THE IMPACT OF QUBIC NONLINEARITY OF SPRING FORCE ON THE VIBRATIONS OF THE DUFFING OSCILLATOR ROBERT KOSTEK University of Technology and Life Sciences Summary The paper shows the impact of qubic non-linearity of spring force on the vibrations of the Duffing oscillator. The comparison leads to the conclusion that, nonlinearity can play a significant role both in case of free and forced vibrations. A number of characteristic phenomena have been observed, i.e. bending resonance peak, and influence of non-linearity on the natural frequency. Received results shows clearly that non-linearity should not be neglected if accurate models are required. Keywords: nonlinear, vibration, Duffing, oscillator 1. Introduction All mechanical systems and bodies possessing mass and elasticity are capable of vibration [11]. Thus, machines are capable of vibration; they generate some vibrations under internal and external forces. Vibrations can be characterised as linear or non-linear. Linear vibrations and systems are described by linear differential equations, which are simple to solve. A system of linear differential equations can be considered as a matrix equation; moreover, the principle of superposition holds [11]. Because of that mechanical system linear modelling is the most common approach. In contrast to this, non-linear differential equations are difficult to solve; thus, non-linear modelling is not very common in machine dynamics. There is a number of difficulties coupled with non-linear approach: the principle of superposition does not hold, analytical solutions can be found only for simple systems, which in turn leads to the application of numerical methods. In spite of the difficulties, non-linear modelling gives an opportunity to receive excellent agreement between experimental results and results of simulation because mechanical systems are usually non-linear. That is the reason why non-linear models are applied in machine dynamics. The most common phenomena observed in non-linear systems are: the influence of vibration amplitude on the natural frequency, bending resonance peaks, instability, and chaos. It should be mentioned here that the phenomena are not observed in linear systems. Within mechanical systems are interactions which introduce non-linear forces. They are coupled with: clearance, friction, contact of bodies, and plastic deformation of bodies, to name a few [3]. They make vibrations of the systems non-linear, which can be result of failure. For instance, a large amplitude of vibrations introduces non-linear spring characteristic, which can be coupled with plastic deformation or slack joint. Therefore, the difference between linear and non-linear vibrations can be treated as a failure mode. That makes the issue of non-linear vibration important in the context of machine diagnostics [1].
98 POLISH ASSOCIATION FOR KNOWLEDGE MANAGEMENT Series: Studies & Proceedings No. 35, 21 2. Model The Duffing oscillator is a classical issue of non-linear dynamics. The oscillator consists of a non-linear spring, linear damper and mass (Fig. 1). The vibration of the Duffing oscillator is described by the following equation: 3, (1) where: m denotes mass, y displacement, c damping coefficient, k stiffness coefficient, E non-linear coefficient, P amplitude of excitation, a my + cy + ky + Ey = P cos( ω t + ϕ) ω frequency of excitation, t time, and ϕ phase angle. The equation (1) after dividing by m can be transform into the following form: 2 3 y + 2hy + ω y + εy = G cos( ωt +,(2) a ϕ where: h denotes damping term, ω un damped natural frequency, ε non-linear term, and G amplitude of excitation. The second form of the equation is often used in non-linear dynamics: ) Fig. 1. Model of the Duffing oscillator
Robert Kostek The impact of qubic nonlinearity of spring force on the vibrations of the Duffing oscillator 99 3. Simulation of vibrations Free vibrations are widely applied in practice because their interpretation is simple. An analysis of the vibrations gives an opportunity to find the natural frequency; moreover, if mass is known, then stiffness and damping coefficients can be identified. The process is very simple in the case of linear vibrations; thus, it has been applied in diagnostics. Some change of natural frequency, stiffness or damping coefficients can be interpreted as a failure mode [2,4,8,12]. The method has been applied e.g. in diagnostic of factory chimneys and drilling platforms [2,5,6,9,1]. The qubic non-linearity of the spring force influences free vibrations; thus, it can lead to misinterpreted results. To illustrate the issue, non-linear and linearised solution were presented in Fig. 2. The linearised solution neglects the nonlinear coefficient, which is not large. The comparison of received solutions shows that the periods of the linearised and non-linear vibrations are different, and the amplitudes are different, which show that the impact of the non-linear term is significant. Fig. 2. Free vibration of the Duffing oscillator received for the following values of parameters: m=1, c=,1, k=1, E=-,15, P a =, y =, y =1,2 Forced vibrations of the Duffing oscillator have been simulated with the Runge-Kutta method, both for non-linear and linearised model. The numerical method gives an opportunity to calculate transient vibrations, which reflect intermediate states of machines. An analysis of received results leads to a conclusion that the impact of the non-linear term is significant and cannot be neglected in the considered case (Fig. 3). Fig. 3. Forced vibration of the Duffing oscillator received for the following values of parameters: m=1, c=,1, k=1, E=-,15, P a =,1, ω=,9, ϕ =/2, y =-1, y =
1 POLISH ASSOCIATION FOR KNOWLEDGE MANAGEMENT Series: Studies & Proceedings No. 35, 21 The resonance is a classical issue in machine dynamics. The main resonance of the Duffing oscillator can be solved with the first harmonic method. Then, the solution is described then by the following formulas [7]: y = Acos( ωt),(3) [( G 2hω tgφ =,(5) 2 2 2 ω ω +, 75εA 2 2 2 2 2 2 2 2 ω ω +,75εA ) + 4h ω ] A =,(4) where: A denotes amplitude of the resonance. An approximate analytical solution of the nonlinear equation is presented in Fig. 4 black circles. The transient vibration (Fig. 3) forced by the harmonic force after some period changes into steady state vibration. That gives an opportunity to determine a resonance characteristic of the Runge-Kutta method. In order to determine the characteristic, numerical calculations were made for a number of frequencies of excitation. Received amplitudes of steady state vibration and frequencies of excitation were recorded and then presented. The amplitude of vibration has been defined as a maximum variation of displacement from the position of equilibrium. The results of these calculations are presented in Fig. 4 red line. Finally, a linearised solution was calculated and presented in Fig. 4 blue line. An analysis of received results leads to a conclusion that the impact of the non-liner term is significant near the main resonance. Approximate analytical and approximate numerical solutions show good agreement, but the numerical solution should be treated as the most accurate because the first harmonic method neglects higher harmonics, while the linearised solution does not reflect the resonance characteristic well. That shows clearly that the nonlinear term cannot be neglected in the considered case. Fig. 4. Resonance characteristic of the Duffing oscillator received for the following values of parameters: m=1, c=,1, k=1, E=-,15, P a =,1
Robert Kostek 11 The impact of qubic nonlinearity of spring force on the vibrations of the Duffing oscillator 4. Conclusions Some failures introduce to mechanical systems non-linear interactions forces, thus nonlinearity can be treated as a failure mode. That is the reason why nonlinear dynamics is applied in diagnostics and advanced identification of mechanical systems. In the article vibrations of the Duffing oscillator were investigated. The oscillator contains non-linear qubic term of spring force, which can approximate: clearance in system, plastic deformation of spring, or micro slips of contact interfaces. In the article degressive characteristic of the spring was studied, which models e.g. micro slips of contact interfaces. In spite of the fact that the value of the nonlinear term adopted to the calculation was not large, its has a significant impact on: time history of free vibration, natural frequency, the amplitude of vibration, time history of forced vibration, and resonance characteristics. That leads to conclusion that the non-linear term cannot be neglected in the considered system. Nevertheless, in practice often linearised models are applied because analysis and interpretation of linear vibration is relatively simple. Bibliography 1. Batko, W., D browski, Z., Kici ski, J., Zjawiska nieliniowe w diagnostyce wibroakustycznej, Wydawnictwo Naukowe Instytutu Eksploatacji, 28. 2. Cempel, C. Podstawy wibroakustycznej diagnostyki maszyn, WNT 1982. 3. Diertich, M., Koca da. M., Korytkowski. B., Ozimowski. W., Stupnicki. J., Szopa. T., Podstawy konstrukcji mazyn, WNT 1995. 4. Godeau, A.J., Deleuil, G.E., Doris, C.G., Heas, J.Y., Statistical analysis of nonlinear dynamic response of fixed structures to random waves, fatigue evaluation, Offshore Technology Conference, 2 5 May, Houston, Texas 24. 5. Han, S.M., Benaroya, H., Vibration of a compliant tower in three-dimensions, Journal of Sound and Vibration, vol. 25, pp. 675 79, 22. 6. Krakowski, M., Elektrotechnika teoretyczna: T.1 Obwody liniowe i nieliniowe. Wyd. 3 PWN, 1999. 7. Ochelski, S., Metody do wiadczalne mechaniki kompozytów konstrukcyjnych, WNT 24 8. Spidsøe, N., Karunakaran, D., Nonlinear dynamic behaviour of jack-up platforms. Marine Structures, vol. 9 pp. 71 1, 1996. 9. Thompson, J.M.T., Stewart, H.B., Nonlinear dynamics and chaos. 2nd Ed. Wiley 22. 1. Thomson, W.T., Dahleh, M.D., Theory of vibration with applications, Prentice-Hall, 1993. 11. Vandiver, J.K., Detection of structural failure on fixed platforms by measurement of dynamic response, Journal of Petroleum Technology, vol. 29, pp. 35 39, 1977.
12 POLISH ASSOCIATION FOR KNOWLEDGE MANAGEMENT Series: Studies & Proceedings No. 35, 21 WPŁYW SZE CIENNEJ NIELINIOWO CI SIŁY SPR YSTO CI NA DRGANIA OSCYLATORA DUFFINGA Streszczenie Artykuł ten prezentuje wpływ sze ciennej nieliniowo ci, siły spr ysto ci na drgania oscylatora Duffinga. Drgania nieliniowe zostały porównane z drganiami zlinearyzowanymi. Analiza wyników skłania do wniosku, e nieliniowo mo e odgrywa znacz c rol zarówno w przypadku drga swobodnych jak i wymuszonych. Zaobserwowano charakterystyczne zjawiska takie jak: zaginanie piku rezonansu i zmiana cz sto ci własnej w funkcji amplitudy drga. Uzyskane wyniki potwierdzaj, e nieliniowo nie mo e by pomijana, je li dokładne modele drga s wymagane. Słowa kluczowe: nieliniowe, drgania, oscylator Duffinga *This paper is a part of WND-POIG.1.3.1--212/9 project. Robert Kostek University of Technology and Life Sciences Faculty of Mechanical Engineering