M. Eissa * and M. Sayed Department of Engineering Mathematics, Faculty of Electronic Engineering Menouf 32952, Egypt. *

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1 Mathematical and Computational Applications, Vol., No., pp. 5-6, 006. Association or Scientiic Research A COMPARISON BETWEEN ACTIVE AND PASSIVE VIBRATION CONTROL OF NON-LINEAR SIMPLE PENDULUM PART II: LONGITUDINAL TUNED ABSORBER AND NEGATIVE Gϕ&& AND G n ϕ FEEDBACK M. Eissa * and M. Sayed Department o Engineering Mathematics, Faculty o Electronic Engineering Menou 395, Egypt. * moh_6_@yahoo.com Abstract- In part I [] we dealt with a tuned absorber, which can move in the transversally direction, where it is added to an externally excited pendulum. Active control is applied to the system via negative velocity eedback or its square or cubic value. The multiple time scale perturbation technique is applied throughout. An approximate solution is derived up to second order approximation. The stability o the system is investigated applying both requency response equations and phase plane methods. The eects o the absorber on system behavior are studied numerically. Optimum working conditions o the system are obtained applying passive and active control methods. Both control methods are demonstrated numerically. In this paper, a tuned absorber, in the longitudinal direction, is added to an externally excited pendulum. Active control is applied to the system via negative acceleration eedback or via negative angular displacement or its square or cubic value. An approximate solution is derived up to the second order approximation or the system with absorber. The stability o the system is investigated applying both requency response equations and phase plane methods. The eects o the absorber on system behavior are studied numerically. Optimum working conditions o the system are extracted when applying both passive and active control methods. Keywords- Spring-pendulum, Absorber, Active and passive control.. INTRODUCTION Vibrations and dynamic chaos are undesired phenomenon in structures. They cause disturbance, discomort, damage and destruction o the system or the structure. For these reasons, money, time and eort are spent to get rid o both vibrations and noise or chaos or to minimize them. One o the most eective tools or passive vibration control is the dynamic absorber or the damper or the neutralizer []. Eissa [3] has shown that a non-linear absorber can be used to control the vibration o a non-linear system. Also, he has shown that the non-linear absorber widens its range o applications, and its damping coeicient should be kept minimum or better perormance [4]. Cheng-Tang Lee et al. [5] demonstrated a dynamic vibration absorber system, which can be used to reduce speed luctuations in rotating machinery. Eissa and El-Ganaini [6,7] studied the control o both vibration and dynamic chaos o both internal combustion engines and mechanical structures having quadratic and cubic nonlinearties, subjected to harmonic excitation using single and multi-absorbers. Active constrained layer damping (ACLD) has been successully utilized as eective means o damping out the vibration o various lexible structures [8-3]. A variable stiness vibration absorber without damping is used or controlling the principal mode o a

2 5 M. Eissa and M. Sayed vibrating structure. The optimal vibration absorber is also utilized or controlling higher mode [4]. Another approach o active damping o mechanical structures is the hybrid system, which is a combination o semi-active and active treatments, in which the advantages o individual schemes are combined, while eliminating their shortcomings [5]. Active damping o mechanical structures can be utilized using piezoceramic sensors and actuators [6-7]. The vibration o rotating machinery is suppressed by eliminating the root cause o the vibration system imbalance [8]. In the present paper, a tuned absorber, which can move in the longitudinal direction, is added to an externally excited pendulum, which is described by a second order non-linear dierential equation having both quadratic and cubic non-linearties, subjected to harmonic excitation. Active control is applied to the system via negative acceleration eedback or via negative eed back o angular displacement or its square or cubic value. The multiple time scale perturbation technique is applied throughout. An approximate solution is derived up to the second order approximation or the system with absorbers. The stability o the system is investigated applying both requency response equations and phase plane methods. The eects o the absorber on system behavior are studied numerically. Optimum working conditions o the system are obtained applying both passive and active control methods. Both control methods are compared numerically.. MATHEMATICAL MODELING The considered system is shown in Fig.. As reported in part I, the kinetic and potential energies are given in the ollowing orms respectively: T = M l & ϕ + m [ u& + ( l+ l ) ] 0 + u & ϕ () 4 U = Mg l( cos ϕ) + mg( l+ l0 + u)( cos ϕ) + ku + k 3u 4 ()

3 Vibration Control o Non-Linear Pendulum, Part II 53 Applying Lagrangian equations and taking into account the eects o linear viscous damping and external excitation on the main system, the ollowing dierential equations o motion are obtained: && ϕ+ c & ϕ+ ω sin ϕ+ β{[( l+ l ) + u] u&& ϕ+ ( l+ l + u) u& & ϕ+ gu sin ϕ} = cosωt (3) 0 0 u&& + c u& + ω u + α u ( l + l + u ) ϕ& + g ( cos ϕ ) = 0 (4) 3 0 whereα is the spring stiness non-linear parameter, ω& ω are the natural requencies, c & c are the linear damping coeicients o the pendulum and absorber respectively, is the orcing amplitude andω is the orcing requency o the pendulum. It is assumed that both u and ϕ are small, and the whole motion is a planer one. Due to these assumptions, both (sinϕ) and ( cos ϕ ) can be written in the orm: 4 3 cos ϕ ϕ ϕ! + ϕ, sinϕ ϕ (5, 6) 4! 3! The damping coeicients and the orcing amplitudes are assumed to be in the orm: cn = ε cˆ n, = ε ˆ n =,. (7) where ε is a small perturbation parameter and 0 <ε <<. Eqns. (3) and (4) can be re-written in the orm: && ϕ + ε cˆ ϕ& + ω s i n ϕ + β { [ ( l + l ) + u ] u && ϕ + ( l l ˆ 0 + u) u&& ϕ+ gu( ϕ ϕ )} = ε cosωt 6 (8) 4 3 u&& + ε cˆ u& + ω u + α u ( l + l0 + u ) & ϕ + g ( ϕ ) = 0 4 (9) Assuming the solution o equations (8) and (9) to be in the orm ϕ( t ; ε ) = εϕ( T 0, T ) + ε ϕ( T 0, T ) +... (0) u ( t ; ε ) = ε u( T 0, T ) + ε u ( T 0, T ) +... () where T = ε n t, ( n = 0,). The derivatives will be in the orms where n d d = D0+ εd +..., = D0 + ε D0D + K (, 3) dt dt D =, n = 0,. Equating the similar powers o ε in both side s yields. n Tn ( D + ω ) ϕ = 0 (4) 0 ( D + ω ) u = 0 (5) 0 ( D0 + ω ) ϕ = D0 Dϕ cˆ D0ϕ β( l+ l0) ud 0ϕ β( l+ l0 ) D0u Dϕ β gu ˆ exp( ) ϕ + iω T0 (6) ( D0 + ω ) u = D ˆ 0Du cd0u + ( l+ l0 )( D0ϕ ) gϕ / (7) The general solutions o Eqns. (4) and (5) can be written in the orm ϕ = B exp( iω T ) + cc (8) 0

4 54 M. Eissa and M. Sayed u = B exp( iω T ) + cc (9) 0 where B and B are complex unctions in T, which can be determined rom eliminating the secular terms at the next approximation, and cc represents the complex conjugates. Substituting rom Eqns. (8) and (9) into Eqns. (6) and (7) and eliminating the secular terms, then the irst-order approximation is obtained as: ϕ = Q exp( iω T ) + Q exp( i( ω + ω ) T ) + Q exp( i( ω ω ) T ) + Q exp( iω T ) + cc (0) u = R exp( iω T ) + R exp( iω T ) + R exp( iω T ) + R + cc () where Qi, Ri ( i=,,3, 4) are complex unctions in T and cc represents the complex conjugates. From the above-derived solutions, the reported resonance cases are: a- Primary resonance () Ω ω () Ω ω b- Sub-harmonic and super-harmonic resonance () Ω ω () Ω ω / c- Internal or secondary resonance () ω ω () ω ω d- Simultaneous or incident resonance Any combination o the above resonance cases is considered as simultaneous or incident resonance. 3. STABILITY OF THE SYSTEM Using the simultaneous primary resonance conditions Ω= ω+ εσˆ, ω = ω + εσˆ (where σ= εσˆ and σ = εσˆ are called detuning parameters) and eliminating the secular terms leads to solvability conditions. ˆ iωd B = icˆ ωb + exp( iσˆ T ) () g iω D B ˆ ˆ = icω B B exp( iσ T ) (3) Putting bˆ n Bn = exp( iη n ), b ˆ n = εbn n =, (4) where b n & η n are the steady state amplitudes and the phases o the motions respectively. Substituting rom equation (4) into equations ()-(3) and equating the real and imaginary parts we obtain: cb b& = + sin( θ ) ω (5) bθ& = bσ + c o s( θ ) ω (6) cb gb b& = sin( θ ) 4ω (7) g b b ( θ& + θ& ) = b ( σ + σ ) cos( θ ) 4ω (8) θ = ˆ σ T η and θ = ˆ σ T + η η where

5 Vibration Control o Non-Linear Pendulum, Part II 55 The periodic motions are obtained when a& n = θ& n = 0. Hence, the ixed points o Eqns. (5)- (8) are given by c b + s in ( θ ) = 0 (9) ω bσ + c o s ( θ ) = 0 (30) ω cb gb sin( θ ) = 0 4ω (3) g b b ( σ + σ ) cos( θ ) = 0 4ω (3) There are three possibilities in addition to the trivial solution. They are: () b 0, b = 0 () b 0, b = 0 (3) b 0, b 0 Table () illustrates the corresponding requency response equation or each case: Table () Frequency response equation or each case. No Case Frequency response equation (FRE) c b b 0, b = 0 b σ + = 0 4 4ω 3 b 0, b = 0 b 0, b 0 c b b ( σ + σ ) + = 0 4 c As b 0 then ( σ + σ ) + = 0 4 c b b σ + = 0 4 4ω c b g b 4 6ω b ( σ + σ ) + = 0 * This equation gives nothing, this means that we will never get b = Stability o the ixed points To analyze the stability o the ixed points, one lets bn = bn 0 + b n, θ n = θ n 0 + θ n (33) where a n0, θ n0 are the solutions o Eqns. (9)-(3). Inserting Eq. (33) into Eqns. (5)- (8) and keeping only the linear terms in a n, θ n, we get cb b& = + θ cos( θ 0 ) (34) ω cb g b& = { b sin( θ 0 ) + b0θ cos( θ 0 )} (35) 4ω *

6 56 M. Eissa and M. Sayed & b σ = sin( ) (36) θ θ θ b 0 0 ω b0 & bσ g bσ θ = { b cos( θ ) b0θ sin( θ0 )} + θ sin( θ ) (37) b ω b0 b0 ω b0 The stability o a particular ixed point with respect to perturbations proportional to exp ( λ T ) depends on the real parts o the roots o the matrix. Thus a ixed point given in Eqns. (34)-(37) is an asymptotically stable i and only i the real parts o all roots o the matrix are negative. To study the stability o the ixed points corresponding to case (), we let b = θ = 0 in Eqns. (34)-(37), and obtain the eigenvalues λ = ( L ) ( ) + L4 ± L + L4 + 4( L L3 L L 4 ) (38) c σ where L =, L = cos( θ ), L ω 3 = and L4 = sin( θ ) (39) b ω b And hence the ixed points are unstable i and only i L L 3 > L L 4 (40) Otherwise they are stable. To study the stability o the ixed points corresponding to case (), we let b = θ = in Eqns. (34)-(37), and obtain the eigenvalues 0 λ = ( L + L ) ± ( L + L ) + 4( L L L L ) (4) Where = c g L, cos( ) σ g L = b θ, L7 = and L8 = b sin( θ ) (4) ω b 4ω b And hence the ixed points are unstable i and only i L 6 L 7 > L 5 L 8 (43) Otherwise they are stable. For the stability o the ixed points corresponding to case (3), the eigenvalues are given by the equation 4 3 λ + R λ + R λ + R 3λ + R 4 = 0 (44) Where R, R, R 3 and R 4 are unctions o the parameters ( b, b, ω, ω, σ, σ,, θ, θ ). According to the Routh-Huriwitz criterion, the necessary and suicient conditions or all the roots o Eqn. (44) to possess negative real parts is that R >0, R R -R 3 >0, R 3 (R R -R 3 )- R R >0, R 4 4 >0 (45) 4. RESULTS AND DISCUSSIONS Results are presented in graphical orms as steady state amplitudes against detuning parameters and as time history or the response or both system and absorber. A good criterion o both stability and dynamic chaos is the phase-plane trajectories,

7 Vibration Control o Non-Linear Pendulum, Part II 57 which are shown or some cases. In the ollowing sections, the eects o the dierent parameters on response and stability will be investigated. Also dierent primary resonance cases are studied and discussed. 4.. System stability Fig. a, shows the eects o the detuning parameter σ on the steady state amplitude o the main system b or the stability irst case, where b 0 and b = 0. It can be seen rom the igure that the maximum steady state amplitude occurs at primary resonance when Ω ω. Fig. b shows that the steady state amplitude o the main system is a monotonic decreasing unction to the natural requency ω. Fig. c shows that the steady state amplitude o the main system is a monotonic increasing unction to the excitation amplitude. Fig. d shows that the steady state amplitude o the main system is a monotonic decreasing unction to the damping coeicient c. Figure 3, shows the eects o the detuning parameter σ on the steady state amplitude o the absorber b or the stability third case, where b 0 and b 0. It can be seen rom the igure that the maximum steady state amplitude occurs at internal resonance when ω ω. Fig. 3b shows that the steady state amplitude o the absorber is a monotonic increasing unction to the steady state amplitude o the main system b. Figs. (3c-3e) shows that the steady state amplitude o the absorber is a monotonic decreasing unction in the natural requency ω, detuning parameter σ and damping coeicient c.

8 58 M. Eissa and M. Sayed Fig. 4 shows the eect o the non-linear parameter α on the main system and absorber. From the igure, we can see that the steady state amplitude o the main system is a monotonic increasing unction or α 5 and or increasing value we obtain the saturation phenomena. Also the steady state amplitude o the absorber is a monotonic decreasing unction in the non-linear parameter α and or increasing value we obtain the saturation phenomena.

9 Vibration Control o Non-Linear Pendulum, Part II 59 α α 4.. Passive control In the ollowing section we will discuss the eects o the absorber on pendulum response, stability and dynamic chaos at the worst resonance case. This case is the primary one, where Ω ω. Fig. 5 illustrates both the response and the phase plane or this case. The steady state response without absorber in this case Ω ω is about 30%, o the excitation amplitude; the system is stable and ree o dynamic chaos. Eects o the absorber: Figs. (6a-6b) illustrate the results when the absorber is eective or the dierent resonance cases. They are Ω ω and Ω ω. Simultaneously the ratio ω /ω is varied between zero and 6, i.e., 0 ω /ω 6. It can be seen or the irst case shown in Fig. 6a that the eectiveness o the absorber E a is about. Best results or the absorber were obtained when 3ω ω. Fig. 6b shows that the absorber is ineective as it may increase the amplitude with its maximum amplitude value occurs when ω ω. A common eature or all cases is the occurrence o saturation phenomenon.

10 60 M. Eissa and M. Sayed 4.3. Active control Active control is applied to improve the behavior o the simple pendulum at the primary resonance case Ω ω. First case, we considered negative acceleration eedback. The equation o motion in this case is: && ϕ + ε cˆ & ϕ + ω sin( ϕ ) + G && ϕ = cosωt (46) Where G is the gain. Here, we are concerned with the eect o the gain G on the pendulum response. From Fig. 7a, we can see that the steady state amplitude is a monotonic decreasing unction in the gain and it is decreased to about % o the steady state amplitude, and more increase o the gain G leads to saturation phenomena. For second case vibration is controlled via negative angular displacement eedback or its square or cubic value. The equation o motion in this case is: n && ϕ + ε cˆ & ϕ + ω sin( ϕ ) + Gϕ = cosωt (47) Three cases will be considered, when n=, and 3. For n=, the amplitude is increasing up to G=0.. Then or the region 0.<G<. the system is unstable. For the region. G.3 the system is stable with increasing amplitude. When 3.4 G, the system is stable with decreasing amplitude as shown in Fig. 7b. This means that G should be greater the 0 to control the system where E a =7, at saturation beginning. For n=, the amplitude is increasing up to G=0.. Then or the region 0.<G<0.4 the system is unstable. For the region 0.4 G 0.8 the system is stable with increasing amplitude. For the region 0.9 G<.3 the system is unstable, and or.3 G.4 the system is stable with increasing amplitude. When.8 G, the system is stable with decreasing amplitude as shown in Fig. 7c. This means that G should be greater the 0 to control the system where E a =3, at saturation beginning. It is clear that with active control, care should be taken because the system may be lead to instability instead o reducing the amplitude. For (n=3), Fig. 7d, shows that or G the steady state amplitude is a monotonic increasing unction in the gain G and it is increased to about 300% o the steady state amplitude. For G> the steady state amplitude is a monotonic decreasing unction and it is decreased to about 30% o the steady state amplitude.

11 Vibration Control o Non-Linear Pendulum, Part II 6 5. CONCLUSIONS From the ormer results, the ollowing may be concluded. - The steady state amplitude o the main system is a monotonic increasing unction in the excitation amplitude. - The steady state amplitude o the main system is a monotonic decreasing unction in its natural requency ω and damping coeicient c. 3- For passive control the eectiveness o the absorber or the system is about E a = when Ω ω, ω ω and best results or the absorber is when 3ω ω. 4- Non-eective absorber is obtained when Ω ω, ω.5ω or Ω ω. 5- The vibration o the system can be controlled actively via negative angular displacement eedback, which can be used to reduce the amplitude o the system to 5% o the original value. 6- For all cases o active control, occurrence o saturation phenomena is noticed. REFERENCES. M. Eissa, M. Sayed, A Comparison between active and passive vibration control o non-linear simple pendulum, Part I: transversally tuned absorber and negative Gϕ& n eedback, Mathematical & Computational Applications, 37-49, J. D. Mead, Passive Vibration Control, John Wiley & Sons, M. Eissa, Vibration and chaos control in I. C engines subject to harmonic torque via non-linear absorbers, ISMV, Second International Symposium on Mechanical Vibrations. Islamabad, Pakistan, M. Eissa, Vibration control o non-linear mechanical system via a neutralizer, Electronic Bulletin No 6, Faculty o Electronic Engineering Menou, Egypt, July, Cheng-Tang Lee et al., Sub-harmonic vibration absorber or rotating machinery, ASME Journal o Vibration and Acoustics, 9, , M. Eissa and El-Ganaini, Part I, Multi-absorbers or vibration control o non-linear structures to harmonic excitations, ISMV Conerence, Islamabad, Pakistan, M. Eissa and El-Ganaini, Part II, Multi-absorbers or vibration control o non-linear structures to harmonic excitations, ISMV Conerence, Islamabad, Pakistan, 000.

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