Pressing speed stability control of a special ceramic roller bearing press based on fuzzy adaptive PID

Abstract

A special ceramic roller bearing press (SCRBP) is developed to press two bearings efficiently and precisely at the same time. A speed control mathematical model of the bearing press is built to obtain stability bearing pressing speed. The fuzzy adaptive PID controller of the bearing pressing speed of SCRBP is designed. The simulation model is also built. Fuzzy adaptive PID control is compared with conventional PID control. By simulation analysis, the simulation results show that adjusting time of fuzzy adaptive PID control is short, and its overshoot is very small, almost coincides with the set pressing speed. Moreover, fuzzy adaptive PID is suitable for the pressing speed control of the bearing pressing speed system with step interference signal. The pressing stability speed is obtained by fuzzy adaptive PID control.

Keywords

Special ceramic roller bearing press mathematical model fuzzy adaptive PID pressing speed

1. Introduction

The ceramic material roller has many advantages, such as high wear resistance and high life span. It is widely used in belt conveyor. Ceramic roller is mainly composed of roll body, mandrel shaft, bearings, sealing rings, etc., as shown in Fig. 1. However, due to high brittleness and high hardness of ceramic materials, the processing of ceramic roller is difficult. The ceramic roller assembly is mainly completed by manual or semi-automatic means, the processing quality and efficiency are low [1, 2, 3, 4]. There are many problems need to be improved. After mandrel shaft is installed in roller body, both ends of the mandrel shaft need an interference pressed bearing to roller body to support the mandrel shaft and meet the requirement of concentricity [5, 6, 7]. The bearing press can be done by assembly equipment, temperature difference assembly, or pressuring-loading with machines, etc. The assembly equipment is to knock the bearing in place, its work efficiency is low. The temperature difference assembly principle is to use the physical properties of the material shaft’s thermal expansion which causes the internal stress and deformation of the materials. This method would affect the assembly accuracy of the parts. Pressure-loading is a popular method of present. Compared with the temperature difference assembly, it is less affected by the environment. Compared with assembly equipment, it has less damage to ceramic body due to the specialized equipment and clamps [8, 9, 10, 11].

Figure 1.

Ceramic roller.

Using traditional bearing press machine, only one bearing is pressed to one side of roller body at a time. After the bearing is preloaded, another bearing is pressed on the other side of roller body again. This method needs a long time, also needs to guarantee the precision. Therefore, two pressing machines are often used to press a pair of bearings at one time. They are placed symmetrically on both sides of the roller. However, this method decreases the repeated utilization rate of the equipment and the synchronization accuracy. Moreover, the ceramic roller body is brittle, and the ability to resist impact during bearing pressing is weak, resulting in lower pressing efficiency and pressing accuracy [12]. Therefore, a high-efficiency special ceramic roller bearing press (SCRBP) is designed to press two bearings on both sides against the ceramic roller body at the same time, which is shown in Fig. 2.

SCRBP mainly consists of a bed frame, two spindle systems, and two clamping devices. The spindle system can finish spindle displacement setting, displacement adjustment for push rod of the spindle system, and the ceramic roller mandrel positioning. The clamp device mainly comprises two parts, a roller body supporting mechanism and a mandrel lifting mechanism. After the ceramic roller is supported by the supporting mechanism, the lifting mechanism lifts the mandrel to the center of the roller body, and the pressure rods at both ends of spindle system move to fix the mandrel, and then the lifting mechanism is down to the lowest point to prevent interference when the bearings are pressing. Finally, the spindle moves to press the bearings on the roller body [13]. Pressing bearings process of SCRBP is seen in Fig. 3.

Figure 2.

Special ceramic roller bearing press.

Figure 3.

Pressing process of the special ceramic roller bearing press.

The press speed and its stability of SCRBP directly affect the pressing quality. When the push rod begins to contact with the ceramic roller body, the pressing pressure suddenly increases and changes because of the change of pressing force. As the pressure in the system changes, the leakage of the hydraulic components changes. The leak flow of the speed control system causes the pressing speed to fluctuate. In addition, the increase in the temperature of the hydraulic oil causes the viscosity of the oil to decrease, which also exacerbates system leakage and reduces the pressing speed. The changing pressing speed may increase the friction coefficient and increase the stress of the contact surface, and affect the roller processing accuracy. Therefore, the speed control transfer function of SCRBP is established, and the speed controller based on fuzzy adaptive PID control is designed to obtain stability pressing speed.

Figure 4.

Schematic diagram of bearing press speed control system.

2. Mathematical model establishment of control system

Speed control system of SCRBP mainly includes an oil hydraulic pump, an oil hydraulic cylinder, an electro-hydraulic proportional control valve, a speed sensor, a data acquisition card. The system control schematic is shown in Fig. 4. The pipeline of the whole hydraulic system is short and thick, so the friction loss inside the pipeline, the influence of the flow quality of the pipeline dynamics can be neglected. The speed of the oil hydraulic cylinder is not high, so internal leakage and external leakage can be regarded as laminar flow.

2.1 Mathematical model of oil hydraulic cylinder

The area ratio of outlet oil chamber to inlet oil chamber for the oil hydraulic cylinder is [14],

$\displaystyle m=\frac{A_{2}}{A_{1}}$ (1)

Where, $A_{1}$ is the rod-less cavity area (m ${}^{2}$ ), $A_{2}$ is the rod cavity area (m ${}^{2}$ ).

The force balance equation of the oil hydraulic cylinder is,

$\displaystyle p_{1}A_{1}-p_{2}A_{2}=F_{L}$ (2)

With Eqs (2) and (3), the load pressure is deduced,

$\displaystyle p_{L}=\frac{F_{L}}{A_{1}}=p_{1}-p_{2}\frac{A_{2}}{A_{1}}=p_{1}-p% _{2}m$ (3)

The pressure-flow equation of valve port 1 (see in Fig. 4) is,

$\displaystyle Q_{1}=C_{d}wx_{v}\sqrt{\frac{2}{\rho}({p_{s}-p_{1}})}$ (4)

Where, $C_{d}$ is the proportional valve flow coefficient, $w$ is the proportional valve opening area gradient (m), $\rho$ is the hydraulic oil density (Kg/m ${}^{3}$ ).

The pressure-flow equation of valve port 3 (see in Fig. 4) is

$\displaystyle Q_{2}=C_{d}wx_{v}\sqrt{\frac{2}{\rho}p_{2}}$ (5)

Then, Eq. (6) is obtained,

$\displaystyle\frac{Q_{2}}{Q_{1}}=\frac{A_{2}v}{A_{1}v}=\sqrt{\frac{p_{2}}{p_{s% }-p_{1}}}$ (6)

Where, $Q_{1}=A_{1}v$ , $Q_{2}=A_{2}v$ , $v$ is the piston rod speed (m/s).

Then,

$\displaystyle p_{s}=\frac{1}{m^{2}}p_{2}+p_{1}$ (7)

With Eqs (3) and (7), it can be obtained that,

$\displaystyle p_{1}=\frac{p_{L}+m^{3}p_{s}}{1+m^{3}}$ (8) $\displaystyle p_{2}=\frac{m^{2}({p_{s}-p_{L}})}{1+m^{3}}$ (9)

Then, the flow $Q_{1}$ of valve port 1, the flow $Q_{2}$ of valve port 3 are calculated,

$\displaystyle Q_{1}=C_{d}wx_{v}\sqrt{\frac{2}{\rho}\frac{p_{s}-p_{L}}{1+m^{3}}}$ (10) $\displaystyle Q_{2}=C_{d}wx_{v}\sqrt{\frac{2}{\rho}\frac{m^{2}({p_{s}-p_{L}})}% {1+m^{3}}}$ (11)

According to the load flow definition, the load flow is,

$\displaystyle Q_{L}=C_{d}wx_{v}\sqrt{\frac{2}{\rho}\frac{p_{s}-p_{L}}{1+m^{3}}}$ (12)

The Flow gain $K_{q0}$ and flow pressure coefficient are calculated,

$\displaystyle K_{q0}=\frac{\partial Q_{L}}{\partial x_{v}}=C_{d}w\sqrt{\frac{2% }{\rho}\frac{p_{s}-p_{L}}{1+m^{3}}}$ (13) $\displaystyle K_{co}=-\frac{\partial Q_{L}}{\partial p_{L}}=C_{d}wx_{v}\frac{% \sqrt{\frac{2}{\rho}\frac{p_{s}-p_{L}}{1+m^{3}}}}{2({p_{s}-p_{L}})}$ (14)

According to the linearized flow formula of the proportional valve, the following equation is given,

$\displaystyle Q_{L}=K_{qo}x_{v}-K_{co}p_{L}$ (15)

Rod-less cavity flow continuity equation is,

$\displaystyle Q_{1}=\frac{V_{1}}{\beta_{e}}\frac{{dp}_{1}}{dt}+\frac{dV_{1}}{% dt}+c_{ec}p_{1}+c_{ic}({p_{1}-p_{2}})=\frac{({V_{01}+A_{1}y})}{\beta_{e}}\frac% {{dp}_{1}}{dt}+A_{1}\frac{dy}{dt}+({c_{ec}+c_{ic}})p_{1}-c_{ic}p_{2}$ (16)

Rod cavity flow continuity equation is,

$\displaystyle Q_{2}=-\frac{V_{2}}{\beta_{e}}\frac{{dp}_{2}}{dt}+\frac{{dV}_{2}% }{dt}-c_{ec}p_{2}+c_{ic}({p_{1}-p_{2}})=-\frac{({V_{02}-A_{2}y})}{\beta_{e}}% \frac{{dp}_{2}}{dt}+A_{2}\frac{dy}{dt}-({c_{ec}+c_{ic}})p_{2}+c_{ic}p_{1}$ (17)

Where, $V_{1}$ is the rod-less cylinder volume (m ${}^{3}$ ), $V_{2}$ is the rod cavity volume (m ${}^{3}$ ), $c_{ec}$ is the external leakage coefficient of the oil hydraulic cylinder (m ${}^{5}$ N/s), $c_{ic}$ is the leakage coefficient in the oil hydraulic cylinder (m ${}^{5}$ N/s), $\beta{}_{e}$ is the equivalent elastic modulus of oil (N/m ${}^{2}$ ).

Given the initial volume of rod-less chamber and rod chamber for the oil hydraulic cylinder are equal, i.e., $V_{01}=V_{02}=\frac{1}{2}V_{t}=V_{0}$ , then, $A_{1}y\ll{V_{t}}\mathord{\left/{\vphantom{{V_{t}}2}}\right.\kern-1.2pt}2$ and $A_{2}y\ll{V_{t}}\mathord{\left/{\vphantom{{V_{t}}2}}\right.\kern-1.2pt}2$ . Simplified Eqs (16) and (17), the following Eq. (18) is available.

$\displaystyle\frac{Q_{1}+mQ_{2}}{1+m^{2}}=A_{1}\frac{dy}{dt}+\frac{V_{0}}{({1+% m^{2}})\beta_{e}}\frac{{dp}_{L}}{dt}+\frac{c_{ec}+c_{ic}}{1+m^{2}}p_{L}-\frac{% p_{2}-mp_{1}}{1+m^{2}}c_{ic}$ (18)

Considering the load flow definition, Eq. (18) is changed into the following style,

$\displaystyle Q_{L}=A_{1}\frac{dy}{dt}+\frac{V_{t}}{2({1+m^{2}})\beta_{e}}% \frac{dp_{L}}{dt}+c_{tc}p_{L}+c_{tco}p_{s}$ (19)

Where,

$\displaystyle c_{\textit{tco}}=\frac{m^{2}({m^{2}-1})}{({1+m^{2}})({1+m^{3}})}% \cdot c_{ic},c_{tc}=\frac{1+m}{1+m^{3}}c_{ic}+\frac{c_{ec}}{1+m^{2}}.$

The hydraulic cylinder stress balance equation is,

$\displaystyle A_{1}p_{1}-A_{2}p_{2}=A_{1}p_{L}=M\frac{d^{2}y}{dt^{2}}+B_{c}% \frac{dy}{dt}+ky+F_{L}$ (20)

Laplace transform can be obtained for Eqs (15), (19), and (20).

$\displaystyle Q_{L}=K_{q0}x_{v}-K_{c0}p_{L}$ (21) $\displaystyle Q_{L}=A_{1}sy+\frac{V_{t}}{2({1+m^{2}})\beta_{e}}sp_{L}+c_{tc}p_% {L}+c_{tc0}p_{s}$ (22) $\displaystyle A_{1}p_{L}=Ms^{2}y+B_{c}sy+ky+F_{L}$ (23)

The valve displacement $x_{v}$ is defined as the input variable, the hydraulic cylinder speed $\dot{Y}$ is defined as the output variable, the transfer function is,

$\displaystyle\frac{\dot{Y}}{x_{v}}=\frac{\frac{1}{A_{1}}k_{q0}s}{\frac{V_{t}M}% {2({1+m^{2}})\beta_{e}{A_{1}}^{2}}s^{3}+\left({\frac{V_{t}B_{c}}{2({1+m^{2}})% \beta_{e}{A_{1}}^{2}}+\frac{k_{ce0}M}{{A_{1}}^{2}}}\right)s^{2}+\left({\frac{V% _{t}k}{2({1+m})\beta_{e}{A_{1}}^{2}}+\frac{B_{c}k_{ce0}}{{A_{1}}^{2}}+1}\right% )s+\frac{k_{ce0}k}{{A_{1}}^{2}}}$ (24)

Where, $k_{ce0}=k_{c0}+c_{tc}$ .

For the hydraulic speed-control-system, the elastic load is very small, therefore, $\frac{k_{ce0}B_{c}}{A_{1}^{2}}\ll 1$ can be ignored. Then, the Eq. (24) can be simplified as,

$\displaystyle\frac{\dot{Y}}{x_{v}}=\frac{\frac{k_{q0}}{A_{1}}}{\frac{s^{2}}{{w% _{h0}}^{2}}+\frac{2\varsigma_{h0}}{w_{h0}}s+1}$ (25)

Where,

$\displaystyle w_{h0}=A_{1}\sqrt{\frac{2({1+m^{2}})\beta_{e}}{V_{t}M}},% \varsigma_{h0}=\frac{1}{2}\left({\frac{k_{ce0}}{A_{1}}\sqrt{\frac{2({1+m^{2}})% \beta_{e}M}{V_{t}}}+\frac{\beta_{c}}{2A_{1}}\sqrt{\frac{2V_{t}}{({1+m^{2}})% \beta_{e}M}}}\right).$

2.2 Proportional amplifier

The proportional amplifier is to enlarge the error between the detected voltage signal and the set voltage signal and converts it into a current signal that controls the electromagnet [14]. Usually, the proportional amplifier is considered as a proportional link. The gain is represented as $K_{a}$ . The mathematical model of the proportional amplifier is,

$\displaystyle G_{a}(s)=\frac{I(s)}{U(s)}=K_{a}$ (26)

2.3 Speed sensor

The function of the speed sensor is to detect the hydraulic cylinder piston rod speed, converts it into a voltage signal, and sends it to the comparator to achieve closed-loop control of the bearing pressing speed. The gain of the speed sensor is expressed as $K_{fv}$ . The speed sensor transfer function is,

$\displaystyle\frac{U(s)}{\dot{Y}(s)}=K_{fv}$ (27)

2.4 Electro-hydraulic proportional directional valve

The proportional valve function is to control the oil hydraulic cylinder speed by controlling the opening of the valve port, so that the bearing pressing speed is in dynamic balance. The proportional valve transfer function is [15],

$\displaystyle G_{q}(s)=\frac{x_{v}(s)}{I(s)}=\frac{K_{q0}}{\frac{s^{2}}{{w_{q}% }^{2}}+\frac{2\varsigma_{q}s}{w_{q}}+1}$ (28)

Where, $K_{q0}$ is the proportional directional valve flow gain, $w_{q}$ is the phase bandwidth of the directional valve, $\varsigma_{q}$ is the directional valve damping.

2.5 Bearing pressing speed system transfer function

From Eqs (25)–(28), the transfer function block diagram of the bearing pressing speed-control-system is obtained, which is seen in Fig. 5. The closed-loop transfer Eq. (29) of the bearing pressing speed control system is obtained.

$\displaystyle G(s)=\frac{{K_{a}{K_{q0}}^{2}K_{fv}}\mathord{\left/{\vphantom{{K% _{a}{K_{q0}}^{2}K_{fv}}{A_{1}}}}\right.\kern-1.2pt}{A_{1}}}{\left({\frac{1}{{w% _{q}}^{2}}+\frac{4\varsigma_{q}\varsigma_{h0}}{w_{q}w_{h0}}+\frac{1}{{w_{h0}}^% {2}}}\right)s^{2}+\left({\frac{2\varsigma_{q}}{w_{q}}+\frac{2\varsigma_{h0}}{w% _{h0}}}\right)s+{K_{a}{K_{q0}}^{2}K_{fv}}\mathord{\left/{\vphantom{{K_{a}{K_{q% 0}}^{2}K_{fv}}{A_{1}}}}\right.\kern-1.2pt}{A_{1}}}$ (29)

Figure 5.

Block diagram of the system transfer function.

3. Bearing pressing speed controller design

Bearing pressing speed control system is a nonlinearity and time-varying system. The pressing speed varies when the external load is changed. To obtain steady pressing speed, the parameters need to be adjusted frequently [16]. The conventional PID control algorithm is often cumbersome to set its parameters under such complicated conditions, and it does not achieve the desired effect on the bearing pressing speed control process. With the advancement of intelligent control technology, various control methods have emerged, such as fuzzy control and artificial neural network. Fuzzy adaptive control is based on fuzzy mathematics and does not require precise mathematical models, it is better for control of nonlinear systems [17, 18]. Combining the advantages of conventional PID control and fuzzy control, a bearing pressing speed controller is designed by fuzzy adaptive PID method to ensure the speed stability in the bearing pressing.

3.1 Speed controller structure design

The speed controller adopts two-input and three-output fuzzy PID control. The bearing press machine detects the current pressing speed value continuously, and the control system compares the current value with the set value to obtain the speed deviation $e$ and the speed deviation change rate $e c$ which are used as the fuzzy control system. The target parameters are corrected by the fuzzy rules built in the control system, and the control variables $\Delta K_{i}$ , $\Delta K_{d}$ , $\Delta K_{p}$ are output. The three control variables are integrated with the three parameters of conventional PID to form a new control quantity parameter for the pressing speed control. The speed controller structure is obtained and shown in Fig. 6.

Figure 6.

Speed controller structure.

3.2 Fuzzy processing of input/output variables

3.2.1 Division of fuzzy space

The fuzzy controller cannot recognize precise input/output variables, so input and output variables need to be fuzzy variables. Generally speaking, the finer the division of input and output variables, the higher the control accuracy. However, it makes the algorithm more complex and impacts the response of the control speed [19]. According to the actual situation and the existing experience, the input and output variables are divided into seven levels, namely {negative big (NB), negative middle (NM), negative small (NS), zero (Z0), positive small (PS), positive middle (PM), positive big (PB)}, respectively to describe $e$ , $e c$ , $\Delta K_{i}$ , $\Delta K_{d}$ , $\Delta K_{p}$ .

3.2.2 Determination of fuzzy domain

The range of variables is the basic domain of variables. The basic domain of speed deviation $e$ is [ $-$ 2, 2], the basic domain of speed deviation change rate $e c$ is [ $-$ 2, 2], the basic domain of output variable $\Delta K_{p}$ is [0, 2], the basic domain of output variable $\Delta K_{i}$ is [0, 3], the basic domain of output variable $\Delta K_{d}$ is [0, 0.6]. According to the requirements of the pressing speed of the bearing press and the frequency of the system, the domain is divided into seven levels.

$\displaystyle e=\left[{{\begin{array}[]{*{20}c}{-2}&{-1.33}&{-0.67}&0&{0.67}&{% 1.33}&2\\ \end{array}}}\right]$ $\displaystyle ec=\left[{{\begin{array}[]{*{20}c}{-2}&{-1.33}&{-0.67}&0&{0.67}&% {1.33}&2\\ \end{array}}}\right]$ $\displaystyle\Delta K_{p}=\left[{{\begin{array}[]{*{20}c}0&{0.33}&{0.67}&1&{1.% 33}&{1.67}&2\\ \end{array}}}\right]$ $\displaystyle\Delta K_{i}=\left[{{\begin{array}[]{*{20}c}0&{0.5}&1&{1.5}&2&{2.% 5}&3\\ \end{array}}}\right]$ $\displaystyle\Delta K_{d}=\left[{{\begin{array}[]{*{20}c}0&{0.1}&{0.2}&{0.3}&{% 0.4}&{0.5}&{0.6}\\ \end{array}}}\right]$

3.2.3 Determination of membership function

Trapezoid membership function, triangle membership function and Gauss membership function are three commonly used membership functions [20]. Compared with trapezoid membership function and Gauss membership function, the mathematical expression of triangle membership function is relatively simple, and its shape is just related to the slope of the straight line, which occupies less memory space during operation. Therefore, triangle membership function is widely used in fuzzy PID control. The input and output variables are all controlled by trigonometric membership function. The membership function curve of input/output variables is shown in Figs 7–10.

Figure 7.

$e$ , $e c$ membership function curve.

Figure 8.

$\Delta K_{p}$ membership function curve.

Figure 9.

$\Delta K_{i}$ membership function curve.

3.3 Fuzzy control rules

The speed deviation is serious when the system load is changed suddenly, therefore, the control system response speed needs to be increased. Moreover, the larger $\Delta K_{p}$ , the smaller $\Delta K_{i}$ should be selected to avoid differential oversaturation. In this way, excessive speed deviation momentarily can be avoided making the system uncontrollable. Therefore, $\Delta K_{i}$ should be set to 0, to avoid integral saturation and eliminate overshoot.

Integration has the greatest impact on the final control result. The response speed of the system can be guaranteed by selecting appropriate integration. When the speed deviation is medium, the smaller $\Delta K_{p}$ and the appropriate $\Delta K_{i}$ should be selected.

When the speed deviation is small, the values of $\Delta K_{p}$ and $\Delta K_{i}$ should be increased to ensure better system stability, and at the same time to ensure that the system will not vibrate near the set value. If the speed deviation rate is small, $\Delta K_{p}$ need to select a larger value.

According to the above analysis, the fuzzy control rules of $\Delta K_{p}$ , $\Delta K_{i}\Delta K_{d}$ are listed in Tables 1–3.

Table 1
$\Delta K_{p}$ fuzzy control rules

[height=0.8cm,width=2.4cm]ece	NB	NM	NS	ZE	PS	PM	PB
NB	PB	PB	PM	PM	PS	ZE	ZE
NM	PB	PB	PM	PS	PS	ZE	NS
NS	PM	PM	PM	PS	ZE	NS	NS
ZE	PM	PM	PM	PS	ZE	NS	NS
PS	PS	PS	ZE	NS	NS	NM	NM
PM	PS	ZE	NS	NM	NM	NM	NM
PB	ZE	ZE	NM	NM	NM	NB	NB

Table 2

$\Delta K_{i}$ fuzzy control rules

[height=0.8cm,width=2.4cm]ece	NB	NM	NS	ZE	PS	PM	PB
NB	NB	NB	NM	NM	PS	ZE	ZE
NM	NB	NB	NM	NS	NS	ZE	ZE
NS	NB	NM	NS	NS	ZE	NS	NS
ZE	NM	NS	NS	ZE	PS	NS	NS
PS	NM	NS	ZE	PS	PS	PM	PM
PM	ZE	ZE	PS	PS	PM	PB	PB
PB	ZE	ZE	PS	PM	PM	PB	PB

Figure 10.

$\Delta K_{d}$ membership function curve.

Table 3

$\Delta K_{d}$ fuzzy control rules

[height=0.8cm,width=2.4cm]ece	NB	NM	NS	ZE	PS	PM	PB
NB	PS	NS	NB	NB	NB	NM	PS
NM	PS	NS	NB	NM	NM	NS	ZE
NS	ZE	NS	NM	NM	NS	NS	ZE
ZE	ZE	NS	NS	NS	NS	NS	ZE
PS	ZE	ZE	ZE	ZE	ZE	ZE	ZE
PM	PB	NS	PS	PS	PS	PS	PB
PB	PB	PB	PM	PM	PS	PS	PB

3.4 Fuzziness resolution

The solution to fuzziness is also called the anti-fuzziness process or the precision process. The commonly used accurate calculation methods mainly include the maximum membership function method, the gravity center method and the weighted average method.

3.4.1 Maximum membership function method

Maximum membership function method simply takes the largest membership degree element as the output value of the fuzzy set of all rule reasoning results. In the output domain, if there is more than one output value of the maximum membership function, the average value of all the outputs with the maximum membership is calculated, that is,

$\displaystyle v_{0}=\frac{1}{J}\sum_{j=1}^{J}{vj},vj={\max}_{v\in V}({\mu_{0}(% v)}),J=\left|{\{v\}}\right|$ (30)

3.4.2 Gravity center method

The gravity center method takes the gravity center of the area enclosed by the fuzzy membership function curve and abscissa as the output value of fuzzy reasoning, namely,

$\displaystyle v_{0}=\frac{\int_{V}{{v\mu}_{v}(v)dv}}{\int_{V}{\mu_{v}(v)dv}}$ (31)

In contrast to the maximum membership function, the gravity center method has a smoother output reasoning control.

3.4.3 Weighted average method

The output value using the weighted average method is obtained by the following formula,

$\displaystyle v_{0}=\frac{\sum_{i=1}^{m}{v_{i}}k_{i}}{\sum_{i=1}^{m}{k_{i}}}$ (32)

In the pressing control system, the selection of coefficient $k_{i}$ is based on the actual situation. Different coefficients means that the system has different response characteristics. When $k_{i}$ gets to $\mu_{v}({v_{i}})$ , it will be transformed into gravity center method.

3.5 System simulation and discussion

To verify fuzzy adaptive PID control of speed control of SCRBP, the simulation model of fuzzy adaptive PID controller is built. Fuzzy adaptive PID control simulation model is shown in Fig. 11. Fuzzy adaptive PID control results are also compared with conventional PID control.

3.5.1 Speed without load

The pressing speed rises rapidly to 5 mm/s after the press machine started. Therefore, the input signal of the control system is regard as a step signal. The step signal amplitude is set as 5. The initial value $k_{p1}$ of PID controller is 14, the initial value $k_{i1}$ of integral coefficient is 52.9, and the initial value $k_{d1}$ of differential coefficient is 0.29. Comparing fuzzy adaptive PID simulation results with conventional PID simulation results, the response curve of the bearing pressing speed control is seen in Fig. 12.

Figure 11.

Simulation model of bearing pressing speed control.

Table 4

Conventional PID and fuzzy adaptive PID control results

Methods	Rising time (s)	Adjustment time (s)	Overshoot (%)
Conventional PID	0.08	0.8	10
Fuzzy PID	0.2	0.5	0

Figure 12.

Response curve of bearing pressing speed control. 1. Fuzzy adaptive PID curve; 2. Conventional PID curve.

Figure 13.

Response curve of bearing pressing speed-control-system with sinusoidal interference signal. 1. Fuzzy adaptive PID curve; 2. Conventional PID curve.

From the curve value of Fig. 12, it is seen that fuzzy adaptive PID adjustment time is 0.5 s, while conventional PID adjustment time is 0.8 s. Fuzzy adaptive PID rising time is 0.2 s, and conventional PID rising time is 0.08 s. In terms of overshoot, the overshoot of fuzzy adaptive PID can be controlled to almost 0, while that of conventional PID is about 10%. The comparison results of fuzzy adaptive PID controller and conventional PID controller are listed in Table 4. Seen from the simulation results in Fig. 12 and Table 4, it is obvious that fuzzy adaptive PID controller shortens control time when the bearing pressing speed reaches the balance value, and the overshoot is very small.

3.5.2 Speed in disturbance

Friction and extrusion force are often occurred in bearing press process, which make the press speed fluctuate irregularity, thus, it is difficult to obtain the actual load data in pressing process. In this paper, sinusoidal interference signal and step interference signal are added to the pressing speed control system of SCRBP to simulate the control effect of fuzzy adaptive PID and conventional PID when the load changes slowly or suddenly.

1) Speed with sinusoidal interference signal

The response curve of bearing pressing speed control system with sinusoidal interference signal is shown in Fig. 13, the magnification of the response curve is shown in 14.

Seen from Figs 13 and 14, after the sinusoidal interference signal of 1 Hz is added, the bearing pressing speed fluctuates up and down near the set speed, and the fluctuation amplitude is small. The control effect of conventional PID on bearing press speed is great with bigger fluctuation amplitude. Therefore, fuzzy adaptive PID is more suitable for the control of pressing speed system with sinusoidal interference signal.

Figure 14.

Magnification of the response curve of the bearing pressing speed control system. 1. Fuzzy adaptive PID curve; 2. Conventional PID curve.

Figure 15.

Bearing pressing speed control system with step interference signal. 1. Fuzzy adaptive PID curve; 2. Conventional PID curve.

Figure 16.

Enlarged view of bearing pressing speed control system. 1. Fuzzy adaptive PID curve; 2. Conventional PID curve.

2) Speed with step interference signal

The speed curve of the bearing pressing speed control system with sinusoidal interference signal is shown in Figs 15 and 16.

It can be seen from the Figs 15 and 16, after the step interference signal is added, there is a sudden change in the pressing speed. Under the control of fuzzy adaptive PID or conventional PID, the bearing pressing speed can be stable near the set pressing speed. However, the speed change range and adjustment time are different. Fuzzy adaptive PID overshoot is smaller than conventional PID control, and the adjustment time of fuzzy adaptive PID is less than conventional PID control. Therefore, fuzzy adaptive PID controller is more suited to the bearing pressing speed control with the step interference signal.

4. Conclusions

A high-efficiency special ceramic roller bearing press machine (SCRBP) is developed to press two bearings at the same time. The bearing pressing speed has a direct influence on the bearing pressing accuracy. To obtain the rapid and stability bearing pressing speed, the speed control system mathematical model of the bearing press is built, and a fuzzy adaptive PID speed controller for bearing pressing is designed. The simulation controller model is built and simulated. It is compared with the controller of conventional PID. The simulation results show that fuzzy adaptive PID control shortens the adjusting time, and is better for speed control of SCRBP with step interference signal, which is beneficial to improve the pressing stability.

Footnotes

Acknowledgments

The authors would like to express heartfelt thanks to the technology research project of Hebei Provincial Department of Science and Technology (grant number 17211906D), the Nature Science Foundation of Hebei Province (grant number E2019402436), Key Laboratory of Intelligent Industrial Equipment Technology of Hebei Province (Hebei University of Engineering), Handan Key Laboratory of Intelligent Vehicles.

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Pressing speed stability control of a special ceramic roller bearing press based on fuzzy adaptive PID

Abstract

Keywords

1. Introduction

2.1 Mathematical model of oil hydraulic cylinder

3.1 Speed controller structure design

3.2.1 Division of fuzzy space

3.2.2 Determination of fuzzy domain

3.2.3 Determination of membership function

Table 1 Δ ⁢ K p fuzzy control rules

3.4.1 Maximum membership function method

3.5.1 Speed without load

1) Speed with sinusoidal interference signal

2) Speed with step interference signal

Footnotes

Acknowledgments

References

Table 1
$\Delta K_{p}$ fuzzy control rules

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