Static and dynamic characteristics of machine components in contact

Applied Acoustics 23 (1988) 1-16

Static and Dynamic Characteristics of Machine Components in Contact M. P. C h u n g , a A. C. S i n g h a l b a n d W. H. C h e n a a

Department of Mechanical Engineering, National Central University, Chungli (Taiwan) h Department of Civil Engineering, Arizona State University, Tempe, Arizona 85287 (USA)

(Received 18 November 1986; revised version received 18 February 1987; accepted 20 February 1987)

SUMMA R Y Static and dynamic characteristics of a machine joint are studied. A simple model of a beam on an elastic foundation and an iterative technique are used to simulate the non-linear behaviour of the static stiffness of the machine joint. Closed-form shape functions for a beam on elastic foundations are obtained by solving the governing differential equation. Numerical results demonstrate that there is always an optimum surface finish condition for the contact surfaces of the machine joint. Analytical values have been correlated with the experimental data.

NOTATION c m E I Kij k Mij Ni(x) n

Constant relating normal compliance and normal pressure Mass per unit length of beam Young's modulus (N c m - 2) M o m e n t of inertia Coefficient of stiffness matrix Static stiffness of contact surface per unit area Coefficient of mass matrix Shape function Constant relating normal compliance and normal pressure 1

Applied Acoustics 0003-682X/88/$03-50 © ElsevierApplied SciencePublishers Ltd, England, 1988. Printed in Great Britain

2

p Pi

M . P . Chung, A. C. S&ghal, W. H. Chen

Excitation force Local coordinate forces k)/EI

fl

= (mo9 2 --

6i 2 O(x)

Displacement in local coordinate Deflection normal to interface Mode shape Frequency of harmonic boundary displacement

INTRODUCTION There has been considerable research effort to establish the static and dynamic characteristics of fixed joints and sliding joints in the case of slideways. Burdekin, Back and Cowley 1 indicated that much of the work was generally concerned with the normal compliance of contacting surfaces and its dependence on surface topography, hardness, contact stresses and material properties. Back, Burdekin and Cowley 2 proposed three methods to determine the contact stress and the normal compliance along the contact surfaces. These three methods are the hydrostatic method, the plate method and the spring method. An empirical equation for representing the nonlinear load-deflection relationships has been used. This equation is given a s 1,2

2 = ce"

(1)

where 2 is the normal deflection (pm), and P is the interface pressure (kgfcm- 2 = 9-8 N c m - 2). The coefficients c and n are constants for the pair of material and surface finish. Oden and Martins 3 determined the elastic stiffness, k, as k = p(l-")/cn

(2)

Thornley et al. 4 proposed an equivalent solid line to represent the static stiffness in the unloading range. In general, the strain-induced friction forces along the contact surfaces can further modify the final pressure distribution and the whole deformation. Efforts are currently under way to gather adequate technical information on the elastomeric friction behaviour of slideways. Singhal and Meng s obtained a closed-form solution for a beam on elastic foundation. In this study, a simple model of a beam on elastic foundations and an iterative technique are used to simulate the non-linear behaviour of the slideways under normal loads. Shape functions for a beam on an elastic foundation have been obtained by solving the governing differential equation. Stiffness and mass matrices are then obtained by using Lagrange's

Machine components in contact

3

equations. The iterative processes have been used to determine the compliances and the contact stresses?-4,6 The results of this study are especially useful in the design of slideways. The shape functions, the stiffness matrices, and the mass matrices can be used to investigate the static and dynamic characteristics of the machine joints. Dynamic responses of machine c o m p o n e n t s are obtained by including the static shape functions that are usually ignored in other analytical methods. 6 The displacements of a machine joint due to normal loads can be minimized by the suitable selection of an appropriate surface finish condition.

DERIVATION OF DYNAMIC SHAPE FUNCTIONS The governing differential equation of m o t i o n for a beam on an elastic foundation is

~4y ~2y + ky = 0 EI~x4 + m-~

(3)

where Elis the rigidity of the beam, m is the mass per unit length of the beam, and k is the foundation modulus or elastic stiffness. For harmonic boundary displacements of frequency 09, the trial solution of eqn (3) is

y(x, t) = ~(x) sin cot

(4)

Substitution of eqn (4) into eqn (3) yields d4~

EI~gx4 - (m~o2 - k)~ = 0

(5)

or

d4O dx 4

flO = 0

(6)

where fl = ~ 4 = mo92_ k. Boundary conditions, as shown in Fig. l(a), are written as • (0) = 6,, • (L) = 63,

~'(0) = 62~ O'(L) 6 4

S

(8)

Depending upon the a m o u n t of the inertia force in the beam, mco2~, and the foundation force, k~, eqn (5) can be solved for three different cases: (1) fl = 0, the inertia force is equal to the foundation force; (2) fl > 0, the inertia force is greater than the foundation force; (3) fl < 0, the inertia force is smaller than the foundation force.

4

M . P . Chung, A. C. Singhal, W. H. Chen

V [~=

L ~_~_

dX~ _.

P3,s3

v

(a)

f

KYdX

(b) Fig. 1.

(a) F o r c e a n d d i s p l a c e m e n t n o t a t i o n f o r a b e a m o n an elastic f o u n d a t i o n . (b) F o r c e s in a b e a m element.

C a s e 1: fl = ~4

= (m~o 2

_

_

k)/El = 0

Equation (6) becomes d4do/dx 4 = 0

(9)

dO(x) = A x 3 + B x 2 + C x + D

(10)

The solution of eqn (9) is

Dynamic coefficients A, B, C and D are determined by substituting eqn (8) into eqn (10). After the process of rearrangement, an alternative form of eqn (10) can be written as do(x) = Nl(x)61 + N2(x)62 + N3(x)63 + N4(x)64

(11)

where Ni(x ) is the dynamic shape function 6 and

C) 2

N2(x) = x ~ - 1

N3(x ) = - 2

N4(x) C a s e 2: fl = ~4

=

(mo92 _

=x2(x--

L \L

(13)

+3

(14)

1)

(15)

k)/EI > 0

Equation (6) becomes d40 ~dx - ~40 = 0

(16)

Machine components in contact

5

T h e s o l u t i o n o f e q n (16) is (l)(x) = A sin c~x + B c o s ~ x + C s i n h ~ x + D c o s h ~ x

(17)

D y n a m i c c o e f f i c i e n t s A, B, C a n d D a r e d e t e r m i n e d b y s u b s t i t u t i n g e q n (8) i n t o e q n (17). A f t e r t h e p r o c e s s o f r e a r r a n g e m e n t , a n a l t e r n a t i v e f o r m o f e q n (17) c a n b e w r i t t e n a s

• (x) = Nl(x)61 + N2(x)6 2 + N3(x)~ 3 -k- N4(x)6 4

(18)

where

1

Ni(x)= ~(aisin~x + bicosex + cisinhex + dicoshex )

(19)

G = - 2 + 2 cos ~L cosh ~L

(20)

a l = sin ~ L c o s h ~ L + s i n h ~ L c o s ~ L

(21)

az = sin ~ L s i n h ~ L + c o s h ~ L c o s c~L - 1

(22)

a 3 = - - sin ~ L - - s i n h ~ L

(23)

a 4 = - cos ~L + cosh ~L

(24)

b l = c o s ~ L c o s h ~ L - sin ~ L s i n h ~ L - 1

(25)

b 2 = c o s ~ L s i n h ~ L - sin ~ L c o s h ~ L

(26)

b 3 = -cos~L

(27)

+ cosh ~L

b,, = sin ~ L - s i n h ~ L

(28)

c l = - sin ~ L c o s h ~ L - s i n h ~ L c o s ~ L

(29)

cz = c o s ~ L c o s h ~ L - sin ~ L s i n h ~ L - 1

(30)

c3 = s i n ~ L + s i n h ~ L

(31)

c4 = c o s ~ L - c o s h ~ L

(32)

d 1 = cos ~L cosh ~L + sin~L sinh ~L - 1

(33)

d 2 = -cos

(34)

~ L s i n h c~L + sin ~ L c o s h ~ L

d3 = c o s ~ L - c o s h ~ L

(35)

d4 = - sin ~ L + s i n h ~ L

(36)

Case 3: fl = ~'* = ( m e 9 z - k ) / E I < 0 Equation

(6) b e c o m e s d4(I) dx 4 + ~40 = 0

(37)

T h e s o l u t i o n o f e q n (37) is O(x) = e'X(A c o s ~ x + B sin

~x) + e- "x(c c o s ~ x + D sin ~x)

(38)

6

M . P . Chung, A. C. Singhal, IV. H. Chen

D y n a m i c coefficients A, B, C a n d D are d e t e r m i n e d b y s u b s t i t u t i n g e q n (8) i n t o e q n (38). A f t e r the p r o c e s s o f r e a r r a n g e m e n t , a n a l t e r n a t i v e f o r m o f e q n (38) c a n be w r i t t e n as (1)(X) : N I ( X ) ~ 1 + N 2 ( x ) ~ 2 + N 3 ( x ) ~ 3 + N,,(x)(54

(39)

where

Ni(x ) =

1

1 e~X(ai c o s e x + b i sin ex) + ~ e - ~X(ci c o s e x + di sin ex)

(40)

H = - e 2~c + e - 2~L + 2 + 4 sin 2 e L

(41)

a 1 = 1 + 2 sin 2 e L + sin 2 e L -- e -2~L

(42)

2 a 2 = - sin 2 e L

(43)

e

aa = e~L( - - c o s e L - sin e L ) + e -~L(cos e L - sin eL)

(44)

a4 = 1 (e~L sin e L - e - ~L sin e L )

(45)

e

b 1 = e - 2~L _ sin 2 e L - c o s 2 e L

(46)

b2 = - 1 ( _ e _ 2 ~ L _ s i n 2 e L + 1)

(47)

e

b a = e~L(cos e L - sin e L ) + e - ~ L ( - c o s e L + 3 sin eL)

(48)

b4 = 1 ( _ e~ L c o s e L + e - ~L(2 sin e L ,t- c o s eL))

(49)

e

c 1 = - e 2~L+ 2 s i n 2 e L c2 = - l s i n 2

sin2eL + 1

eL

(50) (51)

e

c3 = e~L(cos e L + sin eL) + e - ~ L ( - c o s e L + sin e L )

1

(52)

c4 = - ( -- e ~L sin e L + e - ~L sin e L ) c~

(53)

d~ = - e 2~L + c o s 2 e L -- sin e L

(54)

d2 = l ( _ e 2 ~ L + 1 + s i n 2 e L )

(55)

e

d 3 = e~L(cos e L + 3 sin e L ) + e - ~ L ( - c o s e L - sin e L )

(56)

d4 = 1 (e~L(cos e L - 2 sin e L ) - e - ~L c o s eL)

(57)

e

Machine components in contact

7

As indicated in Case 1, the determined d y n a m i c shape functions are the same as the static shape function for an axial loaded b e a m in air. In Case 2, by m a k i n g the term o f the f o u n d a t i o n m o d u l u s zero, the determined d y n a m i c shape functions are the same as the d y n a m i c shape functions for an axially loaded b e a m in air. Therefore, the static shape functions for an axially loaded b e a m in air are used in this case. In Case 3, the determined d y n a m i c shape functions are the same as the static shape functions for an axially loaded b e a m on an elastic f o u n d a t i o n if the frequency o f the h a r m o n i c b o u n d a r y displacements reduces to zero. 6 ANALYTICAL

MODEL

Figure 1 shows a b e a m on an elastic f o u n d a t i o n subjected to bending moments, P2 a n d / ' 4 , and shear forces, P1 and P3" These forces can be related to the c o r r e s p o n d i n g displacements, 61, 62, 63, 64, and accelerations, ~i"1, 6"2, ~i'3, ~'4, by the following discrete e q u a t i o n o f motion: [ M ] { ~ ) + [K]{6} = {P}

(58)

The coefficients o f the stiffness matrix, Ki~, are determined f r o m the static shape functions, Ni and Nj, as

Kit = E1 1"~N:'(x)Nj'(x)dx + k 1"~Ni(x)Nj(x ) d x .)o Jo

(59)

TABLE 1

Static Shape Functions 6

k NI(x )

k~

> mto 2

•,1

~-[e~X(alcos~tx + b I sin ~x)

too) 2

2

(t) 3- 3 (t) 2+ 1 - 2

+ e-~X(c1cos ~x + d I sin ~tx)] N:(x)

z-:_[e~X(azcos ~x + b2 sin ctx) /4 + e- ~X(c2 cos ~x + d 2 sin ~tx)]

x

N3(X)

77[e~X(aacos~x+b3sino~x)/4 + e- ~X(c3 cos ~tx + d 3 sin otx)]

-2

N4(x)

l[e~X(aaCOS~tx+b4sinctx) + e-~X(c4 cos ~tx + d4 sin ~tx)]

X[(L)2 - ( L ) ]

'

Expressions for H, a i, and bl are listed in Table 4.

+ 1

+3 L

(7

8

M.P. Chung, A. C. Singhal, W. H. Chen

TABLE

2

Coefficients 6 of Mass Matrix, M~j k > moo2

k ~
m

[aiajG, + a2-{aibj+ ajb~)G2 + b~biG3 + (aicj + ajcl)G4 + ~(aid~+ aidj + biQ

M~j

+ b/'i)Gs + (bidj + bjbi)G 6 + c f f i 7 + ~2(cidj + cJi)Gs + d,djag] i , j = l, 2, 3, 4 1

G~

~ [e2~L(2 + cos 2~L + sin 2~L) - 3]

G2

~[e2~L(sin2~L-cos2~L)+ 1]

G3

~[e2"L(2-cos2~L-sin2~L)-1]

G4

-I + - -1s i n 2 ~ L 2 4fl

1

!

I 156 22L

4~

[

54 -13L

for all cases

1

G5

~(-cos

G6

- - - - sin 2~L 2 4~

G7

~[e - 2"Llsirl2~L -- COS2~xL - 2) + 3]

G8

- - [e- 2~L(- sin 2o~L - cos 2o~L + 1)] 4~

G9

8~[e 2~L(cos2~L-sin2~tL-2)+l]

1

mL/

2ctL + 1)

1

l

I

22L 4L 2

54 13L

-I3L] -3L2 /

13L

156

-22L /

-3L 2 -22L

4L 2 J

Machine components in contact

9

TABLE 3 Coefficients 6 of Stiffness Matrix, Kij

k> K+j

k < mo~2

mto 2

1

~ [Rita ~+ Rj2bj + gi3c j + Riadj]

Rtl

--2ct3El

Rt2 Rt 3 R1 +

2~t3El 2ot3Ei _2~t3El

R21 R22

0

R23 R24 R3 ~

0 2~t2El 2~t3Ele~L(cosctL+sin~tL)

[- 12 E1 [ 6L L3 -12 6L

_2~2EI ~,

6L 4L 2 --6L 2L 3

-12 -6L 12 -6L

6L ] 2L:

I -6L[ 4L 2 J

+

R32

2~t3Ele ~L(--cos~tL + sin~L)

R33 R34

2otaEle, L(_cosotL+sinotL) 2~3Ele_~L(_cosotL_sin~tL)

[ 156 22L [ 420 [ 54 [-13L

R41

-2~2Ele~LsinotL

for all cases

R42 R43

2~2Ele "Lcos ctL

R4+

--:t2Ele-'Lcos~tL

kL

I

22L 4L 2

54 13L

13L 156 - 3 L 2 -22L

-13L] ~

3L 2

/

[ -22L / 4L 2 J

2~t2Ele-~LsinctL

where Elis the rigidity of the beam, and k is the static stiffness of the contact surface per unit area. The coefficient of mass matrix, Mij, is determined as

Mij = j~L mNi(x)Nj(x) dx

(60)

where m is the mass per unit length of beam. The selection of static shape functions in eqns (59) and (60) depends on whether the inertia force in a beam is greater than, equal to, or smaller than the foundation force. 6 If the inertia force is greater than or equal to the foundation force, the static shape functions selected are the same as those of a beam in air. In the case where the inertia force is smaller than the foundation force, the static shape functions selected are the same as those of a beam on an elastic foundation. Static shape functions are listed in Table 1. Coefficients of mass matrices and stiffness matrices are listed in Tables 2 and 3, respectively.

M. P. Chung, A. C. Singhal, W. H. Chen

10

TABLE 4 E x p r e s s i o n s 6 for H, ai, a n d b i

(K/4EI) I/4 H

- (e z~L + e - :,z) + (2 + 4 sin 2 ~tL)

a~

1 + 2 sin 2 ctL + sin 2ctL - e - 2~L

a2

2 - sin 2 ~tL ~t

a3

e'L(-cos~tL-sin~L)+e-'L(cos~L-sin~tL) 1

a,

- ( e "L sin ctL - e ,L sin ctL) Ct

bt

e 2~L_ s i n 2 ~ L - c o s 2 ~ L

b2

-(-c-2"L-

b3

e'L(cos~L-sin~L)+e-'L(-cos~L

b4

-[--e~Lcos~tL+e-~L(2sinctL+cos~tL)] ~t

c~

- - e 2"L + 2 sin 2 ~tL - sin 2~tL + 1

c2

--

c~

c'L(cos ~L + sin ~tL) + e-~L( --COS ctL + sin ~tL)

c,

1 - (-e~Lsin~L +e-~£sin~L)

1 sin 2~L + 1)

+ 3sin~L)

I

1

d1

-

sin 2 ~L

e 2~L

+ COS 2~L -- sin ~L

1 d2

- [ - e 2 a L + (1 + sin 2~L)]

d3

&t'(cos~L+3sin~L)+e-'L(-cos~L-sin~L) 1

d,

JelL(cos ~tL - 2 sin ~L) - c ~L cos ~L]

Machine components in contact

11

N U M E R I C A L RESULTS In Fig. 2, the analytical values have been compared with experimental data and results obtained from the hydrostatic, plate and spring methods. 2 The analytical values compare well with the experimental data. Figures 3-6 show the effects of the thickness and the length on the compliances and the contact pressures. Figures 7 and 8 show the dynamic responses and the corresponding contact pressures for the contacting joints. Figures 9 and l0 show the effects of constant c on the compliances and contact pressures. When roughness of the contact surfaces increases, the compliances increase and the contact pressures decrease. There is always an optimum value of the constant c to minimize the compliance and the contact pressure. The optimal value of the constant c is about 0-9 for the proposed structure.

F=20Kgf

. . ~

~, ::t

--

THIS STUDy •

L_-,r..jOOmm t =120mm

"" *

x

METHOD PLATE METHOD SPRING METHOD EXPERIMENTAL RESULTS (1)

HYDROSTATIC

< 0.. c~l -

~m"~0

x

x

ll4gf =9.g N

x

125 250 37'5 500 DISTANCE ALONG BEAM (ram) (a)

F=10I~gf L ~ J t=120mm L=500mm E ZO'LLI uJ

~cu

x

ling f=9"8N

715

DISTANCE ALONG BEAM (ram) (b) Fig. 2.

Comparison between numerical results and experimental results.

M. P. Chung, A. C. Singhal, W. H. Chen

12

F=!2Kgf ~L~ -~

A - t = 20mm B - t =40mm C - t =60ram

r'~-. I

~

,,-I

/

F.- c")-

(...)

.

~

.

r°0 Fig. 3.

5() I00 150 2(~ 2,~ 300 DISTANCE ALONG BEAM

Effect of thickness on displacement of machine componen~

~i

A -t = 2Omm B -t =40mm C - t =60mm

O0mm A

~_ t)O'D

t.r) "~" if)el klJ

~.)

f =9.8N

o

50

100 150 200 250 300 DISTANCE ALONG BEAM (mm~

Fig. 4. Effect of thickness on contact stress of machine component.

13

Machine components in contact

.... L : 200ram ---- L= 300ram

F:I2 Kgf E

L= 400mm

~I_

\

//

'~ L ~;

'\

--~J

,

5~,-

go

i

75 150 225 3~)0 DISTANCE ALONG B E A M (turn)

Fig. 5. Effectof length on displacement of machine component,

F=I2 F,g#

- - - L= 200ram

L= Z~mm

Eo. .u--*

~

). W

U') 0 " ~

'

~Q)

~\~ 1KgI=9.SN o

7's

150

2'25

3bo

DISTANCE ALONG B E A M ( m m )

Fig. 6. Effectof length on contact stress of machine component.

M. P. Chung, A. C. Singhal, W. H. Chen

14

~]I

c>_ 04 ~t ~o

"

-4 300ram

E ~ ~

F=(50,30SIN2/Ift) Kgf f =100 Hz o C=0.4 n=0.5 A c=0.6 + c = 0.8

]~I=9.SN °0.0

'

0'.6

'

1'2

'

118

'

2.'4

'

3'0

'

3'.6

'

TIME (x 10-2 SEC) Fig. 7.

Dynamic response at centre of machine component.

~ ~ ~ ~

° 1

~

F =(50÷30SIN2nft) Fgf f = 100 HZ n=0.5

Q) Z O r •

o .

8t

0.0 Fig. 8.

0'.6

1'2 1'.8 2~, TIME ( x l 0 -2 SEC)

c =0.4 c=O.6

+ c =0.8 1Kgf =9.8N 3.0 3.6

Contact stress at centre of machine component.

Machine components in contact

15

ol I Kgf = 9.8 N 300mm d6,0

' 0.4

' 0'.8 '

SURFACE

I'.2 '

PARAMETER

1.6 ' 2~3 C VALUE

Fig. 9. Optimal surface finish condition in static case.

oO

O-

.M~ma O

O-

"

1Kgf-- 9.8N °° 0.0

0

08

SURFACE

Fig. !0.

~O0m; i

PARAMETER

1.6

20

C VALUE

Optimal surface finish condition in dynamic case.

CONCLUSION A method for the design of machine components in contact has been developed. Dynamic responses are obtained by including the static shape functions that are usually ignored in other analytical methods. The closed-form shape functions for a beam on an elastic foundation have been obtained by solving the governing differential equation. Stiffness and mass matrices are obtained by using Lagrange's equations.

16

M. P. Chung, A. C. Singhal, W. H. Chen

Several configurations have been examined for surface roughness effects, length and thickness effects. When roughness of the contact surfaces increases, the normal compliance increases and the contact stress decreases. There is always an optimal value of the surface parameter, c, to minimize the normal compliance and the contact stress. The optimal value of the surface parameter, c, is 0"9 for the proposed structure.

REFERENCES 1. M. Burdekin, N. Back and A. Cowley, Experimental study of normal and shear characteristics of machined surfaces in contact, J. Mech. Engng Sci., 20 (3) (1978), pp. 129-32. 2. N. Back, M. Burdekin and A. Cowley, Pressure distribution and deformations of machined components in contact, J. Mech. Engng Sci., 15 (1973), pp. 993 I010. 3. J.T. Oden and J. A. C. Martins, Models and computational methods for dynamic friction phenomena, Comput. Meth. Appl. Mech. Engng, 52 (1985), pp. 527-634. 4. R. H. Thornley, R. Connolly, M. M. Barash and F. Koenigsberger, The effect of surface topography upon the static stiffness of machine tool joints, Int. J. Machine Tool Design Res., 15 (1) (1965), pp. 57-74. 5. A. C. Singhal and C. L. Meng, Junction stresses in buried jointed pipelines, J. Transport. Engng, ASCE, 109 (3) (1983), pp. 450 61. 6. M. P. Chung, Seismic analysis of buried pipeline networks, Thesis presented to Arizona State University, in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 1984.