Numerical and Experimental Approach for Pattern Formation Phenomena

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Numerical and Experimental Approach for Pattern
Formation Phenomena

• Yoshiaki Terumichi
• Sophia University, Japan

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Contents

• 1. Introduction
• 2. Mechanism of Self Excited Vibration due to
  Time Lag
• 3. Corrugation Development on Rotating Disk with
  Rolling Contact
• 4. Corrugation Development on Flexible Beam with
  Rolling Contact
• 5. Summary

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Rail Corrugation
1. Introduction
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Self-excited vibration due to time lag in rolling
contact
Pattern Formation in Grinding
Corrugation Simulator for Railway System
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Mechanism of Pattern Formation Phenomena in
Mechanical Vibration
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2. Mechanism of Self Excited Vibration due to
Time Lag
Assumptions -Grinder motion is neglected. -Work
piece motion is restricted in normal
direction. -Contact rigidity is given by
Hertz theory.
Analytical model for grinding
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Eigen Value Analysis 1
Unstable
Stable
Countable but infinite eigen values
Real and imaginary part of eigen values
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Eigen Value Analysis 2
In general, wavy surface on work piece is
developed with polygon number by maximum real
part. It is slightly smaller than the natural
frequency neglected the effect of time lag.
Polygon number of work piece
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3. Corrugation Development on Rotating
Disk with Rolling Contact
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a. Numerical approach

• ACenter of gravity
• B Center of gravity
• QAContact point on disk A
• QBContact point on disk B
• O-XYFixed Frame
• OA-xAyALocal frame fixed at center of gravity A
• OB-xByB Local frame fixed at center of gravity B
• OC-xCyC Local frame fixed at contact point on
  disk A
• OD-xDyD Local frame fixed at contact point on
  disk B
• ?ARadius of disk A
• ?B Radius of disk B
• ?A Rotation displacement of disk A
• ?B Rotation displacement of disk B
• dDeformation

Analytical Model
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a1. Deformation d

• Hertzs theory
• Relation between contact rigidity
• and contact force

EA, EBYoungs Modulus ?A, ?B Poison
Ratio PContact force cdDamping coefficient
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a2. Slip and Wear

• Assumptions
• Wear develops on circumference of disk B
• Slip rate
• Wear amount c Constant
• Sum of wear amount
• Radius of disk B

P Contact force
B
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a3. Formulation

• Constraint conditions
• for unilateral contact
• Differential Algebraic Equations
• Constraint force acting on contact point

Configuration of contact between disk A and B
eyUnit vector
mAMass of disk A mB Mass of disk B kSSpring
constant of support cSDamping coefficient ?YLagr
ange Multiplier
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a4. Numerical results
Disk A Material S45C Youngs Modulus
N/m2 2.071011 Poison Ratio 0.28
Radiusm 0.09 Thickness m 0.002
Mass kg 20.0 Disk B Material S45C
Youngs Modulus N/m2 2.071011 Poison
ratio 0.28 Radiusm 0.06 Thickness
m 0.002 Mass kg 4.0 Support
Spring constant N/m 1.23107 Damping
ratio 0.2 Contact Damping ratio 0.2
Wear coefficient 1.210-8

• Rotation velocity
• Disk b2.00m/s
• Slip rate
• 2
• Initial disturbance

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a4-1. Effect of slip rate on self-excited
vibration
100 rounds
300 rounds
500 rounds
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5
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Rotation speed of disk B2.00m/s
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A4-2. Effect of support stiffness
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ks1.25107N/m
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Period of wavy surface
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A4-3. Maximum amplitude of wavy surface
Maximum amplitude versus rotation speed of disk
B(400rounds after, Slip rate 2)
Position of maximum amplitude
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b. Experimental approach
b1. Experimental set up (combined use for
grinding and rolling contact)
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Support spring
Configuration of rolling disks
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b2. Experimental results

• Wavy surface in 954, 1908, 2862, 3816, 4770,
  5724, 6678 and 7632 rounds.
• Frequency 40Hz,650Hz
• Amplitude of the ruggedness increases with
  rotation.
• The outer shape flows.

Corrugation development on rolling disk
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Support stiffness
Support stiffness
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4. Corrugation Development on Flexible Beam with
Rolling Contact
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a. Approach Procedure
Modeling considering beam elasticity, track
flexibility, contact rigidity, slip and wear
development
Finite elements approach

• Flexible multibody dynamics
• Contact mechanics, Tribology

Constraint conditions for unilateral rolling
contact
Wear development
Hertz theory
Formulation
A.N.C Formulation DAE
Simulation for corrugation development
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b. Modeling and formulation
Corrugated surface
Track flexibility
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b1. Disk motion on flexible beam with contact
rigidity

• ? Position vector to Q
• relative position vector
• ? Position
  vector to B
• Position
  vector is expressed
• using
  Absolute Nodal Coordinate.

s
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b2. Wear development

• Assumption for profile of beam surface
• Wear amount is proportional to wear rate
• Wear rate W
• Wear amount z
• Total wear amount after n passages

PContact force
cWear coefficient
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b3. Configuration of disk on worn surface
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b4. Formulation for disk motion
Equation of motion
Constraint conditions
Unilateral contact between beam and disk No-slip
motion
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Constraint condition for unilateral contact
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c. Numerical results for case of one disk

• Initial condition

Parameter of rail
Parameter of wheel
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(during 450 passages)
During 160th passage
Corrugation development until 450 passages
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(after 150 passages with phase difference p/4)
Corrugation development during 151- 450 passages
During 160th passage
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(after 150 passages with phase difference p)
Corrugation development during 151- 450 passages
During 160th passage
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(after 150 passages with phase difference 3p/4)
Phase shift during repeated passages leads to
corrugation development.
Corrugation development during 151- 450 passages
During 160th passage
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d. Numerical results for case of two disks
Disturbance torque -2408Nm
Disturbance torque 2401.55Nm
With slight difference of disturbance torque,
remarkable difference in surface profile occurs.
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Disturbance torque 2401.55Nm
Disturbance torque -2408Nm
Time period when tangential force is larger than
friction force is important factor.
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e. New Type of Corrugation Simulator by
Flexible Track with Elastic Rail
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e1. Configuration of corrugation simulator
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Wheel and moving body
Sub-wheels
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Flexible track with elastic rail and chip
Chip on middle point of rail
Area for corrugation development
Fixed support
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e2. Corrugation development speed0.8m/s,Chip
4mm
Initial state
after 2700 passage
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e3. Condition of corrugation generation and
development
(a) Rigid rail
(b) Flexible rail
?Corrugation developed Corrugation did not
generate. ?Corrugation generated but did not
develop so much.
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e4. Effect of Rail flexibility on Corrugation
development
330Hz

• Corrugation development in various cases of chip
  position. (Chip
  0.1mm, Speed 0.8m/s)

1/2
1/3
Amplitude
120Hz
1/4
Wavy surface
90Hz
Chip position(1/4)
Position after chip(mm)
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Eigen value analysis
Frequency of wavy surface(Experimental results)
Natural frequency of system with disk position
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5. Summary

• Modelling and formulation of motion of two
  rolling disks and flexible track/disk system were
  developed, considering contact rigidity, slip and
  wear development. Profile of worn surface was
  treated as a constraint for unilateral contact.
• In both systems, phase shift during regeneration
  or repeated passages forms the condition of the
  corrugation development.
• Expreimental set up for corrugation development
  in rolling disks system and flexible rail/wheels
  system was built in laboratory. Some numerical
  results revealed the feature of the corrugation
  development in experiment.

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Comparison between corrugation development on
rolling disks and rail/wheel system
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Corrugation development Disturbance is given to
each disk at center of beam.
Corrugation development
Short wave corrugation develops on surface
Slip rate and contact force
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Contact force fluctuation
In-Phase vibration due to contact rigidity
between front and rear disk occurs.
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Corrugation development with phase difference
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1. Introduction

• ?Problems on dynamic behavior of rail/wheel
  systems
• Noise and vibration
• Corrugation and damage on rail surface etc.
• ?Challenging subjects
• ? Modeling and formulation for rail/wheel
  systems
• ? Calculation of contact force fluctuation in
  detail
• ? Demonstration of corrugation development and
• explication its mechanism
• ? Experimental certification for simulation
  validity

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Factors on contact force fluctuation of
rail/wheel systems
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Eigen Value Analysis
(a) without support
(b) with support
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Pattern formation on surface 1
Slip rate
Surface profile after 100 passages
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Friction and tangential force (Front)
Factors of wavy surface
Contact rigidity, rail flexibility, track
elasticity Friction force and tangential force
fluctuation Repeated passage of wheels
Friction and tangential force (Rear)