamic balance is designed to be 100% simultaneously, take single-cylinder compressor for example. Fb1 ¼ Fec þ Fb2 Fb1 Lb1 ¼ Fec Lec þ Fb2 Lb2
(9) (10)
Hence. mb1 ¼
e Lb1 Lec ðmro þ mc Þ 1 þ eb1 Lb2 Lb1
(11)
mb2 ¼
e Lb1 Lec ðmro þ mc Þ eb2 Lb2 Lb1
(12)
To change the dynamic balance is to change the weight of balancers. For example, 80% of dynamic balance is to reduce 20% of the original mass weight of both upper balancer and lower balancer.
2.3.
Fig. 4 e Cross-section of the cylinder. Gas force, eccentric force and vane contact force are acting on the crank part of the crankshaft.
Film force
The crankshaft of a rotary compressor is supported by main and sub bearings, which are hydrodynamic bearings with small clearance, as shown in Fig. 6. Due to the deflection and deformation of the crankshaft, local friction may occur on the journal bearings in extreme situations. Since the oil film in journal bearings is of ‘finite length, dynamic loads and various thickness on axial direction’, finite element method (FEM) is applied to solve the force of oil film.
40
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Fig. 5 e Unbalanced mass forces of rotary compressor. (a) The unbalanced mass forces of single-cylinder compressor. (b) The unbalanced mass forces of twin-cylinder compressor. The reaction forces of oil film between the crankshaft and the bearings are calculated by the Reynolds equations as follows: (Ito et al., 2006) 3 v h vP 1 v h3 vP vh vh ¼ 6u þ 12 þ vqj vt r2j vqj h vqj l2 vz h vz
(13)
h ¼ C 1 þ ε cos qj
(14)
between domain (P > 0) and (P ¼ 0), G1 and G2 are the boundaries at the ends of bearings. 8 G1 : z ¼ þ1 P¼0 > > > < G2 : z ¼ 1 P¼0 > > > : Gc : undetermined vP; P ¼ 0 vn
(15)
where h is the thickness of oil film between journal and bearing, P is oil pressure, qj is circumferential angle of the bearing, h is viscosity of oil given by discharge pressure and temperature, C is radial clearance between journal and bearing, and ε is eccentricity of journal with respect to bearing center. The bilinear four-node rectangle element is used to discretize the oil film and Reynolds Boundary Condition is written as Equation (15), where Gc is the undetermined boundary
Fig. 7 e The finite element model of crankshaft in singlecylinder compressor. The locations of bearing housings are marked by dashed boxes. Unbalanced mass forces of balancers are carried by node16 and node18.
Fig. 6 e Structure of journal bearing Oj is the crankshaft crank's center, and Ob is the bearing housing's center.
Fig. 8 e The finite element model of crankshaft in twincylinder compressor. The locations of bearing housings are marked by dashed boxes. Unbalanced mass forces of balancers are carried by Node21 and Node23.
41
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Table 2 e Nodes to carry forces in finite element model. Forces
Nodes (single-cylinder)
Nodes (twin-cylinder)
16,18 1,2,3 9,10,11,12,13,14,15 5,6,7
21,23 1,2,3 14,15,16,17,18,19,20 5,6,7,10,11,12
Unbalanced mass force of the balancers Oilfilm force of the sub earing Oilfilm force of the main earing Gas force of the compressed gas
3.
Finite element model
4.
The momentum equation of crankshaft is. € þ ½KfXg ¼ ½F ½M X
(16)
with {X} being the displacement vector, [M] the mass matrix, [K] the stiffness matrix and [F] the outer load. Since the crankshaft runs at a relatively low speed, its inertia force (not including inertia force of balance weight and eccentric mass) is much smaller than outer loads and can be ignored. Then the following equation is built. _ ½Kfqg ¼ Fg ðqÞ þ fFo ðfqg; fqgÞg þ Fbal ðqÞ
Coupling method between solid and fluid
Oil film force is the main nonlinear factor in the rotor-journal bearing system. The nodes, which are loaded by oil film force, are defined as nonlinear nodes. For this system with local non-linearity, its governing equation can be rewritten as follows (Xie and et al., 2006)
K11 K21
K12 K22
q1 q2
¼
Fg1 Fg2
þ
0 Fo
(18)
(17)
_ is vector of velocity, where, {q} is vector of displacement, fqg {q} ¼ {y, z, qy, qz}T, Fg(q) is gas force, Fbal(q) is the unbalanced _ is oil film force, q is the mass force of balancers, Fo ðfqg; fqgÞ rotational angle of crankshaft. The Timoshenko beam element is applied to mesh crankshaft. According to the shape of the cross-section, material and structure, the shafts in single-cylinder compressor and twin-cylinder rotary compressor are meshed as shown in Fig. 7 and Fig. 8, respectively. Take Fig. 7 as example: First, the thinner part must be represented by at least one individual element, like element 3 and 8, since stress may concentrate there. Second, as the rotor and crankshaft are made of different materials, element 16 and 17 should be meshed to define the properties. Finally, the structure of bearings are taken into consideration and element 1e2, 9e14 are meshed. The locations of bearing housings are marked by dashed boxes. The nodes to carry the forces are shown in Table 2.
Table 3 e Working conditions and main parameters of structure. Single-cylinder compressor Inner diameter of cylinder (mm) Height of cylinder (mm) Outer diameter of roller (mm) Displacement (cc) Crankshaft diameter (mm) Main bearing length (mm) Sub bearing length (mm) Clearance (mm) Distance Lb1 (mm) Distance Lb2 (mm) Distance Lec (mm)
Twin-cylinder compressor
60
58
28 46
20.28 47.38
32.6 22 67 25 20 124.1 222.1 39.9
35.6 21 67 25 20 147.3 245.3 Lec1 (mm) Lec2 (mm)
38.31 64.81
Fig. 9 e The value and angle of gas force and oil film forces in single-cylinder compressor. (a) forces in X-direction changing with rotating angle, (b) forces in y-direction changing with rotating angle.
42
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where, {q1} is the displacements of linear nodes, {q2} is the displacement of nonlinear nodes. Fg1 is the angular gas force, Fg2 is the radial gas force, Fo is the oil film force. Hence q1 ¼ K1 11 Fg1 K12 q2
(19)
b 2 ¼ Fg2 þ Fo þ b f Kq
(20)
b b ¼ K22 f ¼ K21 K1 where K 11 Fg1 Eq. (18) is a set of nonlinear algebraic equations, Fg2 and b f _ In the are the functions of time, Fo is the function of {q} andfqg. b is determined only by system studied in this paper, matrix K K21 K1 11 K12 ,
rotational speed. Once a certain initial value is given to {q}, we _ with NewtoneRaphson method, hence are able to solve fqg _ at the {q} at next time step. In this way, the value of {q} and fqg every moment can be calculated. With the initial value being updated after every iterative period, the convergent periodical solution can be obtained after repeating the above procedure
Fig. 10 e The value and angle of gas force and oil film forces in twin-cylinder compressor. (a) forces in x-direction changing with rotating angle, (b) forces in y-direction changing with rotating angle.
Fig. 11 e Vulnerable sections of crankshaft in rotary compressor. Wear-out between journal and bearing usually occurs on four vertical planes, namely MBT (main bearing top), MBB (main bearing bottom), SBB (sub bearing bottom) and SBT (sub bearing top).
Fig. 12 e Journal orbits of the bearings in single-cylinder compressor. (a) Orbits of four vulnerable sections at the speed of 2850 rpm. (b) Orbits of vulnerable sections at the speed of 7200 rpm.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
for several times. The NewtoneRaphson method is presented as follows. 9 8 Dy_ > > > = < _> Dz ¼ ½F ½D Dq > > y > > ; : Dqz
½D ¼
q_x2 ¼ q2 ¼
)
( ¼
In fact,
i q_x2 i q2
)
( þ
i Dq_ x2 i Dq2
) (26)
vfFo g _ g vfqx2
is the derivative of oil film force with respect
to the translational velocity of journal in the Jacobian matrix.
vfFo g vfFg vfq2 g v qx2 _
b qi þ Fg2 þ Fo þ b ½F ¼ K f 2 n
iþ1 q_x2 iþ1 q2
(21)
where
(
43
y_2
z_2
qy2
qz2
oT
T
(22)
5.
Results and discussion
(23)
(24)
(25)
so
Fig. 13 e Journal orbits of the bearings for twin-cylinder compressor. (a) Orbits of four vulnerable sections at the speed of 2850 rpm. (b) Orbits of vulnerable sections at the speed of 7200 rpm.
5.1. The dynamic behavior of crankshaft in rotary compressor The dynamic behaviors of crankshafts in single-cylinder and twin-cylinder rotary compressors are analyzed in this paper. The environmental friendly R410A is adopted as refrigerant, with the suction pressure at 1.095 MPa and discharge pressure at 4.185 MPa. Working conditions and main parameters of structure are listed in Table 3. Dynamic behaviors of crankshaft at2850 rpm and 7200 rpm have been analyzed (dynamic balance are designed to be 100%). The gas force and oil film forces are shown in Figs. 9 and 10. The direction of gas force is changing with rotation angle of the crankshaft. For single-cylinder compressor, although the lengths of the main and sub bearings are not the same, they almost take the load averagely, and bearing force has only one peak from 0 to 360 . For twin-cylinder, oil film force has two peaks in one cycle. This is because that even though the main bearing mainly takes the load from upper cylinder while the sub bearing mainly takes the load from lower cylinder, gas force from the other cylinder with a different phase can still influence the bearing. Due to the elasticity of crankshaft, wear-out between journal and bearing usually occurs on four vertical planes, namely MBT (main bearing top), MBB (main bearing bottom),
Fig. 14 e Journal orbits at MTop (motor top section).
44
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SBB (sub bearing bottom) and SBT (sub bearing top), as shown in Fig. 11. Besides, the impact between motor rotor and motor stator may occur at the section of MTop, the top rotor motor. Journal orbit plays a key role in the dynamics of rotary compressor. Journal orbits at both ends of bearings are shown in Figs. 12 and 13. For single-cylinder, at low speed of 2850 rpm, the orbits of journal orbit at SBB and MBB are relatively larger, and the wear-out may occur in these sections where oil film is much thinner. At high speed of 7200 rpm, orbits at SBT, SBB and MBB are smaller, while that at MBT is larger due to the eccentric force of balancers. Conversely, 100% of dynamic balance enlarges the deformation of crankshaft. As a result, wear-out could occur at the upper side of the main bearing at high speed. For twin-cylinder compressor, the shaft whirls two circles in one rotational period, which is different from singlecylinder compressor. Eccentricities of orbits at SBB/MBB are large, which means the oil film is thin and wear-out is possible to occur. As speed increases, the orbits at MBB and MBT grow and almost turn into circles.
Fig. 15 e Maximum eccentricities of journal orbits in single-cylinder compressor. (a) Operating speed is 2850 rpm, (b)operating speed is 7200 rpm.
Orbits at MTop are shown in Fig. 14. When rotating at the same speed, the orbit of the twin-cylinder compressor is much smaller than that of the single-cylinder compressor, which suggests that the rotor-bearing system of a twin-cylinder compressor has a better characteristic of stability. The movements of upper and lower crank parts in twin-cylinder are in opposite phases, so their eccentric forces can counteract to each other. As a result, a pair of balancers with less weight is needed and the orbits of bearing top are smaller. Compare (a), (b) in Figs. 12e14, it can be concluded that the dynamic behavior of crankshaft is determined by its structure (single-cylinder or twin-cylinder), rotational speed and the elasticity.
5.2. Influence of configuration of balancers on the reliability of rotor system In rotary compressor, 100% of dynamic balance may not be the optimum solution since bending deformation of crankshaft is
Fig. 16 e Maximum eccentricities of journal orbits for twincylinder compressor. (a) Operating speed is 2850 rpm. (b) Operating speed is 7200 rpm.
i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 4 7 ( 2 0 1 4 ) 3 6 e4 5
45
detail, including deformation of the crankshaft, journal orbits and etc. 2) For single-cylinder compressor, the main and sub bearings almost take the load averagely and bearing forces only have one peak. However, as to twin-cylinder compressor, bearing forces have two peaks since there are two cylinders, and each bearing mainly takes the load from the closer cylinder. 3) The vulnerable sections of crankshaft vary with the rotational speed. At low speed, the wear may occur at SBB and MBB. At high speed, the wear may occur at MBT and MTop. 4) Dynamic behaviors of the rotor system with different dynamic balances ranging from 60% to 120% had been studied in this paper. 80% of dynamic balance is an optimum solution for rotary compressor because bending deformation of the crankshaft is a more important factor than vibration. Fig. 17 e Orbits of Mtop in single-cylinder and twincylinder compressor with different balancers.
references a more important factor than vibration. Therefore, dynamic behaviors of the crankshaft with different dynamic balances ranging from 60% to 120% are studied in this paper. According to the calculation results, 100% of dynamic balances is the optimum condition for reducing the wear-out of journal bearings at low speed. On the other hand, when the rotor rotates faster, a smaller of dynamic balance could help to decrease the vibration of rotor-journal system. All things considered, 80% of dynamic balancers is the optimum design condition. The maximum eccentricities of the journal orbits are shown in Fig. 15 and 16, respectively. It is found that the trend of wear-out differs. At low speed, maximum eccentricity of journal orbit varies slightly with the increment of dynamic balance. At high speed, maximum eccentricity of journal orbit at MBT increases rapidly with the increment of dynamic balance, and wear-out may occur at upper part of the main bearing. For single-cylinder compressor at high speed, maximum eccentricities of the journal orbits at SBB/MBB decrease with the increment of dynamic balance. For twin-cylinder compressor at high speed, the chance of wearing at SBB is lower, while that at MBB is high. Orbits at motor top section in both single-cylinder and twin-cylinder compressors with different balancers are shown in Fig. 17. When at the same speed, eccentricity of orbit of MTop in single-cylinder compressor is larger than in twincylinder compressor, and become more serious with the increment of dynamic balance.
6.
Conclusions
1) A numerical method to analyze the dynamic behavior of crankshaft in rotary compressor is built up. The elasticity of the crankshaft, the gas force, nonlinear oil film force etc. are considered comprehensively. Based on this method, the dynamic behavior of the crankshaft has been studied in
Brown, R., Addison, P., Chan, A., 1994. Chaos in the unbalance response of journal bearings. Nonlinear Dyn. 5 (4), 421e432. Chang-Jian, C.-W., 2010. Non-linear dynamic analysis of dual flexible rotors supported by long journal bearings. Mech. Mach. Theory 45 (6), 844e866. Chang-Jian, C.-W., Chen, C.-K., 2006. Nonlinear dynamic analysis of a flexible rotor supported by micropolar fluid film journal bearings. Int. J. Eng. Sci. 44 (15e16), 1050e1070. Dufour, R., Charreyron, M., Gerard, M., 1998. Dynamics prediction of refrigerant rotary compressor crankcrankshaft. In: Proceedings of International Compressor Engineering Conference at Purdue. Ehrich, F., 1991. Some observations of chaotic vibration phenomena in high-speed rotordynamics. J. Vib. Acoust. Stress, Reliab. Des. 113 (1), 50e57. Hattori, H., Kawashima, N., 1990. Dynamic analysis of a rotorjournal bearing system for twin rotary compressors. In: Proceedings of International Compressor Engineering Conference at Purdue. Holmes, A., Ettles, C., Mayes, I., 1978. The aperiodic behaviour of a rigid crankshaft in short journal bearings. Int. J. Numer. Methods Eng. 12 (4), 695e702. Ito, Y., Hattori, H., Miura, K., 2006. Numerical analysis on motion of rolling piston in a rotary compressor. In: Proceeding ASIATRIB 2006 KANAZAWA. Kurka, P.R.G., Izuka, J.H., Paulino, K.L.G., 2012. Dynamic loads of reciprocating compressors with flexible bearings. Mech. Mach. Theory 52, 130e143. ve, F., et al., 2003. Balancing of machinery with a flexible Se variable-speed rotor. J. Sound. Vib. 264 (2), 287e302. Wang, Zengli, et al., 2013. Dynamic analysis for the rotor-journal bearing system of a variable speed rotary compressor. Int. J. Refrigeration 36 (7), 1938e1950. Xie, F., et al., 2006. Dynamic analysis of a rotor-journal bearing system of rotary compressor. In: Proceedings of International Compressor Engineering Conference at Purdue. Yanagisawa, T., et al., 1982. Motion analysis of rolling piston in rotary compressor. In: Proceedings of International Compressor Engineering Conference at Purdue. Yanagisawa, T., Shimizu, T., 1985. Leakage losses with a rolling piston type rotary compressor. II. Leakage losses through clearances on rolling piston faces. Int. J. Refrigeration 8 (3), 152e158.