STICK-SLIP MOTION IN PNEUMATIC CYLINDERS DRIVEN BY METER-OUT CIRCUIT

STICK-SLIP MOTION IN PNEUMATIC CYLINDERS DRIVEN BY METER-OUT CIRCUIT Toshinori FUJITA*, Luis R. TOKASHIKI*, Toshiharu KAGAWA* * Tokyo Institute of Technology Precision and Intelligence Laboratory 4259, Nagatsuda, Midori-Ku, Yokohama, 226-8503, Japan (E-mail: fujita@pms.titech.ac.jp) ABSTRACT Pneumatic cylinders may not achieve steady motion in applications where low velocities are required because of the stick-slip effect. However, an effective means to assess the possibility of stick-slip occurrence in pneumatic cylinders has not been reported. Despite friction force is an important parameter on stick-slip motion, there are only few studies on friction force at low velocities and stick-slip motion. In this paper, stick-slip motion in pneumatic cylinders is studied. Friction force is measured and stick-slip motion is simulated. The occurrence conditions of stick-slip are studied analytically using non-dimensional parameters. The influence of each variable on the occurrence of stick-slip motion is clarified. KEYWORDS Pneumatic system, Pneumatic cylinder, Friction, Stick-slip NOMENCALTURE Fluid Power. Forth JHPS International Symposium (C) 1999 JHPS. ISBN4-931070-04-3

INTRODUCTION Stick-slip is observed in many applications at low speeds[1] causing many troubles. In machine tools, stick-slip causes low accuracy and bad finish surfaces. In control systems, stick-slip causes steady errors in positioning tasks. Pneumatic cylinders offer advantages over other actuators as low cost or high/power ratio among others. Meter out circuits are broadly used since constant speed can be attained easily by adjusting the outlet speed control valve. However, the stick-slip effect limits the ability to achieve steady motion in applications where low speeds are required. This paper aims to cover below three topics on the stickslip motion in pneumatic cylinders driven by a meter-out circuit. a) Friction measurements of cylinder driving at low velocities. b) Occurrence conditions of stick-slip motion. c) Influence of variables on stick-slip motion. Friction is an important parameter on stick-slip motion [2]. However, only few papers show actual measurement of friction at low speeds [3][4]. On the other hand, though Okabe et al. [5] studied the stick-slip motion, there is a problem in the method of the analysis, because the viscous frictional force is not included. Especially, occurrence condition and parameters that decide this are not clear. In this paper, simulation of stick-slip motion is carried out according to the friction characteristics found in the experiments. It is shown that stick-slip motion can be described by simulation with good accuracy. The occurrence conditions of stick-slip in pneumatic cylinders are studied analytically using dimensionless parameters. Through the dimensionless parameters, the influence of each variable on stick-slip motion is clarified. responses with or without the stick-slip motion are shown, is example of measured results in Ex. 2. FRICTION FORCE MEASUREMENT Friction force measurement at the seals of pneumatic cylinders is not easy to carry out since several factors are involved [14] These factors can be divided in mechanical and tribological aspects. In this study, tribological aspects are not treated. Within the mechanical aspects are pressure and piston velocity. Since pressures at both sides of the seals do not vary too much, only velocity influence is considered as a first step. In this study, friction is measured indirectly using the equation of motion, from pressure and displacement signals. Displacement was differentiated to obtain velocity and acceleration responses. Since differentiation amplifies the noise present at displacement signals, a FIR low pass filter was used. The raw data was filtered forward and backwards in time to eliminate the filter phase shift. EXPERIMENTAL APPARATUS Fig.1 Experimental Apparatus Fig.1 shows the experimental apparatus. A rodless cylinder of diameter 32[mm] and stroke 600[mm] was set up horizontally. To vary equilibrium velocity and damping of the system, effective area of the outlet speed control valve and load mass were changed. These values, shown in Table 1, were chosen near the critical conditions of stick-slip occurrence. To create similar lubrication conditions in all experiments, cylinder was driven 10 times before making measurements. Since maximum static friction rises with the time piston is stuck at cylinder cover, maximum static friction was maintain almost constant by waiting the same time after the 10th drive. Pressure was measured by semiconductor type sensors and displacement was measured by an optical encoder, resolution 12.5 [Đm]. Fig.2 in which cylinder Fig.2: Experimental and Simulation Results

Tablel Experiment Conditions, Measured Friction Force RESULTS OF FRICTION FORCE MEASUREMENTS Fig.3 shows results of friction force measurement for experiments 2, 8 and 14, in which friction force is plotted against velocity. Results for experiment 2 are shown in two parts for clarity of the figure. For experiments 8 and 14, only results for the first stick-slip cycle are shown. It can be seen in Fig.3 the Stribeck curves which are characteristic for lubricated sliding elements[2]. There are the same lubrication modes for all cases: boundary lubrication, mixed lubrication and elasto-hydrodynamic lubrication. Friction force during the three modes of lubrication has different characteristics. Each of them will be discussed. Boundary Lubrication: It is described by a maximum static friction force. Changes of maximum static friction force with time are not considered in this study. Table 1 shows values of the maximum static friction forces for the initial position Fo and for the first stick mode Fs1. Mixed Lubrication: As it can be seen in Fig.3, it is characterized for a negative slope until a critical velocity ucr. When motion starts, a lubricant film appears between the cylinder wall and seals. This improves the lubrication and consequently, friction decreases. Critical velocity was considered to be 0.06 [m/s]. Elasto-Hydrodynamic Lubrication: It can be described by the sum of Coulomb friction and viscous friction. It can be seen in Fig.3 that for a larger load mass, there is a larger Coulomb friction force. This is because a larger load mass decreases the lubricant film thickness. It was considered a linear variation of the Coulomb friction with load mass. These values are given in Table 1. Due to the viscosity of the lubricant, viscous friction force changes almost linearly with velocity. The viscous friction coefficient was taken to be 64[Ns/m]. It should be noticed that friction force change from static to dynamic is different that the change from dynamic to static. Friction Model BL: Boundary Lubrication. ML: Mixed Lubrication EHDL: Elasto-Hydrodynamic Lubrication Fig.3: Stribeck Curves for Experiment 2, 8 and 14

Cylinder Response and Stick-Slip Motion Air compressibility makes piston to converge to a constant velocity (uo=0.03[m/s]) and makes the discharge chamber of the cylinder to act like a spring. The small equilibrium velocity and the oscillatory nature of the system make velocity to become zero at t= 1.3[s]. At this time, static friction starts to act instead of dynamic friction. Being maximum static friction larger than dynamic friction at t= 1.3[s], driving force is not large enough to surpass the resistance of the maximum static friction and cylinder sticks in this position. Pressure at discharge side decreases since air is discharged from the cylinder without volume change, making the driving force to increase. Motion is restarted when driving force surpasses maximum static friction (t= 1.7[s]). Because maximum static friction at this time is less than at the initial position, velocity overshoot is less and stick-slip does not occur after t= 1. 7 [s]. ANALYSIS OF OCCURRENCE CONDITIONS OF STICK-SLIP MOTION Dimensionless Equations Fig. 2 also provides comparisons with simulation results. It can be seen that simulation results are describing well the cylinder motion at low velocities despite dispersions of the friction force. State change of air is considered adiabatic. To derive dimensionless parameters, basic equations for the cylinder motion used in simulation are made dimensionless [6]. Otherwise, in the analysis, next fact is assumed for the simplicity. The charging side pressure does not change, because the impedance of restriction at charging side is bigger than it at discharging side in the meter-out circuit. Values at equilibrium condition for the discharge side are taken as reference values. Dimensionless equations for air mass flow equation and state equation considering adiabatic change becomes respectively: Analytical Velocity Response. Linearizing Eq.(2), neglecting volume change since velocity is small and neglecting the influence of temperature on the air mass flow since pressure drop is small, Eq. (6) is obtained: From Eq. (3) and Eq. (5), velocity response can be described as a second order differential equation, Where: (7) (8) (4) (5) (6) natural angular frequency of the system. From Eq. (8), it can be deduced that Tf is the natural period of the system under isothermal conditions divided by 2ƒÎ. (9) (1) Where: (10) (2) (11) Equation of motion becomes: (12) Where: (3) Occurrence Conditions of Stick-Slip Motion The condition of stick-slip occurrence can be written as in Eq. (13):

(13) (14) From Eq.(13) and (14): Fig. 4: Occurrence Conditions of Stick-Slip Motion (15) COMPARISON BETWEEN ANALYTICAL AND EXPERIMENTAL RESULTS FOR THE OCCURRENCE OF STICK-SLIP MOTION Fig. 4 shows the occurrence conditions of stick-slip. The curve represents the critical conditions of stick-slip occurrence (Eq.(13) and Eq.(14)). Above this curve is the Stick-Slip Region or region in which stick-slip occurs. Below this curve is the Non Stick-Slip Region or region Fig. 5: Minimum Equilibrium Velocity for Stick-slip Occurrence in which stick-slip does not occur. As it can be seen in Fig. 4, smaller values of and larger values of A* increases the possibility of stick-slip occurrence. This is because smaller damping factors and larger values of A* increase the amplitude of velocity oscillations. (16) Values of Ċ* and Ď* of experiments are plotted in blank circles when stick-slip occurred and blank squares when stick-slip did not occur. It can be seen a good agreement between the analytical curve and the experiments. Despite the experiments conditions were chosen very near the conditions of stick-slip occurrence, the analytical curve predicts well the occurrence of stickslip motion. MINIMUM EQUILIBRIUM VELOCITY FOR STICK-SLIP OCCURRENCE It is of practical interest to know the minimum equilibrium velocity that a cylinder can achieve without stick-slip occurrence. However, the plane Ċ*-Ď* does not provide information about this minimum equilibrium velocity. equilibrium velocity u, can be calculated. The vertical axis in the right side indicates equilibrium velocities for the particular case of the cylinder tested. Thus, the critical curve for stick-slip motion also shows the critical equilibrium velocity.

times and increases times because D is a function of Tf*, and Tf* changes A times from Eq.(4); thus, the initial point (Ċ*,Ď* ) is moved to the point (Ċ*/, Ď*), inside the stick-slip region, which means that the possibility of stick-slip occurrence increases. Following this example, the influence for other parameters can be deduced. Fig.6b indicates that if pressure supply Ps, load mass M and cylinder stroke L makes damping factor to approach toĊ*=0.18, possibility of stick-slip increases. CONCLUSIONS a) Stick-slip motion in a pneumatic cylinder driven by meter out circuit is studied through experiments and analysis. Friction force is measured and has the characteristics found in the different lubrication modes. b) It is found that the occurrence conditions of stickslip motion can be expressed in function of two non-dimensional parameters and that the minimum equilibrium velocity can achieve without stick-slip occurrence. c) Experimental results show that the non-dimensional parameters provide a good means to predict stick slip occurrence. The influence of each parameter of the system on stick-slip occurrence is clarified. Fig.6 : Diagrams Indicating the Influence on Stick-Slip Occurrence when a Variable is increased A times. To solve analytically the differential equations, linearization had to be carried out. To study the effect of the linearization, the linear model (analytical) is compared with the non-linear model. Simulation is carried out without linearizing equations, varying INFLUENCE OF VARIABLES Fig. 6 shows the influence on stick-slip occurrence when a variable is increased A times. Fig.6a indicates that when the difference between static and Coulomb friction FS or diameter D are increased, possibility of stick-slip occurrence increases. When viscous friction coefficient C or effective area of outlet speed control valve Sed are increased, possibility of stick-slip occurrence decreases. As an example, the case for diameter D is explained. When diameter D is increased A times, Ċ* decreases 1/ REFERENCES 1. Matsuzaki A., Hashimoto S.: Theoretical and Experimental Analysis of Stick-slip in Hydraulic Driving Mechanism. Trans. JSME, 1962, 28-194, pp.1394-1404. (In Japanese) 2. Gao C., Kuhlmann-Wilsdorf D.: On Stick-slip and the Velocity Dependence of Friction at Low Speeds. Trans. ASME, J. Tribology, 1990, 112, pp.354-360. 3. Helduser S., Muth A.: Dynamic Friction Measurement Method Evaluated by Means of Cylinders and Valves. Proc. 3rd JHPS Int. Symposium, 1996, pp.271/276. 4. Backe W., Eschmann R.: SSP- A Simulation Program for Pneumatics. Proc. 6th Bath Int. Fluid Power Workshop, 1993, pp.64-78. 5. Okabe S., Kamiya Y., Miura M., Tokoyama Y.: Stick-slip Motion of Pneumatic Cylinder. Trans. JSPE, 1988, 54-01, pp.183-188. (In Japanese) 6. Fujita T., Jang J., Kagawa T., Takeuchi M.: Dynamics of Pneumatic Cylinder Systems. Proc. 3rd JHPS Int. Symposium, 1996, pp.259/264. 7. Belforte G., Raparelli T., Velardocchia M.: Study of the Behavior of Lip Seals in Pneumatic Actuators. Lubrication Eng., 1993, 49-10, pp.775/780.