Control of Chaos in Rate-Dependent Friction-Induced Vibration Using Adaptive Sliding Mode Control and Impulse Damper

Abstract

Two different control methods, namely, adaptive sliding mode control and impulse damper, are used to control the chaotic vibration of a block on a belt system due to the rate-dependent friction. In the first method, using the sliding mode control technique and based on the Lyapunov stability theory, a sliding surface is determined, and an adaptive control law is established which stabilizes the chaotic response of the system. In the second control method, the vibration of this system is controlled by an impulse damper. In this method, an impulsive force is applied to the system by expanding and contracting the PZT stack according to efficient control law. Numerical simulations demonstrate the effectiveness of both methods in controlling the chaotic vibration of the system. It is shown that the settling time of the controlled system using impulse damper is less than that one controlled by adaptive sliding mode control; however, it needs more control effort.

1. Introduction

There exist a lot of works on the theory of friction-driven oscillations in the literature, for example, influence of the belt speed on the system response [1], dynamics of three-block mechanical system with dry friction [2], investigation on the geometry of chaotic attractors for dry friction oscillators [3, 4], influence of parametric and external excitations on a dry friction oscillator dynamics [5], and the dynamic behavior of friction-driven oscillator with an impact damper [6].

The characteristics of the friction force between two surfaces are quite complex and depend on many parameters such as, normal pressure, slip velocity, surface, and material properties [7]. The friction-actuated oscillation is strongly nonlinear, and discontinuous and has nonsmooth process, which is a source of instabilities generating stick-slip, chatter, squeal, and chaos [8, 9]. LuGre friction law is one of the most widely used friction laws which models rate dependency of the friction force by one additional inner variable [10]. In [11], LuGre friction model is applied to the single degree of freedom friction-induced oscillator, and it is shown that the oscillations of this system turned out to be chaotic for most parameter combinations.

Many active control methods have been presented for control of chaotic systems such as nonlinear feedback control [12], drive-response synchronization method [13], adaptive control method [14, 15], variable structure (or sliding mode) control method [16–19], back stepping control method [20, 21], fractional control [22], impulsive control [23], and adaptive sliding mode control [24].

Chatterjee presented a novel method for controlling the vibration, called impulse damper [25]. In this method, mass loaded PZT stack is attached to the system, and applying suitable voltage to the PZT stack leads to impulsive force which can control the system by setting control parameters correctly. This technique was used to control a nonlinear friction-driven oscillator [26] and to control the chaotic vibration of friction-driven oscillator with coulomb friction model [27, 28].

In the present study, two different control methods, namely, impulse damper and adaptive sliding mode control, are used to control the chaotic vibration of a single degree of freedom system subjected to the rate-dependent friction. To control the chaotic vibration of this system using adaptive sliding mode control, an adaptive control law is established which stabilizes the chaotic response of the system.

Since chaotic systems are sensitive to the initial conditions, applying appropriate impulses at suitable positions causes the system to maintain its path within a specified bound. This was our motivation to investigate the effectiveness of impulse damper to suppress the chaotic behavior of the system. The effectiveness of both methods will be investigated in following sections.

2. Mathematical Model

In this section, the mathematical modeling of uncontrolled and controlled system, using adaptive sliding mode control and impulse damper is presented.

2.1. Friction Mathematical Model

Experimental investigations, especially at low sliding speeds, have shown that friction laws which give the friction force as an algebraic function of the underlying parameters (i.e., static friction laws) do not capture all of the observable frictional effects. Among the additional effects is the presliding displacement due to lateral contact elasticity, the increase of static friction with time due to diffusion processes on the interface, and frictional lag in sliding, which stands for the effect of the friction force lagging in time behind changes in relative velocity or normal load. To model the effects, the so-called dynamic friction models have been proposed. The following investigation is based on one of the most widespread models, the LuGre model [10], which allows a comparatively simple representation of the dynamic effects. Based on this model, the friction force is given by 𝑁𝐹=𝑁₀𝜎₀𝑧+𝜎₁𝑧̇+𝜎₂𝑣_𝑟. (1) Governing equation of 𝑧 is 𝑧̇=𝑣_𝑟−𝜎₀𝑣_𝑟𝑔𝑣_𝑟𝑧, (2) where 𝑧 represents the average deflection of the bristles, 𝑣_𝑟 is the relative sliding velocity, 𝑁 is the normal load with a reference value 𝑁₀, and 𝑔(𝑣_𝑟) is the friction force which is formulated as a function of the constant relative sliding velocity 𝑣_𝑟. Moreover, 𝜎₀,  𝜎₁, and 𝜎₂ parameterize the pre-sliding displacement, an internal viscous frictional damping, and a viscous damping contribution due to the relative velocity [11], respectively. For steady sliding, 𝜎₂ vanishes and the friction force reduces to 𝑁𝐹=𝑁₀𝑔𝑣_𝑟sign𝑣_𝑟. (3) Function 𝑔(𝑥) is given as follows: 𝑔(𝑥̇)=𝐹_𝑐+𝐹_𝑠−𝐹_𝑐−𝑥̇exp𝑣_𝑠², (4) where 𝑣_𝑠 is the speed of belt. In this research, sliding block on a rigid belt moving with a constant velocity as shown in Figure 1 is considered.

Figure 1 

The structure of the block on a rigid belt.

Using LuGre friction, the equations of motion are 𝑚𝑥̈+𝑐𝑥̇+𝑘𝑥=𝜎₀𝑧+𝜎₁𝑧̇+𝜎₂𝑣_𝑠−𝑥̇,𝑧̇=𝑣_𝑠−̇𝜎𝑥−⁰𝑣_𝑠−𝑥̇𝑔𝑣_𝑠−𝑥̇𝑧, (5) where the dot denotes differentiation with respect to 𝑡. Introducing the new parameter 𝜏=𝜔_𝑛𝑡, where 𝜔_𝑛=𝑘/𝑚, the following relations will be concluded: 𝑥^′′+𝑐√𝑥𝑘𝑚^′𝜎+𝑥=⁰𝑘𝜎𝑧+¹√𝑧𝑘𝑚^′+𝜎₂𝑘𝑣_𝑠−𝜔_𝑛𝑥^′,𝑧^′=𝑣_𝑠𝜔_𝑛−𝑥^′−1𝜔_𝑛𝜎₀𝑣_𝑠−𝜔_𝑛𝑥^′𝑔𝑣_𝑠−𝜔_𝑛𝑥^′𝑧. (6)(7) It should be noted that the dimension of (6) and (7) is length. In these equations, the prime denotes differentiation with respect to 𝜏, and the values of parameters are set as follows.

𝑚=1 kg,  𝑘=200 N/m,  𝜎₀=300 N/m,  𝜎₁=100 N/m,  𝐹_𝑐=2 N,  𝐹_𝑠=8 N,  𝑣_𝑠=0.5 m/s,  𝐷=0.01.

Here, a relative damping measure 𝐷=(𝑐+𝜎₂)/2√𝑘𝑚 has been introduced to combine the effects of linear damping from both structure and friction.

It is shown that for these values of parameters, the system has chaotic vibration [11].

2.2. Adaptive Sliding Mode Controller (ASMC) Design

In this section, the adaptive sliding mode control is used to control the chaotic vibration. The controlled chaotic system can be rewritten as follows: 𝑥^′′+𝑐√𝑥𝑘𝑚^′𝜎+𝑥=⁰𝑘𝜎𝑧+¹√𝑧𝑘𝑚^′+𝜎₂𝑘𝑣−𝜔_𝑛𝑥^′,𝑧^′=𝑣𝜔_𝑛−𝑥^′−1𝜔_𝑛𝜎₀𝑣−𝜔_𝑛𝑥^′𝑔𝑣−𝜔_𝑛𝑥^′𝑧. (8) It is desired that 𝑥 be adjusted to equilibrium state 𝑥₀, where 𝑥₀ is a given constant. The error states are defined as 𝑒₁=𝑥−𝑥₀,𝑒₂=𝑒^′₁=𝑥^′. (9) To guarantee the stability of the sliding mode, a switching surface 𝑠(𝑡) in the error space is defined as follows: 𝑠(𝑡)=𝑒₁(𝑡)+𝑒₂(𝑡). (10) As well known, when the system operates in the sliding mode, it satisfies the following [29]: 𝑠(𝑡)=𝑠^′(𝑡)=0. (11) Therefore, the following sliding mode dynamics can be obtained as 𝑒^′₁=𝑒₂=−𝑒₁,𝑒^′₂𝑐=−√𝑒𝑘𝑚²−𝑒₁+𝑥₀+𝜎₀𝑘𝑧+𝜎₁√𝑧𝑘𝑚^′+𝜎₂𝑘𝑣−𝜔_𝑛𝑒₂+𝑢. (12) From (8), it can be seen that the internal parameter 𝑧 will converge and be stabilized when (𝑒₁,𝑒₂)→(0,0).

The following function is defined: 𝑓𝑒₁,𝑒₂,𝑧,𝑧^′𝑐=−√𝑒𝑘𝑚²−𝑒₁+𝑥₀+𝜎₀𝑘𝑧+𝜎₁√𝑧𝑘𝑚^′+𝜎₂𝑘𝑣−𝜔_𝑛𝑒₂. (13) Using (13), (12) can be rewritten as 𝑒^′₂=𝑓𝑒₁,𝑒₂,𝑧,𝑧^′+𝑢. (14) The controller is designed as 𝑢=−𝛾𝜂sign(𝑠(𝑡)), (15) where 𝜂=|𝑓(𝑒₁,𝑒₂,𝑧,𝑧^′)|+|𝑒₂| and the adaptive law for parameter 𝛾 is proposed as 𝛾^′=−𝛾−𝜉,𝜉>1. (16) The proposed adaptive control scheme will guarantee the global asymptotic stability of the error and is shown in the following.

2.3. Stability Analysis

In the following, we analyze the stability of the sliding mode dynamics based on the Lyapunov stability theory. The Lyapunov function is selected as 𝑉(𝑡)=(1/2)(𝑠²+𝛾²), which leads to 𝑉^′=𝑠𝑠^′+𝛾𝛾^′=𝑠𝑒^′₁+𝑒^′₂+𝛾𝛾^′=𝑠𝑒₂+𝑓𝑒₁,𝑒₂,𝑧,𝑧^′+𝑢+𝛾𝛾^′≤|𝑠|𝑒₂+𝑓𝑒₁,𝑒₂,𝑧,𝑧^′+𝑠𝑢+𝛾𝛾^′. (17) Solving (16) leads to 𝛾=𝑒^−𝜏+𝜉. Substituting (16) in (17), the following is obtained: 𝑉^′≤|𝑠|𝜂1−𝑒^−𝜏−𝜉+−𝑒^−2𝜏−𝜉𝑒^−𝜏<0. (18) Now, if we define 𝑤(𝑥)=|𝑠|𝜂(1−𝑒^−𝑥−𝜉)+(−𝑒^−2𝑥−𝜉𝑒^−𝑥) and integrating the previous equation from zero to 𝑡, it yields 𝑉(𝜏)≤𝑉(0)−^𝜏₀𝑤(𝜆)(0)(𝜏)+𝑑𝜆⟹𝑉≥𝑉^𝜏₀𝑤(𝜆)𝑑𝜆_𝜏0𝑤(𝜆)𝑑𝜆≤𝑉(0)<∞. (19) Taking the limit as 𝜏→∞ on both sides of (19) yields lim_𝜏→∞^𝜏₀𝑤(𝜆)𝑑𝜆≤𝑉(0)<∞. (20) Now, Barbalat’s lemma is applied as shown in the following.

Lemma 1 (Barbalat’s lemma [30]). If 𝑤∶𝑅→𝑅 is a uniformly continuous function for 𝑡≥0 and if lim_𝑡→∞∫^𝑡₀|𝑤(𝑡)|𝑑𝑡≤𝑉(0)<∞ exists and is finite, then lim_𝑡→∞𝑤(𝑡)→0.

Thus, according to Barbalat’s lemma, it is obtained that lim_𝜏→∞𝑤(𝑡)=lim_𝜏→∞|𝑠|𝜂1−𝑒^−𝜏−𝜉+−𝑒^−2𝜏−𝜉𝑒^−𝜏⟶0. (21) Since lim_𝑡→∞𝑒^−𝜏=0,  𝜂>0, and 𝜉>1, it is concluded that 𝑠(𝑡)→0 as 𝜏→∞.

Thus, the system is in the sliding mode, and using (12), stability of 𝑒₁ and 𝑒₂ is surely guaranteed. Therefore, both 𝑒₁(𝜏) and 𝑒₂(𝜏) converge to zero.

2.4. Mathematical Model of Impulse Damper

In this section, for the purpose of controlling and suppressing vibration of the system, an impulse damper is used (Figure 2). The basic principle of the impulse damper is to generate suitable impulsive force by quickly expanding or contracting the PZT stack actuator. The stack is expanded by applying a suitable positive voltage at the instants of the displacement zero crossing with a positive velocity. In addition, a negative voltage is applied to contract the stack when the zero crossing takes place with a negative velocity. In fact, damping is generated by velocity feedback.

Figure 2 

PZT stack attached to the system as controller.

Applied force by impulse damper can be written as 𝐹_𝑎+𝐶_𝑝𝑋_𝑝, where 𝑋_𝑝 is the elongation of PZT stack and 𝐹_𝑎 is the force produced by the actuator. 𝐹_𝑎 can be found from the electromechanical characteristics of the PZT actuator which is given by the following: 𝑄_𝑝𝑋_𝑝=𝐶_PZT−𝑛𝑑₃₃𝑛𝑑₃₃−1𝐾_𝑝𝑉_𝑝𝐹_𝑎, (22) where 𝑄_𝑝, 𝑋_𝑝,  𝑉_𝑝,  𝐹_𝑎,  and  C_PZT represent charge, elongation, voltage, force, and the effective capacitance of the PZT actuator, respectively. 𝑑₃₃ is the piezoelectric constant of each wafer, and 𝑛 is the total number of wafers in the actuator. 𝑘_𝑝⁻¹ is the compliance of the actuator in short circuit condition and is given by 𝐾_𝑝=𝐸𝐴/𝑙, where 𝐸 is the elastic modulus, 𝐴 is the cross-sectional area, and 𝑙 is the length of the actuator. From (22), the following expression is obtained for the actuator force: 𝐹_𝑎=𝑛𝑑₃₃𝐾_𝑝𝑉_𝑝−𝐾_𝑝𝑋_𝑝. (23) Employing the previous equation and adding it to the right side of (6) as an actuation force which is applied to the system by the impulse damper, one can conclude that 𝑥^′′+𝑐√𝑥𝑘𝑚^′𝜎+𝑥=⁰𝑘𝜎𝑧+¹√𝑧𝑘𝑚^′+𝜎₂𝑘𝑣−𝜔_𝑛𝑥^′+𝛼𝑦+ℎ_𝑒𝑦^′−𝜆²1−𝜆²𝛼𝑉. (24) The equation of motion of the added mass can be written as [25] 𝑟_𝑚𝑦^′′+ℎ_𝑒𝑦^′𝜆+𝛼𝑦=−²1−𝜆²𝛼𝑉−𝑟_𝑚𝑥^′′, (25) where 𝑥=𝑥₁/𝑥₀,  𝑦=𝑥_𝑝/𝑥₀,   𝜔_𝑛=𝑘/𝑚,  𝜉=𝑐/2√𝑘𝑚,  𝑟_𝑚=𝑚_𝑎/𝑚,  ℎ_𝑒=ℎ_𝑝+(𝜆²/(1−𝜆²))𝛼𝛾,  ℎ_𝑝=𝑐_𝑝/𝑚𝜔_𝑛,  𝑉=𝑉_𝑝/𝑉_ref,  𝑉_ref=𝜆²/(1−𝜆²)𝑛𝑑₃₃,  𝛼=𝐾_𝑝/𝐾, and 𝜆²=𝑛²𝑑²₃₃𝐾_𝑝/𝐶_PZT.

Depending upon the problem, the reference displacement 𝑥₀ can be suitably defined. The following values are chosen for the previous mentioned parameters in this study: 𝑟_𝑚=0.1,  𝜆=0.7,  ℎ_𝑝=25,  𝑉_𝑚=4,  𝜀=0.003, and 𝛼=100.

The proposed control law is mathematically recast as [25] 𝑉𝑉=^𝑚(𝑋𝑋|≥𝜀sgn)|2,2𝑉_𝑚𝜀𝑋|𝑋|≤𝜀2. (26)

3. Results and Discussion

Equations of this system are solved numerically. For the numerical analysis, Runge-Kutta’s integration procedure is employed where the initial condition of system is set as (𝑥(0),𝑥̇(0),𝑧(0))=(0,0,0). Figure 3 shows the time history of the mass block displacement, and Figure 4 shows the phase plane of the uncontrolled system. Chaotic dynamics of the system can be clearly seen in these figures.

Figure 3 

Block displacement of the uncontrolled system.

Figure 4 

Phase plane of the uncontrolled system.

In the following, the numerical results are given to confirm the validity of the proposed adaptive sliding mode method. In the numerical simulations, 𝜉 is selected as 𝜉=1.5, and the desired response is set at 𝑥₀=0.0383 (equilibrium point). Synchronization error of the state variables is depicted in Figure 5.

(a)

(b)

Figure 5 

Time response of error states using adaptive sliding mode method.

Phase plane of the controlled system has been shown in Figure 6. As can be seen in this figure, the controlled system is not chaotic.

Figure 6 

Phase plane of the controlled system.

The time response of the switching function 𝑠(𝜏) and the control input 𝑢(𝜏) are shown in Figures 7 and 8, respectively.

Figure 7 

Time response of the switching function using adaptive sliding mode method.

Figure 8 

Time response of the input control using adaptive sliding mode method.

For the purpose of indicating the efficacy of the proposed impulse damper controller, we have examined the system in two different situations. In the first situation, the controller is turned on when the system is at origin, and accordingly, the mass moves towards its equilibrium position at 𝑥=0.0383. In the second situation, controller is off for 𝜏<100; therefore, the mass is vibrating chaotically around its equilibrium position at 𝑥=0.0383. At this moment, the controller is turned on and the effect of the control input on the behavior of the system is evaluated. After that, to show the usefulness of the controller, the controller is turned off, and there is no suppressing signal to damp system vibrations. Figure 9 shows response of the system controlled by the impulse damper when the controller is turned on at 𝜏=0. Figure 10 shows applied force by the impulse damper when the controller is turned on at 𝜏=0. Figures 11 and 12 illustrate the response and control input for the system controlled by the impulse damper when the controller is turned on at 𝜏=100 and is turned off at 𝜏=200.

Figure 9 

Block displacement (controller is turned on at 𝑡=0).

Figure 10 

Applied force (controller is turned on at 𝑡=0).

Figure 11 

Block displacement (controller is turned on at 𝑡=100 and is turned off at 𝑡=200).

Figure 12 

Applied force (controller is turned on at 𝑡=100 and is turned off at 𝑡=200).

In Figure 9, when the controller is turned on at 𝜏=0, response of the system is absolutely fast. The mass reaches its equilibrium position in a very short time. Although previously mentioned quick response and short settling time demonstrate high performance of the controller, it is achieved at the expense of a high actuating force (shown in Figure 10) which is applied by the impulse damper. The achieved result is rational since the mass is at origin and, thus, is far from its equilibrium position. Therefore, impulse damper controller applies big impulses to the mass for the purpose of moving it towards the equilibrium position. In contrast, when the controller is off for 𝜏<100 and thus the chaotic trajectory enters a small neighborhood of the fixed point 𝑥=0.0383 (m) (Figure 10), there is no need for such a high force as in the previous case and impulses with a short amplitude as shown in Figure 12 can control the system effectively. In this case, the distance between the current position of the mass and its equilibrium position is not as far as the first case. In other words, the farther (close) the system is to the goal dynamics, the bigger (small) the weight given to the control effort is. Feasibility of using impulse damper for controlling the system is obvious in these figures. As seen in these figures, impulse damper can damp the vibration of the system absolutely in a short time.

4. Conclusion

In this paper, chaotic vibration control of a single degree of freedom oscillator subjected to a LuGre type friction law was presented. Two different methods were used to control the chaos, ASMC, and impulse damper.

For adaptive control of the chaotic vibration of this system, a switching surface was adopted such that it becomes easy to ensure the stability of the error dynamics in the sliding mode. Then, an ASMC was derived to guarantee the occurrence of the sliding motion. The adaptive laws were derived in the Lyapunov sense to guarantee the stability of the controlled system. Furthermore, impulse damper was also used for controlling the system. The idea is that, since chaotic systems are exponentially sensitive to perturbations, careful choice of even small control perturbations can, after some time, have a large effect on the trajectory location and can be used to guide it. In this method, controlled impulses are generated by expanding and contracting a mass loaded PZT actuator used between the primary system and the secondary mass. Numerical results verified the effectiveness of the proposed methods in controlling the chaotic vibration of the rate-dependent friction-driven oscillator. It was shown that impulse damper decreases the settling time of the controlled system by applying higher control forces in comparison with ASMC.