University of Technology and Life Sciences
Summary
The paper shows the impact of qubic non-linearity of spring force on the vibra-tions of the Duffing oscillator. The comparison leads to the conclusion that, non-linearity 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 cou-pled 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].
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:
)
cos(
3=
ω
+
ϕ
+
+
+
c
y
ky
Ey
P
t
y
m
a , (1) where:
m
– denotes mass,y
– displacement,c
– damping coefficient,k
– stiffness coefficient,E
– non-linear coefficient, aP
– amplitude of excitation,ω
– frequency of excitation,t
– time, andϕ
– phase angle.The equation (1) after dividing by m can be transform into the following form:
)
cos(
2
+
ω
02+
ε
3=
ω
+
ϕ
+
h
y
y
y
G
t
y
, (2) where:
h
– denotes damping term,ω
0 – un damped natural frequency,ε
– non-linear term, andG
– amplitude of excitation.The second form of the equation is often used in non-linear dynamics:
3. Simulation of vibrations
Free vibrations are widely applied in practice because their interpretation is simple. An analy-sis 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 fre-quency, 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 [2,5,6,9,10].
The qubic non-linearity of the spring force influences free vibrations; thus, it can lead to mis-interpreted 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=0,1, k=1, E=-0,15, Pa=0, y0=0, y0=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=0,1, k=1, E=-0,15, Pa=0,1, ω=0,9,
ϕ
= /2, y0=-1, y0=0The 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]:
)
cos( t
A
y
=
ω
, (3) 2 2 2 2 2 2 2 2 00
,
75
)
4
]
[(
ω
−
ω
+
ε
A
+
h
ω
A
=
G
, (4) 2 2 2 00
,
75
2
tg
A
h
ε
ω
ω
ω
φ
+
−
=
, (5) where:A
– denotes amplitude of the resonance.An approximate analytical solution of the nonlinear equation is presented in Fig. 4 – black cir-cles.
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 agree-ment, 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 consid-ered case.
Fig. 4. Resonance characteristic of the Duffing oscillator received for the following values of parameters:
4. Conclusions
Some failures introduce to mechanical systems non-linear interactions – forces, thus nonlin-earity 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 calcula-tion 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
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9. Thompson, J.M.T., Stewart, H.B., Nonlinear dynamics and chaos. 2nd Ed. Wiley 2002. 10. 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
WPŁYW SZEĝCIENNEJ NIELINIOWOĝCI SIŁY SPRĉĩYSTOĝCI NA DRGANIA OSCYLATORA DUFFINGA
Streszczenie
Artykuł ten prezentuje wpływ szeĞciennej nieliniowoĞci, siły sprĊĪystoĞci na drgania oscylatora Duffinga. Drgania nieliniowe zostały porównane z drganiami zlinearyzowanymi. Analiza wyników skłania do wniosku, Īe nieliniowoĞü moĪe od-grywaü znaczącą rolĊ zarówno w przypadku drgaĔ swobodnych jak i wymuszonych. Zaobserwowano charakterystyczne zjawiska takie jak: zaginanie piku rezonansu i zmiana czĊstoĞci własnej w funkcji amplitudy drgaĔ. Uzyskane wyniki potwierdzają, Īe nieliniowoĞü nie moĪe byü pomijana, jeĞli dokładne modele drgaĔ są wymagane. Słowa kluczowe: nieliniowe, drgania, oscylator Duffinga
*This paper is a part of WND-POIG.01.03.01-00-212/09 project.
Robert Kostek
University of Technology and Life Sciences Faculty of Mechanical Engineering
