Bui Thi Thuy, Tran Thi Tram
Hanoi University of Mining and Geology,
Vietnam
ESTABLISHING NONLINEAR VIBRATIONAL DIFFERENTIAL EQUATION OF CAR WITH FRACTIONAL DAMPING
Abstract
Based on the theory of fractional derivative, Newton's second law of motion and some transformation, the present study aims to establish nonlinear vibrational differential equation of car with integer damping. Then, nonlinear vibrational differential equation of car with fractional damping is obtained.
By establishing nonlinear vibrational differential equation of car with fractional derivative, complex structures can be designed logically, technical standard assurance.
Keywords
vibration, nonlinear, car, fractional order, damping.
1. Introduction
Cars are a means of transport that play a very important role in the national economy and are currently widely used in all fields of economy and life. Car vibrations not only affect people (drivers and passengers), transported goods, durability, and safety of movement of cars, but also affect the life of the road. Especially during movement, when the car vibrates, it generates very large dynamic loads that impact the car's chassis system, details, overall structure... affecting the durability and longevity of the car. they. Therefore, studying car vibrations is necessary and useful. One of the important tasks of car's vibration research is to establish and solve differential equations to determine vibration parameters.
The generalization of the concept of derivative Da[ f (x)] to noninteger values of a goes back to the beginning of the theory of differential calculus. In fact, Leibniz, in his correspondence with Bernoulli, L'Hopital and Wallis (1695), had several notes about the calculation of D1/2[ f (x)]. Nevertheless, the development of the theory of Fractional Calculus is due to the contributions of many mathematicians such as Euler, Liouville, Riemann, and Letnikov [1-3].
In recent years Fractional Calculus has been a fruitful field of research in science and engineering [14]. In fact, many scientific areas are currently paying attention to the Fractional Calculus concepts and we can refer its adoption in viscoelasticity and damping, diffusion and wave propagation, electromagnetism, chaos and fractals, heat transfer, biology, electronics, signal processing, robotics, system identification, traffic systems, genetic algorithms, percolation, modeling and identification, telecommunications, chemistry, irreversibility, physics, control systems as well as economy, and finance...
2. Establishing nonlinear vibrational differential equation of car with fractional damping
Linear viscoelasticity is a combination of models: linear elasticity (Figure 1), integer linear viscosity (Figure 2) and fractional linear viscosity (Figure 3). Where o is the stress, s is the strain, E is the elastic modulus when tensile or compressive (characterizing the stiffness of the material), n and c are the viscous resistance
coefficients, D" which are fractional derivatives with time t.
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Fig 1 - Linear elasticity model (& = E.D °s )
Fig 2 - Integer linear viscosity model (a = V.Dls)
Fig 3 - Fractional linear viscosity model (& = c.D"s)
2.1. Integer derivative model
Let y and u be the displacements of the object and the wheel. It is assumed that the stiffness of the shaft lining can be represented by a spring equivalent to the stiffness k3 with the displacement denoted by z. We have Newton's second law of motion
my = -R(t), (1)
Where m is the mass of the object, y(t) is the displacement, and R (t) is the internal force generated inside the viscoelastic object.
Fig 4 - Integer derivative model
Fig 5 - Force analysis
The force value R ( t ) is equal to the total value of the force R acting on the spring ki and the force R2 acting on spring k2 or damping c
R ( t ) = R + R, (2)
R2 R (k3 ) R (c) '
(3)
Where
R,=K(y-u) + k2(y-u) , R2(ki) =k3(y-z), R2{c) = c(z-w), ^ R(t) = R + R2 = R + R2^)
= kl { 7 - U ) + k2 { 7 - U )3 + k3 { 7 - Z ) > From equations (3) and (4) we have
k3 ( v — z) = c(z —u) Substituting equation (5) into (1) we obtain the motion equation
my = -kx (y-u)-k2 (y-u)3 -k3(y-z),
Derive z from above equation
z =
my + kx(y- u) + k2(y- uf
k^
+ У,
(4)
(5)
(6)
(7)
(8)
From equations (8) and (6) we get
k3(y-z) = c(z-u)
-my -kx(y- и) - kn (_y - w)3 = c(z -u)
о -my-kx(y-ii)-k2(y-u)3
my + (kx + k3 ) (y - ù) + 3k2 (у - и)2 (у - й)
Let 7 — u = x we have the differential equation of motion
... k-, .. 3A:? 7. k +L . kk-, k0k, ? + —x + —-xx + —-x + —r =■
m
mc
mc
... k-, .. и H—— и
V Су
(9)
c m 2.2. Fractional derivative model
Let y and u be the displacements of the object and the wheel. It is assumed that the stiffness of the shaft lining can be represented by a spring equivalent to stiffness k3 with the displacement denoted by z. We have Newton's second law of motion
my = -R(t), (10)
Where m is the mass of the object, y{t) is the displacement, and R {t) is the internal force generated inside the viscoelastic object.
Fig 6 - Fractional derivative model
Fig 7 - Force analysis
c
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The force value R (t) is equal to the total value of the force R acting on the spring k and the force R2 acting on spring k2 or damping c
R(t) = R + R2, (11)
R2 = R2 (k3 )= R2 (c), (12)
Where
(13)
(14)
R = ki (y - u) + k2 (y - u) , R2{h)= k3 (y - z), R (c) = cDf (z - и ),
2 (c) ^
^ R ( t ) = R + R = R + R
2 ( кз )
= h (y - u) + k2 (y - u) + h (y - z),
From equations (12) and (13) we have
h ( y - z ) = cDp (z - u ) Substituting (14) into (10) we have the motion equation
fny = -k1(y-u)-k2(y-uf -k3(y-z),
Derive z from above equation
my + ( j — u) + k2(y — uf
к
+ У,
From equations (17) and (15) we get
k(y-z) = cDp (z—u)
—my — k^y — u) — k2[y - w)3 = cDtp
к
к
my + kl(y-u) + k2(y-uf
к1кз
+ У
■cDtpu,
^Dtpy + ^y + ^Dtp(y-uy+^Dtp(y-u) + ^Dtpy + ^(y-u) cm m m тс
+ k2kL(y _ и)3 - к3Dpu = 0.
к
mc m
Let y — u = x we have the differential equation of motion
Dtpx
cm m mc mc
x = —
V
Dfii + ^-i? с ;
(15)
(16)
(17)
(18)
3. Conclusions
In this paper, we used the force analysis method and Newton's second law to establish the nonlinear vibrational differential equation of a car in the case of integer and fractional damping. In the case of integer damping, the nonlinear vibrational differential equation takes the form of a third-order differential equation. In the case of fractional damping, the nonlinear vibrational differential equation contains fractional derivatives.
The authors will use the definitions and properties of fractional derivatives, through analytical and numerical methods to find solutions of established differential equations in subsequent studies. Thanks to that, it is possible to survey the vibrations of car, contributing to setting directions and measures to improve the quality of produced cars.
References
1. K.B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 11 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1974.
2. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.
3. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
4. R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific Publishing, Singapore, 2000.
Abstract
In this paper the dynamic response of Zener oscillator with third order derivative is researched by the Newmark method. First, the approximately analytical solution is obtained. Then, we calculate linear vibration of Zener viscoelastic system.
1. Introduction
The Newmark scheme, originally introduced by Newmark [1959], is a classical time-stepping algorithm popular in structural mechanics codes. It has been modified and improved by many other researchers such as Wilson, Hilber, Hughes and Taylor. However, these methods are only used for the system of second order equations.
Many vibration problems in engineering lead the system of differential equations of third order. In this paper we calculate vibration of Zener system by using established Newmark integration method for calculating vibration of third order system.
2. The Newmark method for the third order systems
The Newmark method is a single-step integration formula. The state vector of the system at a time t«+i = t„ + h is deduced from the already-known state vector at time tn through a Taylor expansion of the displacements, velocities and accelerations.
We get the approximation formulas of displacements, velocities and accelerations of system at time t , to approach solving the system of third order differential equations.
©Bui Thi Thuy, Tran Thi Tram, 2024
Bui Thi Thuy
Doctor at Hanoi University of Mining and Geology,
Vietnam
LINEAR VIBRATION OF ZENER SYSTEM USING NEWMARK METHOD
Keywords:
viscoelastic, third order, dynamic, Newmark, Zener.
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