Friction Between Belt And Pulley

Chapter 24: Friction Between Belt and Pulley

24

Friction Between Belt and Pulley 

Summary



Introduction



Requested Solutions



Analytical Solution



FEM Solutions



Modeling Tip



Input File(s)



Video

446

439 440 440 440

441 444 446

CHAPTER 24 439 Friction Between Belt and Pulley

Summary Title

Chapter 24: Friction Between Belt and Pulley

Contact features

• • • •

Geometry

3-D (units: mm)

(Slightly) changing contact area Curved contact surfaces Deformable-deformable and deformable-rigid contact Friction between deformable bodies • • • • • •

Material properties

Pulley outer radius = 0.55 Pulley inner radius = 0.25 Out of plane pulley thickness = 0.3 In plane belt thickness = 0.05 Out of plane belt thickness = 0.2 Initial angle spanned = /2 rad 13

R ϕ

r2 r1

y z

t x

F

10

E pulley = 1.0 10 Pa E belt = 1.0 10 Pa  pulley =  belt = 0.3

Linear elastic material Analysis type

Quasi-static analysis

Boundary conditions

An 180o section of the pulley is modeled, which is clamped along the inner radius using “glued” contact conditions. On both ends of the belt, load-controlled rigid bodies are defined and connected to the belt using “glued” contact conditions. The forces F and R are external and reaction forces on the control nodes. On the loaded control node we have u x = u y = 0 , while on the other control node u x = u y = u z = 0 .

Applied loads

Point load F y = – 1.0 105 N

Element type

3-D 20-node hexahedral solid elements

Contact properties

Different coefficients of friction between belt and pulley:  = 0.05 ,  = 0.15 and  = 0.25

FE results

Reaction force for each value of the friction coefficient

440 MD Demonstration Problems CHAPTER 24

Introduction A belt is positioned around a pulley such that a 90o section of the pulley is contacted. One end of the belt is fixed; the other end is loaded by a tensile force with magnitude F = 1.0 105 . It is assumed that the material behavior for both the belt and the pulley is linear elastic. Although this problem can be solved by a 2-D approximation, a full 3-D model is chosen here in order to show the characteristic behavior of 3-D parabolic hexahedral elements in a contact analysis involving friction. An analytical solution for the case with Coulomb friction is known.

Requested Solutions Analyses will be carried out for three different values of the friction coefficient:  = 0.05 ,  = 0.15 , and  = 0.25 . With a constant value of the applied load, the reaction force will decrease for increasing values of the friction coefficient. This reaction force is the primary requested quantity, as this can be easily compared with an analytical solution.

Analytical Solution Assuming Coulomb friction between the belt and the pulley, the principle of rope friction according to the EulerEytelwein formula provides a relation between the magnitude F of the applied force, the magnitude R of the reaction force, the angle  spanned by the belt and the friction coefficient  between the belt and the pulley: F R = ------- e

With F = 1.0 105 and  = --- , the theoretical value of the magnitude of the reaction force R is listed in Table 24-1 for 2

various values of the friction coefficient  . Table 24-1

Reaction Force for Various Values of the Friction Coefficient (Theory)

Friction Coefficient 

Reaction Force R

0.05

9.2447x104

0.15

7.9008x104

0.25

6.7523x104

CHAPTER 24 441 Friction Between Belt and Pulley

FEM Solutions Numerical solutions have been obtained with MD Nastran’s SOL 400 for the element mesh shown in Figure 24-1 using 3-D 20-node hexahedral elements. Assuming that the deformations of the pulley are small and localized around the contact area, only an 180o section has been modeled. In total, there are five contact bodies: two deformable and three rigid. The rigid bodies will be used to easily apply the boundary conditions (single point constraints and forces).

load controlled rigid body

fixed rigid body; glued contact

load controlled rigid body

Figure 24-1

Element Mesh applied in MD Nastran Simulation

The first deformable body consists of all elements of the belt, where the second deformable body consists of all elements of the pulley. The body number ID’s of the belt and the pulley are 1 and 2, respectively. These deformable contact bodies are identified as 3-D bodies referring to the BSURF IDs 1 and 2: BCBODY BSURF

BCBODY BSURF

1 1 8 16 24 32 40 48 56 64 72 2 2 82 90 98 106 114 122 130

3D 1 9 17 25 33 41 49 57 65 73 3D 75 83 91 99 107 115 123 131

DEFORM 2 10 18 26 34 42 50 58 66 74 DEFORM 76 84 92 100 108 116 124 132

1 3 11 19 27 35 43 51 59 67

4 12 20 28 36 44 52 60 68

5 13 21 29 37 45 53 61 69

6 14 22 30 38 46 54 62 70

7 15 23 31 39 47 55 63 71

2 77 85 93 101 109 117 125 133

78 86 94 102 110 118 126 134

79 87 95 103 111 119 127

80 88 96 104 112 120 128

81 89 97 105 113 121 129

442 MD Demonstration Problems CHAPTER 24

The first rigid body is a half cylinder described as a NURBS surface and will be used to clamp the grids on the inner radius of the pulley. Its body ID number is 3 and it is identified as: BCBODY

3 0 RIGID NURBS

3D 0.

RIGID 0.

0 0. 1. 0. 0 1 RIG-INNER -7 13 4 4 50 .176777 -.176777 0. .324015 -.029538 .237263 .222631 0. .0306021.24812

1 0.

0 0.

50 0. 0.

0

...

The second and the third rigid bodies are load controlled rigid bodies. A load controlled rigid body is associated with a control grid, which can be used to apply forces and/or single point constraints. In the current analysis, two flat load controlled rigid bodies are used. They will be glued to both ends of the belt and their control grids will be used to prevent a rigid body motion in the basic z-direction, to apply the external force on the belt and to transfer the belt load to the fixed control grid. The load controlled rigid bodies are identified as: BCBODY

... BCBODY

4 0 RIGID NURBS

3D 0. 526 -2 -.2 -.2

5 0 RIGID NURBS

3D 0. 527 -2 .55 .55

RIGID 0. 1 2 .6 .6 RIGID 0. 1 2 -.2 -.2

0. RIG-R 2 .05 .25 0. RIG-F 2 .05 .25

0 1.

0.

1 0.

50 .55 .55

50 .05 .25

0 1.

0.

1 0.

2 .6 .6

50 -.2 -.2

50 .05 .25

2 -.2 -.2

526 0. 4

527 0. 4

...

Note that the control grids have the IDs 526 and 527. The BCTABLE option will be used to indicate: • which grids are to be treated as slave nodes and which as master grids in the multipoint constraints for deformable-deformable contact; • the friction coefficient between the belt and the pulley; • glued contact between the pulley and the half cylinder; • glued contact between the load controlled rigid bodies and the belt. The entries of the BCTABLE option are defined as: BCTABLE

1 SLAVE MASTERS SLAVE MASTERS SLAVE MASTERS SLAVE MASTERS

1 1 2 1 0 5 1 0 4 2 0 3

0. 0

4 0.

.05

0.

0

0.

0. 1

0. 0

0.

0.

1

0.

0. 1

0. 0

0.

0.

1

0.

0. 1

0. 0

0.

0.

1

0.

CHAPTER 24 443 Friction Between Belt and Pulley

The first SLAVE MASTERS combination indicates that the grids of deformable body 1 are treated as slave grids when contact is established with body 2. The friction coefficient is set to 0.05. The other SLAVE MASTERS combinations activate glued contact between the bodies with body ID numbers 1 and 5, 1 and 4, and 2 and 3, respectively. The bilinear Coulomb friction model will be activated using the BCPARA option (FTYPE = 6); this option is also used to indicate that the separation behavior is based on stresses (IBSEP = 4), which is necessary in a contact analysis involving quadratic elements: BCPARA

0

NBODIES 5

IBSEP

4

FTYPE

6

In order to activate the full nonlinear formulation of the 20 node hexahedral elements, the nonlinear property extension of the PSOLID entry is used. For the materials defining the belt (material ID number 1) and the pulley (material ID number 2), this results in: MAT1 MAT1 PSOLID PSLDN1 PSOLID PSLDN1

1 2 1 1 2 2

1.+9 1.+13 1

.3 .3 0

2

0

1. 1.

The nonlinear procedure used is: NLPARM

1 1.e-4

1 1.e-4

1.e-4

FNT 10

1

25

UPW

YES

Here the FNT option is selected to update the stiffness matrix during every recycle using the full Newton-Raphson iteration strategy. Convergence checking is performed based on displacements, forces and work. The error tolerance is set to 10-4 for all criteria. Note that the MAXDIV field is set to 10 to avoid that bisections occur, since too many bisections may increase the overall solution time. The obtained values of the reaction forces are listed in Table 24-2, together with the relative error compared to the analytical solution. The numerical and analytical solutions turn out to be in good agreement. Table 24-2

Numerical Solutions and Relative Errors

Friction Coefficient 

Reaction Force R

Error (%)

0.05

9.2314x104

0.14

0.15

7.9476x104

0.59

0.25

6.8448x104

1.37

444 MD Demonstration Problems CHAPTER 24

Modeling Tip Convergence Behavior A nonlinear analysis involving contact and friction may need several iterations to fulfil the convergence requirements. In such inherently nonlinear analyses, it may be advantageous to increase the number of criteria needed to force a bisection. As discussed above, this number (MAXDIV on the NLPARM option) has been set to 10 instead of the default value 3. The tables below show the convergence behavior with the increased value (Table 24-3) and the default value (Table 24-4). The increased value clearly reduces the overall number of Newton-Raphson iterations and thus the analysis wall time. When looking at Table 24-3, iteration 9 reaches displacement, load and work errors which are within the required tolerances. The extra iterations needed are caused by the fact that some grids of the belt which are initially in contact with the pulley, separate because of tensile contact stresses. After separation of these grids, a new solution with a smaller number of contact constraints has to be found. Table 24-3

Convergence Behavior with MAXDIV=10 (

Load Factor

Step

Iteration

Disp. Error

Load Error

Work Error

1.000

1

1

1.00E+00

1.70E-01

1.70E-01

1.000

1

2

7.76E+00

3.54E-01

1.58E+00

1.000

1

3

6.61E+02

2.31E+01

6.17E+02

1.000

1

4

2.12E+02

1.80E+02

1.30E+04

1.000

1

5

8.61E-02

2.78E+01

7.33E+00

1.000

1

6

3.12E-03

1.70E-01

4.67E-02

1.000

1

7

2.60E-04

4.31E-03

3.50E-03

1.000

1

8

7.87E-06

4.09E-05

1.34E-04

1.000

1

9

3.92E-06

9.30E-07

5.09E-05

1.000

1

10

3.39E+00

1.41E-02

4.30E+00

1.000

1

11

4.26E-02

2.05E-03

6.67E-01

1.000

1

12

2.42E-03

3.31E-02

3.33E-02

1.000

1

13

8.19E-06

2.26E-05

1.30E-04

1.000

1

14

4.93E-06

1.61E-06

6.57E-05

CHAPTER 24 445 Friction Between Belt and Pulley

Table 24-4

Convergence Behavior with MAXDIV=3 (

Load Factor

Step

Iteration

Disp. Error

Load Error

Work Error

1.0000

1

1

1.00E+00

1.70E-01

1.70E-01

1.0000

1

2

7.76E+00

3.54E-01

1.58E+00

1.0000

1

3

6.61E+02

2.31E+01

6.17E+02

1.0000

1

4

2.12E+02

1.80E+02

1.30E+04

0.5000

1

1

1.00E+00

9.36E-02

9.36E-02

0.5000

1

2

8.06E+02

2.96E-01

3.12E+02

0.5000

1

3

5.62E+02

3.36E+01

6.19E+02

0.5000

1

4

8.37E+01

8.70E+01

1.92E+02

0.5000

1

5

3.27E-02

1.91E+00

8.84E-02

0.5000

1

6

8.88E-04

2.22E-02

2.19E-03

0.5000

1

7

1.27E-04

2.24E-04

2.84E-04

0.5000

1

8

2.93E-06

6.83E-06

8.15E-06

0.5000

1

9

1.94E+00

1.02E-02

2.71E-01

0.5000

1

10

2.89E-02

1.31E-03

6.47E-02

0.5000

1

11

3.25E-04

7.79E-03

5.95E-04

0.5000

1

12

2.44E-05

8.00E-06

5.31E-05

1.0000

2

1

5.60E-01

2.26E-01

1.27E-01

1.0000

2

2

1.25E+02

2.32E+02

7.04E+03

0.7500

2

1

1.25E+02

2.32E+02

7.04E+03

0.6250

2

1

1.25E+02

2.32E+02

7.04E+03

0.5625

2

1

1.25E+02

2.32E+02

7.04E+03

0.5312

2

1

3.86E-01

6.06E-01

3.32E-01

...

...

...

...

...

...

...

...

...

...

...

...

0.9688

16

3

4.10E-03

1.92E-02

6.62E-03

0.9688

16

4

7.84E-05

4.16E-04

1.37E-04

0.9688

16

5

9.70E-06

4.13E-06

1.67E-05

1.0000

17

1

3.58E-02

5.91E-03

2.16E-04

1.0000

17

2

4.49E+00

7.24E-01

6.56E+00

1.0000

17

3

3.37E-03

1.27E-02

5.40E-03

1.0000

17

4

6.27E-05

2.93E-04

1.08E-04

1.0000

17

5

7.94E-06

2.83E-06

1.34E-05

446 MD Demonstration Problems CHAPTER 24

Input File(s) File

Description

nug_24_1.dat

Friction coefficient 0.05

nug_24_2.dat

Friction coefficient 0.15

nug_24_3.dat

Friction coefficient 0.25

Video Click on the image or caption below to view a streaming video of this problem; it lasts about 25 minutes and explains how the steps are performed.

Figure 24-2

Video of the steps above