Dynamic analysis of flexible linkages with lubricated joints

Journal of Sound and Vibration (1990) 141(2), 193-205

DYNAMIC

ANALYSIS

OF FLEXIBLE

LUBRICATED T. S. Department

LIU

AND

LINKAGES

WITH

JOINTS Y.

s.

LIN

qf Mechanical Engineering, National Chiao Tung University, Hsinchu 30050, Taiwan, Republic of China (Receiued 22 September 1989)

The dynamic behavior of flexible linkages with lubricated revolute joints is investigated, with consideration of elastic deformation, bearing clearances and hydrodynamic lubrication. An inertial frame based approach to elastic linkage dynamics yields equations of motion in a form amenable to finite element formulations. Non-linear stiffness and damping characteristics are used to obtain impact forces at revolute joints. Pressure caused by hydrodynamic lubrication is calculated by using a finite difference method. Illustrative examples are shown to compare vibratory behaviors of idealized, clearanced, squeeze film lubricated and hydrodynamically lubricated flexible linkages.

1. INTRODUCTION Dynamic analysis of mechanical systems is often conducted under the assumption that links are rigid. Material deformation and damping during motion are routinely ignored, as well as bearing clearance and lubrication. Due to the increasing requirement for high-speed machines and machinery precision, it becomes imperative to treat links in a linkage mechanism as being flexible. In addition, since lubricants are usually interposed between a journal and bearing to minimize friction and wear, the influence of the bearing clearance and lubrication on linkage vibratory behavior is worth studying. Winfrey [l] analyzed the motion of a four-bar linkage by the structural dynamics method. Linear superposition was employed by assuming small link deformation. Imam et al. [2] presented an eigenvalue changing rate method to save computation time. Song and Haug [3] used finite element nodal co-ordinates to generate equations of motion for flexible mechanisms with idealized joints. Simo [4] employed the inertial reference frame to investigate finite strain links that undergo gross motion. Equations of motion ended up in a form more amenable to numerical solution. In these studies [l-4] bearings at linkage joints were idealized: i.e., no clearance and lubrication at joints. In the present study, based on Simo’s formulation, dynamic models are developed for elastic linkages that possess various joint properties. When dealing with linkage deformation, the play due to the bearing clearance, which has the same order of magnitude as the link deformation during motion, cannot be ignored. Dubowsky [5] studied planar and spatial mechanical systems with bearing clearance; however, lubrication effects were not considered in his work. Haines [6] derived equations that describe the conditions at an idealized revolute joint, with clearance but with no lubrication present. Although Rogers [7] considered clearance and lubrication, he treated each link as rigid and ignored hydrodynamic pressures due to wedging actions. In reality, link flexibility, joint clearance and lubrication may interact and affect dynamic performance of linkages significantly. 193 0022-460X/90/170193+

13 $03.00/O

@ 1990 Academic

Press Limited

194

T. S. LIU

AND

Y. S. LIN

The present work not only includes a comparative study on vibratory behaviors among various revolute joint characteristics but also provides an approach to dynamic analysis of elastic linkages with lubricated joints. Four kinds of revolute joint models are treated to compare resultant linkage responses, as follows. (1) Idealized joint-a perfect joint is assumed in which neither clearance nor friction exists and the rotating axes of the journal and bearing remain aligned. (2) Clearance with finite stiffness bearing-two-dimensional clearances lead to surface contact and subsequent impact, and a high-frequency content in the linkage response is anticipated. (3) Lubricated joint with a squeeze film phenomenon-this load-carrying phenomenon arises because a viscous lubricant cannot be instantaneously squeezed out from between two surfaces that are approaching each other; thus, pressure is built up and the load is actually supported by the oil film; if the pulsating or reciprocating loads, as will be shown, are applied for a short enough period, the two surfaces may not meet at all. (4) Joint with hydrodynamic lubrication-many types of machines produce pulsating or reciprocating loads on bearing and bearing surfaces, on which hydrodynamic forces and impact forces take effect in turn; as will be shown, this phenomenon is primarily determined by the relative motion of the journal and bearing under the influence of large gross linkage motion.

2. DYNAMIC BEAM MODEL A beam undergoing gross motion and elastic deformation is illustrated in Figure 1. Unit vectors e, and e, are along inertial frame axes X and Y, respectively. Unit vectors a, and a2 are defined along a floating frame that corresponds to the gross motion of the

-

-

Figure 1. Inertial and floating frames in rotating - -, gross motion; - - -, initial configuration.

beams: X-

Y and a,-+.

-,

X

Deformed

rotating

beam;

beam. Moreover, unit vectors t, and t2 are defined to account for the elastic deformation of the beam, with t2 being parallel to the cross-section of the beam. As shown in Figure 1, the neutral axis and t, do not coincide after deformation because of the shear force effect. Before deformation the position vector of an arbitrary point p in the beam can be written as X,e, + Xze2. After deformation its position vector r is expressed as r=ro+XZt2,

(1)

where r. = (X, + u,)e, + u2e2

(2)

LINKAGE

WITH

LUBRICATED

195

JOINTS

and u, and u2 are components of the elastic deformation in the inertia1 frame. With 8 defined as the rotation angle between tl and e, , one has the co-ordinate transformation

0 1 =

cos sin 8e

e2 6

-sin cos 8

I(1.

(3)

t* t,

The kinetic energy of the beam under investigation is expressed as K = (l/2)

(4)

P(X,, X,)llrll* dX, dX,,

where p(X,, X2) denotes the mass density at (X,, X2). Upon using equations (l)-(3), the kinetic energy becomes K = (l/2)

I

(5)

oL[m(ti,+U,)+I(zj2]dX,,

where m and I represent the mass and moment of inertia in a unit beam length respectively. For a beam with finite strain, the strain, y, can be written in vector form [4] as y=rb-t,=(l+u:-cos fI)e,+(uisin e)e,, where primes denote derivatives with respect to X, , or, in the floating frame, as y = T,tl + rzt2. As a consequence, the strain energy for the beam of interest can be written as &=(1/2) Furthermore,

I0

L[EAT:+~~T:+E1(e’)2]dX,.

(6)

the total potential energy is given by (7)

n = flS -K,,,

where 27,,, is the potential caused by external forces. Equations of motion can be formulated, based on Hamilton’s principle. With W, denoting the energy loss due to damping, Hamilton’s principle for a damped system is “(K-l7)dt=-W,. I 11

(8)

Substituting equations (5)-(7) into equation (8) leads to the equations of motion in matrix form, Mii(X,, r)+Cti(X,,

t)+P(u(X,,

t))=F(X,,

r),

(9)

where

x,+u, u=

u2

t

e

i

represents the instantaneous position of an arbitrary point on the neutral axis, and the internal force P results from the coupling between the gross motion and elastic deformation.

3. BEARING CLEARANCE AND LUBRICATION Most linkage mechanisms, e.g., engine crankshafts and air compressors, have journal bearings at connections. Between journals and bearings are clearances that must be

196

T. S. LIU AND Y. S. LIN

lubricated. The impact between the journal and bearing aggravates the linkage’s highfrequency content and adds to the deformation due to nominal inertia forces and their own vibratory behavior. Resultant linkage responses thus vary rapidly. Moreover, in common practice link deformation is approximately one thousandth of the link length, which is of the same order as the bearing clearance relative to the journal radius. Hence, bearing clearance is crucial to the linkage performance. Due to bearing clearances, impact forces are generated at link joints. The impact process consists of a period of free flight followed by metal-to-metal contact and impact. In the first stage, there is no force acting between the journal and the unlubricated bearing. In the second stage, the direction of the joint force remains parallel to the relative positions for the centers of the journal and bearing. The Coulomb friction on the contact surface can be ignored compared to the magnitude of the impact force. The joint force magnitude upon contact can be obtained from Hertzian contact theory, which is based on the assumption that the dimensions of the contact region between the journal and bearing is much smaller than the bearing radius. The applied force p is related to the bearing radial compression (Yby [7]

a = cdc2-ln (p)l,

(10)

where C, = (k, + kJ/L, ~2 = In {L3 exp C,/[ r, rz( k, + kz)]}, ki = (1 - Vf)/( TEi), and V, E, L, r, and rz denote the Poisson ratio, Young’s modulus, journal radius, bearing length and bearing radius, respectively. During impact, the bearing compressive deformation cy is given by (Y= Ar - C,, where Ar is the relative displacement between centers of the journal and bearing and C, is the clearance. In addition to the elastic force p caused by the material deformation, material damping yields a force F,. Accordingly, the impact force Fimp acting on the journal and bearing is written as Fi,,=p+F,=ka+c&=k(Ar-C,)+cti,

(11)

where k = p/ a and k and c represent the stiffness and damping coefficients, respectively. As shown in Figure 2, to facilitate numerical solution, the centers of the journal and bearing are denoted as finite element nodal points belonging to two contiguous links, respectively. There would be four degrees of freedom, namely, two translations and two

Figure 2. Finite element nodes representing journal and bearing.

LINKAGE

WITH

rotations, at the coincident nodes if bearing clearance is considered in degrees of freedom. Therefore, the forces, is calculated and applied to

LUBRICATED

JOINTS

197

an idealized joint were assumed. However, since the this study, two separate nodal points result in six joint force Fh, consisting of stiffness and damping the two nodes. The force F,, can be written as

Fh = Fiimp for Ar> C,,

Fh=O

for ArsC,.

(l-2)

The equations of motion (9) are nonlinear partial differential equations. In order to facilitate numerical solution, the Galerkin method is employed [4] to separate the time and space variables, so that the equations of motion become ordinary differential equations in the time domain, Mii+C(I+P=P(t),

(13)

where q is the generalized displacement, and p and F denote internal and external forces, respectively. Upon incorporating X and Y components of the nodal force Fhr equation (13) thus becomes ‘0’

F,,

Ftq sliYi+aj+F+

0

= F,

(14)

- Ftn - Fh, -

b-

where Fhx = Fj, COS8’ Fhy = Fb sin

8’

for Ar> C,,

for Ar s C,,

and 8’ represents the angle between the joint force direction and the global X axis, and is depicted in Figure 3.

Figure 3. Bearing

clearance.

198

T. S. LIU

AND

Y.

S. LIN

hydrodynamic lubrication, generated by the lubricant viscosity and the relative The motion between the journal and bearing, involves the wedge shape, stretch and squeeze film action [8]. Phelan [9] investigated the squeeze film effect by ignoring the hydrodynamic pressure caused by the wedging action. For lubricated bearings, the force applied to the journal and bearing is derived based on the squeeze film phenomenon and the impact model previously described. As the journal flies over the clearance that is filled with lubricant, the force generated by the squeeze film lubrication is of the form [9] Frqu = 3~86~~~~(~I~~~)2(~~,/f)o’2sl~/~

(15)

where CL,L, D, C,, f and j- are the viscosity, bearing length, diameter, clearance, film thickness and time change rate of film thickness, respectively. The film thickness is associated with the velocity u: i.e., f=C,+m-Ar

for u>O,

f=C,+a+Ar

for v
Moreover, I= IuI- &, where u is the relative velocity for the centers of the journal and bearing. Since the deformation rate ci usually is much smaller than u, equation (15) can be rewritten as Frqu = bldlf

1’28,

(16)

where d represents a constant that is determined by the viscosity of the lubricant and the geometry of the journal and bearing. The force p that causes the bearing to deform is due to the squeeze film force FEq,,.Hence, P = F%p

(17)

Equations (lo), (16) and (17) are used to solve for (Y and p. At any instant, either the impact mode or the squeeze film mode applies, depending on whether the film thickness is greater than the clearance or not. Thus, the nodal force can be denoted as F,, = Fv4u for ArG C,,

Fh = 4imp for Ar> C,.

In the semi-empirical formula presented by Phelan the hydrodynamic pressure due to the wedging action is not taken into account. In view of the shortcoming of the Phelan squeeze film model, the pressure distribution at the joint is derived in this study in order to model lubricated joint behaviors. The resultant pressure distribution will be numerically integrated to yield nodal forces that are applied to the journal and bearing. Assume that the lubricant is filled in the clearance and that the temperature, viscosity, and density are constant. The Reynolds equation is

The first term on the right side is the hydrodynamic pressure caused by the film-stretching phenomenon. The second and third terms on the right side represent wedge and squeeze film actions, respectively. In a cylindrical co-ordinate system defined such that x = r6 one has dx = r de. The rotation axis of the journal is denoted as z. As shown in Figure 4, the rotational speeds of the journal and bearing are w, and w2, respectively. U. and U, are the velocities of points A and A’ on the journal. V, represents the relative velocity of the centers of the journal and bearing. It follows that, according to Figure 4, uo= (r+ C)w2, V,=ro,ah/raO-vcos(&,-O),

U, = rw, + v sin (0” - e), h = C, - e, cos 8 - e,. sin 8.

LINKAGE

Figure

WITH

4. Degrees

LUBRICATED

of freedom

JOINTS

199

in beam element.

With these equations substituted into the Reynolds equation (18), it becomes 1

3hz(ex sin 8 - eYcos 0) r*

G

h3 d*p+ h3 d*p ap+ -a9 r2ae2 a2*

(19) with the boundary

conditions:

de, 4=p(e+2~,

4,

p(e,~*I/2)=0,

ap(e, z)/ae =ap(e+277, z)/ae,

m, 0) =P.~,

where a circumferential groove on the bearing is assumed. Note that the pressure of the supplied lubricant at z = 0 equals pF. For a finite length journal bearing, the finite difference method is used to obtain a numerical solution. Integration of the pressure distribution with respect to 0 and z yields F,,,,,, which is the lubrication force exerted on the journal and bearing due to hydrodynamic lubrication. As a result, the joint force F,, in the presence of hydrodynamic lubrication can be formulated as

Fh= hh where the components F lubx

=

for Arc C,,

Fh = Eimp for Ar> C,,

of Fh in the inertial reference frame are expressed as p( 8, Z) COS 8 I

de dz,

&by =

p( 0,

Z)

sin 8 r de dz,

200

T. S. LIU AND Y. S. LIN 4. NUMERICAL

PROCEDURE

Since the beam undergoing gross motion may have finite strain, isoparametric elements are employed. A second order function q(r) = ar*+ br+ c is defined as the generalized displacement in equation (13). As illustrated in Figure 5, each of the three nodes in an element possesses three degrees of freedom. The isoparameter r is equal to -1, 0 and 1 at each node, respectively. After deformation, the displacement of any point in an element

>

3

41 r=

,

44

-I

47

r=l

r=O

Figure 5. Journal and bearing during hydrodynamic lubrication.

and that of the nodal point are related by 41 q2 q3

xl+%

94

0

N,

0

0 N,

00

N, 0 00 0 N2

u2

[

0

0

N2 0

0

NJ

N2 0 00 0 N3

0

q5

N3 0I

q6 q7 e3

49

where N,, N, and N3 are shape functions. Analytical solutions of the Reynolds equations for the hydrodynamic pressure with finite length bearings are not available. Therefore, in this work finite difference methods were used for solving the Reynolds equations. The journal surface can be unfolded into a 2nr long by h wide rectangle. With the rectangular surface divided into n x m grids, the partial derivatives for the pressure distribution are given by the finite difference expressions [lo] Q as=

P(i,j+l)-p(U-1) 248

d*p p(i,j+l)-2p(i,j)+p(i,j-1)

'

-=

ae*

Af3*

,

a*p p(i+l,j)-2p(i,j)+p(i-1,j) AZ* az*

-=

The Reynolds equations with the prescribed boundary of simultaneous linear equations of order n x m. 5. RESULTS

equations thus become a system

AND DISCUSSION

A spring-damper-loaded slider-crank mechanism [7] is illustrated in Figure 6. This linkage mechanism is designed so that when the crank rotates 106 degrees from its initial position neither compression nor tension exists. Although the crank BC cannot rotate a

LINKAGE

Figure

6. Slider-crank

WITH

mechanisms

full cycle, it can rotate between 150 and mass of links AB and BC, respectively. specified as two nodal points in the finite as idealized joints. In Figure 7 is shown a rotation of 106 degrees.

-700

0

’

’

LUBRICATED

’

JOINTS

with spring

201

and damper.

-150 degrees. Points E and F are the centers of To simulate the journal and bearing, point A is element model. All the revolute joints are treated the angular velocity variation of link BC during

’

’ 0.0035

’

’

’

’

I

0 )07

Time (s) Figure

7. Crank

angular

velocity

with idealized

joint.

Assume that the revolute joint B has a clearance of 0.0025 cm. Both the diameter and length of the journal are 0.64 cm. The journal and the bearing are made of steel and babbit, respectively. The impact loss factor is O-02, and the Poisson ratio is 0.3. As shown in Figure 8, when the journal flies across the clearance the impact forces at joint B between links AB and BC are zero and the curve oscillates about the gross angular velocity right after every impact. Angular velocity is reduced further every time the journal and the bearing collide. If the linkages are treated as rigid, the angular velocity versus time curves are stairwise [7]. Assume that the lubricant is filled in the clearance at revolute joint B. The viscosity of the lubricant is 0.07 poise. The force between the journal and the bearing has been calculated by using Phelan’s squeeze film formula. The time response curve for angular velocity is smoother in the lubricated case shown in Figure 9 than in the clearanced case depicted in Figure 8. To investigate hydrodynamic lubrication effects, the force between the journal and the bearing was calculated from Reynolds’ equation. As depicted in Figure 10, due to

I-. S. LIU AND

202

Y. S. LIN

Time (s Figure

8. Crank

angular

velocity

)

with bearing

clearance.

Time(s) Figure

9. Crank

angular

velocity

with squeeze

film lubrication.

hydrodynamic pressure, the response curve appears the smoothest among the various linkage joint results. This validates the importance of hydrodynamic lubrication for machine operation. In Figure 11 are shown variations of the axial strain at point E for link AB with idealized joints. When link AB passes the horizontal position, the slider still tends to move to the right side, causing the link AB to be in compression whereas link BC is in tension. Consequently, the axial strain remains negative. The high-frequency content of the axial strain at E is due to the vibratory deflection of crank BC. In the presence of the bearing clearance, the axial strain in beam AB changes drastically, as depicted in Figure 12. The axial strain responses when the linkage has a joint with the squeeze film phenomenon or with hydrodynamic lubrication are shown in Figures 13 and 14, respectively.

LINKAGE

WITH

LUBRICATED

203

JOINTS

-70001

I,( 107

Time Figure

10. Crank

is )

angular velocity with hydrodynamic

lubrication.

0,005

0.01

0

2

0

-7.5

x 10-5

.G 0 L ;;

-I~5xio-*

0

Time Figure

11. Strain variation

6.

(s ) with idealized

joint.

CONCLUSIONS

To carry out the dynamic analysis of flexible linkages with clearances and lubrication at revolute joints, an effective method has been developed for finite strain links with different lubrication models. Wedge, stretch and squeeze film actions are included in dynamically loaded bearings. Depending on the relative positions of the centers of the journal and bearing during linkage motion, the proposed method can determine whether or not surface contact at a revolute joint occurs. If it does, impact forces can be calculated and applied to the journal and bearing. If not, lubricant forces composed of wedging, stretching and squeeze film actions take effect. Two models are employed for a fluid film bearing. The model that accounts for the squeeze film action is computationally efficient for obtaining the dynamic response. To

204

T. S. LIU

AND

Y. S. LIN

0

t-’

I

I

’

E -75 x10-" 2 G cl L 21

LY, -I.5xlO-' 0

0.005

0.01

Time(s)

Figure 12. Strain variation with bearing clearance.

0~005

Time

(s)

Figure 13. Strain variation with squeeze film lubrication.

be more rigorous in modeling lubricated joint performance, a solution of the Reynolds equation should be included in the algorithm. The Reynolds equation can also be used to calculate the maximum pressure and minimum thickness of the fluid film during motion. The use of lubricants at revolute joints is demonstrated to be an effective way of ensuring better machine performance. The illustrative examples underscore their importance. ACKNOWLEDGMENT

This work was supported by the National Science Council of Taiwan, Republic of China (Grant No. NSC79-0401-E009-01).

LINKAGE

WITH LUBRICATED JOINTS

205

0

-2 9

E

2

-7.5

x

IO“

c

e

‘j

-1.5 Ylo-’ 0

0.005 Time (s)

Figure

14. Strain variation

with hydrodynamic

lubrication

REFERENCES 1973 American Society of Mechanical Engineers, Journal of Engineering for 268-272. Elastic link mechanism dynamics. 2. I. IMAM, G. N. SANDOR and S. N. KRAMER 1973 American Society of Mechanical Engineers, Journal of Engineering for Industry 95(2), 541-548. Deflection and stress analysis in high-speed

1. R. C. WINFREY Industry 93(l),

7.

8. 9. 10.

planar mechanisms with elastic links. J. 0. SONG and E. J. HAUG 1980 Computer Methods in Applied Mechanics and Engineering 24, 359-381. Dynamic analysis of planar flexible mechanism. J. C. SIMO 1985 Computer Methods in Applied Mechanics and Engineering 49, 55-70. A finite strain beam formulation. The three dimensional dynamic problem. Part I. R. S. HAINES 1980 Journal of Mechanical Engineering Science 22(3), 129-136. A theory of contact loss at revolute joints with clearance. S. DUBOWSKY 1987 American Society of Mechanical Engineers, Journal of Mechanisms, Transmissions and Automation in Design 109,87-94. The dynamic modeling of flexible spatial machine systems with clearance connection. R. J. ROGERS and G. C. ANDREWS 1977 American Society of Mechanical Engineers, Journal of Engineering for Industry 99(B), 131-137. Dynamic simulation of planar mechanical systems with lubricated bearing clearances using vector-network methods. A. CAMERON 1983 Basic Lubrication Theory. Chichester, England: Ellis Horwood, third edition. R. M. PHELAN 1970 Fundamentals of Mechanical Design. New York: McGraw-Hill, third edition. J. STOER and R. BULIRSCH 1980 Introduction to Numerical Analysis. New York: SpringerVerlag.