Vibrations of elastic connecting rod of a high speed slider crank mechanism

Vibrations of Elastic Connecting Rod of a The research involved in this paper jails into the area of analytical vibrations applied to planar mechanical linkages.. Specifically, a study

PETER W.JASINSKI

Graduate Student

HOCHONGLEE

Adjunct Associate Professor

Also Employed a t IBM C o r p ,

Endicott, N Y

Mem ASME

GEORGE N.SANDOR

A L C O A Foundation Professor

o f Mechanical Design

Chairman, Division o f Machines and Structures

Fellow ASME

Rensselaer Polytechnic Institute,

Troy, N Y

Vibrations of Elastic Connecting Rod of a

The research involved in this paper jails into the area of analytical vibrations applied to planar mechanical linkages Specifically, a study of the vibrations, associated with an elastic connecting-bar for a high-speed slider-crank mechanism, is made To simplify the mathematical analysis, the vibrations of an externally viscously damped uniform elastic connecting bar is taken to be hinged at each end {i.e., the moment and displace-ment are assumed to vanish at each end) The equations governing the vibrations of the elastic bar are derived, a small parameter is found, and the solution is developed as an asymptotic expansion in terms of this small parameter with the aid of the Krylov-Bogoliubov method of averaging The elastic stability is studied and the steady-state solutions for both the longitudinal and transverse vibrations are found

Introduction

s, IINCE the kinematics of linkages play an important role in machine design, research on the subject is extensive Some

in-vestigators considered elasticity or elastic constraints in linkages

[2, 3]2 and others investigated effects of rigid mass inertia

[4-18] Although a linkage member may have both rigid mass

and flexibility, elasticity and inertia (using harmonic analysis or

graphical methods) have generally been treated separately

Only for simple mechanisms (such as cam-follower systems) have

combined effects been studied [19, 20] Since at high speed a

linkage is subjected to its own inertial forces and suffers elastic

deformation, the combined effects must be fully investigated

Thus, with the speed of machinery constantly increasing, a

detailed mathematical investigation of the vibrations of linkages

is needed To begin this, one naturally turns to the slider-crank

mechanism which is the simplest linkage (Fig 1) For the first

step of the mathematical investigation, a model must be chosen

which represents the important characteristics of an actual

slider-crank mechanism but which lends itself readily to solvability

To accomplish this, the elastic connecting bar in Fig 1 is

as-sumed to be hinged at each end (i.e., the moment and

displace-1 Based on part of a dissertation by P W Jasinski toward

fulfill-ment of the requirefulfill-ments for the Degree of Doctor of Engineering,

Division of Machines and Structures, School of Engineering,

Rens-selaer Polytechnic Institute, Troy, N Y

2 Numbers in brackets designate References at end of paper

Contributed by the Design Engineering Division for publication

(without presentation) in the JOUBNAL or ENGINEERING FOB

IN-DTJSTBY Manuscript received at ASME Headquarters, June 18,

1970 Paper No 70-DE-C

ments vanish at each end) These boundary conditions are satisfied exactly by the elastic bar mounted on a rigid slider-crank mechanism in Fig 2 These boundary conditions for the con-necting bar (displacement and moment being zero at each end) permit investigation more readily Thus the model consisting of

a distributed-mass, externally viscously damped elastic bar with the foregoing boundary conditions is taken as a first approxima-tion for the study of an elastic connecting bar

But even this simplified model results in a fairly complicated mathematical representation The equations governing this system, in which both longitudinal and transverse vibrations are considered, are two simultaneous nonlinear periodically

time-Fig ? M o d e l of a slider-crank linkage w i t h a hinged elastic connecting

bar and a rigid crank

Copyright © 1971 by ASME

Fig 2 M o d e l of a rigid slider-crank linkage w i t h a mounted elastic bar

variant partial differential equations with periodic forcing

func-tions These equations are neither readily solvable with the aid

of classical methods nor readily reducible to the well-known

Hill's (or Mathieu) equation Thus, one is led almost out of

necessity to approximate methods among which the

Krylov-Bogoliubov (K-B for short) asymptotic method of averaging is

foremost T h e K-B method along with t h e Galerkin variational

method enables the solution of the above stated problem to be

written in terms of an asymptotic series once a small parameter is

found and the equations are written in standard form [1] T h e

nonlinear term is assumed small and disregarded Also, a small

amount of external viscous damping is assumed

Introduction of Dimensionless Quantities

The system equations,3 as derived in Appendix 1, are:

'dl V, —

city AE

+ pA

d(f>

tit — v — — aco sili (cot — r/>)

d<j>

= x \ 7~ I + a c°2 G 0 S ( ui ~ 0 ) (1)

vhere

d<t> (d<t>y d*<t> - EI

v" + 2diu'-\'di) v + d¥u + Mv—

+ 1 pA A vt + (x + u) — + aco cos (cot — <j>)

df aco 2 sin (col — <j>) ( 2 )

3 The nonlinear coupling term is disregarded

— — sin cot - sm cot — sma l a 3 cot

Letting

u

the equations (1) and (2) are written

ITT + ( 2 - COS T

AE

v T ~ ( j sm r I v - - — (cos 2r + l ) d

pAV-co" 1

p^-co 2 L 2

+ - C O S T + - — ( C O S 2 T - 1) (4)

v TT — I 2 — cos T I u T — - — (cos 2r + 1) v + I — sm r 1 u

EI f a a + ,-. 2 v mv + —r- v r = -V T sm r + - sm r

l a2

+ - - sm 2r (5)

) • ( ! ) '

where - has been taken to be <K1 and therefore I — are small compared with ( - I and f — J and thus can be neglected when equations (4) and (5) are written

Reduction to Ordinary Differential Equations

T h e displacement and moment are assumed to vanish at each

end of the elastic bar in Fig 1 The functions sin nirri, where n

is an integer, satisfy these boundary conditions and thus the Galerkin variational method can be used to reduce the partial differential equations to ordinary differential equations Sub-stituting

(T, 7]) = a(r) sin 7rr/, v(r, ij) = j8(r) sin -rij (6) into equations (4) and (5), multiplying by sin7nj, and integrating

over t] from 0 to 1, the following is obtained:

a + pi 2 a = e/i + e2</i /3 + j»22/3 = ifc + 62ff-2

(7) (8)

-Nomenclature-u =•• longit-Nomenclature-udinal vibration (in.)

v = transverse vibration (in.)

u = dimensionless longitudinal

vi-bration

v = dimensionless transverse

vi-bration

t = real time (sec)

T = dimensionless time

L = length of elastic bar (in.)

a = length of crank (in.)

a; = spatial coordinate for elastic bar (in.)

1} — dimensionless spatial

coordi-nate r/> = angle defined in Fig 1

co = angular velocity of crank (l./sec)

E = Young's modulus (psi)

I = area moment of inertia about

neutral axis (in.4)

A = cross-sectional area (in.2)

p = mass density (lb-sec2)/in.4

, p22 = dimensionless natural

fre-quencies

a = dimensionless longitudinal

first mode /? = dimensionless transverse first mode

t = dimensionless small

param-eter

^) / = proportionality constants for

viscous damping (lb-sec/

in.2)

e, f — dimensionless proportionality

constants

cii, (h,} _ new dimensionless dependent

hi, h2 ) variables

Si, cii, \ _ new dimensionless dependent

bi, h ) variables as given by second

approx

>, < = "is greater t h a n , " "is less

t h a n "

» , « = "is much greater t h a n , " "is

much less t h a n "

= = "equal by definition"

—• = "approaches"

== = "almost equal t o "

where

Vi = AEir*

pA L2co2'

EIT*

pALW

/ i = - cos T — 2/3 cos r + /3 sin r

ir

(9a)

(9b)

+ ( - Pa + - j &2 sin (p2 + Pl + 1 ) T + ( - Pi - - j (13a)

{Cont.)

ai = - a cos 2 T -\— a -\— cos 2 r ea (9c)

2 2 7T 7T

2

fi — - sin r + 2 d cos r — a sin r (9d)

ir

<jr2 = - /3 cos 2 T + - (3 + - sin 2 T - Af (9e) and the parameters are chosen to satisfy

a e V f IJ

Reduction to Standard Form

If four new unknown dependent variables,

ai(r), ai(r), 6i(r), b2(r) defined bjr t h e relations,

a = ai sin p i r + a2 cos p i r (10a)

a = pidi cos p i r — pia2 sin p L r (106)

(3 = hi sin p2r + 62 cos p2r (10c)

/3 = pin cos j)2r — p262 sin p2T (lOd) are introduced into equations (7) and (8), the following equations

result:

X 62 sin (p2 — pi - l)r

2 sin (pi + 1 ) T

1 T 2

pi L I T

- f 2 P2 ~ 4 ) fei s i l 1 (Pi + P2 ~ 1)^ "" ( 2 ^2 ~ 4 )

- „ ) ) ) + - H i sin (pi + p2 + 1) T

\ 2 V ' 2 + 4 / hl S i" ^ ' ~ P 2 ~~ ^ T + \ 2 P 2 + 4 /

X 61 sin (pi - p2 + 1 ) T

sin (p, — pa — 1) r +

X 62 cos (pi — p2 — 1) r — I 'z Pi + ^ ) *>2 c o s (P1 + Vi + 1) T

' 1 1^

A3 = -P2

62 cos (p, - p2 + 1) r - l - p2

X 62 cos (pi 4- p2 — 1 ) T

1 1

- - sin (p2 — l ) r H— sm (p2 + l ) r

(\ 1

- )ai cos (p2 pi + l ) r + I pi

-(136)

di = — (e/i + e2ai) oos p i r

Pi

d2 = (e/i + f2(/i) sin p i r

Pi

61 = — (e/2 + e2a2) cos p2r

62 = — {efi + e2<72) sin pir

Pi

T h e equations (11) are in t h e form

where4

(11a)

(116)

(He)

( H d )

(12)

cos (p2 + Pi - l ) r + ( - Pi + - J »i cos (p2 pi

-+ \ 2 Vl + 4 / ai C°S ^ + P' + 1') r ~ \ 2 Pl ~ 4 /

sin (pi -f- p2 - 1 ) T - ( - pi + - J a2 sin (pi - p2 +

- ( 2 Pi + A a "- s i n (Pi + Vi + D r - f - P l ~ 4 )

X4 =

P2

X a2 sin (pt — p2 — 1 )r

1 1

- cos (p2 — 1 ) T cos (ps + 1 ) T

(13c)

ai sin (p2 + pi - 1 ) T + I pi -( 2 ^ + ^ a , ,

X 4 r x xi i

2

X 3

xt

y ^

Ft

F2

F3

F,

X ai sin (p2 - pi + 1 ) T + I -+ I - Pi -+ - j ai sin (p2 - pi - 1 ) T

sin (p2 + pi + 1 ) T

l 1

2P l + 4

and

A'i

Pi

2 2

- cos (pi — 1 ) T H— cos (pi + 1 ) T

ir ir

X ch cos (P2 — pi — l ) r + I - pi + - j a2 cos (p2 + pi + 1 ) T

/ I 1 \ / l 1

_ \ 2 Pl ~ 4 j * °0 S P'2 _ P l + 1-) r + V i Pl ~ 4

( 2 V, - 4 ) 61 co, COS (pi — p2 + 1) T -G-0

X 61 cos (p, + p2 — l ) r — ( - p2 + - j 61 cos (pi - p2 — 1 ) T

- ( - P2 + ^ J &i cos (pi + p2 + l )r + ( - Pi - - J

X 62 sin (ps + pi — l ) r + ( - p2 + - j 62 sin (p2 — p, + 1 ) T

F,

Pi

X a2 cos (p2 + pi — 1 ) T

1 3 3

- a2 + — cos (pj + 2 ) T + —- cos (pi - 2 ) r

4 27r 2ir

(13d)

- - cos pit + - Oi sin (2pi + 2)r + - 01 sin (2px — 2)r 7T 8 8 + - a2 cos (2pi + 2 ) T + - a2 cos (2pi — 2 ) T + - o2 cos 2 r

8 8 4

1 1 1 • + - ai sin 2 pir H— a2 cos 2pir eai cos 2pir

4 4 2

4 It is understood that terms like cos (pu — T) are written as cos

(pi - 1) r

1 1

-eai + ~ (Mi sm 2piT

638 / M A Y 1 9 / 1

(14a)

Transactions of the AS ME

Yi - fd + — siii (pi + 2)r + —- sin (pj - 2 ) T 1 3 3

M

T **)*]* <T? J X | = - [V] M (17)

— - sin piT — - ai cos (2p! + 2)r — - ai cos (2px — 2)r

7T 8 8

1 1 1

+ - a2 sin (2pi + 2)r + - a2 sin (2pi — 2)r + - ai cos 2r

8 8 4

— - ai cos 2piT + - a2 sm 2j>iT + - eai sin 2 ptr ea2

4 4 Z ^i

+ - ecii cos 2p1T (146)

Y 3 =

-P'2

1 1 1

- 62 H sin (p2 4- 2)r — - sin (p2 — 2)r

4 7T 7T

where

Vi = J(p u p2)a2 + K(pi, p2) cos (pi — 2)r

+ B(pi, p2)d2 cos (2jh - 2)r (18a)

72 = - J " ( p i , p2)a, + E(p lt p 2 )di cos (2pi - 2)r (18b)

F3 = / ( p2, pi)Su + I?(p2, Pi)S2 cos (2p2 — 2 ) T (18c)

1^4 = —J{.Pi, Pi)Si — 27f(p2, Pi) cos (p2 - 2)r

+ E{pi, pi)b, cos (2p2 - 2 ) T (18d) where5

1 1 1 + - 6, sin (2?;2 4- 2 ) T + - 6i sin (2p2 - 2)r + - 62 cos (2p,

o 8 8

1 1 1

+ 2)r 4- - b 2 cos (2p2 — 2)r + - 62 cos 2r 4- - 6i sin 2p2r

8 4 4 + - 63 cos 2p2r — - fbi cos 2 p2r fbi + - / 62 sin 2p2r

J(y, 2)

+

2 ^ 4 / \ 2 Z 4 2/ + 2 - 1

(l y+ i)(l°-\ , N i - i

(14c)

Yt

Pi

1 1 1

- bi H— cos (p2 — 2)r — - cos (p2 4- 2)r

4 TV IT

bi cos (2p2 4- 2)r bi cos (2p2 — 2)r +

-8 -8 -8

X b2 sin (2p2 + 2 ) T + - 62 sin (2p2 - 2 ) T 4- - 6, cos 2r

8 4

bi cos 2p2T + - b2 sin 2 p2r -\— fbi sin 2p2T

4 4 2 '

IC(?/, «) =

-yz

E(y, z) =

2/2

2 + Z

-1 2 4 -1 2 4

r + —r~r

1 2 - 1 7T Z 4 - 1

4 A 2 ^ 4

(19a)

(196)

— - /62 + - /62 cos 2 piT

y + z - 1

+

1 \ / l 1

(14d)

2V l) \2 Z '

The Method of Averaging the Second Approximation

T h e equations (11) being in t h e form (12) are seen to be in

standard form for application of the K-B method of averaging

[1] As indicated in Appendix 2, the second approximation is

governed b}' t h e equations,

^ ^ [ X ] + £ ^ [ y ] + £ ^

where for a vector function F(x,r) in the form

F ( x , r ) = 2 Fc(x) e xP i® T

e

with the 6 being constant frequencies,

M

- [F(x,r)] 4 F„(x)

F(x,r) = E F» (x)

0^0

exp idr

(15)

(16a)

(16b)

(16c)

I t should be noted t h a t only those terms which can potentially

contribute to — [V] are included in equations (18) Also, the

T

terms (19) are written for the most general case in t h a t the de-nominators were assumed not to vanish For those cases in which they do vanish, the terms with vanishing denominators are simply disregarded as indicated in equation (16c) Thus there

is no possibility of division by zero

Examining equations (13), (14), (18), it is seen t h a t equations (15) must be considered separately for cases which correspond to

different points in the p it p2 plane The four major cases now follow and it should be remembered throughout t h a t 0 < e « 1,

e = 1 , / = 1, p ! > 0 , p2> 0

Case 1 Consider all (pi, p2) which satisfy pi 5^ 1, 2; p2 9^ 1, 2;

p2 5^ pi ± 1, p2 j£ — pi 4- 1 Equations (15) are written as

M

The operator — is known as the averaging operator and ~ as the

r integrating operator T h e vector ^ is written here as

~d,

_ 62_ From equations (13),

fli = e2fi(pi, pi)&2 — - e2cai

d, = — e2fi(pi, p2)ai — - e2ea2

bi = e2Q(p2, pi)b2 — - e2/6i

b2 = — e2fl(p2, pi)bi — - e2/62

(20a)

(20b)

(20c)

(20d) where

fi(2/, 2) = J(y, 2) +

Mi

Equations (20) are easily uncoupled and it is seen t h a t the

solu-6 y and 2 are "dummy variables."

independent of initial conditions

Case 2 Consider all (pi, p2) which satisfy p2 = Pi — 1 and

p2 5^ 1, 2 Equations (15) become

&x =

Q2 =

1 1

2P 2 + i

?2 + 1

1 1

2P 2 + i

Pi + 1

eSi

= 52

+

_4p2 + 1

e2a2

ee2di (21a)

i —J— + H{Vi + l, p2)

_4 p2 + 1

1

•

2

e2di

•:<?dt (216)

&i, d%, b\, 62 ->• 0 as T -*• co

Case 3 Consider all (pi, p2) which satisfy p2 = pi + 1 and

pi ^ 1, 2 Equations (15) are for this case

1 1

2 P l + 4 ,

di = ebi +

Pi

1 1

+ /(Pi + 1, Pi)

1 , 1

A 2P 2 + i

;/e'6i (21c)

Pi

1 1

~ Pi -\—

2 4 „

61 = r~T~ «°i +

_4p + I(pi + 1, pi)

- e^d, (26b)

l_4 Pi + * J

1

-2 P2 +

p2

1

-4 „

- - / e « f c i (26c)

— + /(P2 + 1, Pi)

_ * P 2

e2b,

; / e2 b2 ( 2 1 d )

1 1 , 2P l + i

62 = —7 ten —

Pi + 1 J where

H(», 0 = ^

yz

-G"-i-)G 0

y + « — l

+ (i v+ i)6*~i) G ^ M ^ J )

- /tsb2 (26d)

These equations are very similar to those of case 2 and it is similarty seen, using Routh's criterion, that

d\, di, bi, hi —» 0 as r —»• 00

Case 4 Consider all (pi, p2): which satisfy p2 = —pi + 1 Equations (15) are written for this case as

Hv,*)-+

y + z + i

"2 y+ i)(l'-Q C y -i)(i'-j)

1 1 2 P ' - i

di = ebi +

Pi

— + A(p„ - p , + 1) .4pi

y - z + 1

Equations (21) are in the form

y + z - 1

ezo2

1

(226)

(23)

di =

-1 -1

2P l" i

ebi

Pi

— + A( Pl , -vi + 1)

.4pi

where Z is a matrix defined by the equivalence of equations (21)

with (23) The characteristic equation of the system (21) is seen

to be6

det(XI - Z) = a,X4 + a3X3 + a2X2 + oA + a0 = 0 (24)

The roots of the algebraic equation (24) are called characteristic

values If the real parts of all characteristic values are negative,

the solution is asymptotically stable This is shown with the

aid of Routh's stability criterion [21] which states that the

char-acteristic values all have negative real parts if

ee 2 d\ (27a)

e'ai

- ee2d2 (276)

61 = 2 Pl

1

-4

Pi

.4 1 - Pi + A ( - p , + ] ,P l) e2b2

- - / f26 , (27c)

«o, ai, a2, a3, a* > 0 ai(a3a2 — a4ai) — a32a0 > 0

(25a) (25b)

6 d e t is s h o r t for d e t e r m i n a n t a n d I is used here as t h e i d e n t i t y

m a t r i x

bi =

where

1 1

2^24

" 1 - P i €0

2 —

1 1

_ 4 1 — pi + A(- Pl + 1, p,) e26i

- 2 / e 2 b 2 ( 2 7 d )

A(\),z) = —

yz

- V +

2 J 4 OG-i)

x + 1

2 +

a s in c a s e s 2 a n d 3, t h e e q u a t i o n s (27) a r e i n t h e f o r m

! = ££

w i t h a c h a r a c t e r i s t i c e q u a t i o n

a4X4 + a3X3 + a2X2 + aiX + ao = 0

B u t u n l i k e c a s e s 2 a n d 3, R o u t h ' s s t a b i l i t y c r i t e r i o n ( i n e q u a l i t i e s

( 2 5 ) ) is n o t satisfied for all v a l u e s of pi, e, e, f B u t a d i g i t a l c o m

-p u t e r m a y b e u s e d t o i n v e s t i g a t e t h e s e i n e q u a l i t i e s for t h i s c a s e

I t is f o u n d t h a t t h e r e is a n i n t e r v a l c e n t e r e d a t

G3 for

w h i c h t h e s o l u t i o n s of (27) a r e a s y m p t o t i c a l l y s t a b l e B u t a t

t h e e n d r e g i o n s of t h e l i n e s e g m e n t p2 — —pj + 1, p i > 0, p% >

0, e l a s t i c i n s t a b i l i t y e x i s t s a n d t h e s o l u t i o n g r o w s e x p o n e n t i a l l y

If t h e v i s c o u s d a m p i n g is i n c r e a s e d , t h e l e n g t h of t h e s t a b l e

region is i n c r e a s e d T h u s , a l t h o u g h i t is p o s s i b l e to effectively

r e m o v e t h e e l a s t i c i n s t a b i l i t y b y m a k i n g t h e v i s c o u s d a m p i n g

l a r g e e n o u g h , c a s e 4 f u r n i s h e s t h e o n l y p o s s i b i l i t y for i t s p r e s e n c e

Additional Cases

T h e l i n e s p i , p2 = 1, 2 h a v e b e e n e x c l u d e d from t h e p r e v i o u s

four cases a n d m u s t b e c o n s i d e r e d s e p a r a t e l y W h e n t h i s is d o n e ,

t h e y a r e f o u n d t o b e e l a s t i c a l l y s t a b l e a n d c o r r e s p o n d t o b o u n d e d

s o l u t i o n s of t h e s y s t e m ( 1 5 )

The Improved Second Approximation and the Steady-State

Solutions

A s p r e v i o u s l y s h o w n , for cases 1, 2, 3 ( w h i c h b e s i d e s s o m e

o t h e r p o i n t s e n c o m p a s s t h e p o i n t s pi > 2, p2 > 2 ) t h e s e c o n d a p

-p r o x i m a t i o n s o l u t i o n s di, m, hi, b-i a -p -p r o a c h zero w i t h i n c r e a s i n g

d i m e n s i o n l e s s t i m e T T h u s i t is n o t a difficult t a s k t o e x a m i n e

t h e s t e a d y - s t a t e s o l u t i o n s of t h e i m p r o v e d s e c o n d a p p r o x i m a t i o n

for t h e s e c a s e s A s s e e n i n A p p e n d i x 2, t h e i m p r o v e d s e c o n d a p

-p r o x i m a t i o n x is g i v e n b y

x = I + 6X + e >9 + e2 (x ^ ) X - e" d *

&* X„

w h e r e ^ satisfies e q u a t i o n s ( 1 5 ) L e t t i n g r

(Ji

0 2

61

2 s i n (pi

X p i ( p i

2 cos (pi

X P i ( p i

1 COS (p2

i l l 2 sin (pi + 1)T 1) x p i ( p i + 1)

2 cos (pi -f- 1 ) T

l ) r 1)

• D r

X P l ( p i + 1)

1 cos (p2 + l ) r

X p-lipi — 1) 7T p2(p2 + 1)

1 sin (p-i — l ) r 1 s i n (p2 + l ) r

X p-l(Pi

3 s i n (pi 4- 2 ) r

(29)

(30)

+ „~

1) T p2( p2 4- 1)

sin (pi — 2 ) r 1 s i n p ^

2TT P l ( p , + 2 ) _3_ cos ( pt + 2 ) T

2 x p i ( p i + 2 ) ' 2 x p i ( p i — 2 ) +

2 x p i ( p t cos (pi

2 )

2 ) r

1 cos (p2 + 2 ) r

•r p2( p2 + 2 )

1 s i n (p2 — 2 ) r

1 cos (p2

x p i2

1 cos p i r

x p i2

• 2 ) r

x p 2 ( p2 - 2 )

1 s i n (p2 + 2 ) r

x p2( p2 - 2 ) x p2 (p2 + 2 )

Q ( P K Pa) ( T ^ h ^ T ~ ) + S ( P l ' P 2 ^

V Pi + 2 p , - 2 /

IT c o s ( p i — 2 ) r \

P i - 2 /

Q(pi, P2)

c o s ( p i + 2)T

P I + 2 2Q(P2, P )

) + S(p 1 ,P2)

sin piT

Pi COS piT

cos (p2 + 2 ) T COS (p2 — 2 ) T

+

2Q(P2, P i )

s i n ( p2 + 2 ) T s i n (p 2 — 2)'

Pi

)

(30)

(Cont.)

w h e r e x8S is t h e s t e a d y - s t a t e s o l u t i o n a n d

Q(v, 2)

1 1_

x yz

i * " i 2* + 4

S(y, z) =

2 ^

X 7/2

« + z —

-2 4 -2 4

(31)

(32)

T h e s t e a d y s t a t e first m o d e s for b o t h t h e l o n g i t u d i n a l a n d t r a n s

-v e r s e -v i b r a t i o n s m a y n o w b e f o u n d u s i n g e q u a t i o n s ( 1 0 a ) a n d (10c)

+

x p i \ p i — 1 p i + 1

L.27TP1

/3»»( T ) = e

+ a'

_1 1_

x p2 1_

,7T p 2

+ <3(Pl, P2)

+ t

1

1

+

Pi + 2 pi

-1 -1

h]°° f

+ S(p u p2) - I (33a)

x p iz p i

cos 2 T

a

+

e

- 1 • p2 + 1

+

s i n 2 T (33b)

Examination of the Steady-State Solutions

T h e e x t e n t t o w h i c h t h e a p p r o x i m a t e s t e a d y - s t a t e s o l u t i o n s (33) s a t i s f y t h e e q u a t i o n s (7) a n d (8) is e a s i l y f o u n d W r i t i n g all

of t h e t e r m s i n ( 7 ) a n d ( 8 ) o n t h e l e f t - h a n d s i d e a n d s u b s t i t u t i n g (33), t h e e r r o r r e s u l t s o n t h e r i g h t - h a n d s i d e :

a + p i2 a — e/i — e2</i = e3 1 5 p i2 — p2 2 — 16

x ( p > - D ( P 22 - ~ 4 )

cos 3 T

, (Z p i2 - p2 2 \ , / 4e 1 \ 1

4- - COS T + I — ) Bill T

+ 64

P22 - 2

4 x ( p ,2 - 4 ) ( p2 2 - 1)

2 (pi2 - 1)(P22 - 2 )

x p i2 ( p ,2 - 4 ) ( p , 2 - 1)

cos 4 T

cos 2 T

(-?

+ - (p,2 - 4 ) ( pPa' 2 2 - 1) sin 2 T

+

(p2 2 - 2 ) ( p i2 - ~)

4 x p i2( p i2 — 4 ) ( p2 2 s: ( 3 4 a )

Pi

e>

/3 «/,

-" / l 15p22 - p!2 - 26 \ / 2 / 1 \

_\27T (pi2 - 4)(p22 - 1)7 \7T p22 - 1 /

( 1 9p22 - pi2 - 14 1 p22 - 2 \

^ V27T (pi2 - 4)(p22 - 1) 7T P!2(P22 - 1 ) /

+ e 4 [ / 1 - Pi2 + 4 \ ,

1 — sin 4 r

_\27r (vi 1 - l)(p2 2 - 4 ) 7

, / 1 " Vi % + 4 \ „

T \TT (p,2 - l)(p2 2 - 4 ) / / 4 / pi2 - 4 \

+ u (P 1 - i ) (P, - 4 ) ;c o 8 2 Tj (346)

As expected, the error only contains e to t h e third and higher

orders For pi, p2 S> 2 and 0 < e « 1, the error is seen to be small

and thus the approximate steady-state solutions (33) satisfy the

equations (7) and (8) very well But the error is unbounded near

Pi, Pi — 0, 1, 2 and thus the validity of (33) is questionable near

these values Furthermore, examining Appendix 3, where the

averaging method is applied to an equation in which the exact

solution is known, more light is shed on this T h e equation in

Appendix 3 contains a sinusoidal forcing function with a

fre-quency X Comparing the approximate solution (by the

averag-ing method) to the exact solution, it is seen t h a t the approximate

solution is very good for large X b u t veiy poor for small X I n

fact the approximate solution is unbounded for small X where the

exact solution is bounded for small X So if X is restricted to be

large ( \ » « ) , the method of averaging works well Similarly, the

solutions (33) should be restricted to pi, p2 5>> 2 and it should not

be concluded that a", (3" are unbounded for pi, p2 = 0, 1, 2

The Final Result

If pi, p2 » 2, Q(p1; p2), Q(p2, pi), S(pi, p^ may be neglected in

equations (33) which may be written:

a" (at)

\ T Pi I cos cot 4- e

2 I - — ) cos 2ut

IT p i 2

(3 SS (fat) = e ( ) sin cot +e

\ T Pi

\ T Pi-1

\ T P227

+ <? I - - — ) (35a)

sin 2 cot (35b)

where T has been replaced by u>l Using equations (3), (6), (35),

the steady-state longitudinal and transverse vibrations become

/ 4 1 \ , n

x,t) = t I I L sm —

\7T p i 2 / L

\7T P i 2 /

COS OJ<

7TX / 1 1 \ 1TX

L sin — cos 2 coi + e2 I ) L sin — (36a)

, N ,2 l \r n

v"(x, t) — € [ r I L sin — sm oil

\7T p2 2/ L

+ 6 ; (*L)

\TT p 22 7

L sin 2oii sin — (366)

Li

where pi, p2 S> 2

Conclusion

I t has been shown t h a t a, /3, may be written as

P = (3"-»"» + ft"

where the transient terms atranB, (3trana approach zero with

in-creasing T and the steady-state terms a", (3" are periodic in t

27T

with period Also,

a«s —» 0 as — if pi —• <*> (or u —»• 0)

P i 2

j3" —>• 0 as — if p2 -»- co (or co -*- 0)

p22

T h a t is, the vibrations have small amplitudes for small rotating speeds

The only possibility of elastic instability occurs on the line segment p2 = — pi + 1, pi > 0, p2 > 0 T h e regions of instability and stability here are determined by Routh's stability criterion which leads t o complicated inequalities which depend upon pi, e,

e, f T h e instability regions are restricted to the end regions of the

line segment and the stability region is centered at t h e point

Pi = Pi = - • T h e length of the stable region may be increased by

increasing the viscous damping

For higher approximations t h a n t h e second, many more cases need be considered (i.e., pi, p2 = integers), b u t these added cases will not affect t h e stability results since t h e signs of the critical

terms in t h e stability analysis are dominated by t h e e and

e-terms Also, in higher approximations, additional constant forc-ing terms appear (as they did for pi, p2 = 1, 2) b u t they will be of

the order of e" for the n th approximation Thus additional terms will appear in t h e steady-state solutions obtained from t h e im-proved nt b-approximation b u t they will affect the one derived

here onty slightly (i.e., with terms of e 3 and higher)

The foregoing results were obtained with some restrictions and assumptions Hinged end conditions were assumed at each end (i.e., t h e moment and displacements were assumed t o vanish at each end) T h e nonlinear coupling term was assumed to be small and was dropped A small amount of external viscous

damping was assumed (that is, e, f = 1) Finally, t h e

dimen-sionless parameter e was chosen small relative to 1

This paper has demonstrated the value of the K-B method of averaging for the study of the dynamics of linkages F u t u r e work

in this area m a y include attempts to determine the elastic stability of higher-order linkages using similar asymptotic meth-ods Also, it may be possible to perform a more sophisticated analysis on the slider-crank mechanism This analysis may de-termine the effect of the nonlinear coupling term and may involve more suitable boundary conditions such as a free-end condition or

an end with a concentrated mass present

Acknowledgment

Support under N S F Grant N o GK-4049 awarded to Rens-selaer Polytechnic Institute in response to a proposal submitted

by the second and third authors and sponsored by the Engineering Mechanics Program, Engineering Division of t h e National Science Foundation, is greatly appreciated The authors would also like to express their appreciation to Mrs Frances K Willson for typing t h e manuscript and to Mrs Diane Jasinski for her computer programming aid

References

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of Nonlinear Oscillations, Gordon and Breach, New York, 1961

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Feb 1967, pp 119-125

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Vol 89, No 1, Feb 1967, pp 111-118

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1960, p p 94-11.2

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pp 333-344

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1932

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Vol 4, Blackie & Son, L o n d o n , 1954

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673-675

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-l u n g von M a s s e n k r a f t e n in K u r b e -l t r i e b e n , " Konstruktion, Vo-l 14,

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-teilung in K u r b e l t - r i e b e n , " Z Feinwerktechnik, Vol 64, N o 6, 1960,

pp 195-199

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M a r 1965

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Or-dinary Differential Equations 2, Auflage Springer, Berlin, 1963

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N o s t r a n d , Inc., 1959, p 40

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1967

A P P E N D I X 1 Derivation of the System Equations

T h e d e r i v a t i o n b e g i n s w i t h t h e w e l l k n o w n r e s u l t f r o m d y

-n a m i c s [ 2 2 ] :

dut

a = A + c o X ( < o X p ) + — X 9

at

d2p do

J + 2 " x i (37)

w h e r e

a = a c c e l e r a t i o n s e e n f r o m fixed c o o r d i n a t e s y s t e m

A = a c c e l e r a t i o n of m o v i n g c o o r d i n a t e s y s t e m

to = a n g u l a r v e l o c i t y of m o v i n g c o o r d i n a t e s y s t e m

p ~ p o s i t i o n v e c t o r for p a r t i c l e i n m o v i n g c o o r d i n a t e s y s t e m

T h e a c c e l e r a t i o n of a d i f f e r e n t i a l e l e m e n t of t h e e l a s t i c b a r i n

F i g 1 m a y b e d e t e r m i n e d u s i n g (37) w i t h

A = — a a )2[ c o s ( u i — <j>)\ + sin (coi — 4>)W

dp

dt

d'p d<j)

T h e force a n d m o m e n t e q u a t i o n s for t h e d i f f e r e n t i a l e l e m e n t

( c e n t e r e d a t x) a r e n o w w r i t t e n i n t h e classical w a y [ 2 3 ] :

dP

dx

Q -\ dx — Q — fVi dx = (pAdx)a„

dx

dx

w h e r e pAdx = m a s s of t h e d i f f e r e n t i a l e l e m e n t ,

d.c

Ut = — aio s i n (ut — <j>) + u t — V

(39)

(40)

dt

d<j>

dt

a n d t h e r o t a t o r y i n e r t i a a n d s h e a r d e f o r m a t i o n a r e n e g l e c t e d

E x t e r n a l v i s c o u s d a m p i n g is a s s u m e d in t h e x a n d y - d i r e c t i o n s

S i m p l i f y i n g (39) a n d u s i n g t h e q u a n t i t i e s i n (38), ( 4 0 ) t h e

e q u a t i o n s a r e d e r i v e d :

u tt — 2

v t , + 2

d4

dt

e

pA

d<j>

/d<t>y d2c£ AE

ft — I — - ) u — TV " ~ ~~,T u ** =

r d(j) i f.dd>y

\u, _ „ - _ « , , s i n (ut -<t>)j + x { - )

+ aco2 cos '(ut — </>) ( 4 1 a )

dt \ dt J

d*d> EI + TTTT l H 7 "x df- pA dip

- + (x + u) — -4- au cos (oil — <j>)

pA I dt

AE ,

H 7 (v x V x )x

pA

df

w h e r e ( M A ) , is a n o n l i n e a r c o u p l i n g t e r m

A P P E N D I X 2

The Method of Averaging

S u p p o s e t h a t a n a p p r o x i m a t e s o l u t i o n is d e s i r e d for t h e s y s t e m

of d i f f e r e n t i a l e q u a t i o n s 7

x = eX(x, r ) + 62Y(x, r ) (42)

w h e r e X, Y a r e i n t h e f o r m of F defined b y

F(x, r ) = £ F , ( x ) e x p (idr)

a n d t h e 6 a r e c o n s t a n t f r e q u e n c i e s a n d 0 ' < e <SC 1 D e f i n e t h e

a v e r a g i n g o p e r a t o r — a n d t h e i n t e g r a t i n g o p e r a t o r ~ b y

r

T h u s a = a x \ + a tJ ) h a s b e e n d e t e r m i n e d w h e r e

a x = utt

2TtVl

d<j>

Cty = Vi, + 2 — Ut

, (d<p\

(x + u) I — I - v

d?4

A2

v + (x + u)

— aw2 cos (not — (j>) ( 3 8 a )

d*4>

df

aw2 s i n (OJ« - 0 ) (386)

M

[F] = F„

F = E F<

e x p idr

7 All q u a n t i t i e s are dimensionless

(43o)

(43b)

T h e Krylov-Bogoliubbv (K-B for short) method of averaging [1]

gives

as an approximate solution where <; is defined by

^ ^ [ X 1 + ^ i [ y ] + ^ [ ( x J

T T T 1 _ \ Oi; )*]

(44)

(45)

T h e solution to (45) is known as t h e second approximation and

(44) as the improved second approximation T h e approximate

solution (44) satisfies (42) to the order of e3; t h a t is, if (44) is

sub-stituted into (42), and (45) is considered, the error only involves

terms containing e3 and higher Thus, if the coefficients of e3,

e4, in this error are of the order of 1 or less, t h e error is seen to

be small and the approximation solution (44) is good But if

these coefficients are large (of the order - or larger), the error is

€

large and the approximation (44) is poor

A P P E N D I X 3 Examination of the Method of Averaging

Consider the differential equation

x = — ex — e sin Xr (46)

T h e method of averaging gives t h e approximate solution

cos AT

where c is an arbitrary constant This approximate solution is unbounded in X for small X

T h e exact solution to (46) is

x — c exp ( — er) H—; , ^~ cos XT —' ^ , sin Xr (48)

+ X* €2+ X2

which is bounded in X for small X Comparing the exact solution (48) with the approximate solution (47), t h e method of averaging

is seen to work very well for large X b u t very poorly for small X