Chapter 5: Bilinear Friction Model: Sliding Wedge

5 Bilinear Friction Model: Sliding Wedge

Summary 103

Introduction 104

Analytical Solution 104

FEM Solutions 104

Modeling Tips 107

Input File(s) 107

Video 108

103CHAPTER 5

Bilinear Friction Model: Sliding Wedge

SummaryTitle Chapter 5: Bilinear Friction Model: Sliding Wedge

Contact features • Bilinear stick-slip friction behavior• Deformable-deformable contact• Friction along the contact surface• Comparison of linear and quadratic elements

Geometry

Material properties , ,

, , ,

Linear elastic material

Analysis type Quasi-static analysis

Boundary conditions All displacement components of the nodes in the lower face of the lower wedge are fixed; of two nodes on the upper wedge with contact between upper and lower

wedge

Applied loads Gravity load ; pressure load

Element type 3-D solid with 4 -node linear and 10-node parabolic tetrahedral elements

Contact properties Friction coefficient

FE results 1. Deformed configuration at the end of the second STEP2. Plots of x-displacement of point A

X

Y

Z

4.0

6.0

1.3

0.7

1.0

1.2

g

p

1.0

A

x

y

Eup 2.06710 Pa= up 0.3= up 1 kg m

3=

Elow 2.061110 Pa= low 0.3= low 1 kg m

3= Kspring 119.5 N/m=

uz 0 m=

gy 764.5 N–= px 1250 Pa and 693.375 Pa–=

0.3=

x-displacement (m)

% of load

0 50 100 150 200

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

Quadratic Elements

Linear Elements

MD Demonstration Problems

CHAPTER 5104

IntroductionThis problem verifies and validates the behavior of the bilinear friction model. A more detailed description of the bilinear friction model can be found in the Release Notes for MD Nastran. The fundamental control parameter of this friction model is the so-called relative sliding displacement below which (elastic) sticking is simulated. This parameter can be user-defined by specifying RVCNST on the BCPARA option. Otherwise, MD Nastran determines the default value as a function of the average edge length of the elements in the contact bodies.

This example was originally proposed by NAFEMS as a 2-D large sliding contact and friction example. Here, we use a modified version of the problem: namely 3-D instead of 2-D and an alternating load instead of a linearly increasing load.

A large displacement is expected in this solution but the strains will be pretty small. Assuming the motion as rigid body, it can be predicted analytically as shown in the NAFEMS documentation (NAFEMS Benchmark Tests for Finite Element Modeling of Contact, Gapping and Sliding, 2001).

First, a gravity load is applied to the whole model. Then, a positive pressure is applied as such that point A will

have displacement . The next step, a negative pressure is applied as such that point A will have displacement

. The last step is again an application of positive pressure . The applied pressure will be determined

analytically.

The analysis results are presented with linear and parabolic elements.

Analytical SolutionAssuming a rigid body motion and neglecting the loss of energy due to friction, the relation among the total force on the upper wedge in the x- and y-direction ( and ), the friction coefficient ( ), the wedge angle ( , the total spring

stiffness ( ) and the positive displacement ( ) of the upper wedge is:

With , , , (based on ) and , the total spring stiffness

( ) is . Thus, the applied that correlates with is . This load is applied during the second step.

Alternatively, with the given value of , , and , results in a displacement of the upper wedge

( ). that correlates with this is . This pressure is applied in the third step. The fourth

step is again the introduction of .

FEM SolutionsA numerical solution has been obtained with MD Nastran’s SOL 400 for the element mesh shown in Figure 5-1. The colored regions of the wedges have been identified as contact bodies. Contact body IDs 1 and 2 are identified as a set of elements of upper and lower wedge, respectively as:

px

ux 1 m=

ux 1 m–= px px

Fx Fy

K ux

KFx 1 tan– Fy tan+ +

ux 1 tan– -------------------------------------------------------------------------------=

tan 0.1= 0.3= Fx 1500 N= Fy 3058 N= gy 764.5 N–= ux 1 m=

K 239 N/m px 1250 Pa

K tan Fy Fx 832.8 N–=

ux 1 m–= px Fy Fx 693.375 N–=

px 1250 Pa=

105CHAPTER 5

Bilinear Friction Model: Sliding Wedge

BCBODY 1 3D DEFORM 1 0 .3BSURF 1 42 107 118 132 194 236 239...

and

BCBODY 2 3D DEFORM 2 0 .3BSURF 2 1 2 3 4 5 6 7...

Figure 5-1 Element Mesh applied in Target Solution with MD Nastran

Furthermore, the BCTABLE entries shown below identify that these bodies can touch each other.

BCTABLE 0 1 SLAVE 1 0. 0. 0. 0. 0 0. 0 0 0 MASTERS 2BCTABLE 1 1 SLAVE 1 0. 0. 0. 0. 0 0. 0 0 0 MASTERS 2

Thus, any deformable contact body is simply a collection of mutually exclusive elements and their associated nodes.

To activate contact with Coulomb friction, FTYPE must be set to 6 in BCPARA option (the only supported Coulomb friction model). The contact separation option is based on relative stresses. It is done by setting IBSEP = 4. Furthermore, for model with quadratic element, the genuine parabolic contact has to be set using LINQUAD = -1.

BCPARA 0 FTYPE 6 IBSEP 4 LINQUAD -1

3-D tetrahedral elements are used in this analysis.

PSOLID 1 1PSOLID 2 2 +

The two material properties are isotropic and elastic with Young’s modulus and Poisson’s ratio defined as

MAT1 1 2.06+07 .3 1.MAT1 2 2.06E+11 .3 1.

MD Demonstration Problems

CHAPTER 5106

The nonlinear procedure used for the analysis:

PARAM LRGDSIP 1NLPARM 1 1 FNT UVNLPARM 2 25 FNT UV

Here the FNT option is selected to update the stiffness matrix during every recycle using the Newton-Raphson iteration strategy and the default convergence tolerance for displacement (relative to the incremental displacement) will be used.

The simulation is eventually controlled by the case control section which consists of four STEPS.

STEP 1 LABEL = Gravity Load ...STEP 2 LABEL = Px is 1250 ...STEP 3 LABEL = Px is -694 ...STEP 4 LABLE = Px is again 1250 ...

The deformed structure plot (magnification factor 1.0) is shown in Figure 5-2. After the second step, as seen in Figure 5-2, the upper wedge moves in the x-direction one meter as predicted analytically.

Figure 5-2 Deformed Structure at the End of the Second Step (magnification factor = 1)

The displacement plot of point A, for linear and parabolic elements, is shown in Figure 5-3. It is clearly seen that the upper wedge moves alternately from to and then back to as expected using the analytical

solution. The result of the linear element is nearly the same as that of the parabolic elements. As clearly seen from this figure, during (linear) sticking contact, the displacement of the upper wedge varies linearly.

deformedundeformed

ux 1.0=

ux 1 m= ux 1 m–= ux 1 m=

107CHAPTER 5

Bilinear Friction Model: Sliding Wedge

Figure 5-3 Displacement Plot for Point A (Representing the Displacement of the Upper Wedge)

Modeling TipsIt is very important to have accurate coordinates for those points that are located on the both sides of the contact interfaces. Failure in representing accurate smooth surfaces may lead to unexpected contact behavior. That is why the coordinate of the grid points both for models with linear and parabolic elements are expressed in the extended format of MD Nastran.

Input File(s)

File Description

nug_05a.dat Linear Elements

nug_05b.dat Quadratic Elements

0 50 100 150 200 250 300 350 400

-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0

x-displacement (m)

% of load

x-displacement (m)

% of load

0 50 100 150 200

-0.0006

-0.0004

-0.0002

0.0000

0.0002

0.0004

0.0006

0.0008

0.0010

0.0012

Quadratic Elements

Linear Elements

MD Demonstration Problems

CHAPTER 5108

VideoClick on the image or caption below to view a streaming video of this problem; it lasts about 47 minutes and explains how the steps are performed.

Figure 5-4 Video of the Above Steps

X

Y

Z

4.0

6.0

1.3

0.7

1.0

1.2

g

p

1.0

A

x

y