外文翻譯-合規(guī)和間隙對(duì)機(jī)械傳動(dòng)系統(tǒng)輸出速度的影響（英文）

1、Ramamurthy V. Dwivedula Principal Ideal Institute of Technology, Kakinada 533 003, India e-mail: ram_dula@yahoo.comPrabhakar R. Pagilla1Professor School of Mechanical and Aerospace Engineering, Oklahoma State University,

2、 Stillwater, OK 74078-016 e-mail: pagilla@okstate.eduEffect of Compliance and Backlash on the Output Speed of a Mechanical Transmission SystemA dynamic model to describe the effect of compliance in a transmission system

3、is pre- sented. Analysis of this model shows that it is desirable to use feedback from driver-side of the transmission system. This model is extended to include the effects of both compli- ance and backlash in a mechanic

4、al transmission system. The proposed model considers compliance (which may be either due to the elasticity of the shafts or belt in a belt-pulley transmission system) and backlash appearing in series in a drive system. I

5、n contrast to the classical backlash model which considers both input and output to the backlash as displacements, the proposed model considers (torque) force as input to the backlash and (angular velocity) velocity of t

6、he driven member as the output of the backlash. Thus, the proposed model does not assume that the load is stationary when contact is lost due to backlash width, i.e., momentum of the load is taken into account. Using the

7、 proposed model, a bound on the speed error due to the presence of backlash is derived. Experi- ments were conducted on a rectilinear mass-spring system platform, which has a provi- sion to change the backlash width by a

8、 known value. Experiments were conducted with different backlash widths and a velocity error bound was computed. The error bound obtained from the experimental results agrees with the theoretically computed bound. [DOI:

9、10.1115/1.4005493]1 IntroductionBacklash is one of the most commonly encountered nonlinearities in drive systems employing gears or ball-screws and indicates the play between adjacent moveable parts. Since the action of

10、two mat- ing gears can be represented by the action of one pair of teeth, back- lash is commonly represented by the schematic shown in Fig. 1. When used in the context of mechanical engineering, backlash denotes two sali

11、ent features as shown in Fig. 2: (i) a mechanical hysteresis due to the presence of a clearance (D), and (ii) impact phenomena between the surfaces of the masses (Mm and ML). In Fig. 1, Mm and ML are the masses (inertias

12、) of the driving and driven members, xm and xL are the linear (angular) displacements of the driving and driven members, respectively, from a fixed refer- ence position, and Fm and FL are the driving and load forces (tor

13、- ques). It is a common practice to lump all the mass (inertia) on the driving side into one quantity, Mm, and refer to it as the “motor” and lump all the mass (inertia) on the driven side, and refer to it as the “l(fā)oad.”

14、 The classical backlash model considers the schematic shown in Fig. 1 with input to the backlash as the displacement xm and the output of the backlash as the displacement xL. The input–output characteristics of the backl

15、ash are represented by Fig. 2. The slopes of lines GBC and FED are equal to the speed ra- tio of the gearing in the case of rotary systems. The closed curve BCDEFGB in Fig. 2 represents mechanical hysteresis due to the p

16、resence of clearance D. At points B, D, and G in Fig. 2, the two masses impact and near these points, the input– output plot may not be straight but may “oscillate” with a small amplitude. However, impact may be consider

17、ed to be sufficiently plastic so that points on these lines lie along a curve bounded by the dotted circles shown, before they resume to lie on the straightlines. The classical backlash model resorts to this simplificati

18、on mainly because in large industrial machines, which operate at steady state and do not reverse direction, impact does not arise except during starting/stopping conditions. Also, in smaller machines, the gear and impact

19、 energy are very small. Thus, a plas- tic impact is considered to be a reasonable assumption. Since large industrial machines do not reverse direction many times during their operation, the lines CDE and FGB in Fig. 2 ar

20、e ignored. And this prompted many researchers to erroneously consider the input– output graph of backlash to be represented by the curve FEABC, which is the input–output graph for dead-zone nonlinearity. Also, it may be

21、observed that the backlash characteristics shown in Fig. 2 consider the input to the backlash to be the displacement of theFig. 1 Schematic of backlash1Corresponding author. Contributed by the Dynamic Systems Division of

22、 ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 11, 2008; final manuscript received October 14, 2011; published online April 3, 2012. Assoc. Editor: YangQuan

23、 Chen.Journal of Dynamic Systems, Measurement, and Control MAY 2012, Vol. 134 / 031010-1 Copyright V C 2012 by ASMEDownloaded From: http://asmedigitalcollection.asme.org/ on 02/23/2014 Terms of Use: http://asme.org/terms

24、tight tension control mandates even tighter velocity control, even in the face of backlash. Extensive literature survey on modeling and control of indus- trial speed controlled drives indicates that there is a need for a

25、 simple and effective model of backlash. Besides, it is of practical importance to know the achievable accuracy level in a given drive system with a given control scheme and with a known backlash. This practical consider

26、ation is not addressed in any of the existing literature. Motivated by this practical aspect, subsequent sections present a backlash model and a bound on the achievable accuracy in a given plant with a given backlash. Th

27、e remainder of the paper is organized as follows. Section 2considers the effect of compliance and discusses some aspects of the control scheme. Section 3 presents a model for a transmission system with backlash and compl

28、iance in series. Two cases are considered: (i) compliance due to a shaft and (ii) compliance due to a belt. These two cases are presented in Secs. 3.1 and 3.2, respec- tively. Section 4 presents the error bounds due to t

29、he presence of backlash and Sec. 5 presents the results of the experiments. Sum- mary and directions for future work are presented in Sec. 6.2 Effect of Compliance and Choice of Control SchemeIn many applications, a belt

30、-pulley transmission system is a convenient alternative over a gear transmission system. When the center distance between the driving shaft and the driven shaft is too large for use of a gear-pair, using a belt to transm

31、it motion/ power may be the only practical alternative. Figure 5 shows a schematic of the drive system used in a typical web handling application. The load inertia and damping (JL and bL) arise due to unwinder/winder rol

32、l and friction at the mounting bearings. Typically, the load inertia may be several orders of mag- nitude larger than the inertias of the connecting shafts, pulleys, etc., which are shorter and offer much smaller inertia

33、. On the same note, the inertia of the motor is also much larger than the inertias of the shafts and the pulleys. Because of this, it is plausi- ble to ignore the inertias of the “intermediate” shafts, pulleys etc. and,

34、as a first approximation, lump the motor inertia, load inertia, motor viscous friction, and load viscous friction at appropriate places. Assuming that the transmission of power is taking place on the tight side and trans

35、port of belt material is taking place on the slack side, the dynamics of the belt transport system, may be written assm ¼ ðJm € hm þ bm _ hmÞ þ R1KbðR1hm ? R2hLÞ (1a)R2KbðR1hm ? R2

36、hLÞ ¼ ðJL € hL þ bL _ hLÞ þ sL (1b)where sL is the disturbance torque due to changes in web tension. Figure 6 shows a block diagram representation of the system given in Eq. (1). Note that t

37、he block diagram given in Fig. 6 repre- sents the open-loop system. The two loops appearing in the block diagram represent the interconnections in Eq. (1). In the system shown in Fig. 5, typically, one of the objectives

38、is to regulate/control web speed, as inferred by the load speed, xL ¼ _ hL There are two ways to achieve this objective: (i) by usingthe motor speed, xm, as the feedback and designing the control effort, sm and (ii)

39、 by using the load speed, xL, as the feedback and designing the control effort, sm. These two situations are shown in Fig. 7 below. In many typical web handling systems, the speed/ tension controllers are proportional-in

40、tegral (PI) controllers and thus, this paper also considers a PI control scheme. There are two possible feedback schemes, viz., feedback from load side as shown in Fig. 7(a) and feedback from motor side as shown in Fig.

41、7(b). It has been shown in Refs. [22,23] that, even in the case of zero backlash, the scheme as shown in Fig. 7(b), i.e., feedback from the motor side, is desirable. Also, if the system considered has backlash, the probl

42、em is more accentuated. When contact between the driver (motor) and the driven load is lost, the load speed is solely under the “control” of external disturbance. Using the load speed as feedback will put the closed loop

43、 system under the mode of “chasing” the unknown disturbance. Thus, motor speed feedback is considered in the rest of the paper.3 Backlash Model With ComplianceThis section considers transmission systems in which backlash

44、 and compliance exist, as shown in Figs. 8 and 10. Compliance in the transmission systems is considered to arise either due to the elasticity of shafts on which gears are mounted or due to the belt in belt driven systems

45、. The case of a compliant shaft is considered in Sec. 3.1 followed by the case of a compliant belt in Sec. 3.2.3.1 A Model of Backlash With a Compliant Shaft. To de- velop a simplified model, consider the schematic as sh

46、own in Fig. 8. In this figure, a load (JL) is driven through a compliant shaft (k is the stiffness) and a pair of gears (radii R1 and R2). Usually, the motor (Jm) is mounted near the driving gear, thus the driving shaft

47、may be assumed to be rigid. To avoid jamming of the gears at high speeds, the gears are mounted with a center distance slightly greater than the designed center distance. This gives rise to clearance between the mating t

48、eeth; this clearance is termed “backlash.” To pictorially represent backlash in torsional systems, at least two orthographic views areFig. 5 Schematic of a belt driven transmission systemFig. 6 Block diagram of the belt

49、driven transmission system. R is the speed ratio, R 5 R2/R1.Fig. 7 Two feedback schemes: (a) feedback from load shaft and (b) feedback from motor shaftJournal of Dynamic Systems, Measurement, and Control MAY 2012, Vol. 1