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Tangential acceleration feedback control of friction induced vibration
Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Tangential acceleration feedback control of friction induced vibration Jyayasi Nath, S. Chatterjee n Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India
a r t i c l e i n f o
abstract
Article history: Received 9 December 2015 Received in revised form 6 May 2016 Accepted 11 May 2016 Handling Editor: D.J Wagg
Tangential control action is studied on a phenomenological mass-on-belt model exhibiting friction-induced self-excited vibration attributed to the low-velocity drooping characteristics of friction which is also known as Stribeck effect. The friction phenomenon is modelled by the exponential model. Linear stability analysis is carried out near the equilibrium point and local stability boundary is delineated in the plane of control parameters. The system is observed to undergo a Hopf bifurcation as the eigenvalues determined from the linear stability analysis are found to cross the imaginary axis transversally from RHS s-plane to LHS s-plane or vice-versa as one varies the control parameters, namely non-dimensional belt velocity and the control gain. A nonlinear stability analysis by the method of Averaging reveals the subcritical nature of the Hopf bifurcation. Thus, a global stability boundary is constructed so that any choice of control parameters from the globally stable region leads to a stable equilibrium. Numerical simulations in a MATLAB SIMULINK model and bifurcation diagrams obtained in AUTO validate these analytically obtained results. Pole crossover design is implemented to optimize the filter parameters with an independent choice of belt velocity and control gain. The efficacy of this optimization (based on numerical results) in the delicate low velocity region is also enclosed. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Self-excited oscillation Subcritical Hopf bifurcation Averaging Stick-slip oscillation Pole crossover design
1. Introduction The deleterious aspect of friction-induced instability is common to most mechanical systems, e.g. bearings, transmissions, hydraulic and pneumatic cylinders, valves, brakes, lead screws, oil well drill-strings and wheels and nevertheless in control systems like in various drive systems, high-precision servo mechanisms, robots, pneumatic and hydraulic systems, in position control systems and anti-lock brakes for cars, etc. So, the undesirable self-excited vibrations induced by non-linear character of friction in linearly stable systems has motivated various researchers worldwide to investigate into the inherent laws of friction and unanimously put forth various models that can perfectly imitate the frictional phenomena in original systems as well as effective methods to control this undesirable phenomenon that leads to significant inconvenience. Friction-induced self-excited vibrations are attributed to mainly three mechanisms, namely, the velocity weakening characteristics of friction force which is also known as the Stribeck effect, mode-coupling and sprag-slip instabilities. A detailed review of these basic mechanisms is assembled in [1,2]. n
Corresponding author. E-mail address: [email protected] (S. Chatterjee).
http://dx.doi.org/10.1016/j.jsv.2016.05.020 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Nomenclature A1 ; A2
amplitudes of non-linear oscillations of the system and the filter respectively C viscous damping coefficient C c ¼ Mω non-dimensionalized viscous damping n coefficient F friction force between mass and belt J, J1 Jacobian of the linearized flow around the equilibrium, Jacobian of the non-linear flow around the equilibrium K oscillator stiffness kc gain of power amplifier kc' ¼ kc =X 0 non-dimensionalised gain of power amplifier k1 sensor gain k1 ¼ X 0 k1 non-dimensionalised sensor gain 0 kc ¼ k1 kc non-dimensionalised loop gain oscillator mass M normal load N0 time t vb belt velocity v0 ¼ ωvn Xb 0 Non-dimensionalised belt velocity X displacement of oscillator mass Xc controlled displacement of base N0 X 0 ¼ Mω 2 reference displacement n
xf non-dimensional filter variable y ¼ XX0 non-dimensional displacement of mass z ¼ y y0 coordinate transformation where (y0 ,xf 0 ) is the static equilibrium of linearized system α ; α ¼ α ωn X 0 model parameter that determines the slope of the friction-velocity curve in the low velocity range, non-dimensional form of the same β 4 1; β ¼ β ωn X 0 a very large positive quantity, nondimensional form of the same γ Δμe αv0 ε«1 an infinitesimally small quantity λ 1, λ 2 identical complex conjugate pair of poles used in pole cross-over design μ, Δμ minimum kinetic coefficient of friction, the difference between the static friction coefficient and the minimum kinetic friction coefficient ξf damping ratio of second-order filter τ ¼ ωn t non-dimensionalised time ϕ1 ; ϕ2 phases of non-linear oscillations of the system and the filter respectively ω~ f ; ωf natural frequency of the second-order filter, qffiffiffiffi non-dimensional form of the same K natural frequency of the mechanical system ωn ¼ M
Literary contributions suggesting various passive and active control methods to eliminate or effectively reduce these unwanted friction-induced vibrations are extensive. Among the passive control methods,high frequency tangential excitation for controlling these self-excited vibrations is proposed by Thomsen [3] and is more elaborately studied by Feeny and Moon [4] through experimentation. Chatterjee [5] also uses this technique on sophisticated microscopic models. Use of dynamic vibration absorbers are investigated by Popp and Rudolph [6] and Chatterjee [7]. In the domain of the active vibration control methods, time-delayed feedback control methods have gained a remarkable popularity. Elmer [8] considers the time-delayed state feedback control of normal load. Das and Mallik [9] use an experimental setup of frictiondriven system to propose the efficacy of a time-delayed PD feedback control of forced vibration as well as friction-induced vibrations due to Stribeck effect. Chatterjee [10] presents the time-delayed displacement feedback control of different types of friction-induced instabilities. Chatterjee and Mahata [11] introduces a novel method called ‘recursive time-delayed acceleration feedback control’ where the control force is devised as an infinite weighted sum of the acceleration of the vibrating mass measured at regular intervals. Heckl and Abrahams [12] proposed a method of active control of frictioninduced vibration by superimposition of a tangential force on the friction force. Chatterjee [13] presents a novel active control method by normal load modulation depending upon the state of the oscillatory system. Stribeck effect is considered to cause the friction-induced vibrations and Lyapunov's second method is used to derive the basic control law. Chatterjee [14] makes use of the actively controlled impulsive forces generated by expansion and contraction of a PZT (lead zirconium titanate) to control these self-excited oscillations. von Wagner et al. [15] demonstrated an active control technique of suppressing squeal in floating calliper disc brake by the use of “smart pads”. This paper considers the efficacy of a tangential control in arresting the self-excited vibrations generated due to the velocity-weakening friction force. An archetypal model of a spring-mass-damper system on a moving belt is studied where the acceleration of the mass is fed to a controller which in turn actuates the tangential control through an actuator fitted to the base of the vibratory system. Linear stability analysis and nonlinear analysis of the friction-system is performed by describing the friction phenomena as given by the exponential friction model proposed by Hinrichs et al. [16]. Local and global stability boundaries are constructed in the plane of control parameters. Direct numerical simulations are shown to substantiate the analytically obtained results. Pole crossover optimization is performed to design the controller parameters and their dependence with the system parameters are found henceforth.
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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2. Mathematical model Fig. 1 depicts the phenomenological model of a benchmark problem of friction-induced oscillations in mechanical systems mainly attributed to the Stribeck effect. Here the mass M attached to a spring of stiffness K and a damper with damping coefficient C models the mechanical system. The mass rests on a belt moving with a constant speed of vb and is held in contact by applying a constant normal load N0. The controller receives signal from the acceleration sensor (an accelerometer) attached to the mass and in turn controls the displacement of the base of the spring-mass-damper system by an actuator. The equation of motion of the system with the base control reads as MX″ þ C X 0 X 0c þK ðX X c Þ ¼ F; (1a) where F is the friction force acting on the mass and the prime ‘0 ’ denotes differentiation with respect to time t. The controller shown in Fig. 1 consists of a second-order filter of damping ratio ξf and natural frequency ω~ f and a power amplifier of gain kc where as k1 is the sensor gain. Thus, the controlled base displacement Xc is obtained as
X c ¼ kc xf ;
(1b)
where kc is the control gain and xf is the filter variable (non-dimensional) with the second-order filter dynamics given by
x00f þ 2ξf ω~ f x0f þ ω~ 2f xf ¼ k1 X″:
(1c)
The functional form of the friction force, F incorporating the velocity weakening characteristics is considered to follow the exponential model [16] given as 0 F ¼ N0 μ þ Δμe α jðvb x Þj sgn vb X 0 ;
(2)
where μ and Δμ are the minimum kinetic coefficient of friction and the difference between the static friction coefficient and the minimum kinetic friction coefficient, respectively. α* is a model parameter that determines the slope of the frictionvelocity curve in the low velocity range sgn(.) is the standard signum function returning the sign of the argument. The friction model featured by Eq. (2) can be regularized as 0 F ¼ N 0 μþ Δμe α jðvb x Þj tanh β vb X 0 ; where tanh(β*v) is the continuous functional representation of signum function with β*⪢1. The non-dimensionalised forms of Eq. (1) are written as n o 0 0 _ tanh βðv0 y_ Þ y_ þ cy_ þy ckc x_ f kc xf ¼ μþ Δμe αjðv0 yÞj
(3)
(4)
and x€ f þ 2ξf ωf x_ f þω2f xf ¼ k1 y€ ;
(5)
where the ‘over dot’ denotes differentiation with respect to the non-dimensional time τ ¼ ωn t and other non-dimensional ω~ f 0 vb C quantities are y ¼ XX0 ; c ¼ Mω ; vq 0¼ ffiffiffiffiωn X 0 ; α ¼ α ωn X 0 , β ¼ β ωn X 0 ; ωf ¼ ωn ; kc ¼ kc =X 0 and k1 ¼ X 0 k1 with the natural frequency of n N0 K the mechanical system, ωn ¼ M and the reference length, X 0 ¼ Mω2 . n
Fig. 1. Tangential motion control of a spring-mass-damper system on a moving belt.
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 2. a Bifurcation diagram of the uncontrolled system (kc ¼ 0) with v0 as the bifurcation parameter. b Phase plane plot of the uncontrolled system at v0 ¼0.4 showing unstable limit cycle, stable limit cycle and the stable equilibrium point A.
3. Dynamics of the uncontrolled system The non-dimensional equation of the uncontrolled system derived from Eqs. (1a) and (3) reads as n o y€ þ cy_ þy ¼ μþ Δμe αjðv0 y_ Þj tanh βðv0 y_ Þg;
(6)
The dynamics of the uncontrolled system is studied by constructing a bifurcation diagram (Fig. 2a) in AUTO [17] with v0 as the bifurcation parameter. As the belt velocity is increased, the unstable equilibrium (steady-sliding state) undergoes a subcritical Hopf bifurcation at a critical belt velocity and becomes stable. But above this critical velocity, a stable limit cycle oscillation coexists with the locally stable equilibrium which gets annihilated at the Cyclic-Fold bifurcation point upon collision with the unstable limit cycle generated at the subcritical Hopf bifurcation. Beyond this cyclic-fold bifurcation, the equilibrium becomes globally asymptotically stable. A phase plane plot (Fig. 2b) at v0 ¼0.4 shows the co-existence of a stable limit cycle, an unstable limit cycle and a stable equilibrium at A (0.20183,0). The values of the system parameters used to obtain Fig. 2a and b are c ¼ 0:05; μ ¼ 0:2, Δμ ¼ 0:1; α ¼ 10; β ¼ 1000 and these will remain the same for all the subsequent analyses (both analytical and numerical) unless otherwise stated.
4. Dynamics of the controlled system 4.1. Local stability analysis To analyze the local stability of the equilibrium, a linearized model is derived under the assumption that the vibrating mass is slipping on the belt near the equilibrium position. If the velocity of the mass is less than the belt velocity at every instant i.e., v0 4 y_ , then the possibility of stick-slip vibration is ruled out as the reversal of direction of the relative velocity of sliding gets precluded. As v0 4 y_ , one can substitute tanh βðv0 y_ Þ ¼ 1 in Eqs. (4) and (5) and obtain 0 0 y€ þ cy_ þ y ckc x_ f kc xf ¼ μþ Δμe αðv0 y_ Þ ;
(7)
x€ f þ 2ξf ωf x_ f þ ω2f xf ¼ k1 y€
(8)
Eqs. (7) and (8) have static equilibrium ðy€ ¼ 0; y_ ¼ 0; x€ f ¼ 0; x_ f ¼ 0Þ at y0 ¼ μ þ Δμe follows by introducing a coordinate transformation, z ¼ y y0 0 0 z€ þ cz_ þ z ckc x_ f kc xf ¼ Δμe αv0 1 eαz_
αv0
x€ f þ 2ξf ωf x_ f þω2f xf ¼ k1 z€
; xf 0 ¼ 0 and thus, can be modified as (9) (10)
which can be rewritten in the state-variable form as x_ 1 ¼ x2 ;
(11)
0 0 x_ 2 ¼ x1 cx2 Δμe αv0 1 eαx2 þ kc x3 þckc x4 ;
(12)
x_ 3 ¼ x4 ;
(13)
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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0 0 x_ 4 ¼ k1 x1 k1 cx2 þ Δμe αv0 1 eαx2 þ k1 kc ω2f x3 þ ck1 kc 2ξf ωf x4 : where the state-variables are define as x1 ¼ z; x2 ¼ z_ ; x3 ¼ xf ; x4 ¼ x_ f . The Jacobian matrix of the flow at the static equilibrium (now at the origin, i.e. zero), J is derived as 2 3 0 1 0 0 0 0 6 1 7 αv0 c þΔμe :α kc ckc 6 7 7 J¼6 6 0 7 0 0 1 4 5 0 0 αv0 2 :αÞ k1 kc ωf ck1 kc 2ξf ωf k1 k1 ðc Δμe
5
(14)
(15)
The characteristic equation of the Jacobian matrix obtained in Eq. (15) reads as s4 þ a3 s3 þ a2 s2 þa1 s þ a0 ¼ 0;
(16)
where s is a complex variable and a3 ¼ 2ωf ξf þf ckc a2 ¼ 1 þ ω2f þ 2f ξf ωf kc a3 ¼ 2ωf ξf þ f ω2f a0 ¼ ω2f 0
with f ¼ c þ Δμe αv0 :α and the loop-gain, kc ¼ k1 kc . Without loosing any generality one can set k1 ¼1 for subsequent discussions and hence the cotroller gain is the same as the loop gain. The roots of the characteristic Eq. (16) or the eigenvalues of the Jacobian determine the local stability of the equilibrium. If, all the four roots of the characteristic equation, have negative real part, the static equilibrium of the system is said to be locally asymptotically stable. It is observed that a complex conjugate pair of roots with positive real part migrates transversal to the imaginary axis from the RHS s-plane to the LHS s-plane when a bifurcation parameter (say, the control gain, kc) thereby signifying a Hopf bifurcation. The local stability boundary (corresponding to the Hopf bifurcation point) is constructed in the plane of parameters, namely, the non-dimensional belt velocity v0 and the loop-gain kc in Fig. 3. It is observed that the static equilibrium of the uncontrolled system (corresponding to kc ¼0) is locally unstable in the low velocity range. The control system can stabilize the static equilibrium within a specific range of velocity; though, the static equilibrium in the very low velocity range cannot be stabilized by the control. 4.2. Nonlinear analysis The nature of Hopf bifurcation cannot be ascertained by the foregoing linear stability analysis. Thus, a nonlinear analysis is undertaken and to this end, the method of Averaging is employed. The method of Averaging is a pivotal step towards discerning the amplitude and phase of the friction-induced self-excited oscillation undergone by the system and thus investigates the nature of the global behavior away from the trivial equilibrium. It is assumed that the belt velocity is greater
Fig. 3. The bifurcation parameter v0 is varied along the section AB (kc ¼ 0.4) and CD (kc ¼ 1) and the bifurcation parameter kc is varied along the section EF (v0 ¼0.075) and GH (v0 ¼ 0.08). ωf ¼ 1, ξf ¼ 0:25.
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than that of the mass, i.e., v0 4 y_ . Eqs. (7) and (8) are recast as y€ þω2 y ¼ εh1 y; xf ; y_ ; x_ f x€ f þ ω2 xf ¼ εh2 y; xf ; y_ ; x_ f ;
(17) (18)
where ω is the arbitrary frequency of oscillation of the system, ε is an infinitesimally small parameter and ~ 1 y þ c~ kc x_ f þ k~ c xf þf ðy_ Þ ; h1 y; xf ; y_ ; x_ f ¼ Ω ~ 2 xf 2ξ~ f ωf x_ f y þ c~ kc x_ f þ k~ c xf þf ðy_ Þ h2 y; xf ; y_ ; x_ f ¼ Ω ε and
n o f ðy_ Þ ¼ μ~ þΔμe ~ αðv0 y_ Þ c~ y_ ;
with 2 ~ 1 ; kc ¼ εk~ c ; c ¼ εc~ ; ω2 ωf 2 ¼ εΩ ~ 2 ; ξf ¼ εξ~ f ; μ ¼ εμ; ω 1 ¼ εΩ ~ Δμ ¼ εΔμ~ Now, let the solutions to the Eqs. (17) and (18) be assumed as y ¼ A1 ðτÞ sin ωτ þϕ1 ðτÞ ; y_ ¼ A1 ðτÞω cos ωτ þ ϕ1 ðτÞ xf ¼ A2 ðτÞ sin ωτ þϕ2 ðτÞ ;
x_ f ¼ A2 ðτÞω cos ωτ þ ϕ2 ðτÞ
(19) (20)
where Ai(i ¼1,2) are the amplitudes of oscillations and ϕi are the phases of the oscillations of the system and the filter while the oscillations are assumed to be slowly varying functions of time, τ. Using these two solutions the following 1st order differential equations are obtained ε A_ i ¼ hi y; xf ; y_ ; x_ f cos ωτ þ ϕi ðτÞ (21) ω ε Ai ϕ_ i ¼ hi y; xf ; y_ ; x_ f sin ωτ þϕi ðτÞ ω
(22)
where i ¼1, 2 Implementation of the standard methods of averaging finally leads us to the following amplitude and phase equations, also known as slow flow equations Z 2π=ω ε A_ i ¼ hi y; xf ; y_ ; x_ f cos ωτ þ ϕi dτ (23) 2π 0 _i ¼ ε ϕ 2πAi
Z
2π=ω 0
hi y; xf ; y_ ; x_ f sin ωτ þ ϕi dτ
(24)
Thus, performing mathematical manipulations one can easily derive the following expressions of amplitude and phase relations A_ 1 ¼ g 1 ðAi ; ϕ1 ϕ2 ; ωÞ kc A2 γ ckc A2 c cos ϕ1 ϕ2 sin ϕ1 ϕ2 þ I 1 ðαωAÞ A1 ¼ ω 2 2 2ω A_ 2 ¼ g 2 ðAi ; ϕ1 ϕ2 ; ωÞ ckc A2 γ cA1 A1 sin ϕ1 ϕ2 þ þ I 1 ðωαA1 Þ cos ϕ1 ϕ2 cos ϕ1 ϕ2 ¼ ξf ωf A2 ω 2ω 2 2 _ 1 ¼ g 3 ðAi ; ϕ1 ϕ2 ; ωÞ ϕ kc A2 ω 1 ckc A2 sin ϕ1 ϕ2 cos ϕ1 ϕ2 ¼ 2ω 2A1 2ωA1
¼
ω
2
ω2f
2ω
2
(25)
(26)
(27)
_ 2 ¼ g 4 ðAi ; ϕ1 ϕ2 ; ωÞ ϕ
þ
kc cA1 A1 γ þ cos ϕ1 ϕ2 I 1 ðωαA1 Þ sin ϕ1 ϕ2 sin ϕ1 ϕ2 2ωA2 2ω ωA2 2A2
(28)
where I1 (.) is the modified Bessel's function of order one and γ ¼ Δμe αv0 . The steady state solutions (correspond to limit cycle oscillations of the original system) of the slow-flow Eqs. (25)–(28) can be obtained by solving following sets of equations adopting arbitrary initial guesses g j ðAi ; φ1 φ2 ; ωÞ ¼ 0
where
j ¼ 1; 2; 3; 4
(29)
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 4. Plot of amplitudes of unstable limit cycles for different values of kc for a velocity range selected from the locally stable region. ωf ¼ 1, ξf ¼ 0:25.
Fig. 5. Bifurcation diagram obtained from AUTO showing subcritical Hopf bifurcation at v0 ¼0.0673 and Cyclic-Fold bifurcation at v0 ¼ 0.0767 with kc ¼ 0.4. ωf ¼ 1, ξf ¼ 0:25.
_ ¼ g 5 ðAi ; ϕ1 ϕ2 ; ωÞ For simplification, one can assume so that ϕ1 ϕ2 ¼ ϕ so that ϕ_ 1 ϕ_ 2 ¼ ϕ The above-mentioned equations are solved in MATLAB and the eigenvalues of the Jacobian matrix, J1 in Eq. (30) is determined. 2 ∂g ∂g ∂g 3 1
∂A
6 ∂g 1 6 J 1 ¼ 6 ∂A21 4 ∂g5 ∂A1
1
∂A2 ∂g2 ∂A2 ∂g5 ∂A2
1
∂ϕ ∂g2 ∂ϕ ∂g5 ∂ϕ
7 7 7 5
(30)
The amplitudes obtained from the Eq. (29) and the positive sign of the real parts of the corresponding dominant eigenvalues of the Jacobian matrix (Eq. (30)) suggests the existence of an unstable limit cycle around the static equilibrium. Thus, the Hopf bifurcation of the static equilibrium is established as subcritical. The amplitudes of the unstable limit cycles obtained for different values of kc is plotted in Fig. 4 for a velocity range selected from the locally stable region (see Fig. 3). It is also observed that the unstable limit cycle vanishes beyond particular values of kc and v0. This clearly indicates the existence of a globally stable limit cycle (will be confirmed by numerical simulation). It is apparent from Fig. 4 that the amplitude of unstable limit cycle increases with the increase in control parameters (v0 or kc or both)and collides with the stable limit cycle at the saddle- node bifurcation (or Cyclic-Fold bifurcation) point and both of the limit cycles vanish leaving only the stable static equilibrium. The critical values of kc and v0 for which the unstable limit cycle given by the averaging analysis ceases to exist marks the cyclic fold bifurcation and hence the global stability boundary as depicted in Fig. 3. Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 6. Bifurcation diagram obtained from AUTO at kc ¼ 1 (with v0 as the bifurcation parameter) showing subcritical Hopf bifurcation at v0 ¼ 0.0668 and Cyclic-Fold bifurcation at v0 ¼ 0.0742. ωf ¼ 1, ξf ¼ 0:25.
Fig. 7. Bifurcation diagram obtained from AUTO at v0 ¼ 0.08 (with kc as the bifurcation parameter) showing subcritical Hopf bifurcation at kc ¼ 0.209 and Cyclic-Fold bifurcation at kc ¼ 0.291. ωf ¼ 1, ξf ¼ 0:25.
5. Numerical results The bifurcation diagrams of the control systems are numerically constructed using AUTO as one varies the control parameter, here, the loop-gain kc and non-dimensional belt velocity v0. To this end, four sections AB, CD, EF and GH as depicted in Fig. 3 are selected and the bifurcation diagrams are numerically constructed along these sections. Sections AB and CD correspond to the loop-gain, kc ¼ 0.4 and kc ¼ 1 respectively, along which the bifurcation parameter v0 is varied and the bifurcation diagram is constructed. Bifurcation diagrams are also drawn for varying kc along sections EF (with a fixed v0 ¼ 0.075) and GH (with a fixed v0 ¼0.08). The phase portraits obtained from numerical simulations substantiate the results.
5.1. Bifurcation along AB The bifurcation diagram with respect to the bifurcation parameter v0is constructed at kc ¼ 0.4 in Fig. 5. The trivial equilibrium is seen to lose stability due to a subcritical Hopf bifurcation at a certain value of the non-dimensionalised belt velocity, v0(at v0 ¼ 0.0673). Above this critical value of velocity, a stable limit cycle pertaining to stick-slip oscillations coexists with an unstable limit cycle and the locally stable steady sliding state. As one further increases the velocity, the Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 8. Bifurcation diagram obtained from AUTO at v0 ¼ 0.075 (with kc as the bifurcation parameter) showing subcritical Hopf bifurcation at kc ¼ 0.23 and Cyclic-Fold bifurcation at kc ¼ 0.237. ωf ¼ 1, ξf ¼ 0:25.
Fig. 9. Phase portraits obtained at v0 ¼ 0.075 and (a) kc ¼ 0.22 (b) kc ¼ 0.234 (c) kc ¼ 0.25 with initial conditions (3, 8) (d) kc ¼ 0.25 with initial conditions (0.9, 0.11) (e) kc ¼ 0.46 with initial conditions (0.1, 0.11) (f) kc ¼ 0.46 with initial conditions (10, 8). ωf ¼ 1, ξf ¼ 0:25.
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stable and unstable limit cycle collide at a particular velocity (Cyclic-Fold bifurcation) and vanishes, making the system globally stable beyond that critical belt velocity (v0 ¼0.0767). 5.2. Bifurcation along CD An identical Hopf bifurcation is obtained when v0 is varied along CD at kc ¼ 1 and the bifurcation diagram is shown in Fig. 6. Here, as one increases the velocity, Hopf bifurcation occurs at v0 ¼ 0.0668 and then the equilibrium becomes globally stable after the Cyclic-Fold bifurcation at v0 ¼ 0.0742. 5.3. Bifurcation along GH Fig. 7 shows the bifurcation diagram at v0 ¼0.08 while the bifurcation parameter kc is varied along GH. The equilibrium undergoes a subcritical Hopf bifurcation at a certain value of the control gain as we increase the gain value. Below this critical value of control gain (kc ¼ 0.209), a stable limit cycle solution representing stick-slip oscillations is separated from the locally stable steady sliding state by an unstable limit cycle solution. As one further increases the control gain the stable and unstable limit cycle merge rendering the system globally stable beyond that critical control gain (kc ¼ 0.291). 5.4. Bifurcation along EF The bifurcation diagram (Fig. 8) constructed in AUTO at v0 ¼ 0.075 while the bifurcation parameter kc is varied along EF. It may be seen that after the subcritical Hopf bifurcation at kc ¼ 0.23, the locally stable region is seen to exist for a very little span of control gain-value i.e., up to the Cyclic-Fold bifurcation at kc ¼ 0.237. But the numerical simulation results from the MATLAB SIMULINK model confirm a larger locally stable region and that the globally stable region extends only beyond kc ¼ 0.46. Phase portraits are shown in Fig. 9 substantiating the existence of a stable limit cycle solution (Fig. 9c) along with a stable equilibrium (Fig. 9d) at kc ¼ 0.25 while the system reaches an equilibrium (Fig. 9e and f) for any choice of initial conditions beyond kc ¼ 0.46. The bifurcation results so far obtained are illustrated in the plane of control parameters, v0 and kc, in Fig. 10. Bifurcation results obtained from AUTO satisfies the numerical simulations carried out in the MATLAB SIMULINK model in all the cases except when kc is varied along section EF i.e., at v0 ¼0.075. It can be easily noticed that the MATLAB SIMULINK simulation at v0 ¼0.075 present a better result as it monotonically follows the numerically obtained global stability boundary i.e., the line traced by the Cyclic-Fold bifurcation points. 5.5. Comparison of analytical and numerical results The bifurcation results so far obtained are for four particular sections which are arbitrarily selected. A more elaborate comparison is drawn by studying a greater region of the plane of control parameters. Numerically obtained global stability boundary i.e., the boundary traced by the Cyclic-fold bifurcation points obtained from AUTO and validated by numerical
Fig. 10. Hopf bifurcation points and Cyclic-Fold bifurcation points obtained in AUTO and MATLAB SIMULINK model are shown along with the local stability boundary obtained from linear stability analysis and global stability boundary constructed by the method of Averaging in the plane of control parameters. ωf ¼ 1, ξf ¼ 0:25. ▼ – Hopf bifurcations points obtained from AUTO. ▲ – Cyclic-Fold bifurcation points obtained from AUTO. ■ – Hopf bifurcation point obtained from simulations in MATLAB SIMULINK model at v0 ¼0.075. ● – Cyclic-Fold bifurcation point obtained from simulations in MATLAB SIMULINK model at v0 ¼ 0.075.
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Fig. 11. Comparison of the global stability boundary obtained analytically by the Method of Averaging and numerical global stability boundary traced by the Cyclic Fold bifurcation points from AUTO. ωf ¼ 1, ξf ¼ 0:25.
Fig. 12. Comparison of uncontrolled and controlled systems with v0 as the bifurcation parameter. kc ¼ 0.4. ωf ¼ 1, ξf ¼ 0:25. HB-Hopf Bifurcation (Subcritical), CF-Cyclic-Fold bifurcation.
simulations in MATLAB SIMULINK model and analytical global stability boundary (obtained from the method of Averaging) are shown in Fig. 11 which profoundly establishes the efficacy of the analyses to determine a globally stable region for any choice of the control parameter, i.e., the control gain from the globally stable region effectively results in a stable equilibrium or in other words, a stable steady sliding condition of the equilibrium (X_ ¼ 0) is obtained. In this context of comparing numerical and analytical results, Fig. 12 is presented which draws a comparison between the amplitudes obtained from all the nonlinear analyses (both analytical and numerical) for a particular control gain within a definite range of belt velocity. Fig. 12 is actually a bifurcation diagram ( constructed in AUTO) at kc ¼ 0.4 with v0 as the bifurcation parameter in which the amplitudes of the stable limit cycle is compared to that obtained from the numerical simulations in MATLAB SIMULINK model and the unstable limit cycle amplitude is compared with that obtained by the method of Averaging. The amplitude of the uncontrolled system for the selected velocity range is also shown to incorporate a better understanding of the control effectiveness. From Fig. 12 it is quite conspicuous that though by the implementation of control, one cannot completely eliminate the stick-slip oscillations in the low velocity range, but the amplitude of the same can be significantly reduced. The time- history plot (Fig. 13) obtained from the MATLAB SIMULINK model depicts the reduction of stick-slip amplitude when the control gain kc ¼ 0.4 is applied at 512.98 seconds. The static equilibrium of the system remains fixed at y¼ 0.25273. Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 13. Time-history plot at v0 ¼ 0.064 showing the control efficacy as control gain kc ¼ 0.4 is applied at 512.98 seconds.
Fig. 14. Optimal stability region(shaded) constructed in the plane of control parameters by pole crossover design.
6. Optimization of parameters 6.1. Pole crossover design The filter parameters are optimally chosen by the crossover design method in which the poles of the system are placed in a suitable location in the s-plane so as to achieve faster attenuation of the transient response (closer to the real axis) and a greater relative stability (away from the imaginary axis in the LHS s-plane). Taking the non-dimensional belt velocity and control gain as independently chosen parameters, the filter parameters are optimized for achieving two identical pairs of complex conjugate poles of the closed loop system. Thus, the structural and filter modes are merged by this optimization criterion. The poles of the system, i.e., the four roots of the characteristic Eq. (16) are placed in a position such that the real part of all the roots are equal which is seen to maximize the absolute values of the real parts of the poles. Let the identical complex conjugate pair of poles be placed at λ1 and λ2 where λ1;2 ¼ a 7 ib Thus, the characteristic equation of the system can be written as 2 2 2 2 ðs λ1 Þ2 ðs λ2 Þ2 ¼ s4 4as3 þ 6a2 þ 2b s2 4aða2 þb Þs þ a2 þ b ¼0
(31)
Comparing Eq. (31) with the characteristic equation of the linearized model obtained in Eq. (16) obtained in Section 4.1 one can write 4a ¼ 2p1 þ q
(32)
2 6a2 þ2b ¼ p2 2f p1 þ 1 kc
(33)
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2
4aða2 þ b Þ ¼ 2p1 f p2
(34)
2
(35)
ða2 þ b Þ2 ¼ p2 αv0
p2 ¼ ω2f ,
where p1 ¼ ωf ξf , q ¼ f ckc , f ¼ c þ Δμe :α Solving Eqs. (32)–(35) to eliminate a and b, we get !
2
2 f fq q 2 2 4 ω4f þ 4 f ω3f þ f þkc 6 ω2f þ ðf q 2kc þ 4Þωf þ þkc 1 ¼ 0 2 4 4
(36)
This equation is quadratic in ωf and thus four values of the optimal filter frequency, (ωf)opt are obtained and a contour plot is constructed in the plane of control parameters in Fig. 14. The shaded region satisfies the cross-over optimization criteria by including only the real values of (ωf)opt, (ξf)opt, a and b as well as Re (λ1,λ2) ¼a o0, i.e., the real part of all the merged poles are negative in this shaded region. The suitable value of ðωf Þopt satisfying the stability criterion is chosen and optimal filter damping value, ðξf Þopt and the values of the real and imaginary parts of the identical complex conjugate poles are obtained from Eqs. (37–39). f ωf þ q ðξf Þopt ¼ 2 1 ωf a¼
b¼
2ξf ωf þ q 4
(37)
(38)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2f 2f ξf ωf 6a2 þ1 kc
(39)
2
Now, if one attempts to an optimized value of the control gain instead of assigning arbitrary values as in the previous optimization process it is required that all the poles are merged at the same point on the real axis i.e., all poles are real and equal. Let the identical complex conjugate pair of poles be placed at λ where λ ¼ a. Thus, the characteristic equation in Eq. (31) reduces to ðs λÞ4 ¼ s4 4as3 þ 6a2 s2 4a3 s þa4 ¼ 0;
(40)
4a ¼ 2p1 f ckc
(41)
6a2 ¼ p2 2f p1 þ1 kc
(42)
4a3 ¼ 2p1 f p2
(43)
a4 ¼ p2
(44)
where
Solving Eqs. (40)–(44) to eliminate a and b, we get 7=2 5=2 3=2 2 4c 4cf 4f ωf þ ð16 16cf Þω3f 24cωf 16ω2f þ ð4f þ4cÞωf ¼ 0
(45)
Eq. (45) is solved to obtain real values of (ωf)opt and expressions for (ξf)opt, (kc)opt and a are obtained from the following equations pffiffiffiffiffiffi f ωf 4 ωf (46) ξf opt ¼ 2 2 ðkc Þopt ¼ 1 f ωf 2 þ 4f ωf 3=2 6ωf þ 1
(47)
Table 1 Optimal control parameters at different belt velocities. Non-dimensionalised belt velocity, v0
Optimal filter frequency, (ωf)opt
Optimal filter damping, (ξf)opt
Optimal control gain, (kc)opt
a
0.01 0.02 0.05 0.08 0.1
0.9885 0.9863 0.9797 0.9735 0.9698
2.4109 2.3653 2.2522 2.1677 2.1237
8.0283 7.5317 6.3742 5.5786 5.1876
1.0782 1.0684 1.0438 1.0250 1.0151
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Fig. 15. Comparison of uncontrolled and controlled systems with v0 as the bifurcation parameter. kc ¼ 0.8, (ωf)opt ¼ 0.9779, ðξf Þopt ¼0.5188. HB-Hopf Bifurcation (Subcritical), CF-Cyclic-Fold bifurcation.
Fig. 16. Comparison of numerical (from AUTO and MATLAB SIMULINK model) and analytical (from Averaging method) results with kc as the bifurcation parameter. v0 ¼ 0.02, (ωf)opt ¼ 0.9779,ðξf Þopt ¼ 0.5188. HB-Hopf Bifurcation (Subcritical), CF-Cyclic-Fold bifurcation. The stable limit cycle amplitude of the uncontrolled system at v0 ¼ 0.02 is 0.054.
a¼
f þckc 2ωf ξf 4
(48)
The optimal values of the control gain, filter damping and filter frequency corresponding to different choices of velocity are obtained for aas a negative real numberand tabulated below. It can be observed from Table 1 that with the increase in belt velocities the magnitudes of (ωf)opt, (ξf)opt, (kc)opt and a decreases. 6.2. Efficacy of optimization It is observed that the system can be stabilized at any belt velocity provided the filter parameters are suitably chosen according to the procedure described above. As very low belt velocity region is the most difficult to control, a detailed analysis of that region is enclosed to establish the importance of optimization of parameters. For a belt velocity as low as v0 ¼0.02, a control gain value kc ¼ 0.8 can be selected from the shaded region of Fig. 14 for which an optimal set of filter parameters are obtained as (ωf)opt ¼0.9779 and ðξf Þopt ¼0.5188. Now, for this set of parameter values, two bifurcation diagrams (Figs. 15 and 16) are obtained in AUTO as v0 and kc as the bifurcation parameters, respectively. Fig. 15 compares the results obtained from numerical simulations in MATLAB SIMULINK model with that Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 17. Time-history plot at v0 ¼0.016 showing the control efficacy as control gain kc ¼ 0.8 is applied at 435.983 seconds.
Fig. 18. Variation of optimum filter parameters with v0 and kc.
obtained from the method of Averaging and also depicts a significant reduction in the stick-slip amplitude for the selected velocity range. The bifurcation diagram with the particular set of parameters with bifurcation parameter kc is shown in Fig. 16. Here, too amplitude values obtained from MATLAB SIMULINK model and method of Averaging are compared. The amplitude of stick-slip oscillations at v0 ¼ 0.02 for uncontrolled system is observed to be 0.05335 and thus Fig. 16 also shows an effective reduction in amplitude due to the optimization. Fig. 17 depicts the time-history plot obtained from the MATLAB SIMULINK model showing the reduction of stick-slip amplitude when the control gain kc ¼ 0.8 is applied at 435.983 seconds to an uncontrolled system with belt velocity, v0 ¼0.016.
6.3. Relation between control parameters and optimal filter parameters The dependence of optimum filter parameters on the non-dimensional belt velocity v0 and the control gain kc are obtained and plotted in Fig. 18. One can easily notice that the optimum filter frequency ðωf Þopt does not vary much with v0 but decreases slowly with increasing values of kc. The optimum filter damping ðξf Þopt increases for very low values of v0 but remains constant for higher belt velocity while it increases by leaps and bounds for even a very small increase in control gain.
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7. Conclusions A phenomenological mass on belt model is considered to model stick-slip oscillations and the efficacy of a tangential control by a controller consisting of a second-order filter and power amplifier is extensively studied. Linear stability analysis around the trivial equilibrium is performed to arrive at the local stability boundary which is constructed in the plane of two dominant parameters, non-dimensionalised belt velocity and loop-gain. The complex conjugate eigenvalues obtained after linearization of the system are observed to cross the imaginary axis thus, indicating a Hopf bifurcation. Nonlinear stability analysis of the region is performed to understand the dynamics away from the trivial equilibrium as the belt velocity is considered greater than the velocity of the mass at every instant of time. This only includes slipping oscillations around the static equilibrium. The method of Averaging is employed for the nonlinear analysis and thus, the coexistence of stable limit cycle (obtained from numerical simulations) with the stable equilibrium separated by an unstable limit cycle (obtained from the Averaging analysis) is observed i.e., the Hopf bifurcation is observed to be subcritical in nature. Due to this, the existence of a global stability boundary is investigated and observed to exist within the local stability boundary. Bifurcation diagrams with non-dimensionalised belt velocity and loop-gain as the bifurcation parameters are obtained with AUTO and the numerical simulations carried out in MATLAB SIMULINK model substantiate the analytically obtained results. Finally, a comparison between the results obtained by analytical and numerical methods and simulations are carried out. The filter parameters are optimized by choosing belt velocity and control gain as independent parameters by pole crossover design. An optimal stability region is constructed in the plane of control parameters (belt velocity and control gain). Another optimization procedure is carried out with belt velocity as the only independent parameter to arrive at optimal values of filter frequency, filter damping and control gain. As the low velocity region poses the prime difficulty in control implementation, a set of optimal filter parameters corresponding to a very low velocity is selected from the optimal stability region and a detailed bifurcation analyses is performed with velocity and control gain as the bifurcation parameters. Stable and unstable limit cycle amplitudes obtained from MATLAB model and Averaging method, respectively are compared with that obtained in AUTO. The significant reduction in stick-slip amplitude in the low velocity region is also indicated. Finally, variations of filter parameters with change in control parameters are illustrated to gain an insight into the process of choosing optimal values in different circumstances.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
R.A. Ibrahim, Friction-induced vibration, chatter, squeal, and chaos Part II: dynamics and modeling, Applied Mechanics Reviews 47 (7) (1994) 227–253. E.J. Berger, Friction modeling for dynamic system simulation, Applied Mechanics Reviews, 55, , 2002, 535–577. J.J. Thomsen, Using fast vibrations to quench friction-induced oscillations, Journal of Sound and Vibration 228 (5) (1999) 1079–1102. B.F. Feeny, F.C. Moon, Quenching stick–slip chaos with dither, Journal of Sound and Vibration 237 (1) (2000) 173–180. S. Chatterjee, T.K. Singha, S.K. Karmakar, Effect of high-frequency excitation on a class of mechanical systems with dynamic friction, Journal of Sound and Vibration 269 (2004) 61–89. Karl Popp, Martin Rudolph, Dynamic vibration absorber for friction-induced oscillations, IUTAM Symposium on Chaotic Dynamics and Control of Systems and Processes in MechanicsSolid Mechanics and its Applications, Volume 122, 2005, pp. 419–427. S. Chatterjee, On the design criteria of dynamic vibration absorbers for controlling friction-induced oscillations, Journal of Vibration and Control 14 (2008) 397–415. first published on December 20, 2007. F.-J. Elmer, Controlling friction, Physical Review E 57 (1998) 4903–4906. J. Das, A.K. Mallik, Control of friction driven oscillation by time-delayed state feedback, Journal of Sound and Vibration 297 (2006) 578–594. S. Chatterjee, Time-delayed feedback control of friction-induced instability, International Journal of Non-Linear Mechanics 42 (2007) 1127–1143. S. Chatterjee, P. Mahata, Controlling friction-induced instability by recursive time-delayed acceleration feedback, Journal of Sound and Vibration 328 (2009) 9–28. M.A. Heckl, D. Abrahams, Active control of friction driven oscillations, Journal of Sound and Vibration 193 (1) (1996) 417–426. S. Chatterjee, Non-linear control of friction-induced self-excited vibration, International Journal of Non-Linear Mechanics 42 (2007) 459–469. S. Chatterjee, On the principle of impulse damper: A concept derived from impact damper, Journal of Sound and Vibration 312 (2008) 584–605. Wagner, U. von, Hochlenert, D. Jearsiripongkul, T. Hagedorn, P., Active control of brake squeal via ‘smart pads’, SAE Technical Papers 2004-01-2773. N. Hinrichs, M. Oestreich, K. Popp, On the modelling of friction oscillators, Journal of Sound and Vibration 216 (3) (1998) 435–459. Doedel, E.J. AUTO97, Continuation and bifurcation software for ordinary differential equations, 1998.
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Tangential acceleration feedback control of friction induced vibration Jyayasi Nath, S. Chatterjee n Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, West Bengal, India
a r t i c l e i n f o
abstract
Article history: Received 9 December 2015 Received in revised form 6 May 2016 Accepted 11 May 2016 Handling Editor: D.J Wagg
Tangential control action is studied on a phenomenological mass-on-belt model exhibiting friction-induced self-excited vibration attributed to the low-velocity drooping characteristics of friction which is also known as Stribeck effect. The friction phenomenon is modelled by the exponential model. Linear stability analysis is carried out near the equilibrium point and local stability boundary is delineated in the plane of control parameters. The system is observed to undergo a Hopf bifurcation as the eigenvalues determined from the linear stability analysis are found to cross the imaginary axis transversally from RHS s-plane to LHS s-plane or vice-versa as one varies the control parameters, namely non-dimensional belt velocity and the control gain. A nonlinear stability analysis by the method of Averaging reveals the subcritical nature of the Hopf bifurcation. Thus, a global stability boundary is constructed so that any choice of control parameters from the globally stable region leads to a stable equilibrium. Numerical simulations in a MATLAB SIMULINK model and bifurcation diagrams obtained in AUTO validate these analytically obtained results. Pole crossover design is implemented to optimize the filter parameters with an independent choice of belt velocity and control gain. The efficacy of this optimization (based on numerical results) in the delicate low velocity region is also enclosed. & 2016 Elsevier Ltd. All rights reserved.
Keywords: Self-excited oscillation Subcritical Hopf bifurcation Averaging Stick-slip oscillation Pole crossover design
1. Introduction The deleterious aspect of friction-induced instability is common to most mechanical systems, e.g. bearings, transmissions, hydraulic and pneumatic cylinders, valves, brakes, lead screws, oil well drill-strings and wheels and nevertheless in control systems like in various drive systems, high-precision servo mechanisms, robots, pneumatic and hydraulic systems, in position control systems and anti-lock brakes for cars, etc. So, the undesirable self-excited vibrations induced by non-linear character of friction in linearly stable systems has motivated various researchers worldwide to investigate into the inherent laws of friction and unanimously put forth various models that can perfectly imitate the frictional phenomena in original systems as well as effective methods to control this undesirable phenomenon that leads to significant inconvenience. Friction-induced self-excited vibrations are attributed to mainly three mechanisms, namely, the velocity weakening characteristics of friction force which is also known as the Stribeck effect, mode-coupling and sprag-slip instabilities. A detailed review of these basic mechanisms is assembled in [1,2]. n
Corresponding author. E-mail address: [email protected] (S. Chatterjee).
http://dx.doi.org/10.1016/j.jsv.2016.05.020 0022-460X/& 2016 Elsevier Ltd. All rights reserved.
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Nomenclature A1 ; A2
amplitudes of non-linear oscillations of the system and the filter respectively C viscous damping coefficient C c ¼ Mω non-dimensionalized viscous damping n coefficient F friction force between mass and belt J, J1 Jacobian of the linearized flow around the equilibrium, Jacobian of the non-linear flow around the equilibrium K oscillator stiffness kc gain of power amplifier kc' ¼ kc =X 0 non-dimensionalised gain of power amplifier k1 sensor gain k1 ¼ X 0 k1 non-dimensionalised sensor gain 0 kc ¼ k1 kc non-dimensionalised loop gain oscillator mass M normal load N0 time t vb belt velocity v0 ¼ ωvn Xb 0 Non-dimensionalised belt velocity X displacement of oscillator mass Xc controlled displacement of base N0 X 0 ¼ Mω 2 reference displacement n
xf non-dimensional filter variable y ¼ XX0 non-dimensional displacement of mass z ¼ y y0 coordinate transformation where (y0 ,xf 0 ) is the static equilibrium of linearized system α ; α ¼ α ωn X 0 model parameter that determines the slope of the friction-velocity curve in the low velocity range, non-dimensional form of the same β 4 1; β ¼ β ωn X 0 a very large positive quantity, nondimensional form of the same γ Δμe αv0 ε«1 an infinitesimally small quantity λ 1, λ 2 identical complex conjugate pair of poles used in pole cross-over design μ, Δμ minimum kinetic coefficient of friction, the difference between the static friction coefficient and the minimum kinetic friction coefficient ξf damping ratio of second-order filter τ ¼ ωn t non-dimensionalised time ϕ1 ; ϕ2 phases of non-linear oscillations of the system and the filter respectively ω~ f ; ωf natural frequency of the second-order filter, qffiffiffiffi non-dimensional form of the same K natural frequency of the mechanical system ωn ¼ M
Literary contributions suggesting various passive and active control methods to eliminate or effectively reduce these unwanted friction-induced vibrations are extensive. Among the passive control methods,high frequency tangential excitation for controlling these self-excited vibrations is proposed by Thomsen [3] and is more elaborately studied by Feeny and Moon [4] through experimentation. Chatterjee [5] also uses this technique on sophisticated microscopic models. Use of dynamic vibration absorbers are investigated by Popp and Rudolph [6] and Chatterjee [7]. In the domain of the active vibration control methods, time-delayed feedback control methods have gained a remarkable popularity. Elmer [8] considers the time-delayed state feedback control of normal load. Das and Mallik [9] use an experimental setup of frictiondriven system to propose the efficacy of a time-delayed PD feedback control of forced vibration as well as friction-induced vibrations due to Stribeck effect. Chatterjee [10] presents the time-delayed displacement feedback control of different types of friction-induced instabilities. Chatterjee and Mahata [11] introduces a novel method called ‘recursive time-delayed acceleration feedback control’ where the control force is devised as an infinite weighted sum of the acceleration of the vibrating mass measured at regular intervals. Heckl and Abrahams [12] proposed a method of active control of frictioninduced vibration by superimposition of a tangential force on the friction force. Chatterjee [13] presents a novel active control method by normal load modulation depending upon the state of the oscillatory system. Stribeck effect is considered to cause the friction-induced vibrations and Lyapunov's second method is used to derive the basic control law. Chatterjee [14] makes use of the actively controlled impulsive forces generated by expansion and contraction of a PZT (lead zirconium titanate) to control these self-excited oscillations. von Wagner et al. [15] demonstrated an active control technique of suppressing squeal in floating calliper disc brake by the use of “smart pads”. This paper considers the efficacy of a tangential control in arresting the self-excited vibrations generated due to the velocity-weakening friction force. An archetypal model of a spring-mass-damper system on a moving belt is studied where the acceleration of the mass is fed to a controller which in turn actuates the tangential control through an actuator fitted to the base of the vibratory system. Linear stability analysis and nonlinear analysis of the friction-system is performed by describing the friction phenomena as given by the exponential friction model proposed by Hinrichs et al. [16]. Local and global stability boundaries are constructed in the plane of control parameters. Direct numerical simulations are shown to substantiate the analytically obtained results. Pole crossover optimization is performed to design the controller parameters and their dependence with the system parameters are found henceforth.
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2. Mathematical model Fig. 1 depicts the phenomenological model of a benchmark problem of friction-induced oscillations in mechanical systems mainly attributed to the Stribeck effect. Here the mass M attached to a spring of stiffness K and a damper with damping coefficient C models the mechanical system. The mass rests on a belt moving with a constant speed of vb and is held in contact by applying a constant normal load N0. The controller receives signal from the acceleration sensor (an accelerometer) attached to the mass and in turn controls the displacement of the base of the spring-mass-damper system by an actuator. The equation of motion of the system with the base control reads as MX″ þ C X 0 X 0c þK ðX X c Þ ¼ F; (1a) where F is the friction force acting on the mass and the prime ‘0 ’ denotes differentiation with respect to time t. The controller shown in Fig. 1 consists of a second-order filter of damping ratio ξf and natural frequency ω~ f and a power amplifier of gain kc where as k1 is the sensor gain. Thus, the controlled base displacement Xc is obtained as
X c ¼ kc xf ;
(1b)
where kc is the control gain and xf is the filter variable (non-dimensional) with the second-order filter dynamics given by
x00f þ 2ξf ω~ f x0f þ ω~ 2f xf ¼ k1 X″:
(1c)
The functional form of the friction force, F incorporating the velocity weakening characteristics is considered to follow the exponential model [16] given as 0 F ¼ N0 μ þ Δμe α jðvb x Þj sgn vb X 0 ;
(2)
where μ and Δμ are the minimum kinetic coefficient of friction and the difference between the static friction coefficient and the minimum kinetic friction coefficient, respectively. α* is a model parameter that determines the slope of the frictionvelocity curve in the low velocity range sgn(.) is the standard signum function returning the sign of the argument. The friction model featured by Eq. (2) can be regularized as 0 F ¼ N 0 μþ Δμe α jðvb x Þj tanh β vb X 0 ; where tanh(β*v) is the continuous functional representation of signum function with β*⪢1. The non-dimensionalised forms of Eq. (1) are written as n o 0 0 _ tanh βðv0 y_ Þ y_ þ cy_ þy ckc x_ f kc xf ¼ μþ Δμe αjðv0 yÞj
(3)
(4)
and x€ f þ 2ξf ωf x_ f þω2f xf ¼ k1 y€ ;
(5)
where the ‘over dot’ denotes differentiation with respect to the non-dimensional time τ ¼ ωn t and other non-dimensional ω~ f 0 vb C quantities are y ¼ XX0 ; c ¼ Mω ; vq 0¼ ffiffiffiffiωn X 0 ; α ¼ α ωn X 0 , β ¼ β ωn X 0 ; ωf ¼ ωn ; kc ¼ kc =X 0 and k1 ¼ X 0 k1 with the natural frequency of n N0 K the mechanical system, ωn ¼ M and the reference length, X 0 ¼ Mω2 . n
Fig. 1. Tangential motion control of a spring-mass-damper system on a moving belt.
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Fig. 2. a Bifurcation diagram of the uncontrolled system (kc ¼ 0) with v0 as the bifurcation parameter. b Phase plane plot of the uncontrolled system at v0 ¼0.4 showing unstable limit cycle, stable limit cycle and the stable equilibrium point A.
3. Dynamics of the uncontrolled system The non-dimensional equation of the uncontrolled system derived from Eqs. (1a) and (3) reads as n o y€ þ cy_ þy ¼ μþ Δμe αjðv0 y_ Þj tanh βðv0 y_ Þg;
(6)
The dynamics of the uncontrolled system is studied by constructing a bifurcation diagram (Fig. 2a) in AUTO [17] with v0 as the bifurcation parameter. As the belt velocity is increased, the unstable equilibrium (steady-sliding state) undergoes a subcritical Hopf bifurcation at a critical belt velocity and becomes stable. But above this critical velocity, a stable limit cycle oscillation coexists with the locally stable equilibrium which gets annihilated at the Cyclic-Fold bifurcation point upon collision with the unstable limit cycle generated at the subcritical Hopf bifurcation. Beyond this cyclic-fold bifurcation, the equilibrium becomes globally asymptotically stable. A phase plane plot (Fig. 2b) at v0 ¼0.4 shows the co-existence of a stable limit cycle, an unstable limit cycle and a stable equilibrium at A (0.20183,0). The values of the system parameters used to obtain Fig. 2a and b are c ¼ 0:05; μ ¼ 0:2, Δμ ¼ 0:1; α ¼ 10; β ¼ 1000 and these will remain the same for all the subsequent analyses (both analytical and numerical) unless otherwise stated.
4. Dynamics of the controlled system 4.1. Local stability analysis To analyze the local stability of the equilibrium, a linearized model is derived under the assumption that the vibrating mass is slipping on the belt near the equilibrium position. If the velocity of the mass is less than the belt velocity at every instant i.e., v0 4 y_ , then the possibility of stick-slip vibration is ruled out as the reversal of direction of the relative velocity of sliding gets precluded. As v0 4 y_ , one can substitute tanh βðv0 y_ Þ ¼ 1 in Eqs. (4) and (5) and obtain 0 0 y€ þ cy_ þ y ckc x_ f kc xf ¼ μþ Δμe αðv0 y_ Þ ;
(7)
x€ f þ 2ξf ωf x_ f þ ω2f xf ¼ k1 y€
(8)
Eqs. (7) and (8) have static equilibrium ðy€ ¼ 0; y_ ¼ 0; x€ f ¼ 0; x_ f ¼ 0Þ at y0 ¼ μ þ Δμe follows by introducing a coordinate transformation, z ¼ y y0 0 0 z€ þ cz_ þ z ckc x_ f kc xf ¼ Δμe αv0 1 eαz_
αv0
x€ f þ 2ξf ωf x_ f þω2f xf ¼ k1 z€
; xf 0 ¼ 0 and thus, can be modified as (9) (10)
which can be rewritten in the state-variable form as x_ 1 ¼ x2 ;
(11)
0 0 x_ 2 ¼ x1 cx2 Δμe αv0 1 eαx2 þ kc x3 þckc x4 ;
(12)
x_ 3 ¼ x4 ;
(13)
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0 0 x_ 4 ¼ k1 x1 k1 cx2 þ Δμe αv0 1 eαx2 þ k1 kc ω2f x3 þ ck1 kc 2ξf ωf x4 : where the state-variables are define as x1 ¼ z; x2 ¼ z_ ; x3 ¼ xf ; x4 ¼ x_ f . The Jacobian matrix of the flow at the static equilibrium (now at the origin, i.e. zero), J is derived as 2 3 0 1 0 0 0 0 6 1 7 αv0 c þΔμe :α kc ckc 6 7 7 J¼6 6 0 7 0 0 1 4 5 0 0 αv0 2 :αÞ k1 kc ωf ck1 kc 2ξf ωf k1 k1 ðc Δμe
5
(14)
(15)
The characteristic equation of the Jacobian matrix obtained in Eq. (15) reads as s4 þ a3 s3 þ a2 s2 þa1 s þ a0 ¼ 0;
(16)
where s is a complex variable and a3 ¼ 2ωf ξf þf ckc a2 ¼ 1 þ ω2f þ 2f ξf ωf kc a3 ¼ 2ωf ξf þ f ω2f a0 ¼ ω2f 0
with f ¼ c þ Δμe αv0 :α and the loop-gain, kc ¼ k1 kc . Without loosing any generality one can set k1 ¼1 for subsequent discussions and hence the cotroller gain is the same as the loop gain. The roots of the characteristic Eq. (16) or the eigenvalues of the Jacobian determine the local stability of the equilibrium. If, all the four roots of the characteristic equation, have negative real part, the static equilibrium of the system is said to be locally asymptotically stable. It is observed that a complex conjugate pair of roots with positive real part migrates transversal to the imaginary axis from the RHS s-plane to the LHS s-plane when a bifurcation parameter (say, the control gain, kc) thereby signifying a Hopf bifurcation. The local stability boundary (corresponding to the Hopf bifurcation point) is constructed in the plane of parameters, namely, the non-dimensional belt velocity v0 and the loop-gain kc in Fig. 3. It is observed that the static equilibrium of the uncontrolled system (corresponding to kc ¼0) is locally unstable in the low velocity range. The control system can stabilize the static equilibrium within a specific range of velocity; though, the static equilibrium in the very low velocity range cannot be stabilized by the control. 4.2. Nonlinear analysis The nature of Hopf bifurcation cannot be ascertained by the foregoing linear stability analysis. Thus, a nonlinear analysis is undertaken and to this end, the method of Averaging is employed. The method of Averaging is a pivotal step towards discerning the amplitude and phase of the friction-induced self-excited oscillation undergone by the system and thus investigates the nature of the global behavior away from the trivial equilibrium. It is assumed that the belt velocity is greater
Fig. 3. The bifurcation parameter v0 is varied along the section AB (kc ¼ 0.4) and CD (kc ¼ 1) and the bifurcation parameter kc is varied along the section EF (v0 ¼0.075) and GH (v0 ¼ 0.08). ωf ¼ 1, ξf ¼ 0:25.
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than that of the mass, i.e., v0 4 y_ . Eqs. (7) and (8) are recast as y€ þω2 y ¼ εh1 y; xf ; y_ ; x_ f x€ f þ ω2 xf ¼ εh2 y; xf ; y_ ; x_ f ;
(17) (18)
where ω is the arbitrary frequency of oscillation of the system, ε is an infinitesimally small parameter and ~ 1 y þ c~ kc x_ f þ k~ c xf þf ðy_ Þ ; h1 y; xf ; y_ ; x_ f ¼ Ω ~ 2 xf 2ξ~ f ωf x_ f y þ c~ kc x_ f þ k~ c xf þf ðy_ Þ h2 y; xf ; y_ ; x_ f ¼ Ω ε and
n o f ðy_ Þ ¼ μ~ þΔμe ~ αðv0 y_ Þ c~ y_ ;
with 2 ~ 1 ; kc ¼ εk~ c ; c ¼ εc~ ; ω2 ωf 2 ¼ εΩ ~ 2 ; ξf ¼ εξ~ f ; μ ¼ εμ; ω 1 ¼ εΩ ~ Δμ ¼ εΔμ~ Now, let the solutions to the Eqs. (17) and (18) be assumed as y ¼ A1 ðτÞ sin ωτ þϕ1 ðτÞ ; y_ ¼ A1 ðτÞω cos ωτ þ ϕ1 ðτÞ xf ¼ A2 ðτÞ sin ωτ þϕ2 ðτÞ ;
x_ f ¼ A2 ðτÞω cos ωτ þ ϕ2 ðτÞ
(19) (20)
where Ai(i ¼1,2) are the amplitudes of oscillations and ϕi are the phases of the oscillations of the system and the filter while the oscillations are assumed to be slowly varying functions of time, τ. Using these two solutions the following 1st order differential equations are obtained ε A_ i ¼ hi y; xf ; y_ ; x_ f cos ωτ þ ϕi ðτÞ (21) ω ε Ai ϕ_ i ¼ hi y; xf ; y_ ; x_ f sin ωτ þϕi ðτÞ ω
(22)
where i ¼1, 2 Implementation of the standard methods of averaging finally leads us to the following amplitude and phase equations, also known as slow flow equations Z 2π=ω ε A_ i ¼ hi y; xf ; y_ ; x_ f cos ωτ þ ϕi dτ (23) 2π 0 _i ¼ ε ϕ 2πAi
Z
2π=ω 0
hi y; xf ; y_ ; x_ f sin ωτ þ ϕi dτ
(24)
Thus, performing mathematical manipulations one can easily derive the following expressions of amplitude and phase relations A_ 1 ¼ g 1 ðAi ; ϕ1 ϕ2 ; ωÞ kc A2 γ ckc A2 c cos ϕ1 ϕ2 sin ϕ1 ϕ2 þ I 1 ðαωAÞ A1 ¼ ω 2 2 2ω A_ 2 ¼ g 2 ðAi ; ϕ1 ϕ2 ; ωÞ ckc A2 γ cA1 A1 sin ϕ1 ϕ2 þ þ I 1 ðωαA1 Þ cos ϕ1 ϕ2 cos ϕ1 ϕ2 ¼ ξf ωf A2 ω 2ω 2 2 _ 1 ¼ g 3 ðAi ; ϕ1 ϕ2 ; ωÞ ϕ kc A2 ω 1 ckc A2 sin ϕ1 ϕ2 cos ϕ1 ϕ2 ¼ 2ω 2A1 2ωA1
¼
ω
2
ω2f
2ω
2
(25)
(26)
(27)
_ 2 ¼ g 4 ðAi ; ϕ1 ϕ2 ; ωÞ ϕ
þ
kc cA1 A1 γ þ cos ϕ1 ϕ2 I 1 ðωαA1 Þ sin ϕ1 ϕ2 sin ϕ1 ϕ2 2ωA2 2ω ωA2 2A2
(28)
where I1 (.) is the modified Bessel's function of order one and γ ¼ Δμe αv0 . The steady state solutions (correspond to limit cycle oscillations of the original system) of the slow-flow Eqs. (25)–(28) can be obtained by solving following sets of equations adopting arbitrary initial guesses g j ðAi ; φ1 φ2 ; ωÞ ¼ 0
where
j ¼ 1; 2; 3; 4
(29)
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Fig. 4. Plot of amplitudes of unstable limit cycles for different values of kc for a velocity range selected from the locally stable region. ωf ¼ 1, ξf ¼ 0:25.
Fig. 5. Bifurcation diagram obtained from AUTO showing subcritical Hopf bifurcation at v0 ¼0.0673 and Cyclic-Fold bifurcation at v0 ¼ 0.0767 with kc ¼ 0.4. ωf ¼ 1, ξf ¼ 0:25.
_ ¼ g 5 ðAi ; ϕ1 ϕ2 ; ωÞ For simplification, one can assume so that ϕ1 ϕ2 ¼ ϕ so that ϕ_ 1 ϕ_ 2 ¼ ϕ The above-mentioned equations are solved in MATLAB and the eigenvalues of the Jacobian matrix, J1 in Eq. (30) is determined. 2 ∂g ∂g ∂g 3 1
∂A
6 ∂g 1 6 J 1 ¼ 6 ∂A21 4 ∂g5 ∂A1
1
∂A2 ∂g2 ∂A2 ∂g5 ∂A2
1
∂ϕ ∂g2 ∂ϕ ∂g5 ∂ϕ
7 7 7 5
(30)
The amplitudes obtained from the Eq. (29) and the positive sign of the real parts of the corresponding dominant eigenvalues of the Jacobian matrix (Eq. (30)) suggests the existence of an unstable limit cycle around the static equilibrium. Thus, the Hopf bifurcation of the static equilibrium is established as subcritical. The amplitudes of the unstable limit cycles obtained for different values of kc is plotted in Fig. 4 for a velocity range selected from the locally stable region (see Fig. 3). It is also observed that the unstable limit cycle vanishes beyond particular values of kc and v0. This clearly indicates the existence of a globally stable limit cycle (will be confirmed by numerical simulation). It is apparent from Fig. 4 that the amplitude of unstable limit cycle increases with the increase in control parameters (v0 or kc or both)and collides with the stable limit cycle at the saddle- node bifurcation (or Cyclic-Fold bifurcation) point and both of the limit cycles vanish leaving only the stable static equilibrium. The critical values of kc and v0 for which the unstable limit cycle given by the averaging analysis ceases to exist marks the cyclic fold bifurcation and hence the global stability boundary as depicted in Fig. 3. Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 6. Bifurcation diagram obtained from AUTO at kc ¼ 1 (with v0 as the bifurcation parameter) showing subcritical Hopf bifurcation at v0 ¼ 0.0668 and Cyclic-Fold bifurcation at v0 ¼ 0.0742. ωf ¼ 1, ξf ¼ 0:25.
Fig. 7. Bifurcation diagram obtained from AUTO at v0 ¼ 0.08 (with kc as the bifurcation parameter) showing subcritical Hopf bifurcation at kc ¼ 0.209 and Cyclic-Fold bifurcation at kc ¼ 0.291. ωf ¼ 1, ξf ¼ 0:25.
5. Numerical results The bifurcation diagrams of the control systems are numerically constructed using AUTO as one varies the control parameter, here, the loop-gain kc and non-dimensional belt velocity v0. To this end, four sections AB, CD, EF and GH as depicted in Fig. 3 are selected and the bifurcation diagrams are numerically constructed along these sections. Sections AB and CD correspond to the loop-gain, kc ¼ 0.4 and kc ¼ 1 respectively, along which the bifurcation parameter v0 is varied and the bifurcation diagram is constructed. Bifurcation diagrams are also drawn for varying kc along sections EF (with a fixed v0 ¼ 0.075) and GH (with a fixed v0 ¼0.08). The phase portraits obtained from numerical simulations substantiate the results.
5.1. Bifurcation along AB The bifurcation diagram with respect to the bifurcation parameter v0is constructed at kc ¼ 0.4 in Fig. 5. The trivial equilibrium is seen to lose stability due to a subcritical Hopf bifurcation at a certain value of the non-dimensionalised belt velocity, v0(at v0 ¼ 0.0673). Above this critical value of velocity, a stable limit cycle pertaining to stick-slip oscillations coexists with an unstable limit cycle and the locally stable steady sliding state. As one further increases the velocity, the Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 8. Bifurcation diagram obtained from AUTO at v0 ¼ 0.075 (with kc as the bifurcation parameter) showing subcritical Hopf bifurcation at kc ¼ 0.23 and Cyclic-Fold bifurcation at kc ¼ 0.237. ωf ¼ 1, ξf ¼ 0:25.
Fig. 9. Phase portraits obtained at v0 ¼ 0.075 and (a) kc ¼ 0.22 (b) kc ¼ 0.234 (c) kc ¼ 0.25 with initial conditions (3, 8) (d) kc ¼ 0.25 with initial conditions (0.9, 0.11) (e) kc ¼ 0.46 with initial conditions (0.1, 0.11) (f) kc ¼ 0.46 with initial conditions (10, 8). ωf ¼ 1, ξf ¼ 0:25.
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stable and unstable limit cycle collide at a particular velocity (Cyclic-Fold bifurcation) and vanishes, making the system globally stable beyond that critical belt velocity (v0 ¼0.0767). 5.2. Bifurcation along CD An identical Hopf bifurcation is obtained when v0 is varied along CD at kc ¼ 1 and the bifurcation diagram is shown in Fig. 6. Here, as one increases the velocity, Hopf bifurcation occurs at v0 ¼ 0.0668 and then the equilibrium becomes globally stable after the Cyclic-Fold bifurcation at v0 ¼ 0.0742. 5.3. Bifurcation along GH Fig. 7 shows the bifurcation diagram at v0 ¼0.08 while the bifurcation parameter kc is varied along GH. The equilibrium undergoes a subcritical Hopf bifurcation at a certain value of the control gain as we increase the gain value. Below this critical value of control gain (kc ¼ 0.209), a stable limit cycle solution representing stick-slip oscillations is separated from the locally stable steady sliding state by an unstable limit cycle solution. As one further increases the control gain the stable and unstable limit cycle merge rendering the system globally stable beyond that critical control gain (kc ¼ 0.291). 5.4. Bifurcation along EF The bifurcation diagram (Fig. 8) constructed in AUTO at v0 ¼ 0.075 while the bifurcation parameter kc is varied along EF. It may be seen that after the subcritical Hopf bifurcation at kc ¼ 0.23, the locally stable region is seen to exist for a very little span of control gain-value i.e., up to the Cyclic-Fold bifurcation at kc ¼ 0.237. But the numerical simulation results from the MATLAB SIMULINK model confirm a larger locally stable region and that the globally stable region extends only beyond kc ¼ 0.46. Phase portraits are shown in Fig. 9 substantiating the existence of a stable limit cycle solution (Fig. 9c) along with a stable equilibrium (Fig. 9d) at kc ¼ 0.25 while the system reaches an equilibrium (Fig. 9e and f) for any choice of initial conditions beyond kc ¼ 0.46. The bifurcation results so far obtained are illustrated in the plane of control parameters, v0 and kc, in Fig. 10. Bifurcation results obtained from AUTO satisfies the numerical simulations carried out in the MATLAB SIMULINK model in all the cases except when kc is varied along section EF i.e., at v0 ¼0.075. It can be easily noticed that the MATLAB SIMULINK simulation at v0 ¼0.075 present a better result as it monotonically follows the numerically obtained global stability boundary i.e., the line traced by the Cyclic-Fold bifurcation points. 5.5. Comparison of analytical and numerical results The bifurcation results so far obtained are for four particular sections which are arbitrarily selected. A more elaborate comparison is drawn by studying a greater region of the plane of control parameters. Numerically obtained global stability boundary i.e., the boundary traced by the Cyclic-fold bifurcation points obtained from AUTO and validated by numerical
Fig. 10. Hopf bifurcation points and Cyclic-Fold bifurcation points obtained in AUTO and MATLAB SIMULINK model are shown along with the local stability boundary obtained from linear stability analysis and global stability boundary constructed by the method of Averaging in the plane of control parameters. ωf ¼ 1, ξf ¼ 0:25. ▼ – Hopf bifurcations points obtained from AUTO. ▲ – Cyclic-Fold bifurcation points obtained from AUTO. ■ – Hopf bifurcation point obtained from simulations in MATLAB SIMULINK model at v0 ¼0.075. ● – Cyclic-Fold bifurcation point obtained from simulations in MATLAB SIMULINK model at v0 ¼ 0.075.
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Fig. 11. Comparison of the global stability boundary obtained analytically by the Method of Averaging and numerical global stability boundary traced by the Cyclic Fold bifurcation points from AUTO. ωf ¼ 1, ξf ¼ 0:25.
Fig. 12. Comparison of uncontrolled and controlled systems with v0 as the bifurcation parameter. kc ¼ 0.4. ωf ¼ 1, ξf ¼ 0:25. HB-Hopf Bifurcation (Subcritical), CF-Cyclic-Fold bifurcation.
simulations in MATLAB SIMULINK model and analytical global stability boundary (obtained from the method of Averaging) are shown in Fig. 11 which profoundly establishes the efficacy of the analyses to determine a globally stable region for any choice of the control parameter, i.e., the control gain from the globally stable region effectively results in a stable equilibrium or in other words, a stable steady sliding condition of the equilibrium (X_ ¼ 0) is obtained. In this context of comparing numerical and analytical results, Fig. 12 is presented which draws a comparison between the amplitudes obtained from all the nonlinear analyses (both analytical and numerical) for a particular control gain within a definite range of belt velocity. Fig. 12 is actually a bifurcation diagram ( constructed in AUTO) at kc ¼ 0.4 with v0 as the bifurcation parameter in which the amplitudes of the stable limit cycle is compared to that obtained from the numerical simulations in MATLAB SIMULINK model and the unstable limit cycle amplitude is compared with that obtained by the method of Averaging. The amplitude of the uncontrolled system for the selected velocity range is also shown to incorporate a better understanding of the control effectiveness. From Fig. 12 it is quite conspicuous that though by the implementation of control, one cannot completely eliminate the stick-slip oscillations in the low velocity range, but the amplitude of the same can be significantly reduced. The time- history plot (Fig. 13) obtained from the MATLAB SIMULINK model depicts the reduction of stick-slip amplitude when the control gain kc ¼ 0.4 is applied at 512.98 seconds. The static equilibrium of the system remains fixed at y¼ 0.25273. Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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Fig. 13. Time-history plot at v0 ¼ 0.064 showing the control efficacy as control gain kc ¼ 0.4 is applied at 512.98 seconds.
Fig. 14. Optimal stability region(shaded) constructed in the plane of control parameters by pole crossover design.
6. Optimization of parameters 6.1. Pole crossover design The filter parameters are optimally chosen by the crossover design method in which the poles of the system are placed in a suitable location in the s-plane so as to achieve faster attenuation of the transient response (closer to the real axis) and a greater relative stability (away from the imaginary axis in the LHS s-plane). Taking the non-dimensional belt velocity and control gain as independently chosen parameters, the filter parameters are optimized for achieving two identical pairs of complex conjugate poles of the closed loop system. Thus, the structural and filter modes are merged by this optimization criterion. The poles of the system, i.e., the four roots of the characteristic Eq. (16) are placed in a position such that the real part of all the roots are equal which is seen to maximize the absolute values of the real parts of the poles. Let the identical complex conjugate pair of poles be placed at λ1 and λ2 where λ1;2 ¼ a 7 ib Thus, the characteristic equation of the system can be written as 2 2 2 2 ðs λ1 Þ2 ðs λ2 Þ2 ¼ s4 4as3 þ 6a2 þ 2b s2 4aða2 þb Þs þ a2 þ b ¼0
(31)
Comparing Eq. (31) with the characteristic equation of the linearized model obtained in Eq. (16) obtained in Section 4.1 one can write 4a ¼ 2p1 þ q
(32)
2 6a2 þ2b ¼ p2 2f p1 þ 1 kc
(33)
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2
4aða2 þ b Þ ¼ 2p1 f p2
(34)
2
(35)
ða2 þ b Þ2 ¼ p2 αv0
p2 ¼ ω2f ,
where p1 ¼ ωf ξf , q ¼ f ckc , f ¼ c þ Δμe :α Solving Eqs. (32)–(35) to eliminate a and b, we get !
2
2 f fq q 2 2 4 ω4f þ 4 f ω3f þ f þkc 6 ω2f þ ðf q 2kc þ 4Þωf þ þkc 1 ¼ 0 2 4 4
(36)
This equation is quadratic in ωf and thus four values of the optimal filter frequency, (ωf)opt are obtained and a contour plot is constructed in the plane of control parameters in Fig. 14. The shaded region satisfies the cross-over optimization criteria by including only the real values of (ωf)opt, (ξf)opt, a and b as well as Re (λ1,λ2) ¼a o0, i.e., the real part of all the merged poles are negative in this shaded region. The suitable value of ðωf Þopt satisfying the stability criterion is chosen and optimal filter damping value, ðξf Þopt and the values of the real and imaginary parts of the identical complex conjugate poles are obtained from Eqs. (37–39). f ωf þ q ðξf Þopt ¼ 2 1 ωf a¼
b¼
2ξf ωf þ q 4
(37)
(38)
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω2f 2f ξf ωf 6a2 þ1 kc
(39)
2
Now, if one attempts to an optimized value of the control gain instead of assigning arbitrary values as in the previous optimization process it is required that all the poles are merged at the same point on the real axis i.e., all poles are real and equal. Let the identical complex conjugate pair of poles be placed at λ where λ ¼ a. Thus, the characteristic equation in Eq. (31) reduces to ðs λÞ4 ¼ s4 4as3 þ 6a2 s2 4a3 s þa4 ¼ 0;
(40)
4a ¼ 2p1 f ckc
(41)
6a2 ¼ p2 2f p1 þ1 kc
(42)
4a3 ¼ 2p1 f p2
(43)
a4 ¼ p2
(44)
where
Solving Eqs. (40)–(44) to eliminate a and b, we get 7=2 5=2 3=2 2 4c 4cf 4f ωf þ ð16 16cf Þω3f 24cωf 16ω2f þ ð4f þ4cÞωf ¼ 0
(45)
Eq. (45) is solved to obtain real values of (ωf)opt and expressions for (ξf)opt, (kc)opt and a are obtained from the following equations pffiffiffiffiffiffi f ωf 4 ωf (46) ξf opt ¼ 2 2 ðkc Þopt ¼ 1 f ωf 2 þ 4f ωf 3=2 6ωf þ 1
(47)
Table 1 Optimal control parameters at different belt velocities. Non-dimensionalised belt velocity, v0
Optimal filter frequency, (ωf)opt
Optimal filter damping, (ξf)opt
Optimal control gain, (kc)opt
a
0.01 0.02 0.05 0.08 0.1
0.9885 0.9863 0.9797 0.9735 0.9698
2.4109 2.3653 2.2522 2.1677 2.1237
8.0283 7.5317 6.3742 5.5786 5.1876
1.0782 1.0684 1.0438 1.0250 1.0151
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Fig. 15. Comparison of uncontrolled and controlled systems with v0 as the bifurcation parameter. kc ¼ 0.8, (ωf)opt ¼ 0.9779, ðξf Þopt ¼0.5188. HB-Hopf Bifurcation (Subcritical), CF-Cyclic-Fold bifurcation.
Fig. 16. Comparison of numerical (from AUTO and MATLAB SIMULINK model) and analytical (from Averaging method) results with kc as the bifurcation parameter. v0 ¼ 0.02, (ωf)opt ¼ 0.9779,ðξf Þopt ¼ 0.5188. HB-Hopf Bifurcation (Subcritical), CF-Cyclic-Fold bifurcation. The stable limit cycle amplitude of the uncontrolled system at v0 ¼ 0.02 is 0.054.
a¼
f þckc 2ωf ξf 4
(48)
The optimal values of the control gain, filter damping and filter frequency corresponding to different choices of velocity are obtained for aas a negative real numberand tabulated below. It can be observed from Table 1 that with the increase in belt velocities the magnitudes of (ωf)opt, (ξf)opt, (kc)opt and a decreases. 6.2. Efficacy of optimization It is observed that the system can be stabilized at any belt velocity provided the filter parameters are suitably chosen according to the procedure described above. As very low belt velocity region is the most difficult to control, a detailed analysis of that region is enclosed to establish the importance of optimization of parameters. For a belt velocity as low as v0 ¼0.02, a control gain value kc ¼ 0.8 can be selected from the shaded region of Fig. 14 for which an optimal set of filter parameters are obtained as (ωf)opt ¼0.9779 and ðξf Þopt ¼0.5188. Now, for this set of parameter values, two bifurcation diagrams (Figs. 15 and 16) are obtained in AUTO as v0 and kc as the bifurcation parameters, respectively. Fig. 15 compares the results obtained from numerical simulations in MATLAB SIMULINK model with that Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
J. Nath, S. Chatterjee / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Fig. 17. Time-history plot at v0 ¼0.016 showing the control efficacy as control gain kc ¼ 0.8 is applied at 435.983 seconds.
Fig. 18. Variation of optimum filter parameters with v0 and kc.
obtained from the method of Averaging and also depicts a significant reduction in the stick-slip amplitude for the selected velocity range. The bifurcation diagram with the particular set of parameters with bifurcation parameter kc is shown in Fig. 16. Here, too amplitude values obtained from MATLAB SIMULINK model and method of Averaging are compared. The amplitude of stick-slip oscillations at v0 ¼ 0.02 for uncontrolled system is observed to be 0.05335 and thus Fig. 16 also shows an effective reduction in amplitude due to the optimization. Fig. 17 depicts the time-history plot obtained from the MATLAB SIMULINK model showing the reduction of stick-slip amplitude when the control gain kc ¼ 0.8 is applied at 435.983 seconds to an uncontrolled system with belt velocity, v0 ¼0.016.
6.3. Relation between control parameters and optimal filter parameters The dependence of optimum filter parameters on the non-dimensional belt velocity v0 and the control gain kc are obtained and plotted in Fig. 18. One can easily notice that the optimum filter frequency ðωf Þopt does not vary much with v0 but decreases slowly with increasing values of kc. The optimum filter damping ðξf Þopt increases for very low values of v0 but remains constant for higher belt velocity while it increases by leaps and bounds for even a very small increase in control gain.
Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i
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7. Conclusions A phenomenological mass on belt model is considered to model stick-slip oscillations and the efficacy of a tangential control by a controller consisting of a second-order filter and power amplifier is extensively studied. Linear stability analysis around the trivial equilibrium is performed to arrive at the local stability boundary which is constructed in the plane of two dominant parameters, non-dimensionalised belt velocity and loop-gain. The complex conjugate eigenvalues obtained after linearization of the system are observed to cross the imaginary axis thus, indicating a Hopf bifurcation. Nonlinear stability analysis of the region is performed to understand the dynamics away from the trivial equilibrium as the belt velocity is considered greater than the velocity of the mass at every instant of time. This only includes slipping oscillations around the static equilibrium. The method of Averaging is employed for the nonlinear analysis and thus, the coexistence of stable limit cycle (obtained from numerical simulations) with the stable equilibrium separated by an unstable limit cycle (obtained from the Averaging analysis) is observed i.e., the Hopf bifurcation is observed to be subcritical in nature. Due to this, the existence of a global stability boundary is investigated and observed to exist within the local stability boundary. Bifurcation diagrams with non-dimensionalised belt velocity and loop-gain as the bifurcation parameters are obtained with AUTO and the numerical simulations carried out in MATLAB SIMULINK model substantiate the analytically obtained results. Finally, a comparison between the results obtained by analytical and numerical methods and simulations are carried out. The filter parameters are optimized by choosing belt velocity and control gain as independent parameters by pole crossover design. An optimal stability region is constructed in the plane of control parameters (belt velocity and control gain). Another optimization procedure is carried out with belt velocity as the only independent parameter to arrive at optimal values of filter frequency, filter damping and control gain. As the low velocity region poses the prime difficulty in control implementation, a set of optimal filter parameters corresponding to a very low velocity is selected from the optimal stability region and a detailed bifurcation analyses is performed with velocity and control gain as the bifurcation parameters. Stable and unstable limit cycle amplitudes obtained from MATLAB model and Averaging method, respectively are compared with that obtained in AUTO. The significant reduction in stick-slip amplitude in the low velocity region is also indicated. Finally, variations of filter parameters with change in control parameters are illustrated to gain an insight into the process of choosing optimal values in different circumstances.
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Please cite this article as: J. Nath, & S. Chatterjee, Tangential acceleration feedback control of friction induced vibration, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.05.020i