INTRODUCTION
Scroll compressor is a new type positive displacement compressor which is widely used in refrigeration and chemical engineering due to its many advantages including low volume, light weight, high reliability and stable and sustainable gas transmission. Its excellent properties and bright prospect have gained a lot of attentions. Due to scroll compressor’s unique structure and working principle, during its operation, the crankshaft system will be subject to the gas force and centrifugal inertia force generated by the eccentric rotor. Vibration of crankshaft will be transferred to the casing through the main and minor bearing and causing vibration of the casing and other parts. The damage of vibration is mainly caused by resonance because when the excitation frequency is close to a natural frequency in the system, the amplitude will be increased sharply (Liu et al., 1998). At present, the studies on scroll compressor’s rotor are primarily focused on one single rotor but rarely on the rotor system (Du and Liu, 1999; Hao et al., 2011; Hu et al., 2007; Wang et al., 2012; Xu et al., 2012). As a matter of fact, as a main component of scroll compressor, the rotor system takes on the task of power transmission. Its steady operation is the key to the overall performance of the scroll compressor since it can affect the radial sealing clearance between the fixed scroll and orbiting scroll, the distribution of lubricant film, the eccentric wear of friction pair, the vibration of rotor and oil temperature. Therefore, when studying the subject of crankshaft dynamics it is more reasonable to consider crankshaft, orbiting scroll, main and minor counterweight and belt pulley as a whole. In this study it uses finite element analysis software ANSYS in the modal analysis on scroll compressor’s rotor and analyzes rotor’s critical speed of rotation under rigid support, elastic support and changing bearing’s stiffness to identify its effects on the rotor’s dynamic characteristics. So the research has both theoretical significance and practical value in enhancing reliability and safety of scroll compressor and extending its useful life.
BASIC STRUCTURE OF SCROLL COMPRESSOR’S ROTOR SYSTEM
The rotor system of scroll compressor is composed of crankshaft, orbiting scroll, main and minor counterweight and belt pulley. Crankshaft bearings includes main bearing (rolling bearing) and minor bearing selfaligning ball bearing. Drive bearing is the sliding bearing. The structure is as shown in Fig. 1.

Fig. 1: 
Crankshaft rotor structure diagram of scroll compressor 
BASIC THEORY OF PRESTRESSED MODE ANALYSIS
Based on theory of vibration mechanics, dynamic equation of the multidegree of freedom system can be expressed as follows (Houhaun et al., 2012; Tiansu et al., 2005; Jingwen et al., 1988):
where, {δ} is the unit node displacement, {} is the unit node speed, {} is the unit node acceleration, [M] is the mass matrix, [C] is the damping matrix, [K] is the stiffness matrix, [Q] is the node load matrix.
[Q] represents the node load matrix, is usually a function of time. In a modal analysis it is used to analyze rotor’s nature properties without exciting force. Therefore, {Q} = 0. Considering the effects of rotating centrifugal force from crankshaft’s rotor, stiffness matrix K, representing centrifugal stress it is applied in the structure, the damping matrix is ignored and modal equation for rotor under rotating centrifugal force is obtained:
When the scroll compressor’s rotor is rotating, radial tensile stress is released due to the centrifugal force role. Thus, when performing the modal analysis for the rotor it needs to treat this as the initial stress.
EQUATION OF ROTOR MOTION
Through discretization, the rotor shaft is divided into n segments. Unit length of the shaft is L. There is a lateral displacement v and rotation angle θ at the two ends of each shaft unit. The deflection and rotation angle of each node at the shaft unit are superposed by the four degree of freedom function of shape and position are φ_{1}, φ_{2}, φ_{3}, φ_{4} (Wang et al., 2009):
Without consideration of gyroscopic torque’s effects it can replace the deflection and rotation angle of each node with degrees of freedom and use lateral displacement and speed of the two nodes at the shaft unit to indicate shaft unit’s kinetic energy T and potential energy U, respectively:
When the shaft is divided into n segments, then there will be n+1 nodes and
each node is with two degrees of freedom. Therefore, the generalized mass matrix
of the system is a 2n+2 square matrix.

Fig. 2:  Crankshaft rotor node displacement and angle coordinates of scroll compressor 
Based on the relationship between generalized force and generalized mass it
can use the superposition principle to add the generalized force, corresponding
to gyroscopic torque, to the Lagrange equation and obtain the equation of motion
for the system:
When
It can calculate the critical speed of rotation and the corresponding natural
frequency by solving the equation.
When the scroll compressor’s rotor can be divided into finite elements,
the following points can serve as nodes: the point where the inner and outer
diameter changes, each supporting point, external point of action and the point
where the deformation occurs. There are five degrees of freedom at each node,
including lateral displacement v and rotation angle θ_{z} in plane
xoy, lateral displacement w and rotation angle θ_{y} in plane xoz
and the deformation θ_{x} around axis x, just as shown in Fig.
2.
DYNAMIC COEFFICIENTS OF SCROLL COMPRESSOR’S BEARING AND THE SIMPLIFIED
MODEL FOR MAIN SHAFT BEARING
The main and minor bearing of the scroll compressor are both rolling bearings
which support the crankshaft, absorb the gas force and inertia force from the
compression chamber and reduce frictions.
The dynamic characteristics of the bearings play a key role in rotor dynamics
and yet, up to now, no systematic and complete method and data regarding rolling
bearings’ dynamic characteristics can be for reference.

Fig. 3: 
Crankshaft bearing simplified diagram 
Rolling bearing is a high load hydrodynamic friction pair that not only shows
elastic deformation under solid contact but also bears the influences from hydrodynamic
oil film which is coupled with elastic deformation. As a result, rolling bearing
dynamics is in fact a matter of liquidsolid coupling and requires the solving
of the Reynolds equation and elastic equation. It can only be done under some
simplified assumptions and the calculation and solving of these nonlinear equations
should be through numeric iteration. In order to simplify the problem, normally
in dynamic analysis of rotor it can use statistical data of rolling bearing’s
stiffness, between 5x10^{7}~5x10^{8 }N m^{1}. Film
damper is rather small in the rolling bearing and therefore it will be ignored
during the analysis (Bangchun et al., 2000; Yie
et al., 1987).
Based on the practical engineering application, the bearing support is not
entirely rigid and therefore, elastic effect on the bearing should be taken
into consideration. In this thesis, both horizontal and vertical spring dampers
are used to stimulate bearing’s constraints on crankshaft, if it makes
assuming that each elastic support is composed of two spring dampers (Wang
et al., 2010). The simplified model of the elastic support is shown
in Fig. 3.
ESTABLISHMENT OF FINITE ELEMENT ANALYSIS MODEL FOR ROTOR OF SCROLL COMPRESSOR’S
CRANKSHAFT
Establishment of finite element analysis model for rotor: The solid
model of scroll compressor’s crankshaft is built by 3D modeling software
Solidworks it should be as close to the actual size as possible. In order to
shorten the calculation time and reduce network flow, the belt pulley is simplified
by omitting its race and ignoring small structures like chamfer, screw hole
and oil hole.

Fig. 4: 
Crankshaft rotor simplified physical
model of scroll compressor 
Table 1: 
Rotor parts physical parameters 

After the model is built in the Solidworks, an assembly is formed and material
attributes are given to the corresponding structures. The 3D solid model of
rotor system is as shown in Fig. 4.
Physical properties and meshing of the rotor system model: The 3D model
built by Solidworks will be saved as model.txt and then should be imported to
ANSYS. After that, material’s elastic modulus, poisson ratio, Poisson's
ratio is 0.3 and density will be identified. Physical parameter of the rotor
is listed in Table 1.
When scroll compressor’s rotor is balanced, orbiting scroll’s center
of mass is moved to the center of the crank pin. When performing the finite
element modal analysis, the orbiting scroll will be treated as a homogeneous
disc in order to realize lumped mass and loaded to the crank pin. Meshing will
be completed in software of ANSYS with 45112 nodes and 26602 elements. The simplified
finite element mesh model is as shown in Fig. 5.
Boundary conditions: after the meshing is completed, loading and boundary
constraints need to be added to the rotor physical model. First of all, before
the modal analysis, finite element model will be given an axial rotational velocity
at 293 rad sec^{1} (scroll compressor’s rotating speed is 2800
r min^{1} when it functions normally). After that, prestressed analysis
is done and the results will be used as loading and should be taken into consideration
during the modal analysis. In ANSYS, the crankshaft bearing journal surface
will be given cylinder constraint (Liu and Yang, 2003;
Zhang et al., 2012), where rotation is allowable
but no displacement in direction x and y (radially and tangentially).

Fig. 5: 
Crankshaft rotor meshing model 
Frictionless support will be added to the shaft shoulder of the bearing cone,
where displacement in direction z (axially) is not allowed and gravity load
is added as g = 9.8 m sec^{2}. The constraint model is shown in Fig.
6.
ROTOR MODAL ANALYSIS UNDER PRESTRESSED STRUCTURE
Since highfrequency modes will not have significant effects on the system,
when performing the modal analysis, only the top five frequency modes of the
crankshaft rotor will be calculated.
Modal analysis for scroll compressor’s crankshaft rotor under rigid
support: As shown in Fig. 7 and Table 2,
the first order and second order critical speed of scroll compressor’s
rotor system is 6673.8 and 6710.4 r min^{1}, respectively and the first
and second mode of vibration are located on the crank outlet end. Normally,
rotor should be working at a speed n, where n<0.75n_{cr},_{i},
or 1.4 n_{cr,i}<n<0.7 n_{cr,i+1} and n_{cr,i}
is rotor’s critical speed at i order. Considering rotor’s operational
safety its working speed is set at more than 20% deviated from the critical
speed. Therefore, the rotation speed range for scroll compressor’s rotor
should be lower than 5300 r min^{1} or higher than 8000 r min^{1}.
If taking everything into account, the working speed of scroll compressor can
be identified as below 5300 r min^{1}.
Through the modal analysis under prestressed structure, the first six natural
frequencies and the according modes of vibration under nondamping free vibration
can be obtained for scroll compressor’s rotor. Based on these natural frequencies,
rotor’s critical rotation speed can be calculated, as listed in Table
2.

Fig. 6: 
Rotor boundary condition constraint
model 

Fig. 7 (ae): 
All order modal vibration
modes (a) First step: Bending, (b) Second step: Torsional, (c) Third step:
Tensile, (d) Fourth step: Bending and (e) Fifth step: Bending 
Table 2: 
Crankshaft rotor first five natural frequencies and critical
speed under rigid support 

Rotor’s natural frequency falls between 111.23 and 655.76 H_{Z}
and it can be accelerated with the increasing of order. The rotor’s critical
rotation speed under first order natural frequency is at 6673.8 r min^{1},
much larger than that of the scroll compressor at 2800 r min^{1} which
means that the rotor has become rigid. Rotor’s fundamental frequency is
46.7 H_{Z}, far smaller than the natural frequency of the first two
orders. Therefore, under normal working conditions, lowfrequency resonance
will not occur in rotor system. On the basis of the above analysis, rotor’s
frequency is not in the scope of the scroll compressor’s resonance frequency
and therefore, the crankshaft rotor meets the requirements of vibration safety.
Figure 7 shows the typical vibration mode of scroll compressor’s
rotor at each order. As indicated in the figure, first order and second order
have a close nature frequency and the vibration mode is characterized with tangential
and radial curve and twist at the end of crank pin. Vibration mode of the 3rd
order shows crank pin’s axial stretching. As for fourth and fifth order,
the vibration of modes indicates rotor’s bending and swaying, with symmetrical
deformation of the belt pulley.
In the first five natural frequency and modal shape of scroll compressor’s
rotor, deformation occurs near the bearing constraints. Since crank pin is a
cantilever beam of the crankshaft with certain deflection, orbiting scroll will
be loaded to crank pin in the form of lumped mass, leaving crank pin with a
more serious deformation than other parts. The deformation of crank pin has
a direct effect on the orbiting scroll’s drive bearing.
Bearing stiffness’s effects on rotor’s dynamic characteristics
under elastic support: Based on the calculated damping matrix C and stiffness
matrix K, elastic support is applied to the main shaft bearing. The purposes
of using elastic support are (Shen et al., 2010;
Chen, 2010): (1) Making critical rotation speed meet
the design requirements by adjusting the stiffness of elastic support; (2) Reducing
the dynamic amplitude by taking advantage of deformation caused by elastic support
and allowing damper to absorb the vibration energy from rotor system.
Once a prestressed mode analysis is performed for the first five orders of
scroll compressor’s rotor, the natural frequencies and the according critical
rotationspeeds can be obtained as shown in Table 3.
Table 3: 
Rigid support the crankshaft rotor first five natural frequencies
and critical speed 

Table 4: 
Crankshaft rotor first five natural frequencies and critical
speed under different bearing stiffness 

Based on the above analysis, stiffness of rolling bearing is set as 5x10^{8}
N m^{1}, considering the insignificant effects of damper it will be
ignored and critical rotation speed under different stiffness can be calculated.
If the bearing’s stiffness is altered, then its dynamic stiffness needs
to be considered. Therefore, during the calculation, the dynamic stiffness will
be adjusted in order to analyze its effects on the rotor modal. Table
4 shows rotor’s models of the first orders with dynamic stiffness decreased
and increased by 20%.
As Table 3 indicates, when elastic support is applied, under
the same rotation speed, natural frequencies for the first five orders of rotor
are lower compared with rigid support. With rotor’s normal function, bearing’s
stiffness will be reduced and so is the rotor system’s natural frequency.
According to Table 4a, natural frequencies of fourth and fifth
orders change a lot while the first and second orders show a smaller change
in natural frequencies. If bearing’s stiffness is increased, then the change
of frequency for the first two orders are even less than that when the bearing’s
stiffness is reduced and the remaining 3 orders still show greater changing
in natural frequencies, as indicated in Table 4b.
Under elastic support, the first order critical rotation speed of crankshaft’s
rotor is 5348 r min^{1}. On the basis of the above analysis its working
speed range should be below 4200 r min^{1} or above 6400 r min^{1}.
When stiffness of elastic support is altered, the working speed range changes
will be below 3800 r min^{1} or above 5800 r min^{1} and below
4700 r min^{1} or above 7000 r min^{1}.
CONCLUSION
Using finite element analysis software ANSYS, the prestressed modal for scroll
compressor’s rotor can be analyzed and the natural frequencies and critical
rotation speeds for the first five orders are obtained. Under rigid support,
rotor’s working rotation speed deviates 20% from the critical rotation
speed with the maximum working speed range below 5300 r min^{1}. During
normal function, rotor system’s fundamental frequency is far less than
the first order natural frequency and therefore no resonance will occur. Since
crank pin is the cantilever beam of the crankshaft, serious deformation can
be happen. By changing constraint way of main shaft bearing and using elastic
support instead of rigid support, rotor’s mode of vibration will be much
smaller. The analysis shows that elastic support can not only enable rotor’s
working speed to easily avoid its bending critical speeds but also can reduce
the dynamic amplitude with the deformation it caused and effectively reduce
rotor’s vibration at critical rotation speed. Moreover, under elastic support,
rotor system’s natural frequency will be changed accordingly as decreasing
and increasing of bearing’s stiffness, with a much smaller vibration mode
than under rigid support. The softwarebased modal analysis of scroll compressor’s
rotor under prestressed structure in the study, can lay the foundation of multibody
dynamics and it will provide a guarantee for scroll compressor’s stead
operation.
ACKNOWLEDGMENT
This research is supported by the Natural Science Foundation of China (NSFC)No.
50975132. No.51265026.