Tap chi Khoa h9c va Cong nghe 102 (2014) 096-100 Mathematical Model of Friction in Pneumatic Cylinder Mo hinh toan hpc cua ma sat xy lanh nen Tran Xuan Bo*, Do Viet Long, Tran Khanh Duong Hanoi Umversity ofScience and Technology - No I, Dai Co Viet Str Hai Ba Ti-ung Ha Noi, Viet Nam Received May 16, 2013: accepted: April 22, 2014 Abstract This paper deals with modeling of dynamic friction behaviors of a pneumatic cylinder In previous research, the author have expenmentaly investigated dynamic friction behaviors of pneumatic cylinders in the slidir\g regime and have showed that the measured dynamic friction behaviors can be simulated by the new modified LuGre model with a relatively good accuracy However, the model cannot simulate accurately the effects of frequency of velocity variation on the friction force-velocity characterisitcs of the pneumak cylinders In this paper, the new modified LuGre model is revisied by varying some static parameters wiOi frequency Comparisons between simulation and expenmental results show that the revised model is appropnate Keywords Friction, Dynamic befiavior, Pneumatic cylinder, Matfiematica! model, Frequency Tom tat Bai bao tap trung phat triin md hinh toan hoc dac tinh ma sat ddng cua mdt xy lanh ndn Trong nghien ciru trwdc day, tac gta da khao sat thwc nghiem dac tinh ma sat ddng cOa cac xy lanh nen diiu kien trwtyt va da chi ring nhu'ng dac tinh ma sat dpng dtrac dac cd thi dirr^c md phong vdi m(i\ chinh xac twang doi tdt boi mo hinh ma sat LuGre swa ddi mai Tuy nhien md hinh ma sat khong the md phdng chinh xac anh hudng cua tin si cua thay dii van tdc din cac dac tinh ma sat-van tie ciia xy lanh nen Trong bai bao nay, md hinh ma sat LuGre sira doi mai dtrac thay ddi bai thay ddi cAc thdng si tJnh ciia md hinh vdi tin sd So sanh kit qua md phdng vdi kit qui fhuc nghiem chi ring md binh dwac thay dii la phii hap Tir khoa: Ma sat, Elac tinh dong iu-c hgc, Xy lanh nen, Mo hinh lo^n hpc, T§n s6 Introduction Fricfion is always found in a pneumafic cylinder between the piston/rod seals and the cylinder walls Friction in a pneumafic system may makes the dynamics of the system highly nonlinear and makes the controllability of position and velocity of the piston more difficult It is, therefore, necessary to invesfigate the fricfion behaviors of the pneumatic cylinder and to develop an accurate mathematical model of friction of the system Several mathematical models, which are used to describe the friction behaviors observed in mechanical systems, have been proposed in literature [1-8] and among them, the LuGre model [5] is most widely ufilized The LuGre model, however, cannot simulate well the dynamic friction behaviors of a hydraulic cylinder in the sliding regime as shovra in [9], Yanada and Sekikawa [9] have made a modification to the LuGre model by incorporating lubricant film dynamics into the model and it has been shown that the proposed model, called the modified LuGre model, can simulate the dynamic • Corresponding author: Tel.: (•'•84) 975.873,202 Email: bo tranxuan^hust edu behaviors of friction observed m hydraulic cylinders with a relatively good accuracy Tran et al [10] have shown that the modified LuGre model is valid only in the negative resistaoK regime and cannot simulate the hysteretic behaviors observed Ln hydraulic cylinders in the fluid lubrication regime In addition, they have revised the modified LuGre model by replacing the usual fluid friction term with a first-order lead dynamics and have shown the usefulness of the new modified LuGre model in the entire sliding regime Tran and Yanada [ I I ] have studied the dynaniS friction behaviors of pneumatic cylinders under oscilafing velocity conditions in the sliding regime (v>0), They have shown that when the piston velocity varies sinusoidally without velocity reversals, tbe dynamic fiiction force-velocity charactenstic exhibits a hysteretic behavior (fVictiona! lag) around the steady-state friction characteristic at low velocities In addition, they have shown that the hysteretic loop is expanded to higher velocities when the frequency Of the velocity variation is increased, and its size increases with the driving pressure and decreases with the resistance pressure In order to simulate those dynamic friction behaviors of the pneuniM cylinder, they have applied the new modified L Tap chi Khoa hoc va Cong ngh? 102 (2014) 096-100 model [ I I ] and have shown that the model can simulate the dynamic fiiction behaviors observed experimentally in the sliding regime of the pneumatic cylmder with a relatively good accuracy However, the new modified LuGre model cannot simulate accurately the effects of fi^quency of velocity variation on the friction force-velocity characterisitcs of the pneumatic cylinder constant for lubncant film thickness, v^ is the velocity within which the lubricant film thickness is vaned, and Thp, Thn, TiiQ are the time constants for acceleratio deceleration, and dwell periods, respectively In Eq (7), h < hss corresponds to the acceleration period, h > h„ to the deceleration period In this paper, the new modified LuGre model is revised by taking the effects of frequency into consideration An investigation is also conducted to identify the influence the parameters of the new modified LuGre model on simulation results Comparisons of the results simulated by the revised model with the measured results are made and the usefiilness of the revised model is shovra '•r.=P Friction models 2.1 New modified LuGre model The new modified LuGre model is described as follows [II]dz _ (T^z (1) dt^ g(v,li)