Dr. Mohammad Mashayekhi | Professor of Mechanical Engineering · Title: Microsoft PowerPoint - Presentation6.ppt Author: mohammad@MASHAYEKHI Created Date: 4/25/2009 9:37:06 AM

Introduction to Contact Mechanics

Contact problems: roller bearing and impact of a lorry.


Sheet metal forming Crash of a car against a deformable barrier


Self-contact of a car component during crash,

Crash of a car against a deformable barrier


Contact of tyres with a road surface,


Contact in a Mass Spring System

Energy of the mass spring systemPoint mass supported by spring


The Energy of system

The variation of Energy

The second variation of Π

The minimum of u

The restriction of the motion of the mass by a rigid support


the point mass contacts the rigid surface, a reaction force RN appears.

two cases within a contact problem

Both cases can be combined

Kuhn–Tucker condition


Lagrange multiplier method

the Lagrange multiplier method adds to the energy of the system a term which contains the constraint

the Lagrange multiplier λ is equivalent to the reaction force RN .

for contact: u = h

for does not contact: RN =0

u=mg/k


Penalty method

or an active constraint one adds a penalty term to the energy

the solution

penalty spring due to the penalty term

The variation of the energy


Penalty method

in the contact,

The reaction force

Penetration of the point mass into the rigid support occurs.

the unconstrained solution.

the correct solution obtained with the Lagrangemultiplier method.

Finite Element Analysis of the Contact of Two Bars

Lagrange method

F


Lagrange method


Lagrange method


Lagrange method


Lagrange method


Penalty method

1

2

3

2 0

0

u FEA EAl l

u gEA EAl l

EAl u g

εε ε

ε εε

− − + − = − + −

2 2

2

3

EA EAF g Fl luEA EAl l

ε

ε

+ + =

+

ε →∞

2

21 23 3

EA g F Fllu gEA EAl

+ = = +

Documents

Dr. Mohammad Mashayekhi | Professor of Mechanical Engineering · Title: Microsoft PowerPoint - Presentation6.ppt Author: mohammad@MASHAYEKHI Created Date: 4/25/2009 9:37:06 AM