# Effect of Mechanism Error on Input Torque of Scroll Compressor

Authors: Man Zhao; Shurong Yu; Chao Li; Yang Yu
Journal: International Journal of Rotating Machinery
 (2013)

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## Abstract

Based on the fundamental principle of plane four-bar mechanism, the force on the equivalent parallel four-bar mechanism was analyzed for scroll compressor with mini-crank antirotation, and the formula of input torque was proposed. The change of input torque caused by the mechanism size error was analyzed and verified with  an example. The calculation results show that the mechanism size error will cause large fluctuation in input torque at the drive rod and connecting rod collinear and the fluctuation extreme value increases with rotational speed. Decreasing of the crankshaft eccentricity errors is helpful for reducing the effects of dimension error on input torque but will increase the friction loss of orbiting and fixed scroll wrap. The influence of size error should be considered in design in order to select suitable machining accuracy and reduce the adverse effect caused by size error.


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## Body

## 1. Introduction

With the needs of clean and oil-free pollution compressed air in food, medicine, fuel cells, and other industries, the investigation of oil-free scroll compressor research has been one of hot topics in scroll compressors. The mini-crank antirotation mechanism is often used to achieve the orbiting scroll work under the oil-free lubrication conditions. The movement relationship of mini-crank, orbiting scroll, crankshaft, and the bracket body can be represented by planar four-bar mechanism [1], and the ideal working should be planar parallel four-bar mechanism. But in fact, machining error and running wear are inevitable, such as the orbiting scroll deviating from the ideal state, resulting in excessive friction wear or large tangential leaking [2], also causing the movable linkages uneven movement. The research of the size error on the four-bar mechanism critical rod’s uneven movement and the change of input torque can provide some theoretical foundation for machining accuracy and scroll compressor reliable operation.


## 2. Four-Bar Linkage Mechanism

The structure of scroll compressor with mini-crank antirotation is shown in Figure1. The mini-crank is respectively connected with the orbiting scroll and the bracket body through the bearing. In order to ensure the orbiting scroll's revolving translational motion, the gyration radius of mini-crank and eccentric crankshaft should be in the same size; that is, the plane equivalent four-bar mechanism of scroll compressor, shown in Figure 2, should be the parallel four-bar mechanism.Figure 1
Configuration diagram of scroll compressor.Figure 2
Four-bar mechanism.Figure2 illustrates the four-bar mechanism in the scroll compressor work, l1 is equivalent to the drive crankshaft, l2 is equivalent to the orbiting scroll, l3 is equivalent to the mini-crank, and l4 is equivalent to the bracket body. In ideal working conditions: l1=l3, l2=l4.


### 2.1. Geometrical Theory

Cartesian coordinate system is shown in Figure2; the vector equation is
(1)l1cosθ1+l2cosθ2=l3cosθ3+l4,l1sinθ1+l2sinθ2=l3sinθ3.Equation (1) can be written as
(2)Acosθ2+Bsinθ2=C,(3)Acosθ3+Bsinθ3=D,
where
(4)A=l4-l1cosθ1,B=-l1sinθ1,C=12l2(l22-l32+A2+B2),D=12l3(l22-l32-A2-B2).According to (2) and (3), the output angles θ2 and θ3 of four-bar mechanism can be expressed as
(5)θ2=2arctanB-MA2+B2-C2A+C,θ3=2arctanB-MA2+B2-D2A+D.The output anglesθ2 and θ3 of the mechanism have two solutions, and the exact value can be determined according to the mechanism’s initial installation and the continuity of the movement. When the hinge point B, C, and D are arranged clockwise, M will take the positive sign; when the hinge points B, C, and D are arranged anticlockwise, M  has a negative sign.Taking first derivative and second derivative of vector equation (1), the angular velocity and angular acceleration of rod 2 and rod 3 can be obtained:
(2.1)θ˙2=l1sin(θ1-θ3)l2sin(θ3-θ2)θ˙1,θ˙3=l1sin(θ1-θ2)l3sin(θ3-θ2)θ˙1,θ¨2=-θ˙12l1cos(θ1-θ3)-θ˙22l2cos(θ2-θ3)+θ˙32l3l2sin(θ2-θ3),θ¨3=θ˙12l1cos(θ1-θ2)+θ˙22l2-θ˙32l3cos(θ3-θ2)l3sin(θ3-θ2).For the parallel four-bar mechanism, when the drive rod is running at a constant angular velocity, the following results can be obtained:(7)θ˙2=0,θ¨2=0,θ˙3=θ˙1,θ¨3=0.


### 2.2. Mechanism Motion Analysis

By the scroll compressor structure, the active bars centroid coordinates are(8)xS2=xS1=l1cosθ1,yS2=yS1=l1sinθ1,(9)xS3=l4+l3cosθ3,yS3=l3sinθ3.Respectively, by seeking the first derivative and second derivative of (8) and (9), the angular velocity and angular acceleration of the movable rods can be obtained.


## 2.1. Geometrical Theory

Cartesian coordinate system is shown in Figure2; the vector equation is
(1)l1cosθ1+l2cosθ2=l3cosθ3+l4,l1sinθ1+l2sinθ2=l3sinθ3.Equation (1) can be written as
(2)Acosθ2+Bsinθ2=C,(3)Acosθ3+Bsinθ3=D,
where
(4)A=l4-l1cosθ1,B=-l1sinθ1,C=12l2(l22-l32+A2+B2),D=12l3(l22-l32-A2-B2).According to (2) and (3), the output angles θ2 and θ3 of four-bar mechanism can be expressed as
(5)θ2=2arctanB-MA2+B2-C2A+C,θ3=2arctanB-MA2+B2-D2A+D.The output anglesθ2 and θ3 of the mechanism have two solutions, and the exact value can be determined according to the mechanism’s initial installation and the continuity of the movement. When the hinge point B, C, and D are arranged clockwise, M will take the positive sign; when the hinge points B, C, and D are arranged anticlockwise, M  has a negative sign.Taking first derivative and second derivative of vector equation (1), the angular velocity and angular acceleration of rod 2 and rod 3 can be obtained:
(2.1)θ˙2=l1sin(θ1-θ3)l2sin(θ3-θ2)θ˙1,θ˙3=l1sin(θ1-θ2)l3sin(θ3-θ2)θ˙1,θ¨2=-θ˙12l1cos(θ1-θ3)-θ˙22l2cos(θ2-θ3)+θ˙32l3l2sin(θ2-θ3),θ¨3=θ˙12l1cos(θ1-θ2)+θ˙22l2-θ˙32l3cos(θ3-θ2)l3sin(θ3-θ2).For the parallel four-bar mechanism, when the drive rod is running at a constant angular velocity, the following results can be obtained:(7)θ˙2=0,θ¨2=0,θ˙3=θ˙1,θ¨3=0.


## 2.2. Mechanism Motion Analysis

By the scroll compressor structure, the active bars centroid coordinates are(8)xS2=xS1=l1cosθ1,yS2=yS1=l1sinθ1,(9)xS3=l4+l3cosθ3,yS3=l3sinθ3.Respectively, by seeking the first derivative and second derivative of (8) and (9), the angular velocity and angular acceleration of the movable rods can be obtained.


## 3. Mechanical Models

While four-bar mechanism is working, in addition to the input torque, the force, and  moment acting on mechanism are mainly three types: each movable rod gravity, inertia force, inertia moment; external forces and moment; hinge friction and friction moment.For the equivalent four-bar mechanism of scroll compressor, external loading in radial plane is shown in Figure3; tangential gas force Ft,  radial gas force Fr, rotation torque Mz, Ffd1, Ffd2, the friction  force  of  the  orbiting plate and fixed plate, as well as the gravity, inertia force and inertia moment used to balance the rotor system [3].Figure 3
Schematic diagram of external loading.The geometric parameters of balance weightsmp1 and mp2 are determined by the rotor system balance. The relationship between the mass of the crankshaft pin, the orbiting scroll assembly, and the balance weights is
(10)(m1+m2)l1+mp2rp2=mp1rp1,
where rp1 is the distance between the centroid of counterweight mp1 and the rotation center; rp2 is the distance between the centroid of counterweight mp2 and the rotation center.The orbiting scroll revolving translational motion characteristics shows that any point of the orbiting scroll is in the same movement rule. The direction of the frictional force is opposite to the orbiting scroll gravity (hingeB) moving, and the  force role point is on the overturning moment action  line [4]:
(11)Ffd1=μdFd1,Ffd2=μdFd2.M
sd, the friction torque on the orbiting scroll center (hinge B) caused by the friction forces Ffd1 and Ffd2, is as follows:
(12)Msd=μdRt(Fd1-Fd2)sin(η-γ),
where η is the azimuth angle between the action line and the x-axis; γ is the azimuth angle between the direction of Ffd and the x-axis; Rt is the radius of overturning moment action circle; μd is the friction coefficient of orbiting scroll and fixed scroll.F
d
1 and Fd2 can be obtained by the axial force balance of the orbiting scroll [5]:
(13)Fd1-Fd2=-MmRt.So, (12) can be rewritten as
(14)Msd=μdMmsin(γ-η),
where Mm is the overturning moment acting on the bearing drive surface.


## 4. Input Torque

Excluding the influence of friction hinge deputy, the input torque,Tin, can be considered to have three parts:(1)
Td, inertia torque, to overcome inertia influence to maintain the driving rod  uniform speed;(2)
Tw, gravitational torque,to balance the effect of gravity;(3)
TF, external load torque, to balance the outside load.For the scroll compressor, with uniform speed and conducted rotor system balance design, by the virtual work equation [6], input torque is as follows:
(15)Td=-J2θ¨2δθ2δθ1+J3θ¨3δθ3δθ1,Tw=m3gl3cosθ3δθ3δθ1,TF=Mt-(Mz+Msd)δθ2δθ1,
where Mt=Ft(θ1)l1, which is the resistance moment caused by the tangential gas force acting on the crankshaft rotation center through the orbiting scroll.δ
θ
2 and δθ3 are given by the formulae (2) and (3) variational solution, respectively:
(16)δθ2=l1[l2sin(θ2-θ1)+l4sinθ1]l2[l1sin(θ2-θ1)-l4sinθ2]δθ1,δθ3=l1[l3sin(θ3-θ1)-l4sinθ1]l3[l1sin(θ3-θ1)-l4sinθ3]δθ1.The mini-crank quality(m3) could be negligible in the dynamical analysis, due to the structural characteristics of the scroll compressor. Thus, when the scroll compressor is in a uniform speed, the input torque can be expressed as
(17)Tin(θ1)=Mt-[Mz+J2θ¨2+Msd]δθ2δθ1.When a scroll compressor is in ideal working condition,θ¨2=0, δθ2=0, the input torque is
(18)Tin(θ1)=Mt.Kinematics and kinetics show that the conditions of drive rod doing whole rotary motion [7] are(1)
the sum of the shortest and longest rod length should be less than or equal to the length of the remaining two;(2)
the shortest rod is connected frame bar.The scroll compressor equivalent four-bar mechanism shows that regardless of the existence of rod size error,l1 and l4, rotation consisting deputy vice must be a complete rotation.So,(19)l1+l2≤l3+l4
or
(20)l1+l4≤l3+l2.And when the actual length ofl3 is less than l1, there must be clearances that exist in hinged deputy to ensure  the  drive rod doing whole rotary motion.


## 5. Model Verification

Taking a twin-wraps scroll compressor as an example, we analyzed the effect of size error on input torque, without considering the mechanism clearances and hinge friction. Only take the shortest leverl1 into account.Table1 shows the parameters of the equivalent four-bar mechanism parameters for input torque analysis. The external load changing with the crank angle of the examples of scroll compressor is shown in Figures 4 and 5; where radial gas force Fr and orbiting scroll assembly inertial force Fc are certain values at a certain speed, and the inertial force Fc increases rapidly with the speed. The rotation moment MZ and the resistance moment Mt change with the crank angle θ at the same discipline of tangential gas force Ft. The friction moment Msd increases with the speed quickly but is approximately constant at a certain speed.Table 1
Parameters of the equivalent parallel four-bar mechanism.
Parameters
Values
l
1
(mm)
4.5
l
2 (mm)
115
l
3 (mm)
4.5
l
4 (mm)
115
J
2 (Kg·m2)
0.142Figure 4
The diagram of external force.Figure 5
The diagram of external moment.It can be seen from Figures6, 7, and 8:(1)
as the scroll compression is in ideal working condition, the variation of input torque of scroll compression is the same with the resistance momentMt, as shown in Figure 6 with slight variation. So, the scroll compressor is designed generally without flywheel setting needed;(2)
when the equivalent parallel four linkage mechanism size error occurs, the actual input torque of compressor differs from the ideal condition a lot.Figure 6
The diagram of input torque in ideal working.Figure 7
The diagram of the input torque.Figure 8
The comparison of input torque.First, in the mini-crank reverse running area to the crankshaft, the input torque will deviate and strongly depend on the size error and crankshaft speed. But, under different conditions, the value is insignificant and little change occurs in the amplitude in the region.Second, the input torque will cause great fluctuation as mini-crank rotation direction changes. Comparison shows that the mini-crank changes in the same direction as the crankshaft, extreme fluctuations is the maximum and the higher the speed, the greater extremes.Meanwhile, the scroll compressor structure determines that the more positive deviationl2 (or negative deviation l4), the more extreme fluctuations increase and the more negative deviation l1 (or positive deviation l3), the more extreme fluctuations reduces.


## 6. Conclusions

Based on the fundamental principle of plane linkage mechanism, effect of mechanism error on input torque of a completely dynamic balancing of scroll compressor was analyzed. Mechanism size error has a big influence on the drive rod and connecting rod collinear. The drive crankshaft eccentricity is beneficial for error reduction but causes more friction loss in the wrap of orbiting and fixed scroll. The appropriate precision should be selected, reducing the adverse effect caused by size error.



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*Source: 438127-2013-12-31.xml*