Analysis and synthesis of a slider-crank mechanism with a flexibly-attached slider

Jnl. MechanismsVolume 5. pp. 239-247/Pergamon Press1970/Printedin Great Britain

Analysis and Synthesis of a Slider-Crank Mechanism with a Flexibly-Attached Slider Joseph K. Davidson* Received 25 August 1969

Abstract The mechanism examined is the common slider-crank mechanism connected in series with a spring and mass, where the spring represents elasticity between the mechanism and an inertial load. Analysis of the mechanism is discussed and a method of approximate synthesis that utilizes precision conditions is developed. The driving crank rotates at constant speed. Friction is assumed small. An example problem is worked. Zusammenfassung--Analyse und Synthese Eines Sch ubkurbelgetriebes mit Einem Elastisch Befestigten Gleitglied: J. K. Davidson. Das untersuchte Getriebe ist ein gewShnliches Schubkurbelgetriebe mit in Reihe befestigter Feder und Masse, wo die Feder Elastizit&t darstellt zwischen dem Getriebe und einer Tr&gheitsbelastung. Analyse des Getriebes ist untersucht und eine Methode der angen&herten Synthese ist entwickelt, die Pr&zisionsbedingungen benutzt. Gleichf6rmige Drehung der Antriebskurbel ist angenommen und die Reibung ist als klein angenommen. Ein Beispiel ist durchgerechnet. Pe3loMe--Ananx3HCHHTe3l(pHBOUJHnHO--non3yHHaroMexaHH3MaC3.rlacTW'IHOnpuKpenneHHmMnOJ't3yHOM. H. K. ,~aBHaCOH. Hacneaosanmalt MexaHM3M AB~eTC~t O()blKHOBeHHblM KpaaoLtmnHo-nott3yHHbtMMexaHH3MOM B cepaa npaKpenneuaoll npyx~)mo)lH Maccoil,rae npy~)iHa npeacraaaaeT 3JIaCTHqHOCTbMex£iyMexaHH3MOM a .Hepu~IOHHOII Hal'py3KOll. A H a n H 3 Mexan143Ma H3cneaoaaH a MeTOJI npe6nexxeaHaro C..Te3a pa3aaT, zoxopbtil npHMen.er HpCLIH3HOHHble yCJIOBH#I. I"IpHH~ITO paSHOMepHoc spameHr~e seaymaro z p H s o u m n a Manoe TpeHMe. Pa3pa6oraH npaMep.

1. Introduction TRADITIONALLY mechanisms have been designed by considering that all parts remain rigid throughout the motion. H o w e v e r it is sometimes convenient to build an overload capability into a drive mechanism by introducing an elastic member into it. Also the increasing desire both to minimize weight and to cause mechanical motions in shorter times can produce mechanisms with sufficiently high acceleration forces to substiantially deform the parts. Design methods for flexible mechanisms permit the continued use of lightweight drives in such cases. Some recent work to develop design procedures in flexible mechanisms has been done by Khotin [ 1], Burns and Crossley [2], Hain [3] and Livermore [4]. Khotin's work considers small deflections while the other three works are concerned with large deflections. All four of these papers consider slowly-moving mechanisms or other applica*Assistant Professor, Department of Mechanical Engineering, The Ohio State University, Columbus, Ohio 43210, U.S.A.

239

240 tions where inertia forces are unimportant. Recent work that includes inertia forces in some of the mechanism parts has been done by Dizioglu[5]. M e y e r zur Capellen [6.7]. and Neubauer. Cohen and Hall [8]. The first two works consider lumped parameter vibrations in mechanisms and the last two consider beam vibrations in a mechanism link. T h e mechanism investigated in this paper has the inertia of the elastically-driven slider as the primary loading force. T h e problem is to develop any new' equations necessary to analyze the motion of the mechanism and to develop the synthesis equations for designing the mechanism as a function generator.

2. Basic Position Equation of the M e c h a n i s m T h e mechanism under study is shown in Fig. l. T h e links 2, 3,4 and 5 are considered rigid, but the connection between links 4 and 5 is elastic with linear rate k. The mechanism driver rotates at a constant angular velocity' and it is assumed that steady-state A

C

m

Figure 1: Slider-crank mechanism with flexibly-attached slider. motion is extant. T h e distance x~ represents the actual location of point B at one end of the coupler. Point B ' is on link 5 and when the m e c h a n i s m is motionless it coincides with point B on the coupler. For very large values of k (a rigid mechanism), B and B ' would coincide for all phases of the motion. But in the flexible m e c h a n i s m of Fig. 1, link 5 is elastically attached and has mass m. Therefore, point B' is displaced an amount (-'(--Xr) from its rigid position. Force equilibrium conditions applied to link 5 ) Jeld m.r + k{ .v -- xr) = 0.

i I)

T r i g o n o m e t r y applied to triangles O.,_CB and O.,BA obtains .vre - 2 r . r r cos 0 + r e + a e - F + 2ra sin 0 = 0

(2)

. r , . = r c o s O + l 1--

t3)

from which sin0+

The second solution to equation (2) from the quadratic formula is redundant since it provides .rr for the same mechanism with a different value of 0 and a change in sign of the offset, a. Substituting equation (3) into equation (1) and assuming that link 2

241

rotates at constant angular velocity, co, one gets

x -r~o'x=roJ o" c o s 0 +

1-

sin0+

,

(4)

where oJ~ = kX/~m/oJ and x" = d°-x/dO". Now ifr/l and aH are made so that the quantity r

a

-~sin0+~

is small (e.g. the sum ofr/l and all less than 0.3),

The identity sin~0 = ( 1 - c o s 2 0 ) / 2 and equations (4) and (5) yield

.... [ l( r"- a"- ra r" )]. x +o~o-x=rcoo "-' c o s 0 + 1 4l'-' 2l '~ 12 sin0+~-~_,cos20

(6)

The steady-state solution to equation (6) is

r" a" ra[\ 1 --~oo"~sinO+~ x = l-- 41 _~ + \[ ~rcoe" ] "~cos 0 +-/co~"J

3. E q u a t i o n s

cos 20.

(7)

for A n a l y s i s

The motion of point B and other points on links 3 and 4 can be obtained by using traditional methods of kinematics. Equation (7) provides the position of points on link 5 and their velocity and acceleration are available by differentiating equation (7). The velocity equation is

rwoJo"-

2=

ra/ o)coo" \

r'/ ~OWo2 \

coo---'-;1 s i n 0 + - 7 - [ ~ ) c ° s 0 - - f / ~ ) s i n 2 0 '

(8)

and the acceleration equation is ;~ =

co"co~.....~ r ~ ra/co"coo" \ Woe -- l C O S 0 - - T ~ ) s i n 0 - -

r'-'[ co"wo'~~

l \wo.,_ 4] cos 20.

(9)

Since some flexible mechanism design problems arise from the incapability of rigid mechanism theory to properly predict motion characteristics in highspeed mechanisms, it would be useful to have a criterion that measures the degree of this incapability. Examination of equation (7) shows that the slider motion in the flexible mechanism is the same as in a corresponding rigid mechanism if coo is very large. The value of coo, then, is the criterion, lfoJ 0 is large enough so that

~°°~

(002_ ]

l] < 0-05.

242

the m e c h a n i s m can be c o n s i d e r e d to be rigid for an a c c u r a c y of motion characteristics within 5 per cent. If coo is so small as to cause a deflection greater than desired, the m e c h a n i s m must be c o n s i d e r e d to be flexible.

4. Equations for Synthesis For normal o p e r a t i o n o f the flexible m e c h a n i s m that satisfies the limitations imp o s e d above, transients die out after startup and link 5. the slider, goes through the s a m e m o t i o n each revolution o f link 2. Let xo and 0,, be specified as initial positions of the slider and link 2. r e s p e c t i v e l y T h e n other positions can be specified by the variables u and v so that 0 = 0,,+ u;

x = .v,~-= v.

~ 10)

Substituting equations l l0) into 17). e x p a n d i n g grouping c o n v e n i e n t l y yields Z~-Z.,

cos

.+Z::

sin . + Z 4

cos 2 u ÷ Z : .

the trigonometric functions, and

sin 2u = c

Ill)

where Z, = l - . r , ,

z: = Z:~=

/ .2

~l 2

41

21"

112)

( rco' 1 (cos 0,,-"7 sin 0,,), t,.~oo=- 1/

c13)

, 7cos0o .

i14)

(.O0 2

and r~(

coo'-'

116)

Z:, = - - 41\coo"- - - 4] sin 20,,.

If the a p p r o x i m a t e synthesis technique d e v e l o p e d by F r e u d e n s t e i n [ 9 J is e m p l o y e d , e q u a t i o n (1 1) can be used for specifying desired input and output positions o f the mechanism. Differentiating e q u a t i o n I 1 1) o n c e to obtain a velocity equation yields Z.. sin u - Za

cos

u -+- 2Z4 sin 2u - 2Z:, cos 2 . -

i:

17)

(tJ

Differentiating again, and recalling that oo is taken as constant, yields Z , - cos u + Z.~' sin u + 4Z~ cos 2u + 4Za sin 2u = - =;. (D-

18)

Velocity and acceleration conditions on the m e c h a n i s m o u t p u t can be specified (see e.g. Mc k a r n a n [ 10]) using e q u a t i o n s (17) and (18), respectively.

243 5. S o l u t i o n of S y n t h e s i s E q u a t i o n s

Since there are five of the Zi in equation (11), five precision conditions of position, velocity, and acceleration can at most be specified. T h e s e precision conditions are reflected in specified values of u, v, tu, b, and v. Substituting into equations ( 1 l), (17), and (18) yields five linear equations which can be solved for the unknowns Z~, Z.,, Z3, Z~, and Zs. From these one can calculate five of the quantities, r, l, a, xo, 0o, and COo. The designer must specify either the sixth quantity or else a relationship among some of the quantities (e.g. specify the sum o f r / l and all). 6. The Case of Zero Offset T h e analysis equations presented above work properly for the in-line slider-crank mechanism where the offset a is made zero. T h e synthesis procedure, however, must be altered slightly. For this case (a = 0) the position equation ( 11 ), the velocity equation (17), and the acceleration equation (18) still are valid but the Zi become r 2

•

Z1 = l - Xo -- 4~'

(20)

Zz

\ ~ ; " ~ - 1 1 cos 0o,

(2 1

Z:; =

_(\ ~r o" ) sin /

I"") --

0o,

Z4 = 4 l \ w o Z _ 4 ] cos 20,,

(I 5)

and r'-'( o~0~ Z~. = -4l\oje., " - 4 / s i n

200.

(16)

Because tan0o= tan 2 0 0 -

-- Z3 Z., '

-Z~ Z4 '

and 2 tan 00 tan 20o = I -- tan20o ' the compatibility condition Zs(Z~ "~-- Za 2) = 2Z.,Z3Z4

(23)

must hold. Equation (23) represents one equation in the five Zi variables so that only four more equations among them can occur. This means the maximum number of precision conditions possible for the in-line slider-crank mechanism is four.

24a

Once the five Z~ are obtained the fi~e variables r. /. x,,. 0,,. and ooo can be obtained. Examination of equations (15), (16) and (20) through t22) reveals that one of these variables, or a relation a m o n g them, must be specified by the designer. Specifying the ratio r// is convenient because it assures that the final mechanism ~ill fit the requirement that r// be small. Specif,,ing roe can be convenient in order to avoid harmonic frequencies of the mechanism drive.

7. Harmonic Frequencies of the Mechanism Drive Since it has been assumed that link 2 rotate~, at a constant velocity. ~ = oJt. Then examination of equation t4) reveals that link 5 is excited by a periodic, but nonharmonic, function of time. Since the quantit>

has period 2rr, all harmonic frequencies (oJ, 2aJ, 3co. . . . no~) will be present in the forcing function to some degree. Design of the offset mechanism, then, should include provision that the frequency V'k/m be different from the driving speed itself and positive integer multiples of it. T h e differential equation for the in-line mechanism (a = 0) includes the quantity !

i..2

.

"~'/ 1 --75, sine0 in the forcing function. Since this quantity has period rr, alternate harmonic frequencies 2no) will be present to some degree. T h e remainder of the forcing function will provide the fundamental frequency w. Therefore, design of the in-line mechanism should include provision that the frequency \/kim be different from the driving speed itself and even integer multiples of it.

(2oJ, 4oJ, 6o) . . . .

8. Example Problem As an example consider a design of the mechanism shown in Fig. 1 when the offset. a, is zero. It is desired that during the steady-state motion of the mechanism the output, link 5, move 4 in. away from point O,, while the driving crank rotates 12W cot, nterclockwise. The driving crank is to rotate at 200 rev/min and the load can be represented by a 4-1b weight. T h e problem then is to find the mechanism dimensions and the spring rate, k. The functional relationship between input and output motions is to be v = 9u2/~v~. Specifying the position at the ends and middle of the range and specifying the velocity at the beginning of the range yields the precision conditions: ;;=0,

v=O

H=O,

t=O

u =

L, =

~/3.

u=27r/3,

1-0

t:=4'0.

245 These conditions substituted into equations (11), (17) and (18) yield four linear equations in the Z variables: Z 1 + Z ~ + Z 4 = 0,

(24)

-

(25)

Z:~ -

2Z~

=

0,

ZI +0"5Z.,_+O'866Z:~-0"5Z4 +0"866Z~ = I'0,

(26)

Zt - 0"5Z~ + 0"866Z:~ - 0"5Z~ - 0"866Z~ = 4"0.

(27)

Combining equations (26) and (27) yields Ze = - 3 - 1"732Zs,

(28)

and manipulation of equations (24), (25), (26) and (28) provides Z~ = 0"333.

(29)

Substitution of equations (25), (28) and (29) into equation (23) yields (30)

Zs(Z¢" - 8"083Zs - 5) = O,

from which there are three solutions for Z~: 0, 8.661. a n d - 0 . 5 7 8 . These three solutions in equations (24), (25) and (28) provide values of Z~, Z._,, and Za. The values of all the Z variables are grouped in Table I. The mechanism dimensions for the three solutions can be calculated from equations (15), (16), (20), (21) and (22) when r/l is specified. The results are shown in Table 2. Solution 2 is likely to be impractical because of size. Certainly both solutions 2 and 3 should be avoided because of the proximity of the natural frequency to the second harmonic of the driving speed. Increasing r/l in these two solutions will give only slight improvement within the constraint on r/l (r/l <~ 0.3). Therefore, solution 1 is the Table 1. Values of the Zi for the example

problem Solution 1 2 3

Z~

Z:

Z:~

2.667 - 3 . 0 0 0 0-000 17-666 - 18-001 - 17-321 1-670 - 2 - 0 0 4 1-155

Z4

Z~

0-333 0.333 0-333

0.000 8.661 -0.575

Table 2. Mechanism dimensions for the example

problem

Solution

r (in.)

l (in.)

I 2 3

2-68 19'61 1.835

9-38 78"5 7"34

Variable x. O. (in.) (deg) 6"52 59"6 5"56

180-0 136-1 210-0

oJ~

r//

3"07 2"16 2"21

0-286 0-250 0-250

246 v I i n. o

4

ir,,i,,~!~,,:,,, ~'o

I,,

"\x ',

'

/////~

60°"> /

,

_t_..-----'-

~

60 °

Figure 2. Solution mechanism for example problem. MECHANISM OUTPUT Displacement, Ve

Io

ci

Accelerotion,

DESIRED

v:

t y,';,: ~.

Displacement

:

MOTION

~

-

-

V e l o c i t y

. . . . . . . .

.

.

.

Acceleration

~. . . . . 0 0 0 0 0,4

0

,,w

go u

_c :

ea

Z w

0m

0

(o "~

(.9 o

el

w

>

~0

20

40 INPUT

60

80

ROTATION, u, Deg

I00

120

~ ,~.

z

_g o =

0 I

Figure 3. Motion characteristics for solution mechanism of example problem. design choice for this example. It is depicted in Fig. 2 in the three positions that were specified as precision conditions. From the definition of co0 one finds that k = mooeoJ0e.

!3J)

Substitution of numbers yields a spring rate of 42,5 Ib/in. for this example problem. The solution mechanism's displacement, velocity, and acceleration characteristics over the range of interest are depicted in Fig. 3. Shown there also are the desired motion characteristics that would be present if the mechanism could exactly represent the function t" = 9u-'/w-'. References

[1] K H O T I N B. M. A kinematic analysis of m e c h a n i s m s with account being taken of the elasticity of the bars. Sb Trud. Leningr. Inst. Zhelezn. Transp. 218, 2 1 4 - 2 1 9 (1964). [2] B U R N S R. H. and C R O S S L E Y F. R. E. Kine t os t a t i c s ynt he s i s of flexible link mechanisms. A m . Soc. mech. Enp, rs Paper No. 6 8 - M E C H - 3 6 . Presented at l Oth M e c h a n i s m s Cona~ (1968).

247 [3] HAIN K. Federausgllich von Lasten, [4] LIVERMORE D. F. The determination of equilibrium configurations of spring-restrained mechanisms using (4 x 4) matrix methods. A S M E Trans. J. Engng Ind. 89, 87-93 (1967). [5] D I Z I O G L U B. Dynamische Getfiebesynthese der Kurbelaus~eichgetriebe. Forschung Arb. Geb. Ing Wes. 2~. 1960.37-47. [6] MEYER zur CAPELLEN W. Torsional vibrations in the shafts of linkage mechanisms. A S M E Trans. J. Engng Ind. 89, 126-136 (1967). [7] MEYER zur CAPELLEN W. Biegungsschwingen in der Koppel einer Kurbelschwinge. Osterrich. Ing. Arch. 16. 341-348 11962), [8] N E U B A U E R A. H., JR.. COHEN R. and H A L L . A. S., JR. An analytical study of the dynamics of an elastic linkage. AS M E Trans. J. Engng Ind. 88, 311-317 (1966). [9] F R E U D E N S T E I N F. Approximate synthesis of four-bar linkages. A S M E Trans. 77. 853-861 (1955). [10] M c L A R N A N C. W. Design equations for four-bar function generators. Trans. 7th Conf. Mechanisms, Purdue Unicersi~'. pp. 73-78 (1962).