Vibration Reduction and Stability of Non-Linear. System Subjected to External and Parametric. Excitation Forces under a Non-Linear Absorber

Transcription

1 Int. J. Contemp. Math. Sciences, Vol.,, no., 5-7 Vibration Reduction and Stability of Non-Linear System Subjected to Eternal and Parametric Ecitation Forces under a Non-Linear Absorber M. Sayed *, Y. S. Hamed * and Y. A. Amer ** Department of Mathematics and Statistics, Faculty of Science, Taif University El-Taif, El-Haweiah, P.O. Bo, Zip Code 97, Kingdom of Saudi Arabia eng_yaser_salah@yahoo.com * Faculty of Electronic Engineering, Menoufia University, Menouf 95, Egypt ** Department of Mathematics, Faculty of Science, Zagazig University Zagazig, Egypt Abstract Vibrations are usually undesired phenomena as they may cause discomfort, disturbance, damage, and sometimes destruction of machines and structures. It must be reduced or controlled or eliminated. One of the most common methods of vibration control is the dynamic absorber. In this paper, the non-linear dynamics of a two-degree-of freedom vibration system including quadratic and cubic non-linearities subjected to eternal and parametric ecitation forces is investigated. The system consists of the main system and the absorber which represents many applications in machine tools, ultrasonic cutting process, etc. The vibration of the main system can be controlled applying non-linear absorber (passive control). The stability of the system is investigated using both frequency response curves and phase-plane trajectories. Multiple time scales perturbation method is applied to differential equations describing the system to obtain the analytical solution. All possible resonance cases are etracted. The effects of different parameters of the system are studied numerically. There eist multi-valued solutions which increase or decrease by the variation of some parameters. The solution loses stability on increasing negative value of α. The Numerical simulations show the system ehibits periodic motions and chaotic motions. A comparison is made with the available published work.

2 5 M. Sayed, Y. S. Hamed and Y. A. Amer Keywords: Non-linear vibration; response; stability; resonance; jump phenomenon. Introduction The dynamic response of mechanical and civil structures subject to highamplitude vibration is often dangerous and undesirable. Sometimes controlled vibration is desirable as in ultrasonic machining (USM), as the machining technique is dependent on tool and abrasive particles vibration. It is required to reduce the vibration in the machine head and have reasonable amplitude for the tools. This can be done via dynamic absorber. It has the advantages of low cost and simple operation at one modal frequency. In the domain of many mechanical vibration systems the coupled non-linear vibration of such systems can be reduced to non-linear second order differential equations which are solved analytically and numerically. Queini and Nayfeh [] proposed a non-linear active control law to suppress the vibrations of the first mode of a cantilever beam when subjected to a principal parametric ecitation. The method of multiple scales is applied throughout. The analysis revealed that cubic velocity feedback reduced the amplitude of the response. Asfar [] took material non-linearity into consideration in the analysis of the performance of an elastomeric damper with a spring hardening cubic effects near primary resonance condition applying multiple time scale method. Eissa [] reported that when using a dynamic absorber, its damping coefficient should be kept minimal for better system performance. Eissa [] has shown that for controlling the vibration of a system subjected to harmonic ecitations, the fundamental or the first harmonic absorber is the most effective one. Eissa and El-Ganaini [5, ] studied the control of both vibration and dynamics chaos of mechanical system having quadratic and cubic non-linearities, subjected to harmonic ecitation using multi-absorbers. Kamel and Amer [7] studied the behavior of one-degree-of-freedom system with different quadratic damping and cubic stiffness non-linearities simulating the aial vibration of a cantilever beam under multi-parametric ecitation forces. Song et al. [] investigated the vibration response of the spring mass damper system with a parametrically ecited pendulum hinged to the mass using the harmonic balance method. The stability analysis showed that the area of unstable motion of the system obtained from the third order approimation. Soom [9] and Jordanov [] studied the optimal parameter design of linear and non-linear dynamic vibration absorbers for damped primary systems. They eamined optimization criteria other than the traditional one and obtained small improvements in steady state response by using non-linear springs. However, the presence of the non-linearities introduces dangerous instability, which in some cases may result in amplification rather than reduction of the vibration amplitudes [, ]. Natsiavas [] applied the method of averaging to investigate the steady state oscillations and stability of non-linear dynamic vibration absorbers. He pointed out that proper selection of the system parameters would result

3 Vibration reduction and stability of non-linear system 5 insubstantial improvements of non-linear absorbers and avoid dangerous effects that are likely to occur due to the presence of the non-linearities. Zhu et al. [] studied the non-linear dynamics of a two-degree-of freedom vibrating system having both non-linear damping and non-linear spring using the averaging method. Results showed that the vibration amplitude can be reduced by properly selecting the values of non-linear damper, non-linear spring stiffness and the range of eciting frequency. Lim et al. [5] investigated the behavior of the (USM) hypothesized theoretical model which described by the coupling of two non-linear oscillators. The method of multiple scales has been used to solve the equations to first order perturbation. The theoretical results showed that controlled variations in the softening stiffness can have a significant effect on the overall non-linear response of the system, by making the overall effect hardening, softening, or approimately linear. Eperimentally, it has also been demonstrated that coupling of ultrasonic components with different non-linear characteristics can strongly influence the performance of the system. Eissa and Amer [] simulated the vibration of a second order system to the first mode of a cantilever beam subjected to both eternal and parametric ecitation at primary and sub-harmonic resonance. They analyzed the system using the method of multiple scales. They reported that the vibration of the system can be controlled by adding a feedback cubic non-linear term. They reported also that there is a threshold value for the linear damping coefficient which can be applied to control the system vibration. Nayfeh [7] compared application of the method of multiple scales with reconstitution and the generalized method of averaging for determining higher-order approimations of three single-degree-of-freedom systems and a two-degree-of-freedom system. He showed that the second-order frequency-response equation possesses spurious solutions for the case of softening nonlinearity. El-Bassiouny [] investigated the effects of quadratic and cubic non-linearities in elastomeric material dampers on torsional vibration control. The multiple time scale is used to solve the stability equations at primary resonance. He showed that the elastomeric damper reduced the vibration of the crankshaft effectively. Eissa et al. [9, ] presented the tuned absorber in both transversely and longitudinal directions of a simple pendulum which designed to control one frequency at primary resonance. The multiple time scale perturbation technique is applied throughout. They demonstrated the effectiveness of the absorber for passive control. They reported that the vibration of the system can be controlled actively via negative velocity feedback, which can be used to reduce the amplitude of the system. Amer [] investigated the coupling of two non-linear oscillators of main system and absorber representing ultrasonic cutting process subjected to parametric ecitation forces. A threshold value of main system linear damping has been obtained, where vibration can be reduced dramatically. This threshold value can be used effectively for passive vibration control, if it is economical. The multiple time scale perturbation technique is applied throughout. A threshold value of linear damping has been obtained, where the system vibration can be reduced dramatically.

4 5 M. Sayed, Y. S. Hamed and Y. A. Amer Hamed et al. [-5] studied USM model subject to multi-eternal or both multi-eternal and multi-parametric and both multi-eternal and tuned ecitation forces. The model consists of multi-degree-of-freedom system consisting of the tool holder and absorbers (tools) simulating ultrasonic machining process. The advantages of using multi-tools are to machine different materials and different shapes at the same time. This leads to time saving and higher machining efficiency.. Passive Control Using a non-linear tuned mass absorber connected to the main system, a model of a two degree-of-freedom oscillator under consideration is shown in Fig., from the principles of the mechanics, the derived equations of motion can be written in the forms: && +ω +εα +εα +εζ & +εζ & +εζ( & & ) εα +εα ( ) + εα ( ) = εf cosω t + ε F cosω t () 5 && + ω ( ) + εβ ( ) + εβ ( ) + εζ ( & & ) = () m k F cos Ωt + F cos Ω t m c k c Fig.. Schematic diagram of the model with absorber. where k ω = + k k, α = k, α = c, ζ = c, ζ = c, ζ = k, α =,, m m m m m m m k k f j k c k k α =, α 5 =, Fj =, ω =, ζ =, β =, β = m m m m m m m c, c are the damping coefficients of the main system and absorber, c the coupling damping coefficient, m, m the mass of the main system and absorber,

5 Vibration reduction and stability of non-linear system 55 Ω j ( j=, ) forcing frequencies, ω, ω natural frequencies, k, k linear stiffness,, displacement of the main system and absorber, & j, && j derivatives of w. r. t time, ε small perturbation parameter, f j the forcing amplitudes, α i, β j non-linear parameters (i =,,5), ζs damping coefficient (s =,, ).. Method of Analysis A first-order uniform solution of equations ()-() is sought using the method of multiple scales [] in the form: ( t; ε ) = ( T, T) +ε ( T, T) +... () ( t; ε ) = ( T, T) +ε ( T, T) +... () where T o = t is fast time scale, which is associated with changes occurring at the frequencies ω j and Ω j and T = ε t is the slow time scale, which is associated with modulations in the amplitudes and phases resulting from the non-linearities and parametric resonance. In term of T o and T the time derivatives became d D D... dt = +ε +, d = D + ε DD+... (5) dt Where D n differential operators; Dn = / Tn ( n=,). Substituting equations () and () into equations ()-() and equating the coefficients of same power of ε in both sides, we obtain: Order ε : ( D +ω ) = () ( D +ω ) =ω (7) Order ε : ( D +ω ) = F cos( Ω T ) + F cos( ΩT ) D D α α ζd ζ( D ) ζ( D D ) +α α ( + ) α5( + ) () ( D +ω ) = D D +ω β( + ) β ( + ) ζ( D D ) (9) The general solution of equations ()-(7) can be epressed in the form iωt = Ae + cc () iωt iωt = Ae + A ω ( ω ω ) e + cc () where A and A are unknown functions in T, which can be determined from eliminating the secular terms at the net approimation, and cc stands for the conjugate of the preceding terms. The particular solutions of equations ()-(9) after eliminating the secular terms are given by:

6 5 M. Sayed, Y. S. Hamed and Y. A. Amer = Ue + Ue + Ue + He + He iωt i ( Ω +ω ) T i ( Ω ω ) T iωt iωt + H e + H e + H e + H e + H e i ω T iω T iω T i ( ω +ω ) T i ( ω ω ) T 5 7 i ( ω+ω ) T i ( ω ω ) T i ( ω+ ω) T i ( ω ω) T + H e + H9e + He + He + H + cc () iωt i( Ω +ω) T i( Ω ω) T iωt iωt = U e + U e + U e + H e + H e 5 + H e + H e + H e + H e + H e iω T iω T iω T i( ω +ω ) T i( ω ω ) T H e + H e + H e i( ω + ω ) T i( ω ω ) T i( ω + ω ) T i( ω ω) T H e H cc () where U i, i = to, H i, i = to, are comple functions in T. The general solution of and up to the first-order approimation is given by = +ε +..., = +ε Resonance Cases From the equations () to () all possible resonance cases are: (a) Trivial resonance: Ωj ω j =, j =,. (b) Internal resonance: ω nω, ω nω, n =,, (c) Primary resonance: Ω ω, Ω ω (d) Sub-harmonic resonance: Ω ω (e) Combined resonance: Ω ±ω ±ω (f) Simultaneous resonance: Any combination of the above resonance cases is considered as simultaneous resonance.. Stability Analysis The stability of the system is investigated at one of the worst resonance cases (confirmed numerically), which is the simultaneous primary, principal parametric and internal resonance where Ω ω, Ω ω and ω ω. Using the detuning parameters σ, σ and σ such that: Ω =ω +εσ, Ω = ω +εσ and ω = ω +εσ () Substituting equation () into equations ()-(9) and eliminating the secular terms leads to the solvability conditions as iω DA = [ iωζ +α iωζ ( ) ] A iωζ + α + α5( ) AA F i T A F i T i T α5( ) AAA + e σ + e σ + α5( ) A Ae σ (5) i σt i ω D A = i ω ζ A + ω + β ( ) A e + ω A β A A β ( ) AAA () where =ω ( ω ω ), = i ωζ +α +α5( ) ω

7 Vibration reduction and stability of non-linear system 57 and = i ωζ + α + α5aa( ) + α 5A A ( ω ω ) We epress the comple function A n in the polar form as i n ( T ) A n = ( ) a n ( T ) e μ ( n =,) (7) where a n and μ n are real. Substituting equation (7) into equations (5) and () and separating real and imaginary part yields, F F ζ +ζ( ) α5( ) ζ a = sin γ + sin γ a+ sin γ aa a () ω ω ω F F α α5( ) α5( ) μ a= cos γ cos γ + a+ aa cos γ aa ω ω ω ω ω α +α5( ) + a (9) ω a ζ + ζ ωζ = sin γ cos γ a ω a () α ( α+β 5 )( ) ( α+β 5 ) ωζ μ a = + + cos γ + sin γ ω a ω a a ω a ω a () ω( α +α5( ) ) β( ) where = + ω ω and γ = σt μ, γ = σt μ, γ = σt μ + μ () For the steady state solution a n = γ n =. Then from equation () yields μ =σ =σ /=σ, μ = σ σ () Then it follows from equations ()-() that the steady state solutions are given by F F ζ+ζ ( ) α5( ) ζ sin γ+ sin γ a+ sin γ aa a = () ω ω ω F F α α5( ) α5( ) aσ+ cos γ + cos γ + a aa + cos γ a a ω ω ω ω ω [α + α ( ) ]/ ω a = (5) { } 5 [ ] a [ ] a ( ζ +ζ ) / sin γ ( ω ζ / ω )cos γ = () α ( α+β 5 )( ) ( α+β 5 ) ( σ σ) ω ω ω a a a a a +cos γ +ωζ ( ω )sin γ a = (7) [ ] Solving the resulting algebraic equations yields two possibilities for the fied points for each case.

8 5 M. Sayed, Y. S. Hamed and Y. A. Amer Case (): The controller is deactivated ( a, a = ), the frequency response equation can be obtained in the form 9α σα ζ F F FF a a + σ + a a a cos( γ γ ) = () ω ω ω ω ω Case (): the controller is activated ( a, a ), the resulting two equations are obtained F F a + 5a + a + 7aa + aa + 9aa +aa a ω ω ( FF / ω ) a cos( γ γ ) = (9) a + a + a + a a + a a + a a + a = () 5 7 where ( i =,...,7) are given in the appendi. i. Numerical Results and Discussion To study the behavior of the main system, the Runge-Kutta fourth-order method was applied to the equation () governing the oscillating system, after eliminating all parameters corresponding to controller. Fig. shows the steady state amplitude and phase plane of the main system at the primary resonance where Ω ω for the parameters ζ =.5, F=., F =., α =., α =.. In this Figure, we observe that the steady state amplitude is about five of the ecitation force amplitude F. The main system is stable. d dt Time Fig.. System behavior without absorber at primary resonance Ω ω and Ω is a way from ω.

9 Vibration reduction and stability of non-linear system 59 Figs. and show the steady state amplitude and phase plane of the main system at the principle parametric and simultaneous primary and principle parametric resonance respectively where Ω ω and Ω ω, Ω ω. In these Figures, we note that the steady state amplitude is about 9 and of the ecitation force amplitude F respectively. Also the chaotic wave motion will increase. d dt Time Fig.. System behavior without absorber at principle parametric resonance Ω ω and Ω is a way from ω. Fig. 5 shows that the steady state amplitude of the system with absorber at the simultaneous primary and internal resonance Ω ω, ω ω. The effectiveness of the absorber E a ( E a = steady state amplitude of the system without absorber / steady state amplitude of the system with absorber) is about 5, which means that the steady state amplitude is reduced to less than % of the maimum amplitude shown in Fig.. d dt Time Fig.. System behavior without absorber at simultaneous resonance Ω ω and Ω ω. Also, the oscillations of the system and absorber have multi-limit cycle and chaotic, respectively. All solutions of the system and absorber are stable. Fig. shows that the steady state amplitude of the system with absorber at the simultaneous principal parametric and internal resonance Ω ω, ω ω. The effectiveness of the absorber Ea is about, which means that the steady state amplitude is reduced to less than % of the maimum amplitude shown in Fig.. Also, the oscillations of the system and absorber have multi-limit cycle and increasing dynamic chaos, respectively.

10 M. Sayed, Y. S. Hamed and Y. A. Amer d dt Time d dt Time Fig.5. Response of the system and absorber at simultaneous resonance Ω ω and ω ω. Fig. 7 shows that the steady state amplitude of the system with absorber at the simultaneous resonance Ω ω, Ω ω and ω ω. The effectiveness of the absorber Ea is about 7, which means that the steady state amplitude is reduced to less than % of the maimum amplitude shown in Fig.. Also, the oscillations of the system and absorber have multi-limit cycle and increasing dynamic chaos, respectively. d dt Time d dt Time Fig.. Response of the system and absorber at simultaneous resonance Ω ω and ω ω.

11 Vibration reduction and stability of non-linear system d dt Time d dt Time Fig.7. Response of the system and absorber at simultaneous resonance Ω ω, Ω ω and ω ω.. Theoretical frequency and force-response curves Case : The controller is deactivated, a, a = : The analytical analysis is represented graphically by using the numerical methods. The frequency response equation () is a nonlinear algebraic equation, which are solved numerically by using Newton Raphson method. The numerical results are shown in Figures -7. In all Figures, the region of stability of nontrivial solutions is determined by applying the Routh-Hurwitz criterion. The stable and unstable solutions are represented by solid and dotted lines respectively on the response curves. Figs -7 show the frequency-response and force-response curves for the stability first case. In Figs. -, we observe that the solutions is stable for negative values of σ and it has stable and unstable for positive values of σ. Fig. shows the effects of he detuning parameter σ on the steady state amplitude of the system. In this Figure, the response amplitude consists of a continuous curve which is bent to the right and has hardening phenomena. This continuous curve has stable and unstable solutions.

12 M. Sayed, Y. S. Hamed and Y. A. Amer a σ a α = α =. α =. - - σ Fig.. Effects of the detuning parameter σ Fig. 9. Effects of the nonlinear spring stiffness α a - - σ F = F = F = Fig.. Effects of the ecitation amplitude F a σ ω =. ω = ω =.5 Fig.. Effects of the natural frequencies Fig. 9 shows that as non-linear spring stiffness α is increased positively the continuous curve is moved downwards and has decreased magnitudes. It can be concluded that increasing the non-linear spring stiffness α can reduce the amplitude of the system and obtain the effect of reduction of the vibration amplitude. The regions of multi-valued and instability zone are increased and decreased respectively. Also for negative value of non-linear spring stiffness α the response amplitude is bent to the left and the stability region is decreased. The positive and negative values of the non-linear spring stiffness α produce either hard or soft spring, respectively. The steady state amplitude a is a monotonic increasing function in the ecitation amplitude F and for small values of F the single-valued curve is shifted upwards with decreased instability zone, as shown in Fig.. The zones of multi-valued and instability region are increased for large values of F. It is clear that from Fig. for decreasing natural frequency ω the steady state amplitude is increasing and the curve is bent to the right, leading to multi-valued amplitudes and to appearance of the jump phenomenon. From Fig., the steady state amplitude is a monotonic decreasing function in the damping coefficient ζ. When the linear damping factor ζ increase up to., we note that the bending contracts with decreased magnitudes and the multi-valued disappears.

13 Vibration reduction and stability of non-linear system Figs. -7 represent the eternal ecitation-response curves for primary resonance. It is evident from Fig. that the response amplitude has a continuous curve and there eist zone of multi-valued solutions. There eist jump phenomenon and the curve has unstable and stable solutions. From Fig., we observe that for increasing positive value of non-linear spring stiffness α the continuous curve is shifted downwards with decreasing region of instability. For negative values of α, the continuous curve becomes unstable and the region of multi-valued is disappeared. Figs. 5 and show that for increasing values of natural frequency ω and detuning parameter σ, respectively, the curve is shifted upwards and has increased instability region. The region of multi-valued is increased for increased natural frequency, detuning parameters. For negative value of detuning parameter σ, the continuous curve becomes stable and the region of multi-valued is disappeared as shown in Fig.. For increasing value of damping coefficient ζ, we note that the region of multi-valued is disappeared and the continuous curve has a single valued curve which is shifted downwards with increasing instability zone as shown in Fig. 7. a ζ =. ζ =. ζ =. a - σ F Fig.. Effects of the damping coefficient ζ Fig.. Ecitation-response curve of Ω ω 5 a a.5.5. F F Fig.. Effects of the non-linear spring stiffness α Fig.5. Effects of the natural frequency ω

14 M. Sayed, Y. S. Hamed and Y. A. Amer a a.. F F Fig.. Effects of the detuning parameter σ Fig.7. Effects of the damping coefficient ζ Case : The controller is activated, a, a : The frequency response equations (9) and () are nonlinear algebraic equations, which are solved numerically. The numerical results are shown in Figures -. The solid curves denote the stable solutions and the dotted curves denote the unstable solutions. These Figures, show the steady state response-frequency curve for the stability second case, where a, a in the case of simultaneous primary resonance in the presence of three-to-one internal resonance. In these Figures, we observe that the solutions is stable for negative values of σ and it has stable and unstable for positive values of σ. Fig. shows that the effect of the detuning parameter σ on the steady state amplitudes of the system and absorber. In this Figure, we observe that, the amplitudes a and a have one continuous curve, that the continuous curve of the system lies upper than the continuous curve of the absorber. For increasing ecitation force F, we note that the stability magnitudes of the system and absorber are increased as shown in Fig. 9. a a a a σ.5 Fig.. Effects of the detuning parameter σ

15 Vibration reduction and stability of non-linear system 5 Several groups of various parameters are tried out to discuss the vibration property of the system and the selections of the relative parameters are taken by virtue of Ref. []. a F = F = F = a F = F = F = - - σ Fig. 9. Effects of the ecitation amplitude F σ.. In Fig., with increase in the non-linear spring stiffness α, the vibration amplitude of the system and absorber reduces. It can be concluded that increasing the non-linear spring stiffness α can reduce the amplitude of the main mass and obtain the effect of reduction of the vibration amplitude. The system and absorber becomes unstable for negative value of non-linear spring stiffness α. Fig. shows that for increasing linear damping coefficient, we observe that the both system and absorber have decreasing magnitudes. Fig. evaluates the effectiveness of the cubic non-linear damping ζ on the amplitude reduction of the system. From the curves, it can be seen that the non-linear damping has an obvious effect on the amplitude reduction. a α =. α =. α =. a α =. α =. α =. - - σ Fig.. Effects of the non-linear parameter α -. σ.

16 M. Sayed, Y. S. Hamed and Y. A. Amer a ζ =. a ζ =. ζ =. ζ =. - - σ σ.. Fig.. Effects of the linear damping coefficient ζ a ζ =. ζ = σ.. Fig.. Effects of the cubic non-linear damping ζ 5. Conclusions The system consists of the main system and the absorber representing the vibration of many applications in machine tools, ultrasonic cutting process, subjected to eternal and parametric ecitation forces is considered and solved using the method of multiple scale perturbation. A simple and powerful effective method is demonstrated to reduce (control passively) both vibration and dynamic chaos in the non-linear system using non-linear absorber. One of the most common methods of vibration control is the dynamic absorber. Multiple scales perturbation technique is applied to determine approimate solutions of the coupled non-linear differential equations. Steady state solutions and their stability are studied for selected values of different parameters. Frequency response equations are deduced to investigate system stability. It can be seen from the results that the optimal working conditions of the system is principle parametric and internal resonance Ω ω, ω ω. From the above study the following may be concluded. (i) When the controller is deactivated. The steady state amplitude is a monotonic increasing and decreasing function to the ecitation amplitudes F, F and natural frequency ω, damping coefficient ζ, respectively.

17 Vibration reduction and stability of non-linear system 7. The zones of multi-valued and instability region are increased for large values of F.. For increasing positive value of non-linear spring stiffness α the continuous curve is shifted downwards produce hard spring with decreasing region of instability.. For negative values of α, the curve is bent to the left produce soft spring, the region of unstable is increased and the region of multi-valued is disappeared. 5. The region of multi-valued is disappeared for increasing value of damping coefficient ζ, and the continuous curve has a single valued curve which is shifted downwards.. The continuous curve becomes stable and the region of multi-valued is disappeared for negative value of detuning parameter σ. 7. (ii) When the controller is activated. The optimum working conditions for the system are when, Ω ω, ω ω since the steady state amplitude of the system is reduced to % of its maimum value. This means that the effectiveness of the absorber E a =.. The effectiveness of the absorber E a is about 5 and 7 for simultaneous primary and internal resonance Ω ω, ω ω and simultaneous resonance Ω ω, Ω ω, ω ω respectively.. For increasing ecitation force F and decreasing damping coefficient, we note that the stability magnitudes of the main system and absorber are increased.. From the curves, it can be seen that the non-linear damping has an obvious effect on the amplitude reduction of the system, which is a good agreement with Ref. []. 5. Increasing the non-linear spring stiffness α can reduce the amplitude of the main mass and obtain the effect of reduction of the vibration amplitude, which is a good agreement with Ref. [].. Numerical simulations show the system ehibits periodic motions and chaotic motions, which is a good agreement with Ref. []. In comparison with the previous work [] the authors consider the nonlinear dynamic and spring as a cubic one only. The method of averaging is used to obtain the steady state response and the system is subjected to single eternal ecitation force. The stability is studied applying bifurcation and Poincare` map. It is worth to mention that in the reported results assuming the nonlinear term is zero. In our study, the model considered in Ref. [], but subjected to eternal and parametric ecitation forces instead of an eternal only.

18 M. Sayed, Y. S. Hamed and Y. A. Amer Appendi ζ α + α5( ) = + ω ζ +ζ ( ) α = + σ+, ω α 5 ( ) =, ω α5( ) F = cos( γ ω α ( α+β = σ σ ω ω ( α+β)( ) 5 = ω 5, ζ α+α 5( ) α =ζ+ζ 5 [ ( ) ] ( σ+ ω ω 9 α5( ) α + α5( ) α5( ) 7 = + ω ω ω α5( ) α 9 = σ+ ω ω ( α+β 5 ), = ω ζ +ζ α, = + σ σ ω ωζ = ( ), = + ( ω) 7 References. Oueini SS., Nayfeh AH., Single-Mode control of a cantilever beam under principal parametric ecitation, Journal of Sound and Vibration () (999) -7.. Asfar KR., Effect of non-linearities in elastomeric material dampers on torsional vibration control, International Journal of Non-linear Mechanics 7() (99) Eissa M., Vibration Control of Non-linear Mechanical Systems Via Neutralizers, Electronic Engineering Bulletin, No., July 999, Egypt.. Eissa M., Vibration and chaos control in I.C engines subject to harmonic torque via non-linear absorbers, In. Proc of ISMV Conference, Islamabad, Pakistan,. 5. Eissa M., El-Ganaini W., Part I, Multi-absorbers for vibration control of non-linear structures to harmonic ecitations, In. Proc of ISMV Conference, Islamabad, Pakistan,.. Eissa M., El-Ganaini W., Part II, Multi-absorbers for vibration control of non-linear structures to harmonic ecitations, In. Proc of ISMV Conference, Islamabad, Pakistan,. 7. Kamel MM., Amer YA., Response of parametrically ecited one-degreeof-freedom system with non-linear damping and stiffness, Physica Scripta ().

19 Vibration reduction and stability of non-linear system 9. Song Y., Sato H., Iwata Y. and Komatsuzaki T., The response of a dynamic vibration absorber system with a parametrically ecited pendulum, Journal of Sound and Vibration 59() () Soom A., Lee M., Optimal design of linear and non-linear vibration absorbers for damped system, Journal of Vibration, Acoustic Stress, and Reliability in Design 5 (9) -9.. Jordanov IN., Cheshankov BI., Optimal design of linear and non-linear dynamic vibration absorbers, Journal of Sound and Vibration () (9) Rice H.J., Combinational instability of the non-linear vibration absorber, Journal of Sound and Vibration () (9) Shaw J., Shaw SW., Haddow AG., On the response of the non-linear vibration absorber, Journal of Non-linear Mechanics (99) -9.. Natsiavas S., Steady state oscillations and stability of non-linear dynamic vibration absorbers, Journal of Sound and Vibration 5() (99) Zhu SJ., Zheng YF., Fu YM., Analysis of non-linear dynamics of a twodegree-of-freedom vibration system with non-linear damping and non-linear spring, Journal of Sound and Vibration 7 () Lim FCN., Cartmell MP., Cardoni A., Lucas M., A preliminary investigation into optimizing the response of vibrating systems used for ultrasonic cutting, Journal of Sound and Vibration 7 () Eissa M., Amer YA., Vibration control of a cantilever beam subject to both eternal and parametric ecitation, Applied Mathematics and Computation 5 () Nayfeh AH., Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging, Nonlinear Dynamics (5) -.. El-bassiouny AF., Effect of non-linearities in elastomeric material dampers on torsional oscillation control, Applied Mathematics and Computation (5) Eissa M., Sayed M., A Comparison between active and passive vibration control of non-linear simple pendulum, Part I: Transversally tuned absorber n and negative G ϕ& feedback, Mathematical and Computational Applications () () Eissa M., Sayed M., A Comparison between active and passive vibration control of non-linear simple pendulum, Part II: Longitudinal tuned absorber n G ϕ&& and negative G ϕ feedback, Mathematical and Computational Applications () () 5-.. Amer YA., Vibration control of ultrasonic cutting via dynamic absorber, Chaos, Solutions & Fractals (7) Nayfeh AH., Introduction to Perturbation Techniques, John Wiley & Sons, Inc., New York, (9).

20 7 M. Sayed, Y. S. Hamed and Y. A. Amer Hamed, Y.S., EL-Ganaini, W., Kamel M. M.: Vibration suppression in ultrasonic machining described by non-linear differential equations, Journal of Mechanical Science and Technology () (9) -5. Hamed, Y.S., EL-Ganaini, W., Kamel M. M.: Vibration suppression in multitool ultrasonic machining to multi-eternal and parametric ecitations, Acta Mechanica Sinica 5 (9) 5. 5 Hamed, Y.S., EL-Ganaini, W., Kamel M. M.: Vibration reduction in ultrasonic machine to eternal and tuned ecitation forces, Applied Mathematical Modeling. (9) 5-. Received: October,