A frictional contact finite element for wheel/rail dynamic simulations
輪軌動態有限元素分析含摩擦力效應
朱聖浩
國立成功大學土木工程學系特聘教授
軌道運輸技術論壇
An earthquake with a magnitude of 6.8 occurred in Chuetsu region of Niigata Prefecture of Japan on October 23, 2004 at 17:56 causing the derailment of the Shinkansen train (arranged from Yomiuri Newspaper and Internet web-sites)
1. 前言
精確的模擬列車脫軌現象需要準確的輪軌有限元素模型。
Taiwan High Speed Rail slightly derailing in southern Taiwan due to the Jiasian earthquake on March 4, 2010
Resonance
柱子勁度越大脫軌係數越小
Se ism ic lo a d s a nd sp ring s
Y X Z
高速列車受地震的影響
高速列車受基礎沉陷的影響
X Y
Z
X Y
Z Brid g e g ird e r
Ra il
Co nc re te p la te
Pie r
X Y
Z (C)
(A) (B)
(D)
(
a) Deformed bridge near the settlement
(b) Deformed bridge at location B (c) Deformed bridge at location C
Settlement of 6.5 cm (Vangle=0.00217 rad) in the vertical direction
2. 車橋互制之非線性動力有限元分析
含碰撞、分離、及摩擦行為之元素
The element includes a wheel node and a
number of target nodes. The wheel node can move on these target nodes.
K
F
r
B A
1 2
C Fs Fn Fs
Fn D
k1, k2, and k3 between points A and B
a 3D beams to b
simulate rails
Cubic splines to
simulate wheel surface
I
J
Time=0.319 s Time=0.347 s
軌道可用三維梁元素模擬 適當的勁度用以模擬輪軌間 之勁度
使用點對曲線之碰撞元素模擬
Time=0 s
0
0
=
k k
K
C
=
0 0
0 0
K
C
= k 1
2
µ
µ µ
K
CThe sticking, sliding, and separation stiffness matrices at one point are as follows:
碰撞不滑動
滑動含摩擦效應
分離
K
1-2= T
TK
CT Change to the 1-2 coordinate
B A
1 2
C Fs Fn Fs
Fn D
k1, k2, and k3 between points A and B
a 3D beams to b
simulate rails
Cubic splines to
simulate wheel surface
I
J
−
= −
−
−
−
−
2 1 2
1
2 1 2
1
4
K K
K
K K Change to 2-node contact
K
BC= T
4×6T K
4T
4×6Change to contact between B and I-C-J
K
BIJ= T
6×18TK
BCT
6×18Change to 18 by 18 global
stiffness matrix
3 1 2 3 2 1
9 16 2
3
//
*
r
RE f
k
=
車輪和軌道之間的垂直剛度可以使用赫茲接觸理論：
我們使用以下公式來模擬車 輪和導軌之間的剛度：
c
r
a bf
k = +
2再使用三維有限元 素分析計算軌道和 車輪之間的剛度來 評估a，b和c。
X Y Z
B A
1 2 C
Fs Fn Fs
Fn D
k1, k2, and k3 between points A and B
a 3D beams to b
simulate rails
Cubic splines to simulate wheel surface
I
J
3. Accuracy study
The accuracy of the element is studied using
dynamic analyses including a SKS-700 train wheel with a UIC-60-kg rail fixed at the two rail ends.
Symmetry of the wheel
FV
Cv Kv
KH
CH FH=20 kN (Section 3.1)
FH=10.4125kN*Sin(20πt) (Section 3.2) (in Y direction)
A
X Y
Z
Fixed end
Fixed at two sides of the rail 0 0.2 0.4 0.6
Time (s) (Frequency=10 Hz) -80
-40 0 40
Fv(t) (kN)
-71.25 kN (-Z direction)
51.25 kN (+Z direction)
X Y
Z
0 0.1 0.2 Time (s)
0 10 20 30 40
Forces between wheel and rail (kN) Simple FEM, Vertical force (P) Simple FEM, Horizontal force (Q) Complex FEM, Vertical force (P) Complex FEM, Horizontal force (Q)
A wheel and rail contact problem with a constant
horizontal force
0 0.05 0.1 0.15 0.2 0.25
Time (s)
0 1 2 3 4 5
Average Q/P between 0.024 s (2 m for train speed=300 km/h)
Simple (FEM) Complex FEM
A wheel and rail contact problem with a sine wave horizontal force
0 0.1 0.2 0.3 0.4
Time (s) -10
0 10 20 30
Forces between wheel and rail (kN) Average between 0.024 s
Simple FEM, Vertical force (P)
Simple FEM, Horizontal force (Q) Complex FEM, Vertical force (P) Complex FEM, Horizontal force (Q)
A B
0 0.1 0.2 0.3 0.4
Time (s)
0 0.4 0.8 1.2 1.6 2
Average Q/P between 0.024 s (2 m for train speed=300 km/h)
Simple (FEM) Complex FEM
A B
0 0.4 0.8 1.2
Time (s)
0 0.4 0.8 1.2 1.6 2
Average Q/P between 0.024 s
A more complicated simulation was performed using the same model as above. An additional torsion in the X direction is
applied to the beam center. Torsion=183 sin (10*pi*t) kN
