A fuzzy logic neural network algorithm for compressor crankshaft-rolling bearing system optimization

Abstract

Taking the crankshaft-rolling bearing system in a certain type of compressor as the research objective, dynamic analysis software is used to conduct detailed dynamic analysis and optimal design under the rated power of the compressor. Using Hertz mathematical formula and the analysis method of the superstatic orientation problem, the relationship expression between the bearing force and deformation of the rolling bearing is solved, and the dynamic analysis model of the elastic crankshaft-rolling bearing system is constructed in the simulation software ADAMS. The weighted average amplitude of the center of the neck between the main bearings is used as the target, and the center line of the compressor cylinder is selected as the design variable. Finally, an example analysis shows that by introducing the fuzzy logic neural network algorithm into the compressor crankshaft-rolling bearing system design, the optimal solution between the design variables and the objective function can be obtained, which is of great significance to the subsequent compressor dynamic design.

Keywords

Statically indeterminate problem crankshaft-rolling bearing system dynamics simulation multi-objective optimization implicit function

1 Introduction

In general, the main drive system of a piston compressor is a complex system composed of multiple connecting rod mechanisms in parallel, and the dynamics simulation is highly complicated [7 –9]. There are mainly two typical forms: the crankshaft-sliding bearing system and the crankshaft-rolling bearing system. Regarding the dynamic analysis of the crankshaft-sliding bearing system, scholars at home and abroad have carried out a lot of research work. Analyze the influence of the clearance on the load distribution of the bearing. The smaller the game, the larger the contact distribution of the bearing, and vice versa. The research methods adopted are mainly based on the crankshaft-bearing system as the research object to solve the simultaneous equations for dynamic control [2, 6], calculate the Reynolds equation and various constraint equations of the bearing reaction force, and so on. The application of dynamics simulation software (such as ADAMS) to solve the dynamics problems of the crankshaft-bearing system can reduce the programming workload of simulation calculations significantly. Hence, it has become an important method for addressing the dynamic analysis problems of the mechanical systems [1, 5] and has been extensively used in the dynamics of the main drive system for piston compressors [10, 12]. The main drive system of small compressors is mostly crankshaft-rolling bearing system [3], which is mainly composed of a crankshaft, a rolling bearing, a connecting rod, and a piston. The ADAMS software and its external interface program [11, 13] are used to study the dynamics of the straight shaft-rolling bearing system [4]. However, research on the dynamics of the crankshaft-rolling bearing system has yet to be reported. In the design of a compressor, the selection of the angle between the two cylinders still relies on the experience of the engineer at present and lacks a reliable theoretical basis. In this paper, the ADAMS simulation software is first used to study the dynamics of the elastic crankshaft-rolling bearing system in the rated operating conditions and obtain the crankshaft main journal center displacement and the bearing reaction force [14]. On this basis, the angle between the left and right cylinders is taken as the design variable, and the weighted mean of the radial displacement amplitude of the crankshaft journal center is taken as the objective function to carry out dynamics optimization in the ADAMS and obtain the optimal angle value.

2 System model and theoretical basis for solution

Figure 1 shows the ADAMS dynamics simulation model for the crankshaft-rolling bearing system of a compressor. In the modeling, the corresponding unit needs to be changed to nkgmms; and the corresponding points need to be constrained, so that the simulation process can accurately find the corresponding points. The crankshaft is subject to the restraint by two main bearings, the bearing is removed at the bearing restraint and replaced with the bearing reaction forces Fbx and Fby. The bearing reaction forces Fbx and Fby are obtained based on the force and deformation relationship equation for the rolling bearing.

Fig. 1

ADAMS simulation model for the crankshaft-rolling bearing system.

2.1 Equation for the relation between force and deformation of the rolling bearing

2.1.1 Equation for the relation between force and deformation of the rolling bearing

For the relationship between the reaction force of the rolling bearing and the kinematic parameters of the journal shaft center, the bearing reaction force generated by the damping of the rolling bearing is relatively small. Hence, it is neglected to simplify the problem. Only the relationship between the reaction force of the rolling bearing and the radial displacement of the journal center is discussed herein, that is, the relationship between the force and the deformation of the rolling bearing. If the displacement is too small, the bearing will generate more heat, the temperature will increase, and the bearing will be burnt seriously, which will lead to failure. Therefore, it is very important to analyze the relationship between bearing displacement and bearing load distribution.

Figure 2 shows the load distribution diagram of the rolling bearing when the single-row radial ball bearing is loaded within the range of 180°. From the figure, it can be observed that this issue is a typical multiple statically indeterminate problem, which can be resolved based on the force method. The balance equation can be expressed as the following:

Fig. 2

Load distribution diagram of the rolling bearing.

$F_{r} = Q_{max} + 2 Q_{1} cos ψ_{1} + 2 Q_{2} cos ψ_{2} + \dots$ (1)

Due to the constraints of the bearing seat or the frame in the outer ring of the bearing and the journal in the inner ring of the bearing, it is assumed that the deformation is only caused by the contact deformation between the rolling elements and the inner and outer ring raceways, while the original size and shape of the inner and outer rings still remain unchanged. The rolling element on the line of action of the radial force Fr is subject to the maximum load, and its contact deformation is the maximum deformation δmax, which is the deformation δr of the rolling bearing. Hence, the contact deformation between the rolling elements and the inner and outer ring raceways at other positions can be obtained as the following: $δ_{ψ} = δ_{r} cos ψ = δ_{max} cos ψ$ (2)

In the above equation: ψ—the angle between the rolling element and the maximum load rolling element.

Equation (2) shows the geometric relationship of the rolling bearing deformation, which is a crucial basis for solving the force distribution of each rolling element.

From the Hertz elastic theory and the geometric relationship of the rolling bearing with point contact, the elastic approach of the two contact bodies can be obtained as the following:

$δ = 2.79 \times 10^{- 4} \frac{2 K}{π m_{a}} {(Q^{2} \sum p)}^{1 / 3}$ (3)

In the above equation: Q—rolling element load;

∑ρ—the sum of the principal curvatures of the two contact bodies at the contact point;

ma—the short semi-axis coefficient of the contact ellipse;

K—the first type of complete ellipse integral that is related to ellipse eccentricity.

The total approach amount δ should be the sum of the approach amount between the rolling element and the inner ring raceway (that is, the contact deformation) δi and the approach amount δe between the rolling element and the outer ring raceway. Hence, the following can be obtained:

$\begin{matrix} δ = δ_{i} + δ_{e} = k \\ [{(\sum p_{i})}^{1 / 3} Q_{i}^{2 / 3} + {(\sum p_{e})}^{1 / 3} Q_{e}^{2 / 3}] \end{matrix}$ (4)

In the above equation: k—coefficient related to the geometrical dimensions and materials of the rolling bearing.

Equation (4) shows the physical relationship between the force and the deformation of a single rolling element. Through simultaneous equation (1) to equation (4), and based on the symmetry of the rolling elements and (n stands for the number of loaded rolling elements), the analytical expression for the radial force Fr and the radial deformation δr of the rolling bearing can be obtained as the following:

$\begin{matrix} δ_{r} = 2.79 \times 10^{- 4} {(\frac{4.37 F_{r}}{n})}^{2 / 3} \\ [k_{i} {(\frac{4}{D_{g}} + \frac{2}{D_{i}} - \frac{1}{r_{i}})}^{1 / 3} + k_{e} {(\frac{4}{D_{g}} - \frac{2}{D_{e}} - \frac{1}{r_{e}})}^{1 / 3}] \end{matrix}$ (5)

In the above equation: ki, ke—the coefficients related to the geometric dimensions and materials of the rolling bearing;

Dg—the rolling element diameter;

Di, De—the diameters of the contact between the inner and outer rings of the bearing and the rolling elements;

ri, re—the radius of curvature of the rolling element in contact with the inner and outer rings.

After it is introduced to the geometric dimensions of the deep groove ball bearing of 6205, the relationship between the load and the total radial deformation can be obtained through numerical calculation, as shown in Fig. 3 below. It can be easily observed that the relationship between the load on the rolling bearing and the total radial deformation is non-linear, and that the rolling bearing can be regarded as a hard spring with variable coefficient of stiffness.

Fig. 3

Relationship curve between the radial load and the total radial deformation of the rolling bearing.

2.1.2 Reaction force of the bearing

As shown in Fig. 1 above, the components Fbx and Fby of the bearing reaction force in the x and y coordinate axis directions are as the following: ${\begin{matrix} F_{bx} = - F_{x} \frac{Δ x}{r} \\ F_{by} = - F_{r} \frac{Δ y}{r} \end{matrix}$ (6)

In the above equations: $r = δ_{r} = \sqrt{Δ x^{2} + Δ y^{2}}$ ;

Δx, Δy—the displacement of the axis in the direction of the x and y coordinate axes.

2.2 Piston force and pulley axial force

2.2.1 Piston force

A working cycle of a piston compressor includes four processes, that is, suction, compression, exhaustion, and expansion. In general, the indicator diagram is used to demonstrate the relationship between the gas pressure and the volume. The indicator diagram can be measured by the dynamometer. Due to the limitation of conditions, the gas pressure is solved approximately by the gas equation of state. The equation for the process of gas changing from state I to state II is shown in [8] as the following: $p_{1} V_{1}^{m} = p_{2} V_{2}^{m} = C$ (7)

In the above equation m—the process index, for air, m = 1.40

C—constant;

p1, p2—gas pressure in the I and II states;

V1, V2—the gas volume in the I and II states.

The results obtained based on equation (7) are shown in Fig. 4 as the following.

Fig. 4

Theoretical indicator diagram of the piston compressor.

The piston force acting on the top of the piston is as the following: $F_{p} = \frac{π}{4} d^{2} P$ (8)

In the above equation: P—cylinder pressure;

d—piston diameter.

2.2.2 Pulley axial force

The power of the compressor is transmitted by the motor to the crankshaft through the belt drive. As the belt drive needs to be pre-tensioned to work properly, the pre-tensioning of the belt drive will generate an axial force on the crankshaft: $F_{0} = 500 \frac{P_{ca}}{vz} (\frac{2.5}{k_{a}} - 1) + {qv}^{2}$ (9)

In the above equation: v—linear velocity of the belt, m/s;

z—the number of roots;

Pca—the belt drive power, kW;

Ka—the wrap angle coefficient;

q—the mass with unit length, kg/m.

2.3 Multi-flexible body dynamic equation of elastic shaft

According to the Lagrangian dynamics equation, the final form of the flexible multi-body dynamics control equation of the elastic crankshaft expressed in the generalized coordinates is shown in [6, 10] below:

$\begin{matrix} M \ddot{ξ} + \dot{M} \dot{ξ} - \frac{1}{2} {[\frac{\partial M}{\partial ξ} \dot{ξ}]}^{T} \dot{ξ} + K ξ + f_{g} + D \dot{ξ} \\ + {[\frac{\partial ψ}{\partial ξ}]}^{T} λ = Q \end{matrix}$ (10)

In the above equation: $ξ, \dot{ξ}, \ddot{ξ}$ —the generalized coordinates of the elastic axis and its derivative with respect to the time;

$M, \dot{M}$ —the mass matrix of the elastic axis and its first-order partial derivative with respect to the time;

K—the generalized stiffness matrix;

fg—the generalized gravity;

D—the damping matrix;

Ψ—the system constraint equation;

λ—the Lagrange multiplier;

Q—the generalized force matrix.

The finite element model for the elastic crankshaft is established in the ANSYS. No. 189 beam element is adopted as the element type, which has 91 nodes and 23 elements, as shown in Fig. 5 below.

Fig. 5

Finite element model of the elastic crankshaft.

2.4 System modeling and solving method

It is not hard to construct a model of the elastic crankshaft-rolling bearing system by using the large-scale dynamics simulation software ADAMS. Firstly, a finite element model for the elastic crankshaft is established in the ANSYS, and then the geometry, inertia, and modal information of the shaft are transferred to the ADAMS through the modal neutral file to obtain the elastic crankshaft components of the ADAMS. In addition, the bearing reaction force, the connecting rod, the piston, the large pulley, and other components are added to constitute the dynamics simulation model for the elastic crankshaft-rolling bearing system. Its form is shown in Fig. 1 above. The difference is that the rigid crankshaft is replaced with an elastic crankshaft. Among them, the piston force is fitted by the spline function before it acts on the top of the piston.

The bearing reaction force is self-programmed through the ADAMS custom function and prepared into the dynamic link library bearing.dll, which is called in the ADAMS through the external interface. The essence of the solution method in the ADAMS is the iterative solution of simultaneous equation (1) to equation (10), and the raw data used for the calculation are shown in Table 1 as the following.

Table 1
Main performance parameters of a compressor

Matching power 1.5kW Rated working pressure 0.8MPa

Piston stroke 45mm Diameter of the cylinder × number of cylinders φ 50×2

Rated speed 1000r/min Bearing type 6205

Matching power	1.5kW	Rated working pressure	0.8MPa
Piston stroke	45mm	Diameter of the cylinder × number of cylinders	φ 50×2
Rated speed	1000r/min	Bearing type	6205

3 Dynamic optimization design of the crankshaft-rolling bearing system

3.1 Mathematical model for dynamic optimization design

Figure 6 shows the schematic diagram of the compressor mechanism. The main drive system is composed of two crank slider mechanisms in parallel. One of the main defects is that it is impossible to balance the inertial force and the moment of inertia completely. Due to the periodic unbalanced inertial force and gas pressure during the high-speed operation, the vibration and noise are relatively large. Hence, the dynamic optimization design is of important engineering significance. The mathematical model of optimal design is as the following:

Fig. 6

Movement diagram of the compressor mechanism.

(1) Design variables.

The angle between the center line of the cylinder and the y axis can be selected as α1, α2. However, it is highly challenging to use the angles for parametric modeling in the ADAMS. Hence, the center coordinates XA, YA, XB, and YB where the hinge (that is, the piston pin) connects the piston and the connecting rod are selected as the design variables, and the included angle is adjusted through the change of the position of the point.

(2) Constraint condition.

➀ It is shared by the crank, and the length of each connecting rod is equal and remains unchanged.

$\begin{matrix} \sqrt{X_{A}^{2} + Y_{A}^{2}} = \sqrt{X_{B}^{2} + Y^{B 2}} \\ = \sqrt{L^{2} + r^{2} + 2 r (cos α \sqrt{L^{2} - r^{2} {sin}^{2} α} - r {sin}^{2} α)} \end{matrix}$ (11)

In the above equation: r—crank length;

L—length of the connecting rod.

➂ It is characterized by symmetry. XA = XB = X,YA = YB = Y, that is, α1 =α2 =α

➂ In order to prevent the interference between the cylinders, α1 min = 33.7° is taken the minimum included angle between the cylinders, and the initial vertical position of the crank and connecting rod is taken as the maximum included angle, that is, α1max = 78.2°. The range of the corresponding variable XA is 70–108 mm.

The objective function is selected to minimize the weighted sum of the radial vibration response amplitude of the two main bearing journal centers, that is, the following can be obtained: $min (f (X)) = min [λ A_{1} + (1 - λ) A_{2}]$ (12)

In the above equation: A1, A2—the radial vibration response amplitude of the crankshaft journal center at the main bearings 1 and 2;

λ—weight coefficient.

It is impossible to give the analytical expression of the objective function directly, and it is implicit in the dynamic control equation of the crankshaft-rolling bearing system. From the mathematical model for the optimal design, it can be seen that although there are 4 design variables, that is, XA, YA, XB, YB, there are 3 equality constraint conditions. Hence, there is only one independent design variable. Therefore, this is a multi-objective one-dimensional dynamics optimization problem with an implicit objective function, which falls into the category of the optimization design problem of complex mechanical systems.

3.2 Results of optimization calculation

In the ADAMS, the dynamics optimization design model for the crankshaft-rolling bearing is established based on the method of parametric modeling, and the angle of the cylinder center line is expressed by the design variable X. As it is a one-dimensional optimization design problem, it can be solved simply by using the design research module of the parametric analysis in the ADAMS. Figure 7 show the relationship curve between the design variables and the objective function values obtained through the design research. It can be seen that as the design variable is increased from 70 mm to 108 mm, the value of the objective function is simply decreased, that is, the greater the angle between the center line of the cylinder, the better the dynamic performance. The optimal advantage in dynamics is X = 108 mm, and the corresponding cylinder center line angle α= 78.2°. In theory, the symmetrical balanced compressor (that is, α= 90°) has the optimal dynamic performance. However, for a compressor, the consideration of an excessively large included angle can lead to an increase in the lateral volume of the compressor. In a compressor product, the cylinder center line included angle is selected as α= 45°, the corresponding design variable is X = 86.8 mm, and the objective function value is 0.02435 mm. It can be seen that the selected design point has certain rationality.

Fig. 7

Relationship between the design variables and the objective function values.

4 Discussion

When the elastic deformation of rolling bearing is considered, the axis trajectories of the two main bearings are similar, but the steady-state amplitude peak and the bearing reaction peak are different, with the difference of 25.9% and 80.7% respectively. The main bearing 1 and the main bearing 2 of the compressor products used in the calculation are different types of bearings, and the design scheme is reasonable. The steady-state axis trajectory of the main bearing and the peak value of the main bearing reaction force of the elastic crankshaft rolling bearing system are obtained. The influence of the elastic deformation of the bearing on the steady-state amplitude of the main journal center and the peak value of the main bearing reaction force is very significant, which provides a reliable basis for the dynamic performance of the decompressor and the design of the main bearing.

5 Conclusion

With the whole crankshaft-rolling bearing system as the research object, the relationship between the axis motion parameters and the reaction force of the rolling bearing is used to resolve the dynamic issues of the crankshaft-rolling bearing system directly, which is more reasonable in theory. The dynamic behavior of the elastic crankshaft-rolling bearing system of the compressor in the rated operating conditions is studied to obtain the reaction forces of the two main bearings, the radial vibration response change rule of the crankshaft main journal center, and the axis trajectory. Establish an optimal design model, propose a multi-objective dimensional dynamic optimal design, and model the model through ADAMS parameters to find the optimal solution to illustrate the rationality of the research model compressor design optimization.

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