C:\Users\raoh\Desktop\Paper_8.xps The Journal of Engineering Research Vol. 8 No. 2 (2011) 66-73 1. Introduction Friction occurs in all mechanical systems, eg. bear- ings, transmission, hydraulic and pneumatic cylinders, valves, brakes and wheels. In many engineering appli- cations, frictional contact occurs between machine parts and the characterization of contact behavior becomes an important subject in solving tribological problems such as friction induced vibration wear. Issues related to mechanical sealing, performance and life of machine elements, and thermal are few exam- ples. The pioneering work of Greenwood and Williamson (1966) has been utilized by many researchers (Ibrahim and Rivin 1994; Mulhearn and Samuels 1962; Abdo and Farhang 2005, Seabra and Berthe 1987) as a basic for further extension to obtain contact models for general or specific contact prob- lems for mainly elastic contact. On the other hand, the work of Pullan and Williamson (1972) utilized as a basic model for pure plastic contact. In an attempt to bridge the gap between the pure elastic and pure _______________________________________ *Corresponding author’s e-mail: jdabdo@squ.edu.om plastic contact, researchers Halling and Nuri (1972), Greenwood and Tripp (1971), and Abdo and Al- Yahmadi (2004) developed a wide intermediate range of interest where elastic-plastic contact triumph. In many cases the classical friction model cannot capture the characteristics such as Stribeck effect, stick-slip, pre-sliding hysteretic motion, break-away force, which play a significant role in application on high precision positioning control (Abdo et al 2010, Abdo and Al-Yahmadi 2009). The role of friction modeling can be categorized according to whether or not the friction compensation is model-based. Examples of non-model-based compensators include high-gain feedback, and impulsive control. This paper contains four sections including the intro- duction. Section 2 reviews the single state elastoplas- tic friction model given by Dupont et al. (2002). Section 3 presents an adaptive compensator to achieve velocity tracking in the presence of the friction force described by the single state elastoplastic friction model and a proof of it is also given in this section. Adaptive Compensator of Single State Elastoplastic Friction Model A. A. Abouelsouda, J. Abdo*b and R. Zaierb aDepartment of Electronics and Communications Engineering, Faculty of Engineering, Cairo University, Egypt *bDepartment of Mechanical and Industerial Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 33, Postal Code 123, Al-Khoud, Muscat, Oman Received 19 April 2011; accepted 25 September 2011 Abstract: A nonlinear friction is an unavoidable phenomenon frequently experienced in mechanical sys- tem between two contact surfaces. An adaptive compensator is designed to achieve tracking of a desired velocity trajectory in the presence of friction force described by a single state elastoplastic friction model. The adaptive compensator includes an adaptive observer and a computed force controller. The closed loop system is shown to be stable using Lyapunov second method. Simulation results show the effective- ness of the proposed compensator. Keywords: Adaptive compensator, Elastoplastic friction model, Lyapunov second method 67 A. A. Abouelsoud, J. Abdo and R. Zaier Since the single state elastoplastic friction model con- tains a bounded function to describe different friction phases and depends on the immeasurable elastic state, the proposed compensator uses an adaptive observer and a computed force controller. Stability analysis of the proposed adaptive compensator is carried out using Lyapunov second method. Simulation results are given in Section 4 and conclusion is drawn in Section 5. 2. Single State Elastoplastic Friction Model In this section we review the single state elastoplas- tic friction model given by Dupont et al. (2002), based on which we design an adaptive velocity com- pensator to follow a desired velocity trajectory. The rigid body displacement x is composed of elastic (z) and plastic (w) components as, x = z + w (1) Friction models define the elastic dynamics explic- itly, while the plastic displacement w is defined implicitly. Following Dupont et al. (2002) model, the friction force is given by: (2) (3) where zss is defined as (4) and fss (x) is the steady state friction force also called the Stribeck function which is shown in Fig. 1. The Stribeck function fss (x) is bounded from below and above as (5) and (6) (7) Figure 2 shows a typical shape of m (z, m) as given in (Dupont et al. 2002). 3. Adaptive Compensator In this section an adaptive velocity compensator is designed to achieve tracking of a desired velocity tra- jectory vd (t) in the presence of friction force that was described by Eqs. (2 and 3). A similar approach was proposed by Canudas et al. (1995) but with a simple friction model and constant m (z, x). Consider the equation of motion of a single mass m subject to driv- ing force u and friction force fl (8) where v is the velocity of the mass (v = x). Figure 1. Stribeck curve of steady state friction force fss (x) versus rigid body velocity 68 Adaptive Compensator of Single State Elastoplastic Friction Model (9) and in matrix form Eq. (9) can be given as (10) The only available signal for measurement is the velocity v of the rigid body, hence, expressing y = v in terms of the state x leads to (11) where (12) Definition 1 Positive Real (PR) function A strictly proper rational function G(s) is positive real (PR) if G(s) is analytic for Re [s] 0 and Re [G(j ) > 0 for al - < . Definition 2 Strictly Positive Real (PR) function A strictly proper rational function G(s) is strictly positive real (SPR) if G (s- ) is positive real for some real > 0. The Kaman-Yakubovich (KY) Lemma (Narendera and Annaswamy 1989): A strictly proper rational function G(s) with state space realization (A,b,c) G(s) = c (SI - A)-1 is strictly positive real (SPR) if there exists positive definite symmetric matrices P=PT>0 and Q=QT>0, such that PA + AT P = -Q (13) 3.1 Adaptive Observer The adaptive observer is given as (14) Figure 3. Adaptive Compensator Adaptation Law State Observer Controller 1/ms Friction Model 69 A. A. Abouelsoud, J. Abdo and R. Zaier L = (l1 l2)T is the observer gain and l1 and l2 are cho- sen to make the transfer function Go(s) = c (sI - A + Lc)-1 b Strictly Positive Real (SPR). Since the pair (c, A) is observable (Seabra and Berthe 1987) for o 0, we can place the eigenvalues of the matrix (A- Lc) at any desired position. We notice also that the transfer function G(s) = c(sI - A)-1 b has a zero at (- o / 1 ) (transfer function equal to zero), hence by plac- ing the eignevalues of the matrix A-Lc at the location of the zero (- o / 1 ) the remaining eigenvalue is in the left half plane and thus we obtain a strictly positive real function (SPR). A zero of a transfer function is a real or complex number. Depending on Kaman-Yakubovich (KY) Lemma there exist a positive definite symmetric matrices P = PT > 0 and Q = QT > 0, such that P (A - Lc) + (A = - LC)T P = -Q (15) 3.2 Controller The controller u is chosen as (16) (17) > 0 is an adaptation gain. 3.3 Stability Analysis The close loop system given by Eqs. (9), (14),(16) and (17) is analyzed in this section utilizing the Lyapunov Second Method. Consider hence (16) From the selected gain L discussed in section 3.1 to make the transfer function c(sI - A + Lc)-1 b SPR there exist a positive definite symmetric P and Q which satisfies Eq. (13). Substituting for the control u into (3.1) we obtain (17) Consider the Lyapunov function (18) Time derivative of W along-with the solution of Eqs. (16 and 17) is given as (19) where (20) where, 70 Adaptive Compensator of Single State Elastoplastic Friction Model 71 A. A. Abouelsoud, J. Abdo and R. Zaier 0 0.5 1 1.5 2 2.5 3 3.5 4 v vd t sec 1.5 1 0.5 0 -0.5 -1 Figure 6. Rigid body velocity v and desired square wave velocity vd 0 0.5 1 1.5 2 2.5 3 3.5 4 time sec alphaestimate alpha 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 72 Adaptive Compensator of Single State Elastoplastic Friction Model This leads to (21) (22) or (23) To regulate the velocity v to follow vd we assume |v| < VM . By completing the squares the following inequalities are reached 4. Simulation Results The desired velocity function is chosen as vd (t) = 10 sin (10t) this is a standard test signal. The elasto- plastic friction model parameters are given in Table 1. The Stribeck friction function is given by: fss = [ fm + (fM-fm)e-(1000v)2 ] sign(v) (23) zba = 0.7169 Zss (24) The gains of the observer and controller are given in Table 2. It is obvious from Figures 4-7 that the proposed adaptive compensator provides an accurate tracking capability to the desired trajectory while keeping all other signals bounded. Conclusions In this paper a single state elastoplastic friction model is reviewed and an adaptive compensator is pro- posed to track a desired velocity trajectory in the pres- ence of friction force model. The adaptive compen- sator includes an adaptive observer which consists of a state estimator and an adaptation law to one of the unknown functions of the friction model. The con- troller cancels the nonlinearities in the friction model using the estimated state and parameter. The model stability is determined by using Lyapunov analysis. Simulation of the proposed adaptive compensator to track both sinusoidal and square wave signals show almost perfect tracking and signals are bounded in the closed loop system which indicate the proposed com- pensator is effective. References Abdo J, Al-Yhmadi A (2009), The effect of controlled frequency and amplitude of vibration on friction. Solid state phenomena 147:380-385. Abdo J, AL-Yahmadi A (2004), A wear model for rough surfaces based on the ultimate-stress asper- ity concept. Int. J. of Applied Mechanics and Engineering 9:11-19. Abdo J, Farhang K (2005), Elastic-plastic contact model for rough surfaces based on plastic Asperity concept. Int. J. of Non-Linear Mechanics 40/4: 495-506. Abdo J, Tahat M, Abualsoud A, Danish M (2010), The effect of frequency of vibration and humidity on the stick-slip amplitude. 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