Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control

International Journal of Non-Linear Mechanics 43 (2008) 801 -- 812

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Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control Khaled A. Alhazza a,∗ , Mohammed F. Daqaq b , Ali H. Nayfeh c , Daniel J. Inman d a

Department of Mechanical Engineering, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Department of Mechanical Engineering, Clemson University, Clemson, SC 29634, USA Department of Engineering Science and Mechanics, Virginia Tech, Blacksburg, VA 24061, USA d Center for Intelligent Material Systems and Structures, Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24061, USA b c

A R T I C L E

I N F O

Article history: Received 28 October 2007 Received in revised form 24 February 2008 Accepted 17 April 2008

Keywords: Stability Delayed feedback Non-linear vibrations Parametric excitation

A B S T R A C T

Non-linear feedback control provides an effective methodology for vibration mitigation in non-linear dynamic systems. However, within digital circuits, actuation mechanisms, filters, and controller processing time, intrinsic time-delays unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If not well-studied, these inherent and compounding delays may inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering controllers' performance ineffective. In this work, we present a comprehensive investigation of the effect of time delays on the non-linear control of parametrically excited cantilever beams. More specifically, we examine three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback. Utilizing the method of multiple scales, we derive the modulation equations that govern the non-linear dynamics of the beam. These equations are then utilized to investigate the effect of time delays on the stability, amplitude, and frequency--response behavior. We show that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented. © 2008 Elsevier Ltd. All rights reserved.

1. Introduction During the last decade, stability and stabilization of time-delayed systems have received considerable attention in the dynamics and control literature. Past research and funding have established a flourishing new branch of mathematics primarily concerned with the analysis and stabilization of delay-differential equations (DDEs). A variety of analytical, graphical, and numerical methodologies have been proposed capturing and assessing the stability of systems operating with single or multiple time delays. However, most of the proposed methodologies were mainly concerned with characterizing the stability of the free response, and only few efforts dealt with understanding the effect of time delays on the response of externally excited systems. It is indeed well-accepted that the most effective approach towards controlling a non-linear system is non-linear feedback control. Moreover, it is also agreed on that intrinsic time delays exist

∗

Corresponding author. Tel.: +965 799 5510; fax: +965 484 7131. E-mail address: [email protected] (K.A. Alhazza).

0020-7462/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2008.04.010

in the feedback and might unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If unaccounted for, these inherent and compounding delays might inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering the controller performance ineffective. Having said that, there are some cases in which time delays actually augment controllers' design. It has been demonstrated that careful and deliberate selection of delays is capable of producing substantial damping in dynamic systems. As a result, they have been successfully applied to mitigate potentially hazardous oscillations of suspended objects on various types of cranes [1,2] and have also been utilized for active vibration control of beams [3--7]. The effect of time delays is specially prevalent in systems with a very high resonant frequency. Current trends in micro and nonfabrication techniques made such systems abundant. Indeed, microdevices that now operate as sensors, actuators, or switches, have natural frequencies reaching up to the megahertz range. At such high frequencies, even the infinitesimal feedback delays can easily destabilize the device or result in undesirable aperiodic responses. Furthermore, piezoelectrically actuated microcantilever beams (Fig. 1)

802

K.A. Alhazza et al. / International Journal of Non-Linear Mechanics 43 (2008) 801 -- 812

Fig. 1. Schematic drawing of a piezoelectrically actuated beam under parametric excitations.

have been recently implemented in atomic force microscopy and scanning force microscopy and have a strong potential for operating as micro sensors for the detection of biological species, explosives, and pathogens. They are sometimes parametrically excited to realize large amplitudes and higher sensitivities for smaller input voltages. Consequently, it is imperative to study and understand the effect of time delays on the non-linear dynamics of parametrically excited cantilever beams. Next, we present a short review of the relevant literature. For a more detailed review of the recent advances related to delay systems, we refer the reader to [8], and for details on the non-linear dynamics and control of parametrically excited cantilever beams, we refer the reader to [9--13]. Parametric excitation was first observed by Faraday in 1931 [14], and occurs when time-dependent excitations multiply one or more states in either the governing equations of motion or the boundary conditions, or both. A common consequence of these excitations is the principal-parametric resonance. This phenomenon occurs when a harmonic excitation appears as a term multiplying one of the linear states. As a result, even for small excitation amplitudes, large-amplitude motions are produced when the frequency of excitation is close to twice one of the system's natural frequencies. Analyzing the dynamics of structures subjected to principle parametric excitations has received considerable attention [14--16]. In one demonstration, the non-linear-non-planar dynamics of a parametrically excited cantilever beam was extensively studied by Nayfeh et al. [17] and Arafat et al. [18]. More specifically, they presented an extensive analytical, numerical, and experimental analysis of the non-linear dynamics of a cantilever beam subjected ¨ to parametric excitations. In another demonstration, Ozkaya and Pakdemerli [19] investigated the stability and control of a cantilever beam with a parametrically excited base. They used the method of multiple scales and matched-asymptotic expansions to construct non-resonant boundary--layer solutions for an axially accelerating ¨ beam with small bending stiffness. As an extension to this work, Oz ¨ [21] used the method of multiple scales and Pakdemirli [20] and Oz to analytically construct the stability boundaries of parametrically excited tensioned beams under simply supported and fixed--fixed conditions, respectively. Parker and Lin [22] also studied the dynamic stability of an axially accelerating cantilever beam and Yang and Chen [23] investigated the effect of viscoelastic damping on the stability boundaries of parametrically excited cantilever beams. Zavodney and Nayfeh [24] investigated the effect of geometric and inertia non-linearities on the stability of parametrically exited cantilever beams. Using the method of multiple scales, they analytically constructed the dynamic response and analyzed its stability.

Their experimental results were only in partial agreement with their theoretical findings. This motivated the work by Anderson et al. [25] to improve the model proposed by Zavodney and Nayfeh by including the effects of the non-linear quadratic air damping on the system response. Their theoretical results are in excellent agreement with the experimental data. Esmailzadeh and Jalili [26] investigated the parametric response and stability of a cantilever beam with a tip mass. They showed that regions wherein periodic solutions are stable shrink as the tip mass increases. Multi-mode non-linear dynamics of a parametrically excited cantilever beam carrying a tip mass were also studied by various researchers. Kar and Dwivedy [27] studied the non-linear response and stability of a parametrically excited slender beam carrying a lumped mass with 3:1 internal resonance. They showed that the multi-branched non-trivial response curves exhibit saddle-node, pitchfork, and Hopf bifurcations. They also investigated the same problem but with 1:3:5 internal resonances [28,29] and showed that the beam could exhibit chaotic responses, which might be controlled by appropriate variations in the system design parameters. The objective of this paper is to present an investigation of the effect of feedback delays on a non-linear controller applied to mitigate vibrations of piezoelectrically actuated parametrically excited cantilever beams near the principle parametric resonance. The reason for choosing a non-linear controller stems from their effectiveness in mitigating such vibrations [13,30]. A linear delayed-state feedback controller has been proven effective in mitigating vibrations of single-degree-of-freedom systems and has been previously analyzed in [31]. However, its implementation on a distributed-parameters systems, similar to the beam considered here, could be very cumbersome due to the infinite number of eigenvalues associated with linear delays systems. These eigenvalues can easily destabilize the higher vibration modes and degrade the performance of the controller. We organize the rest of the paper as follows: In Section 2, we present a mathematical formulation of the problem and the discritization scheme utilized to obtain a set of non-linear-ordinarydifferential equations that govern the dynamics of the beam in the presence of time delays and parametric excitation. In Section 3, we undertake a non-linear analysis of the equation of motion by utilizing the method of multiple scales and obtaining the modulation equations that govern the local dynamics and stability of the response. As a benchmark, in Section 4, we investigate the dynamics and stability of the uncontrolled beam. In Section 5, we analyze the effect of time delays on the behavior and stability of the response. Finally, in Section 6, we conclude with various observations and comments. 2. Model formulation We base our model on the non-linear-differential-equations of motion for an isotropic inextensible Euler--Bernoulli beam derived earlier by Crespo da Silva and Glenn [32,33]. When only planar motions are considered, the equations of motion and the associated boundary conditions of a uniform cantilever beam subjected to parametric-base excitation and piezoelectric actuation are written as [34,35]

Av¨ + cv˙ + c¯ |v˙ |v˙ + EIviv = −EI[v (v v ) ] −

      s s j 1 2 A v v ds ds 2 jt2 0 l

+ q(s, t) + Aap × [(s − l)v + v ] v=0

v = 0

and and

v = 0 at v = 0

(1)

s = 0, at

s=l

(2)

where v denotes the displacement component along the y-axis; the primes and overdots indicate the derivatives with respect to the arc

K.A. Alhazza et al. / International Journal of Non-Linear Mechanics 43 (2008) 801 -- 812

length s and time t, respectively;  is the beam density; A is the cross-sectional area; c is the coefficient of linear viscous damping per unit length; c¯ is the air drag coefficient; E is Young's modulus of elasticity; I is the moment of inertia about the neutral axis of the beam; and ap is the parametric acceleration of the supported end. Here, q(s, t) is the distributed load applied by the piezoelectric actuator to control the beam and is given by q(s, t) =

j2 M js2

(3)

M = bd31 Ea (ta + tb )Va (t)[H(s − s1 ) − H(s − s2 )]

(4)

where b and ta are the width and thickness of the piezoelectric actuator, respectively; d31 is a piezoelectric constant; Ea is the actuator Young's modulus; tb is the thickness of the beam; Va (t) is the control voltage; H(s) is the Heaviside step function; and s1 and s2 are the starting and ending coordinates of the piezoelectric strip.

We derive a reduced-model for the system under consideration by using the Galerkin procedure in the form v(s, t) =

∞ 

n (s)un (t)

(5)

n=1

where the un (t) are generalized temporal coordinates and the n (s) are the linear orthonormal eigenfunctions of a cantilever beam. They are given by

n (s) = Cn {cosh(rn s) − cos(rn s) − n [sinh(rn s) − sin(rn s)]}

(6)

cosh rn l + cos rn l sinh rn l + sin rn l

(7)

the rn are roots of the transcendental equation 1 + cosh(rl) cos(rl) = 0

2n (s) ds = 1.

(9)

Substituting Eq. (5) into Eq. (1), multiplying the result by n , integrating the outcome over the length of the beam, and using the orthonormal properties of the linear mode shapes, we obtain the following set of non-linear ordinary-differential equations: u¨ n + n u˙ n + 2n un ∞ ∞   =− nij |u˙ i |u˙ j + nijk ui uj uk +

+

0

n [(s − l)i + i ] ds

c

n =

A  l EI EI 4 n  rn n ds = 0 A Al4

n,i,j ∞  i,j,k ∞ 

i,j,k

nijk uk (u¨ i uj + 2u˙ i u˙ j + ui u¨ j ) Pni ui ap + Mn Va (t) n = 1, 2, 3, . . .

i

where

nijk =

 l EI   (  + j  k ) ds A 0 n i j k

(11)

and Mn =

bd31 Ea (ta + tb )  [n (s2 ) − n (s1 )] A

Next, we non-dimensionlize Eq. (10) using the length l of the beam and the inverse of the first-mode natural frequency 1 and obtain

=−

+

∞  n,i,j ∞ 

+

i,j,k ∞ 

2n wn 21

nij |w˙ i |w˙ j +

∞ 

ˆ nijk wi wj wk

i,j,k

ˆ ¨ i wj + 2w ˙ iw ˙ j + wi w ¨ j) nijk wk (w ˆ n Va (t) n = 1, 2, 3, . . . Pˆ n wi ap + M

(12)

i

where the non-dimensional parameters, marked by a hat in Eq. (12), are

ˆ n =

n 1

l2

21

2 Pˆ = ˆ ni nijk = nijk l

Pni

21

ˆ n = Mn . M l2n

(13)

2.2. Non-linear feedback control (8)

and the Cn are determined so that

0

 l

ˆ nijk = nijk

where

 l

Pni =

¨ n + ˆ n w ˙n+ w

2.1. Discretized model

n =

 s   s 1 l n k ds i j ds ds 2 0 0 0  l c¯ nij =  | | ds A 0 n i j

nijk = −

2n =

and

803

(10)

To illustrate the effect of time delays on the effectiveness of the proposed non-linear control methodologies applied to mitigate vibrations of parametrically excited beams, we consider three types of cubic feedback algorithms: position, velocity, and acceleration. The driving voltage of the piezoelectric actuator using these algorithms takes the form ˙ 3n (t − ) and Va (t) = −kw3n (t − ) Va (t) = −kw 3 ¨ Va (t) = −kwn (t − )

(14)

where k is the controller gain and is the inherent system delay. We assume that the excitation frequency is sufficiently close to twice the natural frequency of the first vibration mode and that this mode is not involved in internal resonance with any of the other modes. Moreover, we assume that the higher modes are filtered from the feedback to avoid noise and minimize the effects of higher frequency vibrations. Under such assumptions, one can adopt a singlemode approximation to capture the response of the beam. To obtain the closed-loop equations, we substitute the actuator voltage Va (t) into Eq. (12) and obtain ¨ + w ˙ w ˙ + w3 + 2 (w2 w ˙ + w = − |w| ¨ + ww ˙ 2) w ⎧ 3 −kMw (t − ) Case 1 ⎪ ⎨ ˙ 3 (t − ) Case 2 + Pwap + −kMw ⎪ ⎩ ¨ −kMw3 (t − ) Case 3

(15)

804

K.A. Alhazza et al. / International Journal of Non-Linear Mechanics 43 (2008) 801 -- 812

where all the subscripts and superscripts have been dropped for simplicity of notation. 3. Non-linear beam response

where ⎧ ⎨ −3Ke−i

= −3iKe−i

⎩ +3Ke−i

Utilizing the method of multiple time scales [36,37], we obtain an approximate analytical solution of Eq. (15). To that end, we express the solution in the form

In solving Eq. (24), we find it convenient to express A(T2 ) in the polar form ¯ ) = 1 a(T )e−i(T2 ) (25) A(T ) = 1 a(T )ei(T2 ) A(T

w(T0 , T1 , T2 ) = w1 (T0 , T1 , T2 ) + 2 w2 (T0 , T1 , T2 ) + 3 w3 (T0 , T1 , T2 ) + O(4 )

Now, substituting Eq. (25) into Eq. (24) and separating the real and imaginary parts of the resulting equation, we obtain

(16)

where Tn = n t. To analyze the effect of the principal-parametric resonance, the amplitude of excitation and the controller gain are ordered so that they appear in the same perturbation equation as the damping and non-linear terms. In other words, we let

 = 2   =  ap = 2 f cos( t) K = 2 kM

(17)

where is the excitation frequency and  is a bookkeeping parameter. Substituting Eqs. (16) and (17) into Eq. (15) and equating coefficients of like powers of , we obtain O() :

D20 w1 + w1 = 0

(18)

O(2 ) :

D20 w2 + w2 = −2D0 D1 w1

O(3 ) :

D20 w3 + w3 = − 2D0 D1 w2 − 2D0 D2 w1 − D21 w1

(19)

− (D0 w1 )2 sgn (D0 w1 ) + 2 w1 × [D20 w1 w1 + (D0 w1 )2 ] − D0 w1 + w31 + 12 w1 f (ei T0 + e−i T0 ) ⎧ −K[w1 (T0 − )]3 case 1 ⎪ ⎨ 3 + −K[D0 w1 (T0 − )] case 2 ⎪ ⎩ −K[D20 w1 (T0 − )]3 case 3

(20)

where sgn is signum function defined by  (x) +1 x > 0 = sgn(x) = −1 x < 0 |(x)|

(21)

where cc is the complex conjugate of the preceding term and A(T1 , T2 ) is an unknown complex function that will be determined by imposing the solvability conditions at the next levels of approximation. Substituting Eq. (21) into Eq. (19) and eliminating the secular terms (i.e. terms leading to e±iT1 ) yields

jA = 0 ⇒ A = A(T2 ) jT1

(22)

To investigate the response behavior near the principal-parametric resonance (i.e. ≈ 2), we introduce a detuning parameter, , which characterizes the nearness of the excitation frequency to twice the first-mode natural frequency, and let

= 2 + 2 

(23)

Substituting Eqs. (21)--(23) into Eq. (20) and eliminating secular terms, we obtain

j 2 ¯ A(T ) − iA(T2 ) + (3 − 4 + )A(T 2 )A(T2 ) jT2 2  2 1¯ iT2 +  iT0 2 ¯ + A(T (iA(T2 )eiT0 − iA(T 2 )f e 2 )e ) 2 2 0

− 2i

iT0 −iT0 dT = 0 ¯ × sgn(iA(T2 )eiT0 − iA(T 2 )e )e 0

2

2

2

2

2

1 1 4 ¯ a3 a = − a + fa sin() − a2 + 2 4 3  1 3 ˆ a3 − + a = a + fa cos() + 2 4

(26) (27)

where the primes indicate derivatives with respect to T2 ,  = T2 −2, and ⎧ 3 ⎪ + K sin( ) case 1 ⎪ ⎪ 8 ⎪ ⎪ ⎨ 3 ¯

= − K cos( ) case 2 and ⎪ 8 ⎪ ⎪ ⎪ ⎪ ⎩ 3 − K sin( ) case 3 8 ⎧ 3 ⎪ − K cos( ) case 1 ⎪ ⎪ 4 ⎪ ⎪ ⎨ 3 ˆ

= − K sin( ) case 2 ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎩3 K cos( ) case 3 4 Eqs. (26) and (27) represent the modulation equations of the response and are used to investigate the response and stability of Eq. (15) in the presence of feedback delays and principal-parametric resonance. 3.1. Asymptotic solutions and their stability

and Dn = j/ jTn . The solution of the first order equation, equation (18), can be written as w1 = A(T1 , T2 )eiT0 + cc

2

case 1 case 2 case 3

The nature and stability of the steady-state response is investigated by obtaining the fixed points of Eqs. (26) and (27), and examining their stability as a function of the excitation amplitude, f, detuning parameter, , controller gain, K, and controller delay, . These fixed points occur when a =  = 0 which corresponds to the solutions of 1 4 ¯ a3 = 1 fa sin( ) a +  a2 − 0 0 2 0 3 0 4 0

(28)

ˆ )a3 = − 1 fa cos( ) a0 + ( 43  − + 0 0 2 0

(29)

The trivial solution a0 = 0, represents one solution of Eqs. (28) and (29). For non-trivial solutions, we square and add these equations to obtain 2  1 2 2 1 4 ¯ a3 f a0 =  a0 + a20 − 0 16 2 3 2   3 ˆ a3 + −2a0 − − + (30) 0 4 Eq. (30) is an implicit representation of the steady-state amplitude, a0 , as a function of the frequency detuning parameter, , for a given excitation amplitude, f. This equation can be utilized to study the frequency--response of the beam for a given gain--delay combination. Alternatively, one can rewrite Eq. (30) in the more insightful form

 = e a20 

± (24)

 1 ¯ a3 + 4  ¯ 2 a4 + 32  ¯ − 64 2 a2 − 16 a −2 + f 2 −4 0 0 3 0 3 0 4 9 2 (31)

K.A. Alhazza et al. / International Journal of Non-Linear Mechanics 43 (2008) 801 -- 812

where e denotes the effective non-linearity coefficient ˆ) e = −( 34  − +

Table 1 Geometric and material properties of the beam and piezoelectric actuator

(32)

When e = 0, the frequency--response curves are symmetric around  = 0. As a result, one may conclude that the response is linear. On the other hand, the non-linear system has a softening behavior when e < 0, and hardening behavior when e > 0. It is also worth noting that, for the first vibration mode of a parametrically excited and uncontrolled cantilever beam, the response is of the hardening-type, [38,39]. However, in the presence of the non-linear delayed feedback, the gain--delay combination plays an essential role in determining the behavior of the response. More specifically, by examining ˆ <− Eq. (32), one concludes that the behavior is hardening when ˆ > − ( 3  − ), and linear like when ( 34  − ), softening when 4 ˆ = −( 3  − ).

4

3.1.1. Stability of the trivial solutions It is not very convenient to analyze the stability of the trivial solutions by using the polar form of the modulation equations (26) and (27). Therefore, we express A and A¯ in the following complex form [14] 1 iT A = 12 (p − iq)e 2 2

− 1 iT A¯ = 12 (p + iq)e 2 2

(33)

where p and q are real functions of time. Substituting Eq. (33) into Eq. (24) and separating the real and imaginary parts of the resulting equation yields   4  1 f q− q p2 + q2 p = − q + − 2 4 2 3 4 − 3 2 2 + (34) q(p + q ) + (p2 + q2 ) 8   4  1 f + p− q p2 + q2 q = − q + 2 4 2 3 4 − 3 2 2 ¯ (p2 + q2 ) p(p + q ) +  − (35) 8 where ⎧3 ⎪ ⎪ 8 K[+q cos( ) + p sin( )] ⎨  = 38 K[−p cos( ) + q sin( )] ⎪ ⎪ ⎩3 K[−q cos( ) − p sin( )] 8 ⎧3 ⎪ ⎪ 8 K[−p cos( ) + q sin( )] ⎨ ¯  = 38 K[−q cos( ) − p sin( )] ⎪ ⎪ ⎩3 K[+p cos( ) − q sin( )] 8

case 1 case 2 case 3

or || 

case 3

f2 − 2 4

Piezoelectric actuator (PZT PKI 552) Electromechanical coupling coefficient, d31 (m/V) Modulus of elasticity, E (GPa) Thickness, ta (mm) Width, b (mm)

−270 × 10−12 60 0.5 20

By examining Eq. (38), it becomes evident that the stability of the trivial solutions is independent of either the gain or delay of the controller. This is expected since, in general, the stability of the trivial solutions is only a function of the linearized system which, in this case, does not depend on the controller parameters. 3.1.2. Stability of the non-trivial solutions The stability of the non-trivial solutions can be assessed in the neighborhood of a given fixed point by illustrating whether a small perturbation in the steady-state motion decays or grows. To that end, we perturb the solution around the nominal values a0 and 0 by introducing a time-dependent disturbance in Eqs. (26) and (27), that is a = a0 + a1 (t) and

 = 0 + 1 (t)

(39)

Substituting Eq. (39) into Eqs. (26) and (27), linearizing in the disturbance, and finding the Jacobian of the linearized equations, we get ⎡

1 1 8 ¯ a2 −  + f sin(0 ) − a + 3 0 ⎢ 2 4 3  0  J=⎣ 3 ˆ  − + a0 2 4

⎤ 1 fa0 cos(0 ) ⎥ 4 ⎦ 1 − f sin(0 ) 2

(40)

Again, the stability of the non-trivial solutions can be characterized by finding the eigenvalues of Eq. (40). These eigenvalues are given by the following characteristic equation:

b=

case 2

The trivial solutions are stable, if all the real parts of the eigenvalues,

 f  2 2 + 2

207 7800 350 20 0.7

(41)

where

case 1

, are less than zero; and are unstable, if at least one eigenvalue has a positive real part. Since  > 0, the stability conditions are satisfied 

Beam Modulus of elasticity, E (GPa) Density,  (kg/m3 ) Length, l (mm) Width, w (mm) Thickness, tb (mm)

2 + b + d = 0

and

The stability of the trivial solutions (p0 = 0, q0 = 0) is obtained by finding the eigenvalues of the Jacobian of Eqs. (34) and (35) evaluated at the roots, that is    f   1  − −  −  2 4 2  = 0 (36)   1   f  + − −   4 2 2 or  f 2 − 4 2 1 (37) =− ± 2 4

when

805

(38)

4a0



¯ a2 +  − 4 0

¯ ¯ 2 + 2 )a4 − 8 a3 d = (4 e 0  0  322 ¯ − e a2 + 4 a + − 2 0 3 0 9 2 Therefore, by virtue of the Routh--Hurwitz criterion, the steadystate solutions are locally asymptotically stable if and only if ¯ <

  + a0 4a2 0

and

¯ ¯ 2 + 2 )a4 − 8 a3 (4 e 0 0    322 4 ¯ + − 2 − e a20 + a >0 3 0 92

(42)

3.2. Response of the uncontrolled beam In order to understand the effect of time delays on the performance of the non-linear controllers, we benchmark our findings

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K.A. Alhazza et al. / International Journal of Non-Linear Mechanics 43 (2008) 801 -- 812

0.4

0.3 B 0.25

0.35 E

0.3

0.2

0.25

F

0.2

0.15

D

0.15 0.1

0.1

0.05 0

0.05 E

A −0.01

0

C

D 0.01

0.02

0.03

0

0.04

0.02



B C

A 0.025

0.03

0.035

0.04 f

0.045

0.05

0.055

0.06

Fig. 2. Variation of the response amplitude with (a) the frequency detuning parameter  when f = 0. 025 and (b) the excitation magnitude f when  = 0. 02. Results are obtained for  = 0. 01,  = 0. 01,  = −9. 3, and = −6. 5. Solid line represent stable solutions and dashed lines represent unstable solutions.

against the uncontrolled beam response. To that end, we utilize Eqs. (30), (38) and (40) to study variation of the response amplitude with the frequency detuning parameter  and the excitation amplitude, f, when K = 0. All simulations were obtained for the material and geometric properties listed in Table 1. As illustrated in the frequency--response curve shown in Fig. 2(a), when  < A , only a stable trivial solution exists. As  is increased, the trivial solution loses stability at point A through a supercritical-pitchfork bifurcation giving way to a branch of stable periodic solutions. The amplitude of these stable solutions increases as  is increased further towards point B. At point B, the periodic solution loses stability through a saddle-node bifurcation, and the response amplitude jumps down to point C where only trivial solutions exist. Increasing  beyond point C leads only to the stable trivial solutions. Now, approaching from the left by decreasing , only the trivial solutions exist until we reach point C. Beyond this point, two stable solutions coexist. When the initial conditions are small, the system does not oscillate initially, and the response traces the trivial solutions line CD. This solution loses stability through a transcritical bifurcation at point D giving way to stable non-trivial solutions that quickly encounter a saddle-node bifurcation at point E, leading to a jump to point F. Increasing  further, the amplitude follows the curve FA, where the trivial solutions are reached through a supercritical pitchfork bifurcation at point A. On the other hand, when the initial conditions are large, the response jumps to the non-trivial solutions at point B and follows the curve BA as  is decreased. Examining the overall behavior of the response, one can easily observe a hardening-type non-linearity characterized by the existence of multiple stable solutions when  > 0. This fact can also be deduced by examining the sign (positive in this case) of the effective non-linearity coefficient e for the beam design parameters used in the simulations when K = 0. Fig. 2(b) displays the force--response curve for the uncontrolled beam when =0. 02. Starting from the left when f < fA , only the trivial solution exists. Increasing f beyond point A, two stable solutions coexist. In the absence of large disturbances, only the trivial solutions are maintained as f is increased along the line AB. When f reaches point B, the trivial solution loses stability through a transcritical bifurcation giving way to a branch of stable non-trivial solutions that quickly loses stability through a saddle-node bifurcation, leading to

a jump to point E. Further increase in f leads to higher response amplitudes. On the other hand, when f is beyond fA and the beam is subjected to a large disturbance, the response jumps to the nontrivial large-amplitude periodic solutions. When f is decreased, the amplitude traces the curve ED. At point D, the solution undergoes a saddle-node bifurcation, leading to a jump down to point A where only the trivial solution exists thereafter. 3.3. Response of the controlled beam To demonstrate the effect of time delays on the amplitude, stability, and frequency--response behavior of the beam, we revert to the modulation equations (26) and (27), that describe the amplitude and phase of the solution. By further examination of these equations, it ¯ and ˆ are the two essential parameters that becomes evident that characterize the system response. On one hand, the imaginary part ¯ , affects the response amplitude and hence deterof the feedback, mines whether the feedback produces positive or negative dampˆ , affects ing, and on the other hand, the real part of the feedback, the effective non-linearity coefficient and hence determines whether the system has a hardening- or softening-type behavior. Therefore, by analyzing the variation of the sign and magnitude of these parameters with the controller gain and delay, one can establish good understanding of the effect of time delays on the beam's response. 3.3.1. Effect of feedback delays on the softening-hardening characteristics of the response We begin by studying variation of the effective non-linearity coˆ , see Eq. (32)), with the controller gain K efficient (a function of and inherent system delay . These variations are displayed for the three cases under consideration in Figs. 3(a), (c), and (e). Comparing the three figures, we first note that Fig. 3(a) which corresponds to the position feedback represents a reflection about the K = 0 axis of Fig. 3(e), which corresponds to the acceleration feedback. ˆ = − ˆ . Furthermore, we observe This stems form the fact that 1 3 that, by shifting Fig. 3(a) along the normalized delay axis,  = T , by /2 , one obtains Fig. 3(c) which corresponds to the velocity feedT

ˆ = − 3 K cos( − ˆ = − 3 K cos( ) while back. This is true because 1 2 4 4 /2). Having said that, we only analyze case 1 comprehensively,

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making the adaptation of the analysis to the other two cases a trivial issue. Examining Fig. 3(a), one can observe that the response is hardening for very small values of K regardless of the value of the inherent delay. This is expected, since as shown in Fig. 2(a), the uncontrolled beam has a hardening-type response. However, when K is increased, K 0. 6, the response becomes softening for small values

of , 0. 25, hardening for intermediate values, 0. 250. 8, and reverts back to the softening behavior when 0. 251. 0. This alternating behavior continues as  is increased further because the sign ˆ varies periodically with  and is reversed when K is negative. of Contour plots for the effective non-linearity are also shown in Figs. 3(b), (d), and (f). The darker the shades are, the lower the value of the effective non-linearity is. It is evident that, as the controller

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¯ as function of K and . (b) A contour plot showing the amplitude and stability of the non-trivial solutions Fig. 4. (a) A contour plot showing the sign and magnitude of for  = 0, f = 0. 025,  = 0. 01,  = 0. 01,  = −9. 3, and = −6. 5. Results are obtained for the velocity feedback controller.

Fig. 5. Family of frequency--response curves for the controlled beam response obtained at (a) K = 0. 5, (b) K = 2. 0, and different delays. Results are obtained for f = 0. 025,  = 0. 01,  = 0. 01,  = −9. 3, and = −6. 5. Solid lines represent stable periodic solutions and dashed lines represent unstable periodic solutions.

gain or delay varies, the effective non-linearity changes significantly. Indeed, depending on the gain--delay combinations, the effective non-linearity varies from 0. 44 when the system is uncontrolled to a maximum of 5 and a minimum of −7 when the delays are manifested in the feedback.

real part. This causes the non-linear damping to pump energy into the system producing instabilities and amplifying the response am¯ is equal to  +  and d > 0, Eq. (41) yields the plitude. When a 2 0

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1,2 = ±i d 3.3.2. Effect of feedback delays on the amplitude and stability of the response For the given beam parameters and excitation amplitude and frequency, the amplitude of the non-trivial solutions, and hence the ¯ . This can be response is very much characterized by the value of deduced by examining Eqs. (26) and (41) where it is evident that the ¯ plays an important role in characterizing the stability of the term ¯ <  +  , b is greater than zero yielding eigensystem. When a 2 0

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Since the characteristic equation yields one pair of imaginary eigen¯ cr =  +  . The steady-state solution values at (a0 , 0 ) and a 0 4a2 0 √ ¯ cr if the eigenvalues ±i d have a exhibits a Hopf bifurcation at transversal or a non-zero speed crossing of the imaginary axis. To further check if the transversality condition of the eigenvalues is satisfied, we obtain the eigenvalues of Eq. (41) and differentiate their ¯ , then substitute ¯ cr =  +  into the real part with respect to a 2 0

resulting equation to obtain d1,2 = 2a20 4 ¯ d

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¯ is difFor non-trivial solutions, a0 = 0, and the real part of d1,2 /d ferent from zero. Consequently, the transversality condition is satisfied and the steady-state solutions experience a Hopf bifurcation ¯ cr =  +  and d > 0. This implies that for given K, the when a 2

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¯ with the controller gain, A contour plot, showing variation of K, and delay, , is shown in Fig. 4(a). Fig. 4(a) is compared to the stability diagram of the non-trivial solutions shown in Fig. 4(b). This figure is obtained by solving the modulation equations (26) and (27) and analyzing their stability everywhere in the gain--delay domain at  =0. The diagram demonstrates that, except for very small values of K, the stability of the non-trivial solutions at  = 0 is very much ¯ . Moreover, the shades of the contours characterized by the sign of that represent the amplitude of the response (the darker the shades are the smaller the amplitude is) are in good agreement with the ¯. shades of the contours obtained for 3.3.3. Frequency--response curves The effect of time delays on the frequency--response curves is illustrated next. Velocity-feedback (i.e. case 2) is utilized as the control algorithm. Fig. 5(b) displays a family of frequency--response curves obtained for different time delays and a controller gain, K=0. 5. When the delay is equal to zero, the controller is very effective and the beam experiences a large reduction in the response amplitude when compared to the uncontrolled case, see Fig. 2(a). Since the effective non-linearity is positive (e =0. 4431), the frequency--response curve has a hardening-type behavior. Increasing  to 0. 2, yields a reduction in the controller performance and a substantial increase in the response amplitude accompanied with a significant increase in the hardening non-linearity, e = 0. 799. Moreover, an unstable branch of solutions appears as a result of the non-trivial solutions losing stability via a saddle-node bifurcation. When  is increased to 0. 4, an increase in the hardening behavior is noticed. Further, as a result of the controller pumping energy in to the beam at incorrect time intervals, the non-trivial solutions

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lose stability via a Hopf bifurcation yielding aperiodic responses. The unstable aperiodic solutions and the hardening behavior continue to exist when  is increased to 0. 6. Finally, at  = 0. 8, the controller regains its effectiveness, thereby producing small-amplitude and stable responses. Further, since the effective non-linearity is very small, e = 0. 0864, the response exhibits a linear like behavior. Fig. 5 displays a family of frequency--response curves for K = 2. 0. Similar to the previous case, when  = 0, a large reduction in the amplitude is realized. However, when  is increased to 0. 4, the controller becomes completely ineffective, pumping energy into the system and producing aperiodic responses. Increasing  to 0. 6, changes the behavior of the system to the softening type. At  = 0. 8, the controller regains its effectiveness and produces small-amplitude stable

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periodic responses. Further examination of Fig. 5 reveals that the frequency--response curves are expected to have a linear like behavior near  ≈ 0. 5. This agrees with Fig. 3(a), which illustrates that at  = 0. 5, the effective non-linearity is very close to zero for any value of K. The results obtained via the method of multiple scales, Eqs. (26) and (27), are validated against solutions obtained via a long-time integration of the equation of motion (15). Fig. 6(a) displays a comparison between a frequency--response curve obtained via Eq. (31) and that realized via long-time integration of the full non-linear model. The results show excellent agreement. However, it should be borne

in mind, that these solutions will deviate for larger amplitudes as illustrated in Fig. 6(b). This stems from the fact that the multiplescales solution is based on a small amplitude assumption. 4. Effect of time delays on the controller performance Effect of time delays on the performance of the velocity-feedback controller is demonstrated in Fig. 7. The figure displays variation of the response amplitude and stability with the time delay  at the principle-parametric resonance, i.e.  = 0. The case  = 0 has been extensively analyzed by Oueini and Nayfeh [13] and the controller

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 Fig. 9. Variation of the response amplitude with the feedback delay  for the three non-linear feedback controllers. Results are obtained for  = 0, K = 1, f = 0. 025,  = 0. 01,  = 0. 01,  = −9. 3, and = −6. 5. Solid lines represent stable solutions and dashed lines represent unstable solutions.

was proven very effective for vibration mitigation of parametrically excited cantilever beams. However, as illustrated in Fig. 7, as the time delay increases, the controller performance starts to decrease and completely diminishes at  ≈ 0. 3. At that point, the non-trivial solutions lose stability via a Hopf bifurcation yielding a two-period quasiperiodic cantilever response. Time history of the response just beyond the bifurcation point is illustrated in Fig. 8(b) clearly showing the response aperiodicity. Increasing  further, the amplitude of the unstable solution increases until it reaches a maximum at  ≈ 0. 66 (see Fig. 8(d)). Beyond this point, the amplitude of the unstable solutions starts to decrease and the system regains stability via a reverse Hopf bifurcation at  ≈ 0. 74. As a result, the response becomes periodic but with a larger amplitude (see Fig. 8(e)). Increasing  further, the amplitude of the stable solutions decreases until it approaches that corresponding to  = 0 when  = 1. Performance of the three controllers as function of the time delay is further illustrated in Fig. 9. It is obvious that the performance of the position-feedback controller is the most sensitive to the presence of feedback delays. In fact, a very minute time delay,  < 0. 03, can completely destabilize the beam leading to aperiodic responses. On the other hand, this controller has excellent performance for very large time delays 0. 6 <  < 0. 8. The acceleration feedback, on the other hand, is ineffective for very small and large time delays, but has excellent performance for moderate values 0. 2 <  < 0. 4. The commonly used velocity feedback controller exhibits excellent performance for very small and very large time delays, but is completely ineffective for moderate values. 5. Conclusion We presented an analytical and numerical investigation of the effect of time delays on the dynamic stability and non-linear control of parametrically excited cantilever beams. Three closed-loop cubic non-linear controllers were considered: position, velocity, and acceleration. By utilizing the method of multiple scales, we analyzed the local dynamics of the response for the three cases presented. We observed that, the effect of time delays, can be understood by analyzing the sign and amplitude of two delay-periodic parameters. ¯ affects the response amplitude and hence deterOn one hand, mines whether the feedback produces positive or negative damping,

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ˆ affects the effective non-linearity coeffiand on the other hand, cient and hence determines whether the system has a hardeningor softening-type behavior. We generated contour plots of these parameters in the gain--delay domain and utilized them to assess the amplitude and stability of the closed-loop system. Moreover, we presented a detailed discussion of the effect of time delays on the frequency--response curves and controllers' performance demonstrating that, even the smallest amount of delays, can cause a considerable change in the nature and stability of the response. Furthermore, we compared the performance of the three controllers as a function of the time delay. We found that, for sufficiently large gains, the position feedback is the most sensitive to small time-delays, but has excellent performance for very large time delays. The acceleration feedback, on the other hand, is ineffective for very small and large time delays, but has excellent performance for moderate values. The velocity feedback controller exhibits excellent performance for very small and large time delays, but is completely ineffective for moderate values. References [1] Z.N. Masoud, A.H. Nayfeh, A. Al-Mousa, Delayed-position feedback controller for the reduction of payload pendulations on rotary cranes, J. Vib. Control 8 (2002) 1--21. [2] Z.N. Masoud, M.F. Daqaq, N. Nayfeh, Pendulation reduction on small ship mounted telescopic cranes, J. Vib. Control 10 (2002) 1167--1181. [3] K.A. Alhazza, M.A. Alajmi, Non-linear vibration control of beams using delay feedback controller, in: 12th International Congress on Sound and Vibration, Lisbon, Portugal, July 11--14, 2005. [4] N. Jalili, N. Olgac, Identification and re-tuning of optimum delayed feedback vibration absorber, AIAA J. Guidance Control Dyn. 23 (2000) 961. [5] N. Jalili, N. Olgac, Multiple identical delayed-resonator vibration absorbers for multi-degree-of-freedom mechanical structures, ASME J. Dyn. Syst. Meas. 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