Design and analysis of a novel flexible variable stiffness damping mechanism for grinding electric spindle

Abstract

The purpose of this research is to design a flexible variable stiffness damping mechanism to solve the load impact problem in the grinding process of permanent magnet electric spindle. Firstly, combined with the structure and usage scene of the grinding permanent magnet electric spindle, according to the design principles and technical indicators, a novel flexible variable stiffness vibration damping joint is proposed. Then the stiffness and energy storage model of the flexible variable stiffness damping mechanism is established, and the stiffness and energy storage characteristics of the mechanism are analyzed. The strength and dynamic characteristics of the flexible variable stiffness mechanism are analyzed by Ansys and Adams, respectively. Finally, an experimental platform for the comprehensive performance verification of the flexible variable stiffness damping mechanism is built to verify the dynamic characteristics and damping effect of the mechanism. The research results show that the designed flexible variable stiffness structure has better vibration damping effect and can improve the grinding performance of the permanent magnet electric spindle.

Keywords

Flexible variable stiffness damping mechanism permanent magnet motor spindle grinding operation dynamic characteristics

1. Introduction

Grinding and polishing technology, as an important link in manufacturing production, is widely used in the processing and production of various parts and components such as automobile manufacturing, aerospace equipment, and industrial building materials (Xu et al., 2021; Zhao et al., 2014). The grinding electric spindle operating system of the grinding robot has an impact in the transition stage of startup, speed change, or from the free motion space to the constrained motion space. It is easy to cause coupling vibration between the electric spindle, grinding wheel, and grinding load, and even cause serious damage to the robot body, processing tools, and workpiece surface quality (Zhu and Beaucamp, 2020).Therefore, it is of great significance to explore the vibration reduction method of the grinding operation system and reduce the coupling vibration of the system for the smooth grinding operation of the grinding robot.

At present, there are mainly active compliance control strategies and passive compliance control strategies in robot vibration reduction and compliance control strategies. In terms of active compliance control, Duan et al. proposed an adaptive variable impedance control method for the uncertainty of the external processing environment (Duan et al., 2018). Chaudhary et al. constructed a control model on a 6-DOF robot, and used a force-position hybrid control strategy to reduce external environmental disturbances (Chaudhary et al., 2016). Kong et al. designed an optimized PID controller according to the complex characteristics of the simulated dynamic environment, which enabled the hydraulic drive unit to obtain better force control performance (Kong et al., 2016). Chen et al. compared the open-loop and fuzzy PID control algorithms, and proposed a fuzzy sliding mode control method, which reduced the oscillation period and achieved the expected control effect (Chen et al., 2017). Mohammad et al. used neural network and genetic algorithm to optimize and predict the performance parameters to achieve accurate material removal (Mohammad et al., 2017).

In terms of passive compliance control technology, Dezman M et al. proposed a pseudo-linear variable-ratio lever variable-stiffness actuator (Dezman and Gams, 2018). Mahboubi et al. designed a variable stiffness hand (Mahboubi et al., 2018). From the perspective of energy efficiency, Visser et al. designed a variable stiffness scheme with a simple structure under variable load force, and analyzed its strategy to reduce energy consumption from the perspective of control (Visser et al., 2011). Gao et al. presented a novel compliant robotic gripper with three variable stiffness fingers (Gao et al., 2020). Through a simple planar mechanism, equipped with one standard and one variable-stiffness actuator, a variable-stiffness clamp was designed (Le et al., 2015). Jin et al. developed a variable stiffness flexible actuator and decoupled control algorithm. The adjustment mechanism is small and compact and integrated on the output plate (Jin et al., 2018). Li et al. developed a modular and reconfigurable variable stiffness mechanism based on a four-bar linkage mechanism (Li et al., 2020). Zhu et al. also proposed a reconfigurable variable stiffness joint for lower limb rehabilitation exoskeleton knee drive based on this principle (Zhu et al., 2022). Wolf S et al. designed a humanoid robot shoulder joint based on the energy principle (Wolf and Feenders, 2016). Li et al. adjusted the output stiffness of the actuator by rotating multiple leaf springs synchronously (Xiong et al., 2017). Based on this, they successively developed a rotary-type variable-stiffness actuator and a series-type variable-stiffness actuator for the end gripper (Li et al., 2018). To design a simple and efficient flexible manipulator, Tyler Morrison et al. proposed a novel design concept for manipulator links with variable stiffness (Morrison et al., 2019).Liu et al. designed the Mechanical-Rotary Impedance Actuator. One end of the leaf spring is fixed on the output shaft frame and the other end is free. The stiffness of the actuator is adjusted by a set of screw-slider mechanisms (Liu et al., 2018; Misgeld et al., 2019). Wang et al. used the Archimedes helical disk to realize the adjustment of the action point of the input shaft on the leaf spring, and to adjust the output stiffness of the actuator (Wang et al., 2018). Braun et al. have successively developed a variable stiffness actuator that adjusts the effective cantilever length of the leaf spring by a single motor, and ingeniously combines it with the cam pulling/compressing spring mechanism (Braun et al., 2019). Mekaouche et al. combined a memory alloy spring with a flexible beam to design a variable stiffness compliant mechanism (Mekaouche et al., 2018). Carloni et al. designed a flexible variable stiffness structure by using the electro-active layer material as a leaf spring (Carloni et al., 2018). The switch controls the on-off of the electro-active layer spring to realize the switching of the flexible and rigid state of the spring. In addition, Amirhossein et al. proposed an abstract scheme for adjusting the stiffness by changing the magnetic field strength by using the magnetic field as an elastic element (Memar and Esfahani, 2017).

According to the existing research, it is found that the active compliance control generally has problems such as complex control algorithm and difficult realization. Therefore, we are inspired by the existing passive compliance control scheme. Based on the lever principle, the research content of this topic mainly introduces passive flexible variable stiffness vibration reduction structure for the key link of vibration coupling transmission of grinding robot, as shown in Figure 1. The stiffness and energy storage model of the mechanism is established, and the strength check and dynamic characteristics verification of the mechanism are completed, which provides technical and theoretical reference for the upgrading of traditional industries in my country and the popularization and application of robots.

2. Design and analysis of flexible variable stiffness damping mechanism

2.1. Design principles and technical indicators

The vibration damping mechanism with variable stiffness should not only have a simple and compact physical structure, a wide range of stiffness variation, but also better dynamic performance. Considering that the grinding motor spindle is a rotating machine, a rotary flexible variable stiffness damping mechanism is proposed to be designed, and the design principles are established as follows: (1) Good stiffness characteristics; (2) good environmental coordination; and (3) the control system is simple and efficient.

Figure 2 shows the design principle of the flexible variable stiffness structure, which mainly changes the effective length of the cantilever beam by changing the position of the mobile base. Under the same force F, the greater the effective length of the cantilever beam, the greater the deformation of the output end (point O), and the smaller the output stiffness. Conversely, the greater the output stiffness.

The flexible variable stiffness joint of grinding electric spindle is mainly used in the field of human-computer interaction safety. It can adjust its stiffness according to changes in environmental conditions, and has the ability to change the stiffness of the task. At the same time, it should also adapt to the needs of the installation environment. According to the design principles and practical applications, the main design technical indicators are proposed as shown in Table 1.

Table 1.

Technical indicators of flexible variable stiffness structure.

Weight/kg	Diameter/mm	Length/mm	Stiffness/N•m/rad	Energy storage/J	Accepted load/N•m
≤1	≤150	≤100	150∼1000	0∼10	≤10

2.2. 3D diagram of the mechanism

The three-dimensional diagram of the flexible variable stiffness damping mechanism is shown in Figure 3. The input end of the flexible variable stiffness mechanism is connected to the output end of the permanent magnet electric spindle, and the output shaft of the flexible variable stiffness damping mechanism is connected to the grinding wheel. There are four elastic rods in this mechanism. The stepping motor is the driving source, which drives the smaller gear to rotate and then drives the large gear to rotate. The slider is driven by the connecting rod to move on the elastic rod and the guide rail, so as to change the effective bending length of the elastic rod and achieve the effect of changing the rigidity of the whole mechanism. The guide rail is fixedly connected with the shell.

2.3. Dynamic stiffness calculation

To simplify the model, a group of elastic rod structures in the variable stiffness structure are extracted and calculated, and the schematic diagram is shown in Figure 4, where w is the deflection. θ is the rotation angle. L₀ is the total length of the elastic round rod. L_x is the effective bending length of the elastic round rod, and φ is the relative rotation angle of the structure.

The deflection w and the rotation angle θ of the cantilever elastic rod are expressed as follows,

w = \frac{F {L_{x}}^{3}}{3 E I}

(1)

\tan θ = \frac{F {L_{x}}^{2}}{2 E I}

(2)

where F is the concentrated load on the free end of the cantilever elastic rod. L is the effective bending length of the cantilever elastic rod. θ is the bending angle of the cantilever elastic rod. E is the elastic modulus of the cantilever elastic rod. I is the section moment of inertia of the cantilever elastic rod.

From the geometric relationship in Figure 4, we can get

\tan φ = \frac{ω}{L_{x}}

(3)

The moment of inertia of the section is

I = \frac{π d^{4}}{64}

(4)

The total torque is

T = N L_{x} F \cos θ

(5)

Equation (6) can be obtained from equations (1), (3), (4), and (5).

T = \frac{3 π N E d^{4}}{64 L_{x}} \tan φ

(6)

The total torque T is derived from the relative rotation angle φ of the structure

K = \frac{d T}{d φ} = \frac{3 π N E d^{4}}{64 L_{x} \cos^{2} φ}

(7)

The elastic round rod length scaling factor is α = Lx/L₀, that is

L_{x} = α L_{0}

(8)

Substituting equation (8) into equation (7), the dynamic stiffness calculation expression of the flexible variable stiffness structure can be obtained as

K = \frac{3 π N E d^{4}}{64 α L_{0} \cos^{2} φ}

(9)

when the number of elastic round rods is N = 4. The elastic modulus of the elastic round rod is E = 128 Gpa. The diameter of the elastic round rod is d = 3 mm, and the total length of the elastic round rod is L₀ = 50 mm. When α = 0.10, 0.25, 0.50, 0.75, and 1, respectively, the dynamic stiffness change of the structure is obtained as shown in Figure 5.

Figure 5 shows that the stiffness of the mechanism is minimal when the relative angle does not change. And as the elastic round rod length scaling factor is larger, its stiffness is smaller. The variation range of the theoretical dynamic stiffness value of the variable stiffness structure is (120∼∞) N•m/rad.

When the structural parameters are N = 4, E = 1.28×10¹¹ Pa, d = 0.003 m, L = 0∼0.05 m, and φ = −0.2∼0.2 rad, the stiffness change of the mechanism is shown in Figure 6. As can be seen from Figure 6, the output results are similar to Figure 5. The results show that the designed mechanism can meet the design index.

2.4. Dynamic energy storage analysis of mechanism

The potential energy of the flexible variable stiffness structure is the sum of the deformation energies of all elastic rods, which can be expressed as follows

V = \int_{0}^{φ} T d φ

(10)

Substituting equation (7) into equation (10), the potential energy is expressed as

V = - \frac{3 π N E d^{4}}{64 L} \ln (\cos φ)

(11)

Therefore, the relationship among the dynamic energy storage and stiffness and deflection angle φ can be obtained

V = - K \cdot \cos^{2} φ \cdot \ln (\cos φ)

(12)

when the structural parameters of the mechanism are N = 4, E = 1.28×10¹¹ Pa, d = 0.003 m, L = 0∼0.05 m, and φ = −0.2∼0.2 rad. The energy storage of the system is shown in Figure 7.

It can be seen from Figure 7 that when the relative rotation angle is 0, the energy storage of the system is the smallest. The greater the stiffness of the mechanism, the greater the energy storage capacity of the system. The system can realize the energy storage range of 0∼10 J, which can meet the requirements.

3. Finite element analysis of flexible variable stiffness mechanism

The selection of elastic rod elements is the key factor to realize the basic characteristics of flexible variable stiffness structures. The material properties and structural dimensions of the elastic element have an impact on the maximum torque, stiffness range, rotation angle range, and output torque of the mechanism. The following will describe the material selection and external dimensions of the elastic element. The parameters such as the shape and size of the elastic rod are determined by the finite element simulation of the stress and the overall deformation.

Beryllium bronze has the advantages of high elasticity, high hardness, high strength, and high wear resistance, which is widely used in switches, relays, micro-motor brushes, and various elastic components. Compared with common spring steels such as T9A, 65 Mn, and 60Si2MnA, the beryllium bronze has a higher elastic limit, smaller elastic hysteresis, and higher reliability and life. Therefore, beryllium bronze (C1720) is selected as the material of the elastic rod. Physical parameters of beryllium copper material are shown in Table 2.

Table 2.

Physical parameters of beryllium bronze material.

Densityρ/(g·cm⁻³)	Poisson’s ratio	Hardness HRC	Young’s modulus E/Gpa	Yield strength σ_s/Mpa	Tensile strength σ_b/Mpa
8.3	0.3	38∼44	128	1035	1105

To carry out the finite element analysis of the flexible variable stiffness structure, the finite element analysis model was established based on Ansys analysis software, as shown in Figure 8. The external constraints are configured. The output rotation axis is added with rotation pair constraint (A in Figure 8). The coupling is added with a fixed pair of restraints (B in Figure 8). A load torque of 10 N•m is added at the end of output shaft (C in Figure 8).

The main structural dimensions of the structure are as follows. The diameter of the structural shell is 300 mm and the radial width is 45 mm. The length of the beryllium copper rod is 65 mm, of which the effective deformation length is 50 mm. The section diameter is 2 mm or 3 mm, respectively. According to the data in Table 1, material properties of elastic round rod and other parts are set. The contact mode between the elastic rod and the slider was set as Frictionless contact. The elastic rod sets up a more subdivided grid, and the final model is divided into 38,658 nodes and 18,519 cells. The rest of the grid is divided by default.

3.1. Strength check of elastic round rod

3.1.1. Finite element simulation 1

Four elastic rods are used. The diameter of the elastic rod is set to 2 mm, and loads of 10 N•m, 5 N•m, 4 N•m, and 3 N•m are added, respectively. The simulation results are shown in Figure 9.

From the simulation results, the position where the maximum stress occurs in the mechanism is the fixed position of the elastic rod on the output shaft. Under the loads of 10 N•m, 5 N•m, 4 N•m, and 3 N•m, the maximum stress is 2544 Mpa, 1272 Mpa, 1017 Mpa, and 763 Mpa, respectively. It can be seen from the strength conditions in Table 2 that when four elastic rods with a diameter of 2 mm are used, the maximum load that can be supported is about 4 N•m. The result is obviously not able to meet the requirements, so we increase the diameter of the rod.

3.1.2. Finite element simulation 2

Four elastic rods are used. The diameter of the elastic rods is set to 3 mm, and a load of 10 N•m is added (Figure 10).

Under the load of 10 N•m, the maximum stress is 713.98 Mpa. It can be seen from the strength conditions in Table 2 that when four elastic rods with a diameter of 3 mm are used, the mechanical performance requirements are met. Considering the load and impact conditions applied to the grinding robot, an elastic rod with a diameter of 3 mm was used.

3.2. Stiffness calculation of flexible variable stiffness mechanism

In the finite element deformation simulation and stiffness calculation, the workbench software is used. A certain amount of load is applied to the output shaft, and the deformation of the shell of the variable stiffness mechanism is obtained under the static condition of one end being fixed. Using the geometric relationship, the deflection angle of the mechanism is calculated, and the stiffness value of the mechanism is obtained.

According to the geometric relationship, the relationship of each quantity is obtained.

\sin \frac{φ}{2} = \frac{d}{2 r}

(13)

where r represents the shell radius of the mechanism. d represents the deformation of the node, and φ represents the deflection angle of the node. The relationship among the overall stiffness K of the mechanism, the load T, and the deflection angle φ is

K = \frac{T}{φ} = \frac{T}{2 \arcsin \frac{d}{2 r}}

(14)

When the diameter of the elastic rod is 3 mm and the number is 4, the external constraints are set as shown in Figure 4, and the added load is 10 N•m. By changing the position of the slider, the effective deformation lengths of the elastic rods are 50 mm, 40 mm, 30 mm, and 20 mm, respectively, and the deformation of the shell in four cases is obtained. The simulation results are shown in Figure 11.

According to the simulation results of shell deformation with simulated external load of 10 N•m and radius of 75 mm, combined with equation (14), the overall stiffness value of the mechanism can be calculated, and the obtained data and results are shown in Table 3.

Table 3.

Overall stiffness values of the mechanism under different effective deformation lengths of elastic rods.

Effective deformation length (mm)	Shell node deformation range (mm)	Shell node deformation mean (mm)	Deflection angle (degree)	Overall stiffness (N•m/degree)
20	0.8977∼0.90457	0.901	0.688	14.535
30	1.2672∼1.2726	1.270	0.970	10.309
40	1.8489∼1.8528	1.851	1.414	7.072
50	2.5421∼2.5449	2.544	1.944	5.144

From the obtained results, it can be found that it is basically consistent with the results of theoretical analysis, indicating that the designed structure meets the requirements.

4. Dynamic analysis of flexible variable stiffness mechanism

In order to verify the vibration reduction performance of the flexible variable stiffness structure, a three-dimensional model of the flexible variable stiffness vibration reduction mechanism was established by Solidworks software and imported into Adams software. Periodically changing torque SFORCE_1 = 10 sin (1000 t) is applied at the end of the mechanism. The loading model diagram is shown in Figure 12.

The load loading curve is shown in Figure 12 as a cycle change. The angle change between the measurement input and output is shown in Figure 13.

It can be seen from Figure 13 that the fluctuation amplitude of the red curve at the input end of the flexible variable stiffness mechanism is significantly smaller than the amplitude of the blue curve at the output end of the flexible variable stiffness mechanism, indicating that the designed flexible variable stiffness damping mechanism has a good vibration reduction effect.

5. Performance test experiment

In order to verify the vibration reduction performance of the designed flexible variable stiffness vibration damping mechanism, an experimental bench as shown in Figure 14 was built, which mainly includes permanent magnet electric spindle, coupling, rotational speed torque sensor, flexible variable stiffness vibration damping mechanism, dynamometer, and the controller composition. The permanent magnet electric spindle provides the driving force. The rotational speed torque sensor is used to measure the torque change of the flexible variable stiffness damping mechanism. The damping and buffering effect of flexible variable stiffness mechanism is analyzed by comparing the measured results of front and rear torque sensors. The dynamometer simulates the sudden change of load. The experimental process is as follows. First, the controller is used to control the start of the permanent magnet electric spindle. For safety experiments, when the speed of the permanent magnet electric spindle reaches 800 r/min, a sudden load of 3 N•m and 4 N•m is added to the end. The torque change before and after the flexible variable stiffness mechanism is read through the rotational speed torque sensor. The experimental results are shown in Figure 15.

Figure 15 shows the torque diagrams when the rotational speed is 800 r/min and the load is 3 N•m and 4 N•m, respectively. Sensor 1 is installed before the flexible variable stiffness damping mechanism, and sensor 2 is installed after the flexible variable stiffness damping mechanism. It can be seen from Figure 15 that the curve measured by sensor 2 is smoother than that measured by sensor 1. The results show that the output curve is smoother and the grinding process is more stable after the vibration reduction mechanism is added, which is beneficial to ensure the processing quality. In order to compare the fluctuations of the two curves, the variance from point A to point B in the graph is calculated according to the equation (17)

σ^{2} = \frac{\sum {(X - u)}^{2}}{N}

(17)

where σ² is the population variance. X is the variable. μ is the population mean, and N is the population number of cases.

The variance calculated based on the data shown in Figure 15(a) measured by sensor 1 and equation (17) is 0.0093, and the variance measured by sensor 2 is 0.0011. The variance measured in Figure 15(b) is 0.0094. The variance measured by sensor 2 is 0.0043. It can be seen that the fluctuation of the output torque is smaller and more stable after the damping mechanism is added. This also shows that the designed mechanism has better vibration damping performance. As shown in Figure 15(a), the impact at the maximum peak value can be reduced by 16.8%. As shown in Figure 15(b), it can be reduced by 17.9%. With the subsequent structural improvement and optimization, the effect may be better. At present, the impact vibration can be reduced by 10%∼25% through the damping mechanism combined with the control algorithm (Li et al., 2021; Liu et al., 2020). The simple flexible variable stiffness mechanism can effectively reduce about 17% of the impact in this paper, indicating that the structure has a good impact resistance.

6. Conclusions

According to the design requirements and design indicators, a flexible variable stiffness damping mechanism is designed, and the stiffness and energy storage models of the flexible variable stiffness damping mechanism are established and calculated. Based on the finite element analysis element Ansys and the dynamic analysis software Adams, the rated strength and dynamic performance of the flexible variable stiffness damping mechanism were checked, respectively. Finally, the damping effect of the flexible variable stiffness damping mechanism is verified by experiments. The research results show that the designed flexible variable stiffness structure has better dynamic characteristics, which can effectively reduce the impact of end impact, and provide a new idea for the smooth grinding of the grinding electric spindle.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the National Natural Science Foundation of China (Grant No. 52375250 and No. 52005232), Natural Science Foundation of Jiangsu Province, China (Grant No.BK20201024), Basic Research Program of Xuzhou City (Grant No.KC22023). National innovation and Entrepreneurship training program for college student (Grant No.202210320014Z).

ORCID iDs

Lianchao Sheng

Guo Ye

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