Friction and Lubrication in Mechanical Design Episode 1 Part 3 docx

R I , R2 = radii of cylinders positive when convex and negative when concave E, El + g - = - El, E2 = modulus of elasticity for the two materials Case 7: General Case of Contact Betw

Figure 2.6 Two cylindrical bodies in contact

R I , R2 = radii of cylinders (positive when convex and negative when concave)

E, El + g

- = -

El, E2 = modulus of elasticity for the two materials

Case 7: General Case of Contact Between Elastic Bodies with

Continuous and Smooth Surfaces at the Contact Zone

Analysis of this case by Hertz can be found in Refs 1 and 2 A diagrammatic representation of this problem is shown in Fig 2.7 and the contact area is expected to assume an ellipitcal shape Assuming that ( R I , R;) and (R2, R;)

are the principal radii of curvature at the point of contact for the two bodies respectively, and $ is the angle between the planes of principle curvature for the two surfaces containing the curvatures l/R1 and l/R2, the curvature

consants A and B can be calculated from:

These expressions can be used to calculate the contact parameter P from the relations hip:

The Contact Between Smooth Surfaces 31

Figure 2.7 General case of contact

u l , u2 = Poisson’s ratios for the two materials

E l , E2 = corresponding modulii of elasticity

Case 8: Beams on Elastic Foundation

The general equation describing the elastic curve of the beam is:

8 degrees Figure 2.8 Elliptical contact coefficients

where

k = foundation stiffness per unit length

E = modulus of elasticity of beam material

1 = moment of inertia of the beam

With the notation:

the general sohtion for beam deflection can be represented by:

where A , B, C and D are integration constants which must be determined from boundary conditions

For relatively short beams with length smaller than (0.6/8), the beam

can be considered rigid because the deflection from bending is negligible compared to the deflection of the foundation In this case the deflection will be constant and is:

The Contact Between Smooth Surfaces 33

For relatively long beams with length greater than (5//?) the deflection

will have a wave form with gradually diminishing amplitudes The general solution can be found in texts on advanced strength of materials

Table 2.2 lists expressions for deflection y , slope 8, bending moment A4

and shearing force V for long beams loaded at the center

Case 9: Pressure Distribution Between Rectangular Elastic Bars in

Contact

The determination of the pressure distribution between two bars subjected

to concentrated transverse loads on their free boundaries is a common problem in the design of mechanical assemblies This section presents an approximate solution with an empirical linear model for the local surface contact deformation The solution is based on an analytical and a photo- elastic study [18] A diagrammatic representation of this problem is shown

Table 2.2 Beam on Flexible Supports

in Fig 2.9a The problem is approximately treated as two beams on an elastic foundation, as shown in Fig 2.9b The equations describing the system are:

where

qX = load intensity distribution at the interface (lb/in.)

ZI , = moments of inertia of beam cross-sections

The Contact Between Smooth Surfaces 35

Figure 2.9b Simplified model for two beams in contact

The criterion for contact requires, in the absence of initial separations, the total elastic deflection to be equal to the rigid body approach at all points of contact, therefore:

where a is the rigid body approach defining the compliance of the entire

joint between the points where the loads are applied:

The continuity of force at the interface yields:

Equations (2.1)-(2.4) are a system of four equations in four unknowns This system is now reduced to a single differential equation as follows Adding Eqs (2.1) and (2.2) gives:

Substituting Eq (2.4) into Eq (2.5) yields:

The combination of Eqs (2.3) and (2.4) gives:

Substituting Eq (2.7) into Eq (2.6) yields the governing differential

equation:

where Kc, is an effective stiffness:

With the following notation:

Eq (2.10) may now be rewritten as:

d"

- + 4$" = 4$a d,r4

(2.10)

(2.1 I )

The following are the boundary conditions which the solution of (2.1 I) must

satisfy, provided L 2 C, where L is half the length of the pressure zone: The beam deflections are zero at the center location

The slope is zero at the center

The summation of the interface pressure equals the applied load The pressure at the end of the pressure zone is zero

The moment at the end of the pressure zone is zero

The shear force at the end of the pressure zone is zero

The six unknowns to be determined by the above boundary conditions are the four arbitrary constants of the complementary solution, the rigid body approach a, and the effective half-length of contact C The four constants

The Contact Between Smooth Surfaces 37

and the rigid body approach are determined as a function of the parameter

R = /3t A plot of the rigid body approach versus A is shown in Fig 2.10 At

A = n/2, the slope of the curve is zero For values of R greater than n/2, the values of z1 and z2 become negative, which is not permitted The maximum permitted values of il is then n/2 and the effective half-length of contact is

t = n/(2/3) If n / ( 2 p ) is greater than L, the effective length is then 2L

The expression for the load intensity at the interface is:

(sinh 2;1+ sin 2A)

- sinh Bx cos Px + cosh Bx sin Bx

The normalized load intensity versus position is shown in Fig 2.1 1 for

ge = n/2 This figure represents a generalized dimensionless pressure

distribution for cases where t << L

Figure 2.1 1 Normalized load intensity over contact region

The assumption of a constant contact stiffness can be considered ade- quate as long as the theoretical contact length is far from the ends of the beams For cases where t approaches L, it is expected that the compliance as well as the stress distribution would be influenced by the free boundary As a result, it is expected that the actual pressure distribution would deviate from the theoretical distribution based on constant contact stiffness A proposed model for treating such conditions is given in the following The approx- imate model gives a relatively simple general method for determining the contact pressure distributions between beams of different depths which is in general agreement with experimental pho t oelas t ic investigations

In the model the contact half-length t is calculated from the geometry of the beam according to the formula:

When t << L, the true half-length of contact is equal to t and the corre- sponding pressure is directly calculated from Eq (2.12) or directly evaluated from the dimensionless plot of Fig 2.11

As t approaches L, the effect of the free boundary comes into play and

the constant stiffness model can no longer be justified An empirical method

The Contact Between Smooth Surfaces 39

to deal with the boundary effect for such cases is explained in the following The method can be extended for the cases where C 2 L

Because of the increase in compliance at the boundaries of a finite beam

as the stressed zone approaches it, a fictitious theoretical contact length

l ’ ( e ’ > e) can be assumed to describe a hypothetical contact condition for

equivalent beams with L‘ >> e (according to the empirical relationship given

in Fig 2.12 The pressure distribution for this hypothetical contact condi-

tion is then calculated Because the actual half-length of the beam is L, it

would be expected that the pressure between e’ and L would have to be

carried over the actual length L for equilibrium The redistribution of the pressure outside the physical boundaries of the beam is assumed to follow a mirror image, as shown in Fig 2.13

The superposition of this reflected pressure on the pressure within the boundaries of the beam gives the total pressure distribution

The general procedure cam be summarized as follows:

Calculate l from the equation t = n/(2/3)

Using e and L, find f?’ from Fig 2.12 Notice that for e << L,

ANALYSIS A N D DESIGN OF ELASTIC BODIES IN CONTACT

The general contact problem can be divided into two categories:

Situations where the interest is the evaluation of the contact area, the pressure distribution, and rigid body approach when the system configuration, materials and applied loads are known;

Systems which are to be designed with appropriate surface geometry for the objective of obtaining the best possible distribution of pressure over the contact region

In this section a general formulation is discussed for treating this class of problems using a modified linear programming approach A simplex-type algorithm is utilized for the solution of both the analysis and design situa- tions A detailed treatment of this problem can be found in Refs 18 and 19

The Contact Between Smooth Surfaces 41

2.3.1 The Formulation of the Contact Problem

The contact problems which are analyzed here are restricted to normal surface loading conditions Discrete forces are used to represent distributed pressures over finite areas The following assumptions are made:

1 The deformations are small

2

3 The two bodies obey the laws The surfaces are smooth and have continuous first derivatives of linear elasticity Problem formulation and geometric approximations can therefore be made within the limits of elasticity theory

2.3.2 Condition of Geometric Compatibility

At any point k in the proposed zone of contact (Fig 2.14), the sum of the

elastic deformations and any initial separations must be greater than or equal to the rigid body approach This condition is represented as:

where

&k = initial separation at point k

a = rigid body approach

\ l ’ k ( l ) , kt’k(2) = elastic deformations of the two bodies respectively at point k

Figure 2.14 Zone of contact

2.3.3 Condition of Equilibrium

The sum of all the forces F’ acting at the discrete points (k = 1, , N where

N is the number of candidate points for contact) must balance the applied load (P) normal to the surface The equilibrium condition can therefore be written as:

(2.14)

2.3.4 The Criterion for Contact

At any point k , the left-hand side of the inequality constraint in Eq (2.13)

may be strictly positive or identically zero Defining a slack variable Yk

representing a final separation, the contact problem can be formulated as follows

Find a solution ( F , a, Y ) which satisfies the following constraints:

akj(l), akj(2) = influence coefticients for the deflection of the two bodies respectively

s k j = N x N matrix of influence coefticients

F = N x 1 vector of forces

Y = N x 1 vector of slack variables (or final separation)

e = N x 1 vector of 1’s

E = N x 1 vector of initial separations

= rigid body approach, a scalar

The Contact Between Smooth Suflaces 43

2.4 A GENERAL METHOD OF SOLUTION BY A SIMPLEX-TYPE ALGORITHM

The problem as formulated in Eq (2.15) can be solved using a modification

of the simplex algorithm used in linear programming The changes required

for the modification are minor and are similar to those given by Wolfe [S]

When Eq (2.15) is represented in a tableau form in Table 2.3, the condition for the solution can be stated as:

Find the set of column vectors corresponding to ( F , a, Y ) subject to the

conditions, either F k = 0 or Y k = 0, such that the right-hand side is

a nonnegative linear combination of these column vectors These column vectors are called a basis

For a problem with N discrete points, the number of possible combinations

of these column vectors taken ( N + 1) at a time is:

Because of the very large number of combinations, an efficient method is required for finding the unique, feasible solution The following algorithm proved to be effective for the problem under investigation

The original problem as formulated in Eq (2.15) can be rewritten as:

Subject to the conditions that

Either

where

Z j = artificial variables which are required to be nonnegative (j = 1, , N + 1 )

2 = an N x 1 vector of artificial variables with components Z , , , Z N

-

The above problem can be classified as a linear programming problem [ 131 if

it were not for the condition that either Fk = 0 or Yk = 0 The simplex algorithm for linear programming can, however, be utilized to solve by making a modification of the entry rules

The conditions of Eq (2.15) require some restrictions on the entering

variables Suppose the entering variable is chosen as F, A check must be made to see if the Y, is not in the basis, F, is free to enter the basis The actual replacement of variables is accomplished by an operation

called pivoting This pivot operation consists of N + 1 elementary opera- tions which replace a system by an equivalent system in which a specified variable has a coefficient of unity in one equation and zero elsewhere [ 131

A flow diagram of the modified simplex algorithm is shown in Fig 2.15 Computational experience has shown the simplex-type algorithm to converge to the unique feasible point in at most (3/2)(N+ 1) cycles, the

majrity of cases converge in N + 1 cycles

The simplex-type algorithm for the solution of the contact problem requires less computer storage space when compared to available solution algorithms such as Rosen's gradient projection method [14] or the Frank- Wolfe algorithm [ 151 Only minor modifications of the well-known simplex algorithm are required This algorithm is also readily adaptable to the design problem which is discussed later in this section

EXAMPLE 1 The classical problem of two spheres in contact is consid- ered as an example In this case the influence coefficient matrix S in Eq (2.15) is calculated according to a Boussinesq model as discussed earlier

in this chapter:

The Contact Between Smooth Surfaces

No

Replace the r'*

basic variable by

x, by pivoting on the term a,x,

dk, = distance from point k to pointj in the contact zone

Figure 2.16 shows a comparison between the classical Hertzian pressure distribution and that obtained by the described technique The spheres con- sidered are steel with radii of 1 in and loin., respectively and the applied

load is 1001b The algorithm solution gave a value of 0.000281 in for the rigid body approach which compares favorably with 0.000283 in for the classical Hertz solution