Optimization Analysis of the Wheel Hub Dynamic Balancing Machine Based on Particle Swarm Optimization

The vibration supporting rod of a wheel hub dynamic balancing machine is affected by the alternating dynamic load, and its vibration characteristics have a direct impact on the accuracy of measurements. To solve this problem, first, a particle swarm optimization algorithm (PSO) algorithm is proposed to optimize the dynamic stiffness of the vibration supporting rod. According to the actual material selection, the creep risk of the rod is reduced. Second, three schemes for improving the structure of hub balancing machines are put forward and compared. The displacement response and stress response of scheme A are increased by 30% and 20%, respectively, over those of the original equipment. The displacement response and stress response of scheme C are increased by 31% and 22%, respectively, which is a significant improvement. Schemes A and C are acceptable, but a strong first-mode response remains due to assembly variation.


Instruction
With the development of machinery equipment towards high-speed, lightweight, and large-scale rotation, it is increasingly important to consider the dynamic performance of the rotor support rod [1,2].For dynamic balance equipment, the mechanical components comprise mainly the drive part, transmission part and vibration support part, of which the vibration support part is the main component.The part is located at both ends of the balanced rotor and imparts the actual basic characteristics of a balanced rotor as much as possible.The dynamic performance of vibration support has a great influence on the safety and accuracy of a dynamic balance experiment and needs to be analysed comprehensively.The measurement process of a dynamic balancing machine is a reverse mathematical mapping process from the vibration response of the support structure to the unbalance of the rotor.Its essence is to predict and optimize the balance weight according to the vibration response.It is divided into the following two sub-processes: observation, which obtains the observed vibration measurement value of the spring rod, and restoration, which determines the unbalance based on the vibration measurements.In the process of dynamic balance measurement, the unbalance acts on the support structure in the form of centrifugal force and produces an observable vibration response.In the recovery process, the vibration response is returned to the measured value of unbalancing through a certain mathematical mapping relationship.For the vibration support system used to achieve a wheel hub dynamic balance, Chinese dynamic balancing machines generally use a spring plate or spring rod, which has the limitations of poor plane separation ability and unsatisfactory static couple separation.ZHOU et al. proposed an optimal static couple separation equilibrium method to solve this kind of 2 problem, but this method is not easy to implement [3,4].To overcome the disadvantage of the poor static couple separation effect of hard support double-sided vertical dynamic balancing machines, LI et al. designed three kinds of swing frame mechanisms to highlight the characteristics of static couple separation [5].Based on a study of the correlation effect between the cantilever beam and the simply supported beam structure and according to the principle of the instantaneous motion centre in theoretical mechanics, QIN et al. designed a fully flexible series-connected vibration system, which inherited the advantages of the cantilever beam and simply supported beam structure [6][7][8].Schenck et al. analysed error and low mechanical sensitivity of hard support balancing machines and pointed out that the vibration system design of dynamic balancing equipment should meet the following objectives: Eliminate the influence of principle error and force correlation effect to the greatest extent, and improve the stiffness, mechanical sensitivity and plane separation of the pendulum frame [9].Cao et al. also studied the plane separation error of a hard support double-sided vertical dynamic balancing machine [10][11][12].Kenneth S. studied the measurement system and method of wheel unbalance [13].Yang et al. analysed the dynamic characteristics of the support stiffness of a centrifugal compressor [14].According to the principle of disc rotor dynamic balance measurement and air suspension, to solve the problem of low detection accuracy caused by vibration associated with the mechanical transmission, GAO et al. proposed measurement methods for the air suspension static balance rotor and even balance rotor [15,16].Zhou et al. analysed the support dynamic stiffness and random reliability of a dynamic balancing machine [17,18].At present, research on the dynamic characteristics, such as the dynamic balance mode shape and the unbalanced response of the wheel hub, is limited, with few studies on the structural model, structural parameter optimization, or quantitative or qualitative analysis.GAO et al. used a multiobjective optimization method to optimize a centrifugal fan.Ibaraki S et al. used a neural network algorithm to optimize a centrifugal fan and analysed the influence of structural parameters on the optimization results [19].The measurement accuracy of a dynamic balancing machine is closely related to its geometric model and mechanical accuracy.With the development of dynamic balance calculation methods and signal processing technology, the further improvement of detection accuracy depends on the improvement and optimization of the mechanical structure.
In this paper, a dynamic balancing machine produced by the General Institute of Mechanical Research is taken as the research object.Depending on the diameter range of the wheel hub, the mass and geometric size of the wheel hub can vary greatly.The vibration support is affected by the alternating dynamic load of different masses.Random vibration occurs during the operation of the equipment, and the vibration characteristics affect the stability of the test accuracy.This phenomenon depends on the mechanical structure of the dynamic balancing machine, so the vibration support part and the spindle system are the key research objects.At present, the detection of rotor unbalance is indirectly obtained mainly by measuring the vibration at the support.Therefore, the vibration support part of the dynamic balance detection equipment and the spindle system need to have good dynamic performance.A main shaft vibration support frame fixed end model is constructed, and the modal shapes of the balancing machine and the vibration supporting rod are analysed.To improve the stiffness of the system, reduce the concentrated stress and deformation of the spring rod as the core, and accurately measure the detection accuracy of the dynamic balancing machine, three improvement schemes are proposed and compared with the modal analysis.

Dynamic Balancing Machine
The measurement accuracy of the dynamic balancing machine is closely related to its geometric model and mechanical accuracy.The mechanical part of the dynamic balancing machine is composed mainly of three parts: the driving part, transmission part, and vibration supporting part figure 1.At present, the detection of rotor unbalance is mainly obtained indirectly by measuring the vibration at the support.Therefore, the vibration support structure and spindle structure of dynamic balance detection equipment need to have good force and vibration transmission performance.Due to the measurement of the diameter range of the wheel hub, the mass and geometric size of the wheel hub vary greatly, and the vibration support is subjected to alternating dynamic loads of different masses.The configuration shown in figure 2 can be analysed to test the accuracy, stability, and accuracy of the vibration characteristics.The structure of a dynamic test is shown in figure 1, courtesy of the Beijing City machinery technology development Productivity Promotion Centre.Its hardware system includes a Beckhoff AM8553-1K20-0000 driving motor and two sensors installed on the supporting mechanism.The rotor used in these experiments lies in the motor of an electric tool; its permissible unbalance on one plane is 80 g and balancing speed is 300 r/min.
We selected an already balanced rotor as the experiment target and multiples of permissible unbalance on one plane as trial weights.Each trial weight was positioned on only one balancing plane at the angular positions every other 180°, and the vibration response was recorded.This response was estimated by the proposed method.
The calibration steps are described as follows.
(1) Drive the hub without any trial weight on the plane, and measure the original vibration; (2) Add a calibration weight at 0 degrees on the upper plane, and measure the amount of vibration; (3) Add a calibration weight at 180 degrees on the upper plane, and measure the amount of vibration; (4) Add a calibration weight at 0 degrees on the lower plane, and measure the amount of vibration; (5) Add a calibration weight at 180 degrees on the lower plane, and measure the amount of vibration.

Mathematic Model
The rotor support system of the dynamic balancing machine is a type of forced vibration system driven by an eccentric mass.In the case of neglecting the damping of the rotor support system of the dynamic balancing machine, the centre of mass G of the system is taken as the coordinate origin.When the rotor is balanced, the rotor rotates around the Z-axis at angular velocity W. The axis perpendicular to the Z-axis is the X-axis.The model idealizes the rotor and the supporting structure as a two-degree-of-freedom vibrating system.The moment of inertia about the axis is J.The vibration behaviour can be regarded as translational motion along the X-axis at C and rotation around the centre of mass.The displacement of the left and right supports θ lA and θ lB is the centrifugal According to Newton's second law, the differential equation of motion without considering damping is in equation (1).

( ) ( )
When we take equation (3) with B A l l = as the natural frequencies of translation and oscillating motions of the system, the steady-state response equation is equation (4).
When we take the system as a hard support dynamic balancing machine with The sensor measurements are 1 N and 2 N ; then in equation ( 6).

( ) ( )
When the upper and lower sensors have the same stiffness, we take With this formula, we can obtain equation (8).
The solution can be obtained as equation ( 9).
In the formulas From the above formula, as long as the rotor size and stiffness ratio are determined, the unbalance on the front of the two calibrations can be determined as well.

Establish Finite Element Model
The basic idea of the finite element method is to use a simulated object, namely, a unit connected in a certain way after continuous solution domain discretization into a finite number of elements, to represent the original object to simplify a continuous infinite-degree-of-freedom problem to a discrete finite-degree-of-freedom problem.After the object is discretized, the analysis of the whole object is obtained through the element analysis of each element.
The natural frequency and mode shapes of the spindle are analysed by ABAQUS using the created structural model.In this paper, SolidWorks 2018 is used to construct the structure model in 3D, which is then imported into ABAQUS after converting it to X_T format.In this process, model simplification is first performed.The rolling bearing can be simulated by a spring and damper, and the rolling bearing is thus deleted.The bearing cover of the rolling bearing and the inner and outer sleeve of the positioning bearing are deleted, and the screw hole and some part features that have little influence on the dynamic analysis of the shaft are removed.We then import the simplified model into ABAQUS.The contact surface between the shaft and the rolling bearing is precisely cut out on the shaft.The next step is to define the model material properties and contacts.The material of the upper and lower shafts is defined as 45 steel.Its elastic modulus is E=200GPa, and its density is 7580 kg/m 3 .Poisson's ratio is 0.3.The model adopts a free mesh method.The Rbe2 element is used for simulating bolted connections (figure 2).The bolt connection in the supporting rod is considered to be rigid and reliable.In the finite element analysis, rigid connections are used between the models to fix the parts.The chamfer, small hole, thread, and groove in the support swing frame are ignored.The grid is discretized, and the grid quality is evaluated according to the GB/T 33582-2017 standard [20].The total number of grid elements is 628863, and the number of nodes is 150678, which meets the standard (figure 2).The last step is the apply the modal analysis and loading constraints.In the modal analysis setup, the output mode is set as the 6-order mode, and the stiffness, damping, and gyroscopic effect of the rolling bearing are evaluated.
The modal analysis evaluates in equation (11)(12).The frequency response function is in equation (13). () N rr r rr AA Hs s s s s where, M is the mass matrix, and the unit is kg ; C is the damping matrix, and the unit is m s N /  ; K is the stiffness matrix, and the unit is  is the acceleration response vector, and the unit is 2 / s m ; X is the displacement response vector, and the unit is m ; and f is the exciting force vector, and the unit is N .The four vibration spring rods deform, rotate around the Z-axis, and swing in the X-axis direction.
Swing around the X-axis.
The four vibration spring rods deform and rotate around the Z-axis.The four vibration spring rods deform and rotate around the Z-axis, and the support rods deform.
To analyse the basic dynamic characteristics of the dynamic balancing machine and understand the natural frequencies and vibration modes of each order, it is necessary to carry out modal analysis on the dynamic balancing machine, analyse the vibration supporting rod, observe the deformation form of each order mode, and analyse the influence of the displacement response and stress response of the vibration supporting rod on the system.When the dynamic balancing machine is running, its weight is far greater than the centrifugal force caused by the unbalance of the rotor in the y-axis direction due to the large mass of the rotor; hence, it is assumed that no vibration arises in the y-axis direction.It is required that the swing frame has enough supporting stiffness for the rotor in the design, which can be seen from the results of the modal analysis.The first mode shape lies in the Z-axis direction, and the second-, third-and sixth-order shapes involve rotation around the Z-axis and vibration of the supporting rod.The fourth-and fifth-order vibrations of the box do not influence the measurement or main support stiffness of the dynamic balancing machine, so those vibration modes can be removed.In the dynamic balancing machine experiment, when considering the local low-frequency response, the modal analysis and verification of the spring rod show that there is indeed a low-frequency mode.This phenomenon affects the vibration of the box and the supporting rod in different directions in addition to the measurement of unbalance and the stability of the system (shown in table 1).

Displacement Analysis.
The measurement of dynamic unbalance of the structure is impacted based on the change in the force.XING experimentally studied the low-cycle fatigue characteristics and fatigue creep interaction of turbine rotor structures and materials [21,22].It was found that combined with the wheel hub dynamic balancing machine, the vibration support part under the action of the alternating load has an extended influence on the structural fatigue.If the yield limit of the material is exceeded, the equipment cannot operate normally.It is concluded that the vibration support rod may have undergone creep, but further experimental verification is required.

Stress Analysis.
The stress response of the whole system depends on the dynamic stiffness of the spring rod: Increasing the stiffness of the spring rod reduces the amplitude of the frequency response.In the dynamic balancing machine, the vibration supporting part is the main bearing structure in the rotor dynamic balancing system.Because each dynamic balance involves several steps of starting, braking, and crossing critical speeds, the static and dynamic performance and fatigue life directly affect the reliability and safety of the dynamic balance experiment.The part accuracy directly affects the assembly accuracy of products, but higher precision increases the production difficulty and adversely affects the economics of production.Therefore, reasonable tolerance allocation is particularly important for products, and this allocation is related to the product manufacturability and economy.In the process of equipment design and manufacturing, assembly tolerance design can effectively improve the accuracy of assembly, reduce costs, and improve the assembly success rate.Tolerance analysis is an important means to evaluate the accuracy and function of mechanical products [23], and the assembly tolerance is a key parameter to realize high-precision assembled products [24].In the assembly process of the dynamic balancing machine, the assembly accuracy is not high, so it is considered that the existence of assembly variation leads to the stress-strain response of the spring rod in the first mode.

Dynamic Balancing Machine Optimization
The design and calculation of the vibration system of a dynamic balancing machine is not only the central problem of mechanical structure design but also one of the key problems to determine the performance of the balancing machine.This machine should meet the following two basic requirements: First, the relationship between the excitation force and displacement is linear over the possible linear range; second, the natural frequency of the vibration system and the frequency of the balance speed must meet the prescribed relationship; that is, the hard support dynamic balancing machine has to satisfy n   3 .0  [25].Therefore, the design and calculation of the vibration support part of the balancing machine play a decisive role in the measurement accuracy of the machine.The calculation includes the following three aspects:

Establish an Objective Function
The contrast condition is shown in equation (15).
Elastic element stiffness constraint.To ensure the supporting and limiting effects of elastic elements, the upper and lower limits of the stiffness of elastic elements are limited according to a large number of experiments, which is shown in equation (16)(17). We . The calculated stiffness can be improved in two aspects.First, one can change the material and increase the elastic modulus.Second, one can apply the condition that the material remains unchanged, that the geometric dimensions are derived, and that the combined dimensions of the vibration supporting rod and supporting rod are changed.Three improved schemes are proposed and compared with the modal simulation.However, the above stiffness is used only for reference because the specific actual situation also needs to be combined with a strength, stability, and cost evaluation to provide a comprehensive reference evaluation to be verified experimentally.

Verify and Discuss the Optimized Model
The response of the three kinds of dynamic balancing equipment and the response of the vibration supporting rod are analysed.The frequency response to the diagram, deformation, and stress curve of the improved high-speed dynamic balancing machine is obtained.The analysis is shown in table 2.
In scheme A, an additional vibration support rod is added to the dynamic balancing machine to reduce the vibration amplitude.The support rod has enough support stiffness to accommodate the rotor and to avoid the influence of vibration on the unbalance measurement during start-up.The results show that the elastic modulus of the vibration supporting rod is 210 GPa, the interface diameters of the supporting rod are 15 mm and 11 mm, the density is 7850 kg/m³ , and the Poisson ratio is r = 0.3.
In scheme B, the elastic modulus is 250 GPa, the interface diameters are 16 mm and 12 mm, the density is 7850 kg/m³ , and the Poisson ratio is r = 0.3.
Scheme C is a combination of schemes A and B. An additional vibration supporting rod is added to the dynamic balancing machine.The elastic modulus of the vibration supporting rod is 250 GPa.The interface diameters of the vibration supporting rod are changed to 16 mm and 12 mm.The density is 7850 kg/m³ , and the Poisson ratio is r = 0.3.In scheme A, the peak value of the overall displacement response is changed from 18.64 Hz to 33.859 Hz, which is 30% higher than that of the original equipment.The single peak value changes to a multi-peak value, and the single-system vibration is thus transformed to multisystem vibration.The deformation response of the vibration supporting rod is reduced by 20%, and the effect is improved.In scheme B, after the elastic modulus of the vibration supporting rod is changed, the frequency response changes to 19.375 Hz, which is 0.73 Hz higher than that of the original equipment, corresponding to a minimal change.The change in dynamic stiffness is also small.In scheme C, the deformation response is 34.513Hz, which is 31% higher than that of the original equipment and 0.67 Hz higher than that of scheme A. The deformation response of the vibration supporting rod is increased by 22%.Therefore, schemes A and C are desirable.

Stress Analysis.
In a comparison between schemes A, B, C, and the original equipment, the stress of the spring rod in scheme B changes only minimally, but the scheme A and C stress response changes from single-system vibration to multisystem vibration.

Conclusion
(1) Because the measurement of the diameter range of the wheel hub is 17-22 mm, the mass and geometric size of the wheel hub vary greatly, and the vibration support is subjected to alternating dynamic loads of different masses.When the vibration response is in the first mode, the structure behaves abnormally.In the assembly process of the dynamic balancing machine, the accuracy is not high.The assembly tolerance is the main factor affecting the vibration of the rotor system and vibration support, which is the key research direction planned for future work.
(2) In the dynamic balancing machine, the vibration supporting part is the main supporting part of the rotor dynamic balancing system.Because each dynamic balancing requires several iterations of starting, braking, and crossing the critical speed, the dynamic performance and fatigue life directly affect the reliability and safety of the dynamic balance experiment.Under the action of an alternating load, the fatigue of the structure is affected by the vibration supporting part for an extended time, and the creep phenomenon of the vibration supporting rod can be evaluated.
(3) In the comparison of the three schemes A, B, and C with the original equipment, in scheme C, the displacement response is increased by 31%, and the deformation of the spring rod is reduced by 22%.The displacement response of scheme A is increased by 30%, and the deformation of the spring rod is reduced by 20%.Scheme B results in little change relative to scheme A. Considering economics and practicability, scheme A is preferred.
(4) The stiffness calculated by the PSO optimization algorithm can be improved in the two aspects of material and component geometry.However, the calculated stiffness is useful only for reference because of the specific actual situation; the strength, stability, and cost need be addressed in a comprehensive reference evaluation and then need to be verified through experiments.

Figure 1 .
Figure 1.(a) The structure of the dynamic balancing machine.(b) The wheel dynamic balancing machine.

=
force produced by the unbalance; A k and B k are the dynamic stiffnesses of the upper and lower spring rods, respectively; and K is the dynamic stiffness of the supporting rod.

1 F and 2 F 2 N
are the unbalanced forces converted to the two calibration faces of the workpiece, that 1 N and are respectively the reaction forces at the upper and lower spring rods (from the sensor), and that geometric dimensions.It can be obtained as follows equation(10).

Figure 4 (
a) shows the overall displacement response of the equipment, and figure 4(b) shows the displacement response of the vibration supporting rod.(a) The overall displacement response of the equipment.(b) The displacement response of the vibration supporting rod.

Figure 4 .
Figure 4. Comparison chart of the optimized plan and original equipment displacement response.

Figure 5 (
a) shows the overall stress response of the equipment, and figure 5(b) shows the stress response of the vibration supporting rod.(a) Overall stress response of the equipment.(b) Stress response of the vibration supporting rod.

Figure 5 .
Figure 5.Comparison chart of the optimized plan and original equipment stress response.

Table 1 .
Modal analysis results of the frame structure.

Table 2 .
Modal analysis results of the frame structure.