Porny级数参数之决定与轮胎滚动阻力仿真
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VISCOELASTIC BEHAVIOR AND PRONY SERIES PARAMETERS OF RUBBER Rubber shows viscoelastic behavior which can be measured at a creep test or a stress relaxation test. During a creep test the strain gradually increases under a constant stress, and during a stress relaxation test the stress gradually decreases under a constant strain. These material responses may be described by the
∑ E(t)
=
Eo
⎢⎣⎡1 −
M i =1Βιβλιοθήκη gi (1 −e−t
/τ i
)⎥⎦⎤
(1)
The more Prony series parameters, the more accurately Equation (1) represents the viscoelastic behavior. However, in this paper it has been decided to use only four sets of parameters since the Prony series with three or four sets of parameters can represent the behavior good enough. In an oscillation test, under the sinusoidal strain (or displacement) the stress (or force) follows the strain with a phase shift δ. Thus, the strain and the stress can be represented as follows.
Key words: Prony series parameter, rolling resistance, viscoelastic behavior
INTRODUCTION The rolling resistance of a tire is defined to be the energy loss per the travelling distance, and it can be experimentally obtained from the axle force of a tire rolling on a drum [1]. The energy loss costs fuel consumption, and the rolling resistance of a commercial vehicle tire consumes about a third of energy produced by the engine [2]. Therefore, it is necessary to consider the amount of rolling resistance in tire design, and research on the finite element simulation of the rolling resistance has been conducted [3-5]. The rolling resistance stems mainly from the viscoelastic behavior of rubbers in a tire, which causes hysteretic loss during cyclic loading, and it depends on the rolling speed. However, in the previous research static finite element simulations were conducted, and they did not show the sensivity of the rolling resistance to the rolling speed [3-5]. In this paper viscoelastic behavior was modeled by the relaxation modulus represented by the Prony series, and the Prony series parameters were determined to fit the ratio of the loss modulus to the storage modulus for a wide range of loading frequencies. Then, a tire rolling on a drum was modeled in ABAQUS/Explicit, and dynamic simulations were conducted to obtain the energy loss from an output of the code [6]. Finally, the rolling resistance was determined by dividing the energy loss by the travelling distance. The rolling resistance obtained from the simulations were compared with the test data of 40, 60 and 80 km/h, and it was proven that the simulation results were in a good correlation with the test results.
Abstract The rolling resistance of a tire stems from the viscoelastic behavior of rubber which causes a hysteretic loss during cycling loading. Thus, it is essential to model the viscoelastic behavior appropriately in order to simulate the rolling resistance of a tire. However, when the Prony series parameters determined to fit the creep test data of rubber are used, a simulation results in a unreasonably small rolling resistance because the creep test data represent only the quasi-static material response. Thus, in this paper a methodology to determine the Prony series parameters was developed fitting the ratio of the loss modulus to the storage modulus for a wide range of loading frequencies, and the Prony series parameters of the rubbers used in a tire were determined by using the methodology. Then, a tire rolling on a drum was modeled in ABAQUS/Explicit, and this finite element model was used to obtain the rolling resistance numerically. Since the energy loss could be obtained directly from an output of ABAQUS, the rolling resistance was readily determined from the energy loss divided by the travelling distance. The simulation results were proven to be in a good correlation with the test data of various speeds.
follows.
E* = σ (t) = E′ + iE′′
(4)
ε (t)
The complex modulus can be obtained from the oscillation test at various loading frequencies. In addition, the viscoelastic material response due to a varying strain can be obtained from the following convolution equation.
COMPUTATIONAL MECHANICS WCCM VI in conjunction with APCOM’04, Sept. 5-10, 2004, Beijing, China © 2004 Tsinghua University Press & Springer-Verlag
Determination of Prony Series Parameters and Rolling Resistance Simulation of a Tire
ε (t) = ε0 sin(ωt)
(2)
σ (t) = σ 0 sin(ωt + δ ) = E′ε0 sin(ωt) + E′′ε0 cos(ωt)
(3)
Here, ω is the angular velocity, t is time, E′ is the storage modulus, and E′′ is the loss modulus. The storage modulus and the loss modulus are the real and the imaginary parts of the complex modulus E* defined as
—1—
linear viscoelastic constitutive law in which the relaxation modulus E(t) is described by the instantaneous modulus Eo, and M sets of Prony series parameters gi and τi.
T.-W. Kim1, H.-H. Kim1, H.-Y. Jeong1*, H.-C. Park2*, Y.-H. Kim2, J.-H. Choe2 1 Dept. of Mechanical Engineering, Sogang University, Mapo-Gu, Seoul, Korea 121-742 2 R&D Center, Hankook Tire, Yousung-Gu, Daejun, Korea 305-725 e-mail: jeonghy@sogang.ac.kr
∑ E(t)
=
Eo
⎢⎣⎡1 −
M i =1Βιβλιοθήκη gi (1 −e−t
/τ i
)⎥⎦⎤
(1)
The more Prony series parameters, the more accurately Equation (1) represents the viscoelastic behavior. However, in this paper it has been decided to use only four sets of parameters since the Prony series with three or four sets of parameters can represent the behavior good enough. In an oscillation test, under the sinusoidal strain (or displacement) the stress (or force) follows the strain with a phase shift δ. Thus, the strain and the stress can be represented as follows.
Key words: Prony series parameter, rolling resistance, viscoelastic behavior
INTRODUCTION The rolling resistance of a tire is defined to be the energy loss per the travelling distance, and it can be experimentally obtained from the axle force of a tire rolling on a drum [1]. The energy loss costs fuel consumption, and the rolling resistance of a commercial vehicle tire consumes about a third of energy produced by the engine [2]. Therefore, it is necessary to consider the amount of rolling resistance in tire design, and research on the finite element simulation of the rolling resistance has been conducted [3-5]. The rolling resistance stems mainly from the viscoelastic behavior of rubbers in a tire, which causes hysteretic loss during cyclic loading, and it depends on the rolling speed. However, in the previous research static finite element simulations were conducted, and they did not show the sensivity of the rolling resistance to the rolling speed [3-5]. In this paper viscoelastic behavior was modeled by the relaxation modulus represented by the Prony series, and the Prony series parameters were determined to fit the ratio of the loss modulus to the storage modulus for a wide range of loading frequencies. Then, a tire rolling on a drum was modeled in ABAQUS/Explicit, and dynamic simulations were conducted to obtain the energy loss from an output of the code [6]. Finally, the rolling resistance was determined by dividing the energy loss by the travelling distance. The rolling resistance obtained from the simulations were compared with the test data of 40, 60 and 80 km/h, and it was proven that the simulation results were in a good correlation with the test results.
Abstract The rolling resistance of a tire stems from the viscoelastic behavior of rubber which causes a hysteretic loss during cycling loading. Thus, it is essential to model the viscoelastic behavior appropriately in order to simulate the rolling resistance of a tire. However, when the Prony series parameters determined to fit the creep test data of rubber are used, a simulation results in a unreasonably small rolling resistance because the creep test data represent only the quasi-static material response. Thus, in this paper a methodology to determine the Prony series parameters was developed fitting the ratio of the loss modulus to the storage modulus for a wide range of loading frequencies, and the Prony series parameters of the rubbers used in a tire were determined by using the methodology. Then, a tire rolling on a drum was modeled in ABAQUS/Explicit, and this finite element model was used to obtain the rolling resistance numerically. Since the energy loss could be obtained directly from an output of ABAQUS, the rolling resistance was readily determined from the energy loss divided by the travelling distance. The simulation results were proven to be in a good correlation with the test data of various speeds.
follows.
E* = σ (t) = E′ + iE′′
(4)
ε (t)
The complex modulus can be obtained from the oscillation test at various loading frequencies. In addition, the viscoelastic material response due to a varying strain can be obtained from the following convolution equation.
COMPUTATIONAL MECHANICS WCCM VI in conjunction with APCOM’04, Sept. 5-10, 2004, Beijing, China © 2004 Tsinghua University Press & Springer-Verlag
Determination of Prony Series Parameters and Rolling Resistance Simulation of a Tire
ε (t) = ε0 sin(ωt)
(2)
σ (t) = σ 0 sin(ωt + δ ) = E′ε0 sin(ωt) + E′′ε0 cos(ωt)
(3)
Here, ω is the angular velocity, t is time, E′ is the storage modulus, and E′′ is the loss modulus. The storage modulus and the loss modulus are the real and the imaginary parts of the complex modulus E* defined as
—1—
linear viscoelastic constitutive law in which the relaxation modulus E(t) is described by the instantaneous modulus Eo, and M sets of Prony series parameters gi and τi.
T.-W. Kim1, H.-H. Kim1, H.-Y. Jeong1*, H.-C. Park2*, Y.-H. Kim2, J.-H. Choe2 1 Dept. of Mechanical Engineering, Sogang University, Mapo-Gu, Seoul, Korea 121-742 2 R&D Center, Hankook Tire, Yousung-Gu, Daejun, Korea 305-725 e-mail: jeonghy@sogang.ac.kr