BACKGROUND OF THE INVENTION
Field of the invention
The present invention relates to a method for pneumatic
tire simulation capable of predicting a vibration of a pneumatic tire in consideration
of a physical phenomenon of fluid filled in the cavity of the pneumatic tire.

Background art
Conventionally, computer simulation using a numerical analysis
method, such as the finite element method, has been suggested. This approach enables
the prediction of tire performance without manufacturing prototype tires. In order
to predict vibration performance of a tire, for example, rolling simulation which
makes a tire model rotate on a road model, and obtains a history of vertical force
acting on the tire axis is proposed.

Conventional simulation, however, does not consider influence
of fluid filled in a cavity of the tire. Namely, it is well known that a cavity
resonance of the fluid is generated in the cavity during the tire is running. Accordingly
in order to simulate the vibration performance of the tire accurately, it is necessary
to consider influence of a physical phenomenon of the fluid filled in the cavity.

SUMMARY OF THE INVENTION
It is a main object of the present invention to provide
a method for pneumatic tire simulation which considers influence of the physical
phenomenon of the fluid filled in the cavity, and can accurately simulate the vibration
performance of the tire.

According to the present invention, a method for pneumatic
tire simulation comprises the steps of modeling a tire body having a cavity extending
in a circumferential direction of the tire using finite elements to build a tire
body model, modeling the cavity surrounded by the tire body using finite volumes
to build a cavity model, setting a pneumatic tire model coupled the tire body model
with the cavity model so that a relative distance between an outer surface of the
cavity model and an inner surface of the tire body model does not change, modeling
a road using finite elements to build a road model, and executing a numerical simulation
in which the tire model is made to roll on the road model in a predetermined condition.

BRIEF DESCRIPTION OF THE DRAWINGS

- Fig. 1 is an illustration of a computer device for executing the simulation
method;
- Fig. 2 is a flowchart showing an embodiment of the simulation method;
- Fig. 3 is a perspective view showing an embodiment of a tire body model;
- Fig. 4 is a cross sectional view of Fig.3;
- Fig. 5 is a cross sectional view showing an embodiment of a cavity model;
- Fig. 6 is a cross sectional view showing an embodiment of a pneumatic tire model;
- Figs. 7(A) to 7(C) are schematic diagrams explaining a coupling condition between
the tire body model and the cavity model;
- Fig. 8 is a perspective view showing an embodiment of a road model;
- Fig. 9 is a perspective view showing an embodiment of rolling simulation;
- Fig. 10 is a part of an enlarged sectional view of Fig.9;
- Fig. 11 is a flowchart showing an embodiment of the deformation calculation
of the pneumatic tire model;
- Fig. 12 is a perspective view showing pressure distribution of the cavity model;
- Fig. 13 is a graph showing the relation between vertical force acting on the
tire axis of the pneumatic model and time obtained by the simulation;
- Fig. 14 is a graph showing the result of frequency analysis of Fig.13;
- Fig. 15 is a cross sectional view showing another embodiment of the pneumatic
tire model with a noise damper;
- Fig. 16 is a graph showing the stress-strain curve of the noise damper;
- Fig. 17 is a graph showing the result of frequency analysis of the vertical
force of the pneumatic tire model of Fig.15; and
- Fig. 18 is a perspective view showing pressure distribution of the cavity model
of Fig.15.

DESCRIPTION OF THE PREFERRED EMBODIMENTS
An embodiment of the present invention will be described
as follows based on the drawings.

Fig. 1 shows a computer device 1 used for the simulation
method of the present invention. The computer device 1 comprises a main body 1a,
a key board 1b, a mouse 1c, and a display device 1d. The main body 1a includes disk
drives 1a1 and 1a2, a CPU, a ROM, a memory, and a bulk storage (which are not illustrated).
The bulk storage stores programs which execute the simulation method described below.

Fig. 2 shows one embodiment of the procedure of the invention
simulation method. In the Step S1, a tire body model is set by modeling a tire body
using finite elements capable of numerical analysis.

Fig. 3 is one embodiment of the tire body model 2 which
is visualized in three-dimensional, and Fig. 4 is a cross sectional view thereof
including the tire axis. In the tire body model 2, the tire body to be analyzed
is divided into a finite number of small elements 2a, 2b, 2c .... Each of the elements
2a, 2b, 2c ... can be, but not limited to, rectangular plane elements or three-dimensional
tetrahedral solid elements. Other various elements such as pentagonal and/or hexagonal
solid elements are also employed.

Each elements 2a, 2b, 2c ... is a numerical data capable
of deformation calculation by the computer device 1. Further, the tire model 2 includes
the coordinate values of nodes on each element 2a, 2b, 2c ..., their shapes, and
their properties such as density, modulus and/or damping coefficient. The numerical
analysis includes, for example, the finite element method, the finite volume method,
the finite difference method and the boundary element method. In this embodiment,
as for each element 2a, 2b, 2c ..., the Lagrange element which can move with the
tire body model in a simulating space is employed.

The tire body model 2 has a toroidal shape which comprises:
a pair of sidewall portions 2s; a tread portion 2T interposed therebetween; and
a cavity i surrounded by the tread portion 2T and sidewall portions 2S and being
continuously extending in the circumferential direction of the tire. The cavity
i is a space where fluid such as air or other gas is filled. Further, the cavity
i is defined as a closed and toroidal space surrounded by the inner surface 2i of
the tire body model 2 and an outer surface J of the rim on which the tire body model
2 is mounted. The outer surface J of the rim is given according to the size of the
tire body model 2 and various tire standards.

In order to improve the accuracy of the simulation, it
is preferable that the tire body model 2 comprises a tread pattern including longitudinal
grooves and transverse grooves on the tread portion 2T, but these grooves may be
omitted. In the same manner, reinforcing members inside such as a carcass 2A, a
belt 2B and a pair of bead core 2C, are also preferably modeled into the tire body
model 2. In this embodiment, the tire body model 2 is divided into 80 elements with
respect to the circumferential direction of the tire.

Next, in the Step s2, a cavity model 3 is set by modeling
the cavity i in finite volumes capable of the numerical analysis. Fig. 5 shows a
cross sectional view of one embodiment of the cavity model 3 which is visualized
at the same section of Fig. 4. In the cavity model 3, the cavity i to be analyzed
is divided into finite volumes (Euler finite volumes) 3a, 3b, 3c which are numerical
data capable of deformation calculation such as pressure calculation by the computer
device 1.

Further, the cavity model 3 includes coordinate values
of nodes on each volume 3a, 3b, 3c ..., their shapes, and their properties such
as density and bulk modulus. In this embodiment, each finite volume is numerical
fluid corresponding to a simulating fluid such as air, nitrogen, helium or mixture
gas thereof to be filled in the cavity i of the tire body model 2.

In this embodiment, the cavity model 3 is also divided
into 80 elements with respect to the circumferential direction of the tire.

Further, in this embodiment, the cavity model 3 has the
outer surface 3i corresponding to the inner surface 2i of the tire body model 2.
Accordingly, as shown in Fig. 6, the cavity model 3 can perfectly fill in the cavity
i of the tire model 2 by arranging each center axis of the tire body model 2 and
the cavity model 3 on the same position. Thereby, a pneumatic tire model 1 which
comprises the tire body model 2 and the cavity model 3 is built.

In the pneumatic tire model 1, all of outer nodes 3n1,
3n2 ... which face to the inner surface 2i of the tire model 2 are provided on the
same position corresponding to inner nodes 2n1, 2n2 ... on the inner surface of
the tire body model 2. Namely, outer nodes 3n1, 3n2 ... of the cavity model 3 are
shared with inner nodes 2n1, 2n2 ... of the tire body model 2. However, it is not
especially limited to such an embodiment. For example, outer nodes 3n1, 3n2 ...
of the cavity model 3 may be provided on different position from inner nodes 2n1,
2n2 ... of the tire body model 2.

Further, finite volumes of the cavity model 3 comprise
a plurality of covering volumes 3L each being coupled to the inner surface 2i of
the tire body model 2, and a plurality of mid volumes 3R being surrounded by the
covering volumes 3L.

Here, the tire model 2 may roll and deform during a rolling
simulation with tire load. In this embodiment, only covering volumes 3L being directly
coupled with the inner surface 2i of the tire body model 2 are given deformable
property. Namely, each covering volumes 3L is defined as the Lagrange element which
can move and deform together with the tire body model 2. On the other hand, all
of the elements of the cavity model 3 except the covering volumes 3L are defined
as the Euler elements which can only rotate, but not deform. As shown in Fig. 5,
covering volume 3L of the cavity model 3 has a greater volume than that of mid volume
3R being coupled with inside thereof so that negative deformation of the covering
volume 3L does not generate even when the tread portion 2T of the tire body model
2 greatly deforms.

Especially, covering volumes 3L being coupled with the
inner surface 2i of the tread portion 2T and sidewall portions 2s preferably have
a greater volume than that of the mid volume 3R being coupled inside thereof, because
a large deformation tends to easily occur in the tread portion 2T and sidewall portions
2S of the tire body model 2. In this point of view, the covering volume 3L being
coupled with the inner surface 2i of the tread portion 2T preferably has a radial
length h1 greater than the radial length h2 of the mid volume 3R being coupled inside
thereof, as shown in Fig. 5.

In another embodiment, all of the volumes of the cavity
model 3 may be defined as the Euler elements being basically fixed in the space
for simulation. Namely, cavity model 3 is built as an Euler model. In the Euler
model, movement of fluid among meshes is taken into consideration with the simulation,
and the cavity i of the tire body model 2 is always filled by the numerical fluid
of the cavity model 3. Further, pressure acting into each volume is calculated at
each constant position.

In further another embodiment, the cavity model 3 may be
re-modeled according to the deformation of the pneumatic tire model 1. In this case,
it is preferable that a finite volume in which a great pressure change occurs is
re-modeled into as a smaller volume, and a finite volume in which a small pressure
change occurs is re-modeled into as a greater volume.

Further, the coupling condition for coupling between the
tire body model 2 and the cavity model 3 so that a relative distance between the
outer surface 30 of the cavity model 3 and the inner surface 2i of the tire body
model 2 does not change is defined in the pneumatic tire model 1. Namely, inner
planes and nodes (2n1, 2n2, ... ) of finite volumes on the inner surface 2i of the
tire body model 2 are coupled with outer planes and nodes (3n1, 3n2, ... ) of finite
volumes on the outer surface 30 of the cavity model 3 so that the relative position
between inner surface 2i of the tire model 2 and the outer surface 30 of the cavity
model 3 does not change.

For example, Fig. 7(A) shows a coupling boundary between
the inner nodes 2n1 of the tire model 2 and the outer nodes 3n1 of the cavity model
3 which has an initial distance L (In the present embodiment shown in Fig. 6, however,
the distance L is set to zero). Here, if the inner node 2n1 is displaced toward
inside according to the deformation of the tire model 2, the distance L becomes
shorter as L' as shown in Fig. 7(B). However, according to the boundary condition
above, the positions of nodes 2n1 and 3n1 are adjusted so that the distance L' between
the inner node 2n1 and the outer nodes 3n1 is kept the initial length L.

The positions of nodes 2n1 and 3n1 are determined according
to the deformation balance between the tire model 2 and the cavity model 3 based
on their elastic moduli, etc. For example, since the elastic modulus of the finite
element of the tire model 2 is usually greater than the bulk modulus of the finite
volume of the cavity model 3, the inner node 2n1 of the tire model 2 hardly moves
as shown in Fig. 7(C). On the other hand, the outer nodes 3n1 of the cavity model
3 would mainly be moved so that the relative distance is kept as the initial distance
L. Accordingly, force acting on the tire body model 2 is inputted into the cavity
model 3 through the boundary between the tire body model 2 and the cavity model
3. Namely, the interaction between the tire body model 2 and the cavity model 3
is simulated.

In the same manner, inner nodes 3nJ of the cavity model
3 facing to the outline J of the rim are also coupled with the outline J so that
the relative distance is not changed. In this embodiment, the inner nodes 3nJ of
the cavity model 3 can not move because the outline J of the rim is defined as a
rigid body.

Next, in the step S3, a road model 4 is set using finite
elements 4a, 4b, 4c ... which are rigid elements. Fig. 8 shows one embodiment of
the road model 4 which is visualized in three-dimensional. The road model 4 has
a width w and a longitudinal length RL as necessary for the rotation of the pneumatic
tire model 1.

In one embodiment, the road model 4 may be built so as
to have a smooth surface. In the present embodiment, the road model 4 has a rough
surface simulated as rough asphalt road so that the vibration property of the pneumatic
tire model 1 is emphasized. According to a real asphalt road, a vertical gap between
the lowest and the highest surfaces of the road model 4 is preferably set from 3
to 15 mm, and more preferably 6 to 12 mm. Further, in the present embodiment, a
pitch between the nodes of the elements on the road model 4 is 20 mm.

Next, in step S4, boundary conditions for the simulation
are set. The boundary conditions, for example, include necessary condition for executing
the rolling simulation which makes the pneumatic tire model 1 rotate on the road
model 4 such as internal pressure of the pneumatic tire model 1, friction coefficient
between the tire model 2 and the road model 4, the tire load, the slip angle, the
camber angle and rotating speed.

Further, initial density and bulk modulus are defined into
each finite volume of the cavity model 3, according to fluid which is simulated
filled in the cavity i. In this present embodiment, the initial density and the
bulk modulus are defined as follows under the temperature of 25 degrees C. and the
inner pressure of 200 kPa.

Density: 3.52 kg/m^{3}

Bulk modulus: 423KPa.

Further, the internal pressure is expressed by executing
a static inflating-simulation of applying an uniformly distributed load having the
same value as the internal pressure for simulation on the inner surface 2i of the
tire body model 2. Accordingly, in the present embodiment, the cavity model 3 is
not used as a medium for applying the internal pressure on the tire body model 2.
Therefore, the initial cross section of the cavity model 3 is built according to
the inner surface 2i of the tire body model 2 after the mentioned-above inflating-simulation.
If the cavity model 3 is used as the medium above, a large scale simulation which
comprises the steps of preparing a great number of finite volumes of the cavity
model 3 firstly, and injecting them continuously into the tire body model 2 has
to be executed. This is not preferable because a lot of time is required for the
calculation.

Accordingly, in the present embodiment, each finite volume
of the cavity model 3 has zero pressure even when the inner pressure has been applied
on the tire body model 2. However, when the deformation of the tire body model 2
occurs by contacting and rotating on the road model 4, the deformation is given
to each volume of the cavity model 3 as force according to the coupling condition
between the tire body model 2 and the cavity model 3. Thereby, the pressure changes
in the cavity i of the tire body model 2 can be expressed using the relative pressure
based on the initial pressure of each finite volume of the cavity model 3. In the
present invention, other conditions for executing the numerical simulation are as
follows.

Running speed of pneumatic tire model: 80km/H

Internal pressure: 200kPa

Slip angle: 0 degree

camber angle: 0 degree

Static friction coefficient: 1.0

Dynamic friction coefficient: 1.0

Next, as shown in Figs. 9 and 10, deformation calculations
(simulation) of the pneumatic tire model 1 are performed (Step S5). Namely, the
state where the pneumatic tire model 1 rolls on the road model 4 is calculated for
every small time increment by using the computer device 1. In this embodiment, the
pneumatic tire model 1 is made to roll on a stationary road model 4. However, it
is possible that a pneumatic tire model 1 with a free rotating tire axis is driven
by friction force from a moving road model 4 in contact with the tread of the tire
body model 1.

Fig. 11 shows one embodiment of a flowchart showing the
deformation calculation of the pneumatic tire model 1. In Step S51, the deformation
calculation of the tire body model 3 after the time increment &Dgr;t is performed
first. The deformation calculations of the tire body model 2 in the present example
are performed by finite element method using the equation as follows.
$$f=m\stackrel{a}{x}+c\stackrel{a}{x}+\mathit{kx}$$

where

f: external force matrix

m: mass matrix

*ẍ* = acceleration matrix

c damping matrix

*ẋ* = velocity matrix

k = stiffness matrix

*x* = displacement matrix

Namely, in the deformation calculations, the mass matrix m, the stiffness matrix
k and the damping matrix c of the elements are defined based on the material characteristics
of each element of the tire body model 2 such as density, modulus, damping coefficient
and the like. Next, such matrices are combined to form the matrix of the entire
system to be simulated. Then, applying the above-mentioned other conditions, the
equation 1 is defined, and is calculated using the computer device 1.

The explicit time integration method is employed in the
simulation in this present example. According to the explicit method, the moment
that the load acts on each model is taken as time zero, and the time is divided
into small increments so as to find the displacement of the model at each point
in time. Each initial time increment &Dgr;t in the deformation calculation for
the tire body model 2 and the cavity model 3 is preferably set from 10 to 100 µs.
In order to determine the time increment &Dgr;t, for example, the propagating
time of the stress wave of each element is calculated first. After that, the time
increment &Dgr;t is set up by multiplying the minimum of the propagating time
by a safety factor such as 0.9 or less.

By deforming the tire body model 2, acceleration is inputted
into the cavity model 3 through the boundary between the tire body model 2 and the
cavity model 3. Accordingly, in the step S52, pressure of each finite volume of
the cavity model 3 is calculated using equation as follows.
$$\frac{1}{{k}_{f}}\stackrel{a}{p}=\frac{\partial}{\partial x}\left(\frac{1}{{\mathrm{\&rgr;}}_{f}},,\frac{\partial p}{\partial x}\right)$$

where "K_{f}" is bulk modulus defined into each finite volume of the cavity
model 3, "p" is the relative pressure based on the initial pressure of each finite
volume of the cavity model 3, "p_{f}" is density each finite volume of the
cavity model 3, and "x" is position of each finite volume of the cavity model 3.

Next, force to be applied into the tire body caused by
the cavity model 3 is calculated (step s53). The deformation of the tire body model
2 calculated in the step S51 is taken into consideration with the pressure calculation
of the cavity model 3 in the step S52. Therefore, in the step S52, pressure change
after the time increment &Dgr;t of the cavity model 3 can be calculated. Further,
the difference between the force calculated before the time increment &Dgr;t and
the force calculated after the time increment &Dgr;t of the cavity model 3 is
applied to the next deformation calculation (step S51) of the tire model 2 as an
external force when "NO" is selected in the step S56.

Next, the stress wave propagation time of each finite element
of the tire body model 2 is re-calculated based on its size, density and stiffness
(Step S54). Then, based on the minimum value of this stress wave propagation time,
the time increment for the next deformation calculation is set (Step S55). In the
present embodiment, the minimum value of the stress wave propagation time or the
value which is calculated by multiplying the minimum of the stress wave propagation
time by the safety factor (<1.0) is employed.

Next, it is checked whether the predetermined duration
of simulation time has been elapsed or not (Step S56). When "NO" is selected in
the step S56, the process goes back to the Step S51 to perform one more calculation
by adding the newly set time increment. When "Yes" is selected in the Step S56,
the deformation calculation of the pneumatic tire model 1 is terminated and the
process goes to the Step S6.

In the deformation calculation described above, acceleration
according to the deformation of the tire body model 2 is inputted from the inner
nodes 2n1, 2n2 ... of the tire body model 2 to the outer nodes 3n1, 3n2 ... of the
cavity model 3. On the other hand, force caused by the deformation of finite volumes
of the cavity model 3 is inputted into the inner nodes 2n1, 2n2 ... of the tire
body model 2 from the outer nodes 3n1, 3n2 ... of the cavity model 3. Accordingly,
it becomes possible for the cavity model 3 to calculate pressure changes in the
acceleration on the tire model 2 due to changes in the position or shape of the
tire model 2. Therefore, it is possible to analyze the pressure distribution in
the cavity model 3 during the tire body model 2 is running. Further, by checking
the changing of the pressure distribution in time history, it is also possible to
analyze the flow of fluid filled in the cavity I of the tire body model 2. The flow
of fluid is one of physical parameters corresponding to the sound and vibration
of the cavity i.

For the tire body model 2, it becomes possible to calculate
a new deformation due to the reaction force received from the cavity model 3.

Repeating these calculations can analyze the changing contact
situation between the tire body model 2 and the cavity model 3, while considering
their interaction.

Next, as shown in Fig.2, data or information on the tire
body model 2 and/or cavity model 3 is outputted through the simulation (Step 6).
The output of the calculation results may contain various physical parameters, and
those are sequentially memorized in the computer device 1. Accordingly, it is possible
to use the results as numerical data or visualized data such as a chart and a graph.

Usually, required time increment to keep the accuracy of
the deformation calculation of the cavity model 3 is large enough compared with
the tire body model 2. Therefore, it is not necessarily to calculate pressure change
of the cavity model 3 to every time increment calculated from the tire body model
2. For example, in order to reduce time for simulation without deteriorating accuracy
of the result, one pressure calculation of the cavity model 3 may be performed every
two or more times of the deformation calculations for the tire body model 2 are
performed.

Fig. 12 shows a visualized example of the simulation result
of pressure distribution at one moment of the pneumatic tire model 1 being rolling
with the direction "A" on the road model 4. As shown in Fig. 12, the tire body model
2 is drawn as a section taken along the tire equator to show the cavity model 2
outside. Further, a darker part shown in Fig. 12 has a higher pressure. The result
shows that a pair of regions B and C with higher pressure in the cavity model 3
appear on both sides of the tread portion contacting on the road model 4.

Fig. 13 shows a visualized example of the simulation result
of time history of vertical force acting on the tire axis of the pneumatic tire
model 1 during its running. Fig. 14 is a graph showing the result of frequency analysis
of Fig.13. In Fig. 14, a solid line shows the result of example 1 according to the
present invention simulation, and a dotted line shows the result of reference simulation
which uses only a tire body model without the cavity model. Other conditions are
the same between the example 1 and the reference. As shown in Fig. 14, a sharp peak
D with a frequency of about 250 Hz is simulated in the example 1. Generally, this
peak D is well known as cavity resonance which is generated in a tire cavity. Accordingly,
it is confirmed that the pneumatic tire model 1 according to the present invention
can simulate even the cavity resonance into the simulation.

In recent years, a pneumatic tire and noise damper assembly
is proposed by
Japanese patent No. 3,612,059
, for example. The assembly comprises a pneumatic tire and a noise damper
made of spongy material and being attached to an inner surface of the tire so as
to extend in the circumferential direction of the tire. By using the present invention,
the effect of the assembly was tested.

A pneumatic tire model 1a shown in Fig.15 was modeled for
the assembly above. The pneumatic tire model 1a comprises: the tire body model 2;
the cavity model 3; and a noise damper model 6 which is attached to the inner surface
of the tire body model 2 and continuously extends in the circumferential direction
of the tire. The noise damper model 6 has a cross section having a pair of projecting
parts and a groove interposed therebetween. Further, a condition in which the relative
distance does not change is defined in the boundary surface between the noise damper
model 6 and the cavity model 3.

A stress-strain property shown in Fig. 16 and density of
25.0 kg/m^{3} are defined into each finite element of the noise damper model
6.

Fig. 17 shows the result of frequency analysis of the vertical
force of the pneumatic tire model 1a (example 2) from the rolling simulation executed
with the same condition of example 1 described above. In Fig. 17, a solid line shows
the result of example 2, and a dotted line shows the result of the example 1 without
noise damper model 6. As seen in Fig. 17, it is clear from the simulation result
of the example 2 that the peak of about 250 Hz corresponding to the cavity resonance
is disappeared by the noise damper.

Fig. 18 shows a visualized example of the simulation result
of pressure distribution at one moment of the pneumatic tire model 1a being rolling
with the direction "A" on the road model 4. As shown in Fig. 18, the tire body model
2 is drawn as a section taken along the tire equator to show the cavity model 2
outside. Further, a darker part shown in Fig. 18 has a higher pressure. The result
shows that the area with higher pressure in the cavity model 3 is smaller than the
example 1.