Calculo de Levas

ca

•

es1

(cAmco

',,

COMMERCIAL CAM CO., INC.

SUBSIDIARY OF EMERSON ELECTRIC CO.

1444 SOUTH WOLF ROAD

WHEELING,

IL. 60090 U.S.A.

, \

I '

a manual for

•

engineers

designers

and draftsmen

By CLYDE H. MOON, P.E.

Staff Engineer - Dixie Cup, Division Of American Can Company

"Published by" Commercial Cam Division, Emerson Electric Company

PREFACE

This manual is intended for the engineer, designer or draftsman engaged in the ~ sign of moderate speed machinery in which cams are important elements. It presents an orderly design procedure for cams having fairly rigid follower systems. No at-tempt has been made to cover high speed systems in which deflection and vibration effect the efficiency of the operation.

In general it follows the practice of accepted authorities. Tabular data is used ex-tensively to minimize tedious mathematical operations. Design examples are pt~ sented to illustrate the use of the tables and equations.

It contains certain novel features. The method of detennining the radius of curvature, to the best of the writer's knowledge, has not been heretofore in the literature. Graphical methods for pressure angles and radii of curvature have been simplified to an orderly stepped procedure.

It is hoped that this manual will unravel some of the mysteries of cam design to the uninitiated.

Clyde H. Moon, P. E. Dec. 20, 1961 Easton, Penna.

CONTENTS

Section A. Fundamental Mechanics.

''

B. Cam Systems.

''

c.

Cam Nomenclature

,,

D. Basic Curves

''

E.

Displacement, Velocity and Acceleration Tables for Basic Curves

"

F.

Cam Size Uetermination

''

G.

Force Analysis

"

II. Con tact Stress

"

J.

Cam Design Summary

''

K.

Cam Curve Syntbesi s

''

L. Dwell-Rise-Return-Dwell Cam Usiag Basic Carves

''

M.

Cam Curve with Terminal Velocity

"

N. Modified Trapezoid and Modified Sine Carves

''

P. Cam Pronle Determination

"

R.

Cylindrical Cams

.

"

s.

_{Effective Weight of Cam Follower Systems.}

"

T. Polynomial Cam Curves Index

SECTION A

A-1

Fundamental Mechanics

Introduction. Mechanics is a broad subject, but, fortunately, for cam design the band of topics is narrow. The subject matter in this section provides the neces-sary fundamentals. As the principles are basic, the references tabulated at the end of the section have been freely used and credit is hereby given.

Vector and Scalar Quantities. Vector quantities have both magnitude and direction. Examples are displace-ment, velocity, acceleration, and force. Scalar quanti-ties have magnitude only. Examples are weight, mass, and volume. Vector quantities are shown in bold face type, thus: A, B, etc. Scalar magnitudes of vectors ~re shown in italics; thus, A is the scalar magnitude of vector A.

Addition of vectors creates a resultant vector. This may be done by the parallelogram method; Fig. A-I (a), or the triangle method, Fig. A-A-I(b), which show R as the resultant of the addition of vectors A and B. Three or more vectors may be added by the polygon method, Fig. A-1 (c).

Vectors may be subtracted to give a resultant. To subtract vector B from vector A, the sign of B is re-versed, which is the same as reversing its direction, and then added vectorially to A. See Fig. A-1 (d).

A vector quantity may be resolved into components along any chosen axis as shown in Fig. A-1 ( e).

Scalar quantities are added and subtracted alge-braically.

Displacement. Displacement is a vectorial quantity, occurring as translation or rotation or a combination of both. A moving body has translation when every line in the body remains parallel to its original position. It has rotation when all points travel in circles about an axis of rotation. Complex motions can be resolved into translation and rotation.

I I I I I c;... _ ___:....:.,..__.., I { Q)

A+B=R

A-B=R

A+B=R

FIG. A-I

A translating body has linear displacement, Ye loci t y, and acceleration.

A rotating body has angular displacement, velocity, and acceleration.

Velocity. Velocity is the rate of change in displace-ment. When velocity is constant, equal increments of displacement occur in equal increments of time, i.e.:

v

=

y/t. A point has variable velocity when displace-ments vary in magnitude, direction, or both, in equal time increments. The instantaneous velocity of such a point is determined by the derivative of displacement with respect to time: v

=

dy/dt. A point may hav<' several different concurrent velocities which may be combined vectorially into a resultant. A resultant velocity can be resolved into components along chosen axes.

Acceleration. Acceleration is the rate of change in velocity. For a point moving in a straight line \\'itlt constant acceleration, the relations between displace-ment, velocity, and acceleration are

Vr

+

V0 (l) y = - 2 - t ai! (2) y = vt

+-0 2 v,

=

v.

+

at (3) v,2 Vo2

+

2ay (4) v, - Vo (5) a=

where t

=

interval of time; y = displacement; v0

=

velocity at start of interval; v ,,. = velocity at end of interval; a

=

acceleration.

A+B+C+D=R

I I I

________ ....,' (e)

Vx+Vv=V

A-2

_{Fundamental Mechanics}

When acceleration varies in magnitude, direction or, both, the instantaneous acceleration is the derivative of velocity with rPspect to time: a

=

dv/dt

=

d2_y/dt2• A point may also have several different, concurrent, accelerations which may be combined vectorially into a resultant acceleration; and a resultant acceleration may be resolved into components along chosen axes.

Pulse. Pulse (or jerk) is the rate of change in acceler-ation. It is the derivative of acceleration with respect of time; i.e.: p

=

da/dt

=

d3_y/dt3_•

Rotation. A radian is an angle which subtends an arc equal in length to the radiu~ of t hf' arr.

Thus 211" radians = 360°; 1 rad'ian = .57.:3°.

In Fig. A-2(a), consider a point moving at a con-stant speed (l') in a circular path. As the point is con-tinuously changing direction, the velocity is not constant.. The change in velocity is indicated by the vectorial subtraction of V1 from V2. See Fig. A-2(b).

Acceleration, designated as normal acceleration, is directed toward the center with a magnitude a

=

v2_/r.

I

,1,

I I \ I I \ I \ I \ I \

r

I \ I \ I \ I \ I \ (0)

(b)

FiG. A-2

For constant angular acceleration, the relationship of angular displacement, velocity, and acceleration are as follows: <I> WF

+

Wo 2

t

(6) a.t2 (7) <I>

wt+-

0 2 Wp W0

+

at (8) Wp2 _Wo2

+

_2a.<J> (9) Wp - Wo (10) a =

-where t = interval of time; <I>

=

angular displacement in radians; w0

=

angular velocity at start of time interval in radians per time unit; wp

=

angular velocity at end

of time interval; a

=

angular acceleration in radians per unit time squared.

The instantaneous values of variable angular velocity and acceleration are determined by the derivatives with respect to time: w

=

d<J>/ dt and a

=

dw/ dt.

The relationships between linear and angular expres-sions are s

=

r<f, (11) l'

=

rw (12) an v2

- =

rw2

₌

_{l'W (13)} r a,

=

ra. (14)

wheres = arc length of circular path; an

=

normal com-ponent of acceleration; a1 = tangential component of ac-celeration; r = radius of circle; <I>, w, a, v as noted be-fore.

Force and Mass. The concept of force and mass is provided by Newton's laws of motion.

I. A body maintains its state of rest or uniform

mo-tion unless compelled by some force to change that state.

2. An unbalanced force acting on a body accelerates the body in the direction of the force. The accelera-tion produced is directly prr,portional to the force and inversely proportional

to

the mass of the body. 3. To every action or force there is an equal and

op-posite reaction.

According to the second law, if an unbalanced force

F imparts an acceleration a to a body, a different force F1 will impart a different, but proportional, accelera-tion a1. That is:

ma

(15)

where mis the mass of the body.

If a body falls freely, the unbalanced force is the weight (W) and the acceleration is g (386 in./sec. 2

).

Substituting W for Fi and g for a1

F

a

w

g and

w

F = - a = ma (16) g

Force is a vector quantity; so two or more forces can be added vectorially to produce a resultant, which is the single force that will give the resultant acceleration. When the resultant force is zero, the body is in equilibrium and no change occurs in the state of rest or motion of the body.

References

1. Theory and Problems of Engineering Mechanics. McLean

and Nelson. Schaum Publishing Co., N. Y.

2. Physical Mechanics. G. H. Logan. Machine Design. April, 1956.

SECTION B

Cam Systems

Basic Elements. A cam system consists of four ele-ments: cam, follo"·er, follower system, and drive.

A cam is a mechanical part which impa'rts a pre-scribed motion to another part by direct contact. It may 1emain stationary, translate, or rotate.

The follower is the element directly contacting the cam. It may be of various shapes.

The follower system includes all the elements to which motion is imparted by the cam. They may be connected directly to the follower, or through linkage and gearing. The follower and follower system may translate or oscillate.

The drive consists of the prime mover, gears, cam shaft, etc. which impart motion to a rotating or trans-lating cam, or to the follower system of a stationary cam.

Cam systems may be designed in a variety of physi-cal forms. A number of the more commonly used forms, classified according to cam and follower types, are presented in this section. These are based largely on the more comprehensive catalog in the reference at the end of the section.

In this manual, only the open or disk cam, the closed or face-groove cam, and the cylindrical cam will be dis-cussed, but the principles set forth are equally applicable to the other types.

Cam and Follower Classification

I. SEQUENCE OF FOLLOWER OPERATION. The three common types are shown diagrammatically in Fig. B-1.

(a)

(b)

(a) Dwell-rise-dwell cam (D-R-D), Fig. B-l(a). This is the most common type. It has a dwell at the beginning and end of the rise.

(b) Dwell-rise-return-dwell cam (D-R-R-D), Fig. B-1 (b). There is no dwell between the rise and return.

(c) Rise-return-rise cam (R-R-R), Fig. B-l(c). There are no dwells. This type has little appli-cation, as the motion is more adapted to an ec-centric.

•

3 6 0 -DWELL: ....,.,,__RISE RETURN (c) ...._. ...._. ...._. R I S E ...._. ...._. ...._... ...._..._.RETURN ...._. ...._. ...._. -FIG. B-1 2. FOLLOWER SH.\PE

(a) Roller follower, Fig. B-2. This is the most commonly used follower. Pressure angles should be low to prevent jamming.

(b) l{nife-edge follower, Fig. B-3. This is of simple form but edge wears rapidly.

(c) Flat-face follower, Fig. B--1. This type rnn h<' used with a steep cam, as it will not jam. D<'-flection or misalignment can cause high surface stress.

(d) Spherical-face follower, Fig. B-;>. Rndius<'d face compensates for deflection or misalign-ment.

FIG. B-2 FIG. 9-3

FIG. B-4 FIG. 8-5

3. FOLLOWER MOTION.

(a) Translating follower, Fig. B-2. The follow<'l' moves in a straight line.

(b) Swinging arm follower, Fig. B-6. The follower moves in a circular arc.

4. FOLLOWER POSITION.

(a) On-center follower, Fig. B-2. Linc of follower motion passes through axis of cam rotation. (b) Offset follower, Fig. B-7. Line of follown

mo-tion does not pass through axis of cam rot at ion. Off set should be in direction that reduces force components tending to cause jamming.

8-2

ROTATING FOLLOWER S'\'STEM FIG. 8·8

Ca111

Syste111•

: r · ~

FIG. B-9 FIG. 8-10

5. CAM ~VloTIO~. _6. CAM SHAPE.

(a) Rotating cam, Figs. B-2, 7, 10, 11. Rotation is usually of constant speed.

(h) Translating cam, Figs. B-12, 13. The cam usually rC'ciprocatcs in straight line motion.

(ct)

Stationary cam, Figs. B-8, 9. The follower system rotates or translates.

FIG. B-11

(a) Open cam (also known as disk or plate cam), Figs. B-2, 6, 16 show examples. Translating or swinging arm follower must be constrained to maintain contact with cam profile.

(b) Closed cam (also known as face-groove or posi-tive cam), Fig. B-10. The groove constrains follower to positive action.

(c) Cylindrical cam, Fig. B-11. The roller follower operates in a groove cut on the periphery of a cylinder. Follower may be translating or swinging arm type.

CAM MOTION

CAM MOTION

.

..

FIG. 8-12

FIG. B-15 FIG. B-16

FIG. B-13 FIG. B-14

(d) Wedge cam, Fig. B-12. Cam usually has reciprocating straight line motion.

(e) Flat-plate cam, Fig. B-13. This is positive-drive version of the wedge cam.

(f) Inverse cam, Fig. B-14. Kormal functions of cam and follower are reversed. The most common example is the geneva motion.

(g) Roller-gear drive cam, Fig. B-15. Projected ridge contacts dual roller followers. The Fer-guson indexing cam is an example.

7. FoL'LOWER CoxsTRAIXT.

(a) Gravity constraint, Fig. B-2. Weight of fol-lower system is sufficient to maintain contact. (b) Spring constraint, Fig. B-16. The spring must

be strong enough to maintain contact.

(c) Positive constraint, Figs. B-10, 11, 13. The groove maintains positive action.

Reference

1. Basic Cam Systems. H. A. Rothbart. Machine Design.

SECTION C

C-1

Cam Nomenclature

The Displacemrnt Diagram is a rectangular coordi-nate layout of the follO\ver motion in one cycle of cam operation. The rise of the cam is shown as the ordi-nate, the length of the abscissa being arbitrarily chosen. The abscissa is dividPd into equal cam angles or time divisions. A sketch of the displacement diagram is the first step in the devf.'lopmf.'nt of the cam profile. Ree Fig. C-I(a).

The Velocity and Acceleration Diagrams are coordi-nate layouts of the magnitude of thf.' velocities and ac-cPlf.'rations. Typical diagrams are shown in Figs. C-1 (b)

and C-1 (c).

r

_..

,., ~-===---1~--+---=-+---,_J_

MO··

The Transition Point is the point of maximum veloc-ity.

The following definitions apply to Fig. C-2:

The Cam Profile is the working surface of cam in contact with the follower. In a closed or grooved cam there is an inner profile and an outer profile.

The Base Radius (Rb) is the smallest radius from the cam center to the cam profile.

The Trace Point is the center of the roller follower. The Pitch Curve is the path of the trace point. This curve is usually determined first and the cam profile established by tangents to the roller follower.

The Minor Radius (R0 ) is the smallest radius from the cam center to the pitch curve ..

The Major Radius (RN) is the largest radius from the cam center to the pitch curve.

The Pressure Angle ( 'Y) is the angle at any point be-tween the normal to the pitch curve and the instantane-ous direction of the follower motion. This angle is important in cam design because it represents the steepness of the cam profile. Too steep a contour can cause jamming of the follower system in its guides, therefore, cams with radial followers are usually de-signed with a maximum pressure angle of approxi-mately 30 degrees.

With a swinging arm followN the pressure angk is less important as it would be impossible to jam the sys-tem unless the line of action went through the pivot of the follower arm.

The Pitch Point is the point of maximum pressure angle. This is the start of df'sign for minimum ram size. On cylindrical cams it is coinrident with the transition point. On disk cams, because of the distor-tion resulting from com·erting the displacPment dia-gram to radial divisions, it does not coincidr with the transition point. Howewr. for practical purposes these points on most cams can be assumed at the same point.

The Pitch Radius (RP) has its center at thf' cam axis and passes through the pitch point. This radius is used for calculating a cam of minimum size for a given pressure angle.

The Radius of Curvature (Re) at any point of the pitch curve is the radius of a circle, tangent to the curve, whose curvature is the same as that of the pitch curve at that point.

FIG. C-2

TRACE POINT

~-i..--~---'1:--MAJOR R4DtUl PITCH RM>tUS

Reference

1. Cams--De~ign, Dynamics and Accuracy. H. A. Rothbi

SECTION D

0-1

Basic Curves·

Nomenclature

J JinC'ar ncc<'IC'ration of follower (in./scc.2)

1· ii11rar ,·rlority of follo,,·<'r (in./sec.)

h total Ji:-;p]acrnwnt of follower (in.)

!J di:-;plac·f'mf'11t of follom'r nt any point (in.)

p a11gnbr di:,;phH'<'Illf'nt of cam for displacement h (radia11s)

e :rn~nhr di:-:pl:1<'f'Illf'11t of cam for displacement y

( r:HI i:111:-:)

tim<' for C':rn1 to rotatr through fJ (sec.)

7' = tinw for <":tm to rotntc throu~h {3 (sec.)

Classification. Ba:-;i<· <'111'\'C's are' primarily of two C'!:1:-:--<•:--: sirnpl<· pnly11omial and trigonometric. The :--impl<· polyllrnni:tl <·111T<'s i11d11de the constant velocity or :--tr:iiµ:ltt li,l('. tlH· C'011st:111t ncrC'lcration or parabolic, :111d t 111· ,·1tl>i<· 1·11n <",. ( )11!~· the ronstn.nt ,·elocity and C'oll:--t :t 111 :1<T<'l<·r:1 t irn1 1·1u·,·C's "·ill he discussed. The

1 ri1.!:<l11onwl ri" <·111T<'s indndC' thr harmonic, the

cy-('lnid:tl. t IH' do11lil<· h:mnonir. an<l thr elliptical. Only

11l<' h:1rmn11i<' :,nd ryeloidnl "·ill he discussed.

Constant Velocity. SC'<' Fig. D-1. This cur\'e has a :--t r:1iµ;ht !in<' displa<·rm<'nt din.gram. It has uniform displa<"<'ll1<'1lt, !'Onstant Yelocit~r, and zero acceleration . . \t the• tNmi11als there is the impractirahle rondition of i11sta11t:UH'Olls rhange in velocity, resulting in

theo-r<'t i1·all? i11fi11itC' ncc·elerations. This rondition makes

this 1·111TP undC'sirable except in combination with other <0

11n·l'S. ( 'harnderistics of the constant velocity curve are: y~ Displacement: 1J h -fJ {3 IJ h Velocity: v =

t

=

T

~\cceleration: a

=

0 (1) (2) (3)

L ____ -- -

- --- _______

VELOCITY_---., 0 1 _{• _ _ _ _ _ _ _ _ _ .ACCEI.ERAT!Q!L.}

--;;i

_~ H 8· FIG. 0-1

e

1

Constant Acceleration. See Fig. D-2. This curve, also known as paraholic or gravity curve, has constant positive and negative accelerations. It has an abrupt change of acceleration at the terminals and the transi-tion point, which makes it undesirable except at low speeds. It provides the lowest acceleration of all curves for a given motion. In combination with other curves it can he used to advantage. For constant acceleration, the following equations are valid:

l'

=

l'o

+

at (4)

v2 V02

+

2ay (5)

y V0t

+

0.5at2 (6)

.II 0.5(vp

+

v0)t (7) where v0

=

initial velocity; Vp

=

final velocity; a

=

acceleration; y = displacement.

FIG. 0·2

In terms of linear and angular displacements, its characteristics are as follows:

Cam angle fJ from zero to 0.,>/3:

Displacement: _y

₌

2h(~)'

Velocity: v Acceleration: 4h(fJ/{3) T 4h a=

-T"

Cam angle fJ from 0.5/j to fj:

Displacement: y = h - 21.(1 -

;r

Velocity: v = 4h [l - (B/fj)) T 4h Acceleration: a

=

T"

(8) (9) (10) (11) (12) (13)

Construction of the displacement diagram is shown in

Fig. D-3.

Line AB represents the number of degrees through which the cam rotates to produce the desired displace-men t of the cam follower. It is divided into the desired number (N) of equal parts. Line AC represents to scale the displacement of the cam follower. Any line AD is drawn and divided into N parts of units 1, 3, 5, 7 -7, 5, 3, 1. Lines parallel to CD are drawn inter-secting line AC. The intersection of horizontally pro-jected points on AC with vertically propro-jected points on AB locates the necessary points for the displacement curve.

0-2

Basic Curves

"t=.-.t:::::::::....~l~__..JL_~4L-~L5~-L~~7~~.·

Fl8. 0-3

CONSTRUCTION DIAGIWI-CONSTANT ACCELIEIIATION

Harmonic. See Fig. D-4. This curve is a definite improvement over the previous curves. It has a smooth continuous acceleration but has a sudden change at the dwell ends when used in a dwell-rise-dwell cam. This is objectionable at high speeds. In combination with other curves it is valuable for use in a dwell-rise-return-dwell cam.

The characteristics of this curve are:

Displacement: y = 0.5h( I - cos.-~) (14) 0.5rh[sin r(fJ/{3)] v

=

T (15) Velocity: Acceleration: 0.5r 2_{h[cos ,... (fJ/{3)}

I

a= (16)

r2

8 • -..._ ACCELERATION fJ FIG. 0-4 - ~ ,

·--._J

Construction of the displacement diagram is shown in Rig. D-5.

Line AB represents the number of degrees through which the cam rotates to produce the desired displace-ment of the follower. It is divided into the desired number (N) of equal parts. Line AC represents to scale the displacement of the cam follower. A semi-circle with radius equal to one-half the displacement is drawn as shown and divided into N equal parts. The intersection of horizontally projected points of the semi-circle with vertically projected points on AB lo-cates the necessary points for the displacement curve.

Fl8. 0-5

Cycloidal. See Fig. D-6. The cycloidal curve is de-veloped from the path of a point on a circle which is rolled on a straight line. It is the most popular cun·e for high speeds, as there is no sudden change in accelera-tion at the dwell ends. It has the lowest vibration, wear, and stress of all the basic curves.

The characteristics of this curve are:

y

Displacement:

.

y = h - - - -

[(J

sin 21r(8/f3)]

{3 21r (17)

V e oc1ty: 1 . v

=

h[l - cos 21r(8/{3)]

T (18)

A cce erat1on: l . a

=

h[2,.- sin 21r(8/{1)]

_r2

(19)

CYCLOIDAL CURVE

c D

~~~--:::..-::----;:----:4~~5:-~~,~---:;1~--:!e'

FIG. 0·7

CONSTitUCTION DIAGRAM-CYCLOl~L

Construction of the displacement diagram is shown in Fig. D-7.

Line AB represents the number of degrees through which the cam rotates to produce the desired displace-ment of the follower. It is divided into the desired number (N) of equal parts. Line AC represents to scale the displacement of the cam follower. With A as a center, a circle whose circumference is equal to AC is drawn, and divided into N <>qHal parts. From the projection of these points on AC, lines are drawn parallel to AD, intersecting the vertical projections of the divisions of AB. These intetsections locate the necessary points for the displacement curve.

Reference

1. Cams-Design, Dynamics and Accurac~·, H. A. Rothbart.

John Wiley & Sons, Inc. 1956.

SECTION E

E-1

Displacement, Velocity and

Acceleration Tables for Basic

Curves

a l' h /3 y () Nomenclature

linenr acf'clerntion (in./sec. 2 )

linear wlocity (in./sec.)

total displucrrnr11t of follower (in.)

angular dispbrPment of cnm for displacement h (deg.)

displacement at any point (in.)

angular <lisplacc'nw11t of cnm for displacement !J

(deg.)

dispbc·C'rn<'11t factor :1t nny point \·dof'ity C'Odli<'iC'nt nt any point af'crlnntion C'OrtfiC'ient at :my point

RP~I of cnm shaft

Basic Curve Factors. To ~irnplif>· determination of the c·am profi]p and thP n·locity :rn<l n1-celcration of follo,Yer, point-hy-point factors ha\·e been calculated from the <'qunt io11s for the drnraf'teristics of each ha sic cun-e. These faC'tors arc C'Ontaincd in Tables E-1

through E-4. In addition, Tables E-.5 and E-G contain factors for the modified trapC'zoi<l and modified sine

curves, which ha\·e not yet heen discussed. These com-bination cun·e~ "·ill he presented in Section X.

The tabulated ,·alm~s arc dimensionless, the earn angle factor going from Oto 120; the displacement fac-tor from Oto 1.

Equations. Using these factors, or coefficients, the characteristics of any basic curve ean be stated in generalized form. Displacement: y

=

K h Velocity: v

=

Crh(6: )

(G.V)

2 Acceleration: a = Cah

-{3

(1) (2) (3)

The tables a1Hl 1•1p1:.1tions are useful in many ways. They will he Hs<>d for calculating profiles, pressure angles, radii of f'lllTatur<', and inertial forces.

In Figs. E-1 through E-6 are. shown characteristic diagrams of the basic curves in terms of the coefficients.

E-2

Basic Curve Tables

TABLE

E-1.

_{CONSTANT VELOCITY FACTORS}

Pt. K

Cv

Co

Pt. K

Cv

Ca

Pt. K

cv

Ca

Pt. K

Cv

Ca

0 0.00000 1.0000 Zero 30 0.25000 1.0000 Zero 60 o. 50000 1.0000 Zero 90 0.7)000 1.0000 Zero

l o.oo833 31 0.25833 61 o.5o833 91 0.75833 2 0.01667 32 0.26667 62 o. 51667 SQ 0.76667 3 0.02500 33 0.27500 63 0. 52500 ₉₃ 0.77500 4 0.03333 34 0.28333 64 0.53333 9'i 0.78333 5 o.o4167 35 0.29167 65 0. 54167 9) 0.79167 6 0.05000 36 0.30000 66 0.55000 ₉₆ o.8oooo 7 0.05833 37 o.3o833 67 0.55833 ₉₇ o.8oS33 8 0.06667 38 0.31667 68 0.56667 ₉₈ 0.81667 9 0.07500 39 0.32500 69 0.57500 99 0.82;00 10 0.08333 4o 0.33333 70 0.58333 100 0.83333 11 0.09167 41 0.34167 71 0.59167 101 o.84167 12 0.10000 42 0.35000 72 0.60000 la? o.8)000 13 o.1o833 43 0.35833 73 o.6o833 103 0.85833 14 0.11667 44 0.36667 74 0.61667 104 o.86667 15 0.12500 45 0.37500 75 0.62500 105 0.87500 16 0.13333 46 0.38333 76 0.63333 166 0.88333 17 0.14167 47 0.39167 77 o.64167 107 0.8)1167 18 0.15000 48 o.40000 78 0.65000 lo8 0.90000 19 0.15833 49 o.40833 79 0.65833 109 0.90833 20 0.16667 50 o.41667 8o 0.66667 110 0.91667 21 0.17500 51 o.42500 81 0.67500 111 o.~:.iO<) 22 0.18333 52 o.43333 82 0.68333 112 0.93333 23 0.19167 53 o.44167 83 0.69167 113 o.94167 24 0.20000 54 o.45000 84 0.70000 114 o.<;6000 25 o.2o833 55 0.45833 85 o.7o833 11'.) 0.95833 26 0.21667 56 0.46667 86 0.71667 116 0.96667 27 0.22500 57 o.47500 ₈₇ 0.72500 117 0.97500 28 0.23333 58 0.48333 88 0.73333 118 0.98333 29 0.24167 59 o.49167 89 0.74167 119 0.99167 30 0.25000 6o o. 50000 90 0.75000 120 1.00000

K

CONSTANT VELOCITY

Cv

I. 0 ·~r--,----,--,,----,----r----r~----r~--r----.--r~--r--r--,----r--r---r---r-,---::i::;:==---ti .1

·~ i . . . -0.5·--

---~y-~--

-

~-~-~~-~~

--

-

..

-

--

-

--

- ,,... - 1- - - - - - I. 0

---o~---~---b..~~-~~~~+--+~f---t---t~t--t---t~+--+--1r--t--t-~t-;--t~t--r-~~o

0 10 20 30 40 50 60 70

80

90 IQ() 110 120

Pt. 0 l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 2j 26 21 28 29 30 K Cv 0.00000 0.0000 0.00014 0.0333 0.00056 0.0667 0.00125 0.1000 0.00225 0.1333 0.00347 0.1667 o.oo;oo 0.2000 0.00681 0.2333 o.oo889 0.26b7 0.01125 0.3000 0.01389 0.3333 0.01681 0.3667 0.02000 o.4000 o. 02347 o.4333 0.02722 o.4667 0.03125 o. 5000 0.03556 0.5333 o.o4014 0.5667 0.04500 0.6000 0.05014 0.6333 o.0;556 0.66b7 0.0612:; 0.1000 0.06722 0.7333 0.07347 o. 7667 o o8ooo 0.8000 0.08681 0.8333 0.09389 o.8667 0.10125 0.9000 0.10889 0.9333 0.11681 0.96b7 0.12500 1.0000

K

•

0 10 1.0 0.5

,,,,,

...

0

,_,,,,

5~r

0 - 5.0 0

.

10

Basic Curve Tables

TABLE

E-2.

CONSTANT ACCELERATION FACTORS

Co Pt. K Cv Ca Pt. K Cv Ca Pt. K 4.000 30 0.12500 1.0000 4.000 60 0.)0000 2.0000 -4.000 90 0.87500

..

_{31 0.13347 1.0333}

..

_{bl C.51653 1.9667 91 0.88319}

..

_{32 o.1~222 1.0667}

..

_{62 o. 53278 1.9333 ~ 0.89111}

..

_{33 0.15125 1.1000}

..

_{63 o.,4875 1.9000} 93 0.89875

..

_{34 0.160:,b 1.1333}

..

_{64 0. 56444 1.86b7} 94 o. 90611

..

_{35 0.17014 1.1667}

..

_{b5 0. 57986 U~333} 95 0.91319

..

_{36 0.13000 1.2000}

..

_{60 o. 59500 1.aooo 96 o.~ooo}

..

_{37 0.19014 1.2333}

..

_{t.;7 O.b0986 1.7667 g-r 0.926;3}

..

_{38 0.20056 1.2667}

..

_{68 o.62444 1. 7333} 98 0.93278

..

_{39 0.21125 1.3000}

..

_{b9 0.63~75 1.7000} 99 0.93875 If _{40 0.22222 1.3333}

..

70 0.65278 1.6667 100 o. 94441~ If _{41 0.23347 1. 3667}

..

71 0.6665'3 1.6333 101 o. 94986

..

_{42 0.24500 1.4000}

..

_{72 0. 68000 1.6000 102 0.95500}

..

_{4~ 0.25681 1.4333}

..

_{73 0.69319 1.5667} 103 0.95986

..

_{44 0.26889 1. 4667} If 74 o. 70611 1. 5333 .:.04 0.96444 If 45 0.28125 1.5000 If 75 0. 71875 1.5000 105 0.96875

..

_{46 0.29389 l.::>333}

..

_{76 o. 73111 1.4667 106 0.97278} If 47 0.30681 l.::,6b7

..

77 0.74319 1.4333 107 o. g-,653

..

_{48 o. 32000 1.6000}

..

_{78 0.75500 1.4000 lo8 0.98000}

..

_{49 0.33347 1.6333}

..

_{79 0.76653 1.3667 109 0.98319}

..

_{c;o 0.34722 1. 66b7}

..

_{80 0. 77778 1.3333 110 o. 98611}

..

_{51 o. 36125 1.7000}

..

_{81 0.78875 1.3000 111 0.98875}

..

_{52 0.37556 1. 7333}

..

_{82 o. 79944 1.2667 112 0.99111}

..

_{53 0.39014 l. 76t>7}

..

_{83 o.8o986 1.2333 113 0.99319}

..

_{54 o.40500 1.Booo}

..

_{84 o. 82000 1.2000 114 0.99500}

..

_{55 o.42014 1.8333}

..

_{85 o. 82986 1.1667 115 0.99653}

..

_{5b 0.43556} l. 8667

..

86 0.83944 1.1333 116 o.9g-r75

..

_{57 o. 45125 1.9000}

..

_{87 0.84875 1.1000 117 o. 99875}

..

_{58 o.46722 1. 9333}

..

_{88 0.85778 l. ei:>61 118 0.99944}

..

_{59 o.48J47 1.9667}

..

_{89 0.86653 1.0333 119 c.99986}

..

_{60 o. 50000 2.0000} If 90 0.87500 1.0000 120 1.00000

CONSTANT ACCELERATION

20 30

40

50 60 70

80

90

1()0

--

! ' ,

-

.,,

-

---

-

---

_ .... I

-~

-

__

...

· " I

--

-•i..

-f\J'i

---

_"t...,..,,'"

---

~--

---

,,,,

~

--

...

--

-

-

I

..

---

I I

--

-

_·-

~

--

,_

i -1 i I I

!Ca

I . I

-

-20

30

40

50

60

70

80

90

100

FIG. E-2

Point Position

E-3

Cv Ca 1.0000 -4.ooo 0.')667 0.9333 0.9000 o.8667 0.8333 o.8ooo 0. 7667 0.7333 0.1000 0.6667 0.6333 0.6000 o. 5667 0.5333 0.5000 o.4667 o.4333 o.4000 0.3667 0.3333 0.3000 0.2667 0.2333 0.2000 0.1667 0.1333 0.1000 o. ei:>61 0.0333 0.0000

Cv

'

110 120 2.0 1.0 ... ,_

_{.... __}

0 110 120

E-4

Pt.

0 l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 K 0.00000 0.00017 0.00069 0.00154 o.oce74 o.00428 o.oo615 o.oo837 0.01092 0.01381 0.017()4 O,c:205~ 0.024~7 o.c:2868 0.03321 0.038<>6 O,o4323 o.o4871 0.05450 o.o6Q59 o.o6698 0.07368 o.o8o66 0.08793 0.09549 0.10332 0.11142 0,11979 o.12SJ+3 0.13731 0.14644 1.0

K

•

0

0.5

Cv

ca

0.0000 4.9348 o. o4ll 4.9331 o.o822 4.9281 0.1232 4.9196 0.1642 4.9078 0.2050 4.8926 0.2457 4.8741 0.2862 4.8522 o. 3266 4.8270 0.3667 4.7985 o.4066 4. 7666 o.4461 4.7316 o.4854 4.6933 o. 5243 4.6518 o. 5629 4.6o71 o.6o11 4.5592 0.6389 4.5oee o.6762 4.4541 0.7131 4.3970 0.7495 4.3368 0.7854 4.2737 0.8207 4.2076 0.8555 4.1387 0.8897 4.o669 0.9233 3.9924 0.9>62 3.9150 0.9885 3.8351 l.c:2(2 3.7525 1.1511 3.6673 1.o813 3.;796 1.1107 3.489; 10 20

--

i

-

,_-0

1.1""

5.0

-

-....

I"""--,..

-0

Basic Curve Tables

TABLE E-3. HARMONIC FACTORS

Pt.

K Cv Ca

Pt.

K

cv

Co 30 0,14644 1.1107 3.4895 6o o. 50000 l.57o8 0.0000 31 0.15582 1.1394 3.3969 61 0.51309 1.5703 -0.1292 32 0.16543 1.1673 3.3020 62 o. 52617 1.5686 -0.2582 33 0.17527 1.1945 3.2o49 63 o. 53923 1.566o -0.3872 34 0.18543 1.2207 3.1056 64 0.55227 1.5622 -0. 5158 35 0.19562 1.2462 3.0041 65 o. 56527 1.5574 -0.6441 36 0.20611 1.2865 2.9006 66 o. 57822 1. 5515 -O.Tfl9 37 0.21679 1.2945 2.7951 67 o. 59112 1. 5469 -0.8993 38 0.22768 l. 3174 2.6877 68 o.6o396 l. 5365 -1.0260 39 0.23875 1.3393 2.5784 69 0.61672 1.5274 -1.1520 40 0.25000 1. 36o4 2.4674 70 o.62941 1. 5173 -1.2772 41 0.26142 1.38o4 2.3547 71 o.64201 1.5061 -1.4015 42 0.27300 1.3996 2.2404 72 0.65451 1.4939 -1. 5250 43 0.28474 1.4178 2 .1245 73 0.66691 1.4807 -1.6473 44 0.29663 1.4350 2.0072 74 0.67919 1.4665 -1. 7685 45 o.3o866 1. 4564 1.8884 75 0.69134 1. 4564 -1.8884 46 o. 32081 1.4665 1.7685 76 0.70337 1.4350 -2.0072 47 0.33309 l.48o7 1.6473 77 o. 71)26 1.4178 -2 .1245 48 o. 34549 1.493~ 1.5250 78 0.12100 1.3996 -2.24o4 49 0.35799 1.5061 1.4015 79 0.73858 1.38o4 -2.3547 jO _{0.37059 1.5173 1.2772 Bo 0.75000 1. 36o4 -2.4674} 51 0.38328 1.5274 1.1520 81 0.76125 1.3393 -2. 5784 52 0.396()4 1. ;365 l.026o 82 _{o. 77232 1. 3174 -2.6877} 53 o.40888 1.5469 0.8993 83 o. 78321 1.2945 -2.7951 54 o.42178 1. 5515 0.7719 84 _{0.79389 1.286; -2.9006}

55 o.43478 1. 5574 o.6441 85 o.8o438 1.2462 -3.0041 56 o. 44773 1.5622 o. 5158 86 _{0.81466 1.2207 -3.1056}

57 o.46o77 1. 56oO 0.3872 87 o.82473 1.1945 -3.2o49 58 o.47383 1.5686 0.2583 88 0.83457 1.1673 -3,3C20 39 o.48691 1.5703 0.1292 _{89 o.84418 1.1394 -3,3969} 60 0.50000 1. 57o8 0.0000 90 0.85356 1.1107 -3.4895

7rr_

-z..

HARMONIC

30

40

50 60

70

80

90 T

,_

---

,.-

--

---

l ,,.

-

_.

-

_---

---v"'i

..

-

, I

---

-

... _"'"

--

.

~-

...

~---

~ I ~

-

I I ~

·-

r-... ... ... r,,.... ... ... ii- ...

"""'

~a

-i',.... ...

Pt.

K Cv Co 90 0.85356 1.1107 -3· 4895 91 0.86269 1.o813 -3,5796 92 0.87157 1.0511 -3.6b73 93 o.88o21 1.0202 -3.7525 94 0.88858 0.9885 -3.8351 95 0.89668 0.9562 -3,9150 96 0.90451 o. 9233 -3· 9924 97 0.91207 0.8897 -4.0669 98 0.91934 0.8555 -4.1387 99 0.92632 0.8207 -4.2076 100 0.93302 0.7854 -11.2737 101 0.93941 o. 7495 -4.3368 102 0.94550 0.7131 -4.3970 103 0.95129 0.6762 -4.4541 lo4 0.95677 0.6389 -4.;o82 105 0.96194 o.6o11 -4. 5592 106 0.96679 0.5629 -4.6o71 107 0.97132 0.5243 -4.6518 1o8 0.97553 o.4854 -4.6933 109 0.97941 o.4461 -4.7316 110 0.98296 a.4066 -4. 7666

.

111 0.98619 0.3667 -4. 7985 112 0.989()8 0.3266 -4.8270 113 0.99163 0.2862 -4.8;22 114 0.99385 0.2457 -4.8741 115 0.99572 0.2050 -4.8926 116 o. 99726 0.1642 -4.9078 117 0.99846 0.1232 -4.9196 118 0.99931 o.o822 -4.9281 119 0.99983 o.o411 -4.9331 120 1.00000 0.0000 -4. 9348

Cv

t

100 110 120 2.0 ~

--

"'"-...

_,...,

1.0 ~ ...

-

-

_....

-0 ,-.

.

1 ... _

-5.0

0

c~

10

20

30

FIG. E-3

50

60

10

80

Point Position

-

i,....__

r--

--90

100 110 120 Corrected Match 1962

Pt.

K

Cv

Ca

Pt.

0 0.00000 0.0000 0.0000 30 l 0.00001 0.0014 o. 3289 31 2 0.00003 0.0055 0.6568 32 3 0.00010 0.0123 o. 9829 33 4 0.00024 0.0218 1. 3o63 34 5 0.00048 0.0341 1.6262 35 6 0.00082 o.0489 l. 9416 36 7 0.00130 o.o664 2.2517 37 8 0.00194 o.o865 2.5556 38 9 0.00275 0.1090 2.8525 39 10 0.00375 0.1340 3.1416 40 11 o. 00499 0.1613 3.4221 41 l2 O.oo645 0.1910 3.6931 42 13 O.oo817 0.2228 3. 9541 43 14 0.01018 0.2569 4.2043 44 15 0.01246 o. 29'29 4.4429 45 16 O.O'.l.5o6 0.3309 4.6693 46

iJ

0.01799 0.3707 4.8830 47 0.02124 o.4122 5.o832 48 19 0.02486 o.4554 5.2695 49 20 0.028j84 o. 5000 5.4414 50 21 0.03320 o. 546o 5. 5984 51 22 0.03794 0.5933 5.7400 52 23 o.04309 o.6416 5.8659 53 24 o.o4864 0.6910 5.9757 54 25 o.0546o 0.7412 6.o691 55 26 O.o6100 0.7921 6.1459 56 27 o.o6781 o.8436 6.2059 57 28 0.07505 0.8955 6.2488 58 29 o.o8274 o. 9477 6.2746 59 30 0.09()85 1.0000 6.2832 6o

-z~

. / ; 10 20

30

-

. v , _,

,,,,

~

....

... i-.,1 ...

-._-

--

.

-~·

~ 6.0

,,,

/ / 4.0

'

2.0

_,

/

I 0

/

0

10

20

30

t

Co

FIG. E-4

Basic Curve Tables

TABLE

E-4.

CYCLOIDAL FACTORS

K

Cv

ca

Pt. K

Cv

ca

0.09()85 1.0000 6.2832 6o 0.50000 2.0000 0.0000 0.09940 1.0523 6.2746 61 0.51666 1.9986 -0.3289 o.1o839 1.1045 6.2488 62 0.53331 1.9945 -o.6;68 0.11781 1.1564 6.2059 63 o. 54990 1.¢77 -0.9829 0.12766 1.2079 6.1459 64 o. 56642 1.9782 -l. 3o63 0.13794 1.2588 6.0691 65 o. 58286 l. 9659 -l.6262 0.14864 1.3090 5.9757 66 o. 59918 1.9511 -1.9416 0.15975 l. 3584 5.8659 67 0.61536 1.9336 -2.2517 0.17128 1.4o67 5.7400 68 0.63140 1.9135 -2.5556 0.18320 l. 4540 5. 5984 69 o.64725 1.8910 -2.8525 0.19550 l. 5000 5.4414 10 0.66291 1.8660 -3.1416 o.2o820 1. 5446 5 .2695 71 0.67835 1.8387 -2.4221 0.22124 l. 5878 5.o832 72 0.69355 1.8o90 -3.6931 0.23465 1. 6293 4.8830 73 0.7o849 l. 7772 -3. 9541 0.24840 1.6691 4.6693 74 0.72316 1. 7431 -4.2043 0.26.?46 l. 7071 4.4429 75 0.73754 1.7071 -4.4429 0.27684 L 7431 4.2043 76 o. 7516o 1.6691 -4.6693 0.29151 1.m2 3. 9541 77 o. 76535 1.6293 -4.8830 o.3o645 1.809() 3.6931 78 0.77876 1.5878 -5.o832 0.32165 1.8387 3.4221 19 o.7918o 1.5446 -5 .2695 0.33709 1.8660 3,1416 8o o.8o450 1.5000 -5.4414 o. 35275 1.8910 2.8525 81 o.81£i8c 1.4540 -5.5984 o.3686o 1.9135 2.5556 82 0.8287~! l.4o67 -5.7400 o. 38464 l. 9336 2.2517 83 o.84025 1.3584 -5.8659 o.4oo82 l. 9511 l. 9416 84 0.85136 l.3Q90 -5. 9757 o.41714 1.9659 . l. 626.? 85 o.862o6 1.2588 -6.o691 o.43358 1.9782 1. 3o63 86 0.87234 1.2079 -6.1459 o.45010 1.9BT7 0.9829 87 0.88219 l.1564 -6.2059 o.46o69 1.9945 0.6;68 88 0.89161 1.1045 -6.2488 o.48334 J..9986 o. 3289 89 0.9()06o 1.0523 -6.2746 0.50000 2.0000 0.0000 90 0.90915 1.0000 -6.2832

CYCLOIDAL

40

50 60 70

80

90

--

-

..

-

.,

-.... ,_

--

-- loo'

---~,-i., ~

-

... ~

--I ~

"-"

..--

-

""'---

-

---...

_.

"'-

'·,"-"

\.

'"Ca

'

,..,

·,

'\.

I I

'

'

~

'

_·~

...

Pt.

K 90 0.90915 91 0.91726 92 o. 92495 93 0.93219 94 0.93900 95 o. 94540 96 0.95136 97 0.95691 98 o.962o6 99 0.9668<> 100 0.971116 101 0.97514 102 0.97876 103 0.98201 lo4 0.98494 105 0.98754 lo6 0.98982 107 0.99183 lo8 0.99355 109 0.99501 llO 0.99625 111 0.99725 112 o.998o6 113 0.99870 114 0.99918 115 0.99952 116 0.99916 ll7 0.99990 118 0.99997 119 0.99999 120 1.00000 IOO 110 I ' ' ._ ...

--·-

...

I I

1/

.

,

4

_/.,,

.• .

E-5

Cv

Ca 1.0000 -6.2832 0.9477 -6.2746 0.8955 -6.2488 o.8436 -6.2059 o. 7921 -6.1459 0.7412 -6.o691 0.6910 -5.1_R57 o.6416 -5.8659 0.5933 -5.74oo o.546o -5.5984 0.5000 -5.4414 o.4554 -5.26~ o.4122 -5.o832 0.3707 -4.8830 0.3309 -4.6693 0.2929 -4.4429 0.2569 -4.2043 0.2228 .3.9541 0.1910 -3.6931 0.1613 -3.4221 0.1340 -3.1416 0.1090 -2.8525 o.o865 -2.5556 o.o664 -2.2517 o.o489 -1. 9416 0.0341 -1.6262 0.0218 -1.3o63 0.0123 -0.9829 0.0055 -0.6568 0.0014 -0.3289 0.0000 0.0000

Cv

12b

2.0

1.0 0

J

0

.,,

-2.0 c, -4.0 0 I

...

. / -6.0

-40

50 60

70

80

90

100

110

120

Point Position Corrected Sept. 1963

E-6

Pt. 0 l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 K 0.00000 0.00001 0.00005 0.00016 0.00037 0.00073 0.00120 0.00198 0.00293 O.oo413 0.00561 0.00738 0.00946 0.01186 0.0l46o 0.01767 o.021o6 0.02483 0.02894 o.0333r 0.0381 o.o4321 o.o4864 0.05445 0.06061 o.o67o6 0.07386 o.o8100 o.o8847 0.09630 O.lo451

K

J

0 1.0 0.5 0

Baai c Curve Tables

TABLE E-5. MODIFIED TRAPEZOID FACTORS

Cv

Ca

Pt. K

Cv

Ca

Pt. K

cv

ca

Pt. K

Cv

0.0000 0.0000 30 O.lo451 1.0000 4.8881 6o 0.50000 2.0000 0.0000 90 0.89549 1.0000 0.0021 0.5110 31 0.11300 1.0407 4.8881 61 0.51667 l,9'179 -0.5110 91 0.90370 o. 9593 o.oo85 1.0163 32 0.12181 1.o815 4.8881 62 0.53328 1. 9915 -1.0163 92 0.91153 0.9185 0.0190 1. 5105 33 0.13100 1.1222 4.8881 63 o. 54985 1.9810 -1. 5105 93 0.91900 0.8778 0.0336 1.9882 34 0.14053 1.1629 4.8881 64 o. 56623 1.9664 -1.9882 94 0.92614 0.8371 0.0521 2.4440 35 0.15036 1.2037 4.8881 65 0.58255 1.9479 -2.4440 95 0.93294 o. 7963 0.0743 2.8731 36 0.16o57 1.2444 4.8881 66 0.59873 1. 9257 -2.8731 96 0.93939 0.7556 0.0999 3,27o8 37 0.17113 1.2851 4.8881 67 0.61467 1.9001 .3.27o8 97 o. 94555 0.7149 0.1287 3.6326 38 0.18203 l. 3259 4.8881 68 0.63036 1.8713 .3.6326 98 0.95136 0.6741 0.16o4 3,9546 39 0.19323 1.3666 4.8881 69 o.64586 1.8396 . 3. 9546 99 0.95679 0.63.34 0.1945 4.2333 40 0.2o471 1.4073 4.8881 70 0.66101 1.8o55 -4.2333 100 0.96187 0.,927 o.23o8 4.4655 41 0.21669 1.4481 4.8881 71 0.67592 1. 7693 -4.4655 101 0.96666 0.5519 0.2688 4.6489 42 0.22891 1.4888 4.8881 72 0.69053 1. 7312 -4.6489 102 o.971o6 o. 5512 o.3o81 4.7813 43 0.24147 1. 5295 4.8881 73 o. 70476 1. 6919 -4.7813 103 0.97517 o.4705 o. 3483 4.8613 44 0.25443 1.5703 4.8881 74 o. 71873 1.6517 -4.8613 104 0.97894 o.4297 0.3890 ~.8881 45 0.26767 1.6110 4.8881 75 o. 73233 1.6110 -4.8881 105 0.98233 0.3890 o.4297 4.8881 46 0.28126 1.6517 4.8613 76 0.74557 1. 5703 -4.8881 1o6 0.98540 0.3483 o.4705 4.8881 47 0.29524 1.6919 4.7813 TI 0.75853 1.5295 -4.8881 107 0.98814 o.3o81 0.5112 4.8881 48 0.30947 L 7312 4.6489 78 O.TI109 1.4888 -4.8881 lo8 0.99054 0.2688 0.5519 4--.8881 49 o . .324o8 1. 7693 4.4655 79 0.78331 1.4481 -4.8881 109 0.99262 o.23o8 o. 5927 4.8881 50 0.33899 L8o55 4.2333 8o o. 79529 l.4o73 -4.8881 110 o. 99439 0.1945 0.6334 4.8881 51 0.35414 1.8396 3.9546 81 o.8o677 1.3666 -4.8881 111 0.99587 0.16o4 0.6741 4.8881 52 0.36964 1.8713 3. 6326 82 0.81797 1. 3259 -4.8881 112 o.m01 0.1287 0.7149 4.8881 53 0.38533 1.9001 3.27o8 83 0.82887 1.2851 -4.8881 113 0.99802 0.0999 0.7556 4.8881 54 o.40127 l. 9257 2.8731 84 0.83943 1.2444 -4.8881 114 0.99874 0.0743 0.7963 4.8881 55 o.41745 L 9479 2.44-40 85 o. 84964 1.2037 -4.8881 115 0.99927 0.0521 0.8371 4.8881 56 o.43377 1.9664 1.9882 86 o.85947 1.1629 -4.8881 116 0.99963 0.0336 0.8778 4.8881 57 o.45015 1.9810 1. 5105 87 0.86900 1.1222 -4.8881 117 0.99984 0.0190 0.9185 4.8881 58 o.46672 1.9915 1.0163 88 0.87819 1.o815 -4.8881 118 0.99995 o.oo85 0.9593 4.8881 59 0.48333 l.9'179 0.5110 89 0.88700 1.()407 -4.8881 119 0.99999 O.Oa?l 1.0000 4.8881 6o 0.50000 2,0000 0.0000 90 0.89549 1.0000 -4.8881 120 1.00000 0.0000

MODIFIED TRAPEZOID

10 20 30 40 50 60 70 80

90

100 110 --

_.,,--

_

..

,.._

-

-...

--

-_.,- ~ _.,,,,.

--

_...

l.~ _{__}

..

...-

\C.

._...

--

_....

~ -....

....

-~

-

...

----,,,.

__...

...__

,,,.

--

-....

....

--

--Ca

-4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8881 -4.8613 -4.7813 -4.6489 -4.4655 -4.2333 -3.9546 -3.6326 .3.27o8 -2.8731 -2.4440 -1.9882 -1. 5105 -1.0163 -0.5110 0.0000

Cv

i

120 2.0 1.0 >

u

0

•Cycloid-. . - Constant Ac eel.____,. .. cycloid~ ... cycloid ....

~

Constant Accel.~Cycloid__.,

5.0 I 0

I

-5.0 0

c~

,

/ I J

,

~-

-

-·-

-

·-

-

-

-

...

, ' I I

!

10 20 30

40

FIG. E-5

i .... !

\

\.

'

.

"

\

\Ca

_J

'

.

\ I

\

)

_'

\

/ , ,

,

..

_-

_--

_-·

..~

...

-

·--·

50 60 70 80

90

100 110 120 Point Position

Basic Curve Tables

TABLE E-6. MODIFIED SINE

Pt. K Cv

Co

Pt. K

Cv

Ca

Pt. K 0 0.00000 0.0000 0.0000 30 0.11718 1.0997 4.7874 6o 0.50000 1 0.00000 0.0024 o. 5778 31 0.12650 1.1392 4.638o 61 o. 51466 2 0.00005 0.0096 1.'1493 32 0.13616 1.1778 4. 5829 62 o. 52931 3 0.00018 0.0215 1. 7o82 33 0.14613 1.2156 4.4722 63 o. 54393 4 0.00042 0.0380 2.2484 34 0.15642 1.2524 4. 3561 64 0.55851 5 o.ooo83 0.0589 2.7640 35 0.16701 1.2882 4.2347 65 0.57304 6 0.00142 o.o84o 3.2493 36 0.17788 1. 3229 4.lo81 66 o. 58750 7 0.00224 0.1130 3.6989 37 0.18905 L 3566 3. 9765 67 0.6o178 8 0.00331 0.1455 4.lo81 38 0.20049 1. 3892 3.8401 68 0.61611 9 o.oo467 0.1813 4.4723 39 0.21220 1.4206 3.6989 ()9 0.63035

lC o.oo634 0.2199 4.7874 40 0.22417 1.45o8 3.5553 70 o.64442 11 o.oo834 0.2610 5.0501 41 0.23638 1.4798 3.4034 71 0.65835 12 o.01o69 o. 3040 5.2574 42 0.24883 1.5075 3.2493 72 0.67214 13 0.01341 0.3484 5.4072 43 0.26150 1. 5340 3. 0912 73 0.68577 ,.4 0.01650 0.3939 5.4977 44 0.27439 1. 5590 2. 9294 74 0.69924 b 0.01998 o.4399 5.5280 45 0.28748 1.5828 2. 7640 75 0.71252 lb 0.02389 0.4860 5.5246 46 0.30076 1.6051 2. 5952 76 0.72561 l7 o.028o8 o. 5320 5.5145 47 o. 31423 1.6260 2.4233 77 0.73850 18 0. 03210 o. 5778 5.4977 48 o. 32786 1. 6455 2.2484 78 0.75117 19 0.03771 o. 6236 5.47<+2 49 0.34165 1. 6635 2.0708 79 0.76362 20 0. ()4310 o. 6691 5.4440 50 0.35558 1.68oo 1.8907 80 0.77583 21 o.04886 0.7143 5.4072 51 0.36965 1.6950 1.1082 81 0.7878() 22 0.05500 o. 7')92 5.3638 52 0.38383 1.7~ 1. 5237 82 o. 79951 23 6.06151 o.8o36 5.3138 53 0 39812 1. 72o4 1.3373 83 0.81095 24 0.06839 o. 8477 5.2574 54 0.41250 1. 7307 1.1493 84 0.82212 2;; o. 07564 0.8912 5 .1946 55 o.42696 1. 7395 o.9j99 8') 0.83299 2b o.o8324 0.9343 5 .1255 56 o. 411149 1. 7467 O 7693 86 o.84358 27 0. 09120 o. 9767 5.0501 57 o.45607 1. 7523 o. 5778 87 0.85387 28 0.09952 1.0184 4.9685 j8 o.47o69 1.7564 0.3856 88 0.86384 29 o.1o818 1.0594 4.88o9· j9 o.48534 1. 7588 0.1929 89 0.87350

30 0.11718 1.0997 4.7874 6o 0. 50000 1.7596 0 0000 90 o.88282

K

i

MODIFIED SINE

1.0 0 10

20

30

40

50

60

70

._-

....

-

-·

~-,.

.

...

-v ... ,,,

--

--__ ,~

--_

...

_J ~ 0.5 -... '~

---

,

----~ ~ 0

-

....--

~ ~cycloid Harmonic 6.0 _{..._}

-~

----.

-I

...

' ... ~ ~ I

I.

..

'-i

_'·

Co

I

_~

'

0

.

"'

-

..

FACTORS

Cv

Ca

Pt. 1.7596 0.0000 90 1.7'588 -0.1929 91 1.7564 -0.3856 92 1. 7523 -0. 5778 93 1.7467 -0.7693 94 1.7395 -0.9599 9) L 7307 -1.1493 96 l . 7204 -1. 3373 91 L 7o84 -1.5237 98 1._6950 -1. 7o82 99 1.6800 -1.8907 100 l. 6635 -2.07o8 101 1.6455 -2.2484 102 1.6260 -2.4233 103 l. 6051 -2. 5952 ¥)4 1. 5828 -2.7640 10') 1. 5590 -2. 9294 106 l. 5340 -3.0912 107 1.5075 -3.2493 1o8 1.4798 -3.4034 109 l. 45o8 -3-5553 110 1.4206 -3.6989 111 1. 3-'392 -3. 8401 112 1. 3566 -3. 9765 113 1.3229 -4.lo81 114 1.2882 -4.2347 115 1.2524 -4.3561 116 1.2156 -4.4722 117 1.1778 -4. 5829 118 1.1392 -4.6880 119 1.0997 -4.7874 120 80

90

100

~--~

...iill"'li,;,..

-,_

1-.

-""-

_~""'

E-7

K

Cv

Ca

o.88282 1.0997 -4.7874 0.89182 1. 0594 -4.8809 o. 9()048 1.0184 -4.9685 0.90880 0.9767 -5.0501 0.91676 0.9343 -5.1255 o. 924 36 0.8912 -s.1946 0.93161 o.8477 -5.2'574 0.93849 o.8o36 -5.3138 o. 94500 o. 7592 -5.3638 o. 95114 0.7143 -5.4072 0.95690 0.6691 -5.4440 o. 96229 0.6236 -5.4742 0.96730 O.j778 -5. 4977 0.97192 o. '.:1320 -5.5145 o. 9'7611 0.4860 -5.5246 0.98002 10.4399 -5.5280 0.98350 o. 3939 -5.4977 0.98659 0.3484 -j.4072 0.98931 0.3040 -5.2'574 0.99166 0.2619 -j .0501 0.99366 0.2199 -4.7874 0.99533 0.1813 -4.4722 0.99669 0.1455 -4.1081 o. 99TI6 0.1130 -3.6989 0.99.'.358 o.o84o -3.2493 0.99917 0.0589 -2.7640 0.99958 0.0380 -2.2484 0.99982 0.021) -l. 7o82 0.99995 0.0096 -1.1:.93 1.00000 o.oa:>4 -0.'.;1778 1. 00000 . 0. 0000 0.0000 110

Cv

t

120 2.0 I .0 ~ ....

"

~ ... 0 : Cycloid ... j I .,..._

I

""""'

,

I

...

. I _...

I

l ...

_

---

_·-

.-,' -6.0 O 10 20

30

40

50 60

70

80

90 100 110 120

ct

FIG. E .. 6

Point Position

SECTION F

F-1

Cam Size Determination

Basic Considerations. Usually the first consideration

in designing a ram is its physical size. A cam of mini-mum size may be desired, or a cam of a fixed size may have to be evaluated. The determining fac.tors for a cam of minimum size are the maximum pressure angle, the least radius of curvature, and the cam shaft diame-ter.

FIG. F-1

FIG. F-2

Pressure Angles. As may be seen in Figs. F-1 and

F-2, the greater the pressure angle, the greater the side thrust on the follo,Yer. Too much side thrust may re-sult in jamming the follower rod in its guides. For this reason, it is customary to limit the maximum pressure angle to approximately 30 degrees for cams with trans-lating followers. If the follower is on center, pressure angles may be easily determined mathematically. The mathematics for an off-center or swinging arm follower are more complex, but pressure angles may be found by inspection from a layout, or by graphical methods which will be described later. Pressure angles depend on the velocities of the cam and follower. From Fig. F-3, the tangent of the pressure angle equals the quotient of the follower velocity and the cam velocity.

Vr

tan 'Y

=

-i'c

(1)

Converting this equation to terms of displacements of follower and cam, this becomes

57.3Crh

tan 'Y

=

-R,,J3 (2)

where Rn

=

radius to the reference point on the pitch curve, and the other symbols as in section E nomencla-ture. This equation is valid at any point on the pitch curve with an on-center translating follower.

FIG. F-3

Pitch Radius. To determine the pitrh radius of a

cam of minimum size, the following equations are con-venient:

For a cycloidal, constant acceleration or modified trapezoid curve,

For a harmonic curve,

For a modified sine curve, 200h (3 I.57h (3 176h RP= -(3 (3) (4)

(S)

These equations give the pitch radius of a cam with a maximum pressure angle slightly less than 30 degrees.

Radius of Curvature If, as shown in Figs. F-4 and F-.S, the radius of curvature (Re) is held constant and the radius of follower (r1) increased, the cam profile will

eventually become undercut, and the follower will not follow the prescribed motion. To prevent this, the least radius of curvature must be substantially greater than the radius of the follower. A rigorous calculation of the radius of cun'at.ure is quite involwd. However, the follO\ving- method gives sufficiently . · ~qrate results for an on-center translating motion. Gra.phical ....,rth-ods for all types are described later.

I

I

FIG. F-4

FIG. F-5

In Fig. F-6, for a very small section AB of a cam pitch c'urve, it is reasonable to assume that the center of the radius of curvature lies very near the intersection of normal lines AP and BP. It is also reasonable to as-sume that, for the small section involved, arcs AB and AD are substantially equal. With these assumptions:

RA sin M =

R,

sin a

F-2

Cam Size Determination

FIG.

F-6

From the geometry of the layout

a

=

'YA - 'YB

+

AO

where -y A and 'YB are the pressure angles at reference points A and B. Therefore,

R,

= RA

si~ AfJ

sm a

(6)

If a is positive, the center of

R,

lies inside the pitch curve periphery; if a is negative, the center lies out-side the periphery. This method gives good accuracy near the terminals of the curve, where the increment BD is quite small. Accuracy decreases as the transition point is approached. In this area, more nearly accu-rate results may be obtained by a better approximation of arc AB. AD

RA

sin A() BD

Ra - RA

AB

VAD

2

+

_BD2

R,

AB sin a (7)

The least radius of curvature occurs at or near the point of maximum negative acceleration. Calculation of the radius at this point provides an evaluation of the cam size as compared with the diameter of the follower.

Cam Shaft Diameter. Usually before the cam has

been designed the cam shaft diameter has been de-termined from stress and deflection factors. The cam must have a hub of sufficient size to accommodate this shaft and its key. The base radius of an open cam profile must be greater than the hub radius. This is also necessary on closed cams where the hub is on the grooved side.

Graphical Methods. Procedures for graphically

de-termining pressure angles, pitch radii, and radii of curvature are described under Figs. F-7 through F-12. The methods are as accurate as the draftsmanship.

Pressure Angles-Nomenclature

h total linear displacement of follower (in.) b length of swinging arm (in.)

r radius-cam axis to reference point (in.) x offset of follower center from cam axis (in.)

C v coefficient of velocity at reference point

ct, total angular displacement of swinging arm (deg.)

/3 total angular displacement of cam (deg.)

"Y cam pressure angle (deg.)

On-Center Translating Follower. See Fig. F-7.

To determine the pressure angle at anv reference point A when linear and angular displac;ments and radius are known,

1. Draw AB equal to 57.3C J,,/{J in direction of fol-lower motion.

2. Draw AC equal and perpendicular to radius r(OA) in direction of cam rotation.

3. Draw BC to complete triangle ABC. 4. Draw AD perpendicular to BC.

5. Measure pressure angle -y.

To determine radius to the pitch curve for desired pressure angle at any reference point A when linear and angular displacements are known,

I. Draw AB equal to 57.3C J,,/(J in direction of fol-lower motion.

2. Draw AD, indefinite in length, at desired angle

-y with AB so that BD will be in direction of cam rotation.

3. Draw BC, indefinite in length, perpen~icular to AD.

4. Draw AC perpendicular to AB in direction of cam rotation.

5. Scale AC, which is the required radius.

Cam Size Determination

_F-3

Offset Translating Follower. See Fig. F-8(a).

To determine pressure angle at any reference point A when linear and angular displacements and radius are known,

1. Draw AB equal to 57.3C0h/{3 in direction of fol-lower motion.

2. Draw AC equal and perpendicular to radius r(OA) in direction of cam rotation.

3. Draw BC, completing triangle ABC. 4. Draw AD perpendicular to BC. 5. Measure pressure angle 'Y.

rx~B

I

I

o ~

-<a>

FIG. F-8

(b)

To determine radius to pitch curve for desired pres-sure angle at any reference point A, when linear and angular displacements are known, Fig. F-8(b),

1. Draw AB' equal to 57.3C 0h/{3, turned 90° from line of follower motion, and in direction of cam rotation.

2. Draw AD', indefinite in length, at desired angle

-y with AB'.

3. Draw B'O, perpendicular to AD', intersecting vertical cam axis at 0.

4. Scale OA, which is the required radius.

Swinging Arm Follower. See Fig. 9.

To determine pressure angle at any reference point A when angular displacements of arm and cam, radius, arm length, and relative position of cam axis O and fulcrum

Q

are known, Fig. 9(a),

1. Draw AB equal to C0b<1>/f3 perpendicular to QA in direction of follower motion.

2. Draw AC equal and perpendicular to OA in direc-tion of cam rotadirec-tion.

3. Draw BC completing triangle ABC. 4. Draw AD perpendicular to BC. 5. Measure pressure angle -y.

8

Q_

Q

To determine radius to pitch curve for desired prPs-sure angle when angular displacement of arm and cam, arm length, and relative position of cam axis O and fulcrum

Q

are known, Fig. 9(b),

CASE I. When relation of fulcrum Q to either the horizontal axis X-X, or the vertical axis Y-Y is kno"·n. Fig. F-9(b).

1. Draw AB' equal to Cb<t,/{3, turned 90° from line of follower motion, and in direction of cam rota-tion.

2. Draw AD', indefinite in length, at desired angle 'Y

with AB'.

3. Draw B'O, perpendicular to AD' intersecting axis X-X or Y-Y at 0.

4. Scale OA, which is the required radius.

CASE

II.

When relation of fulcrum Q to cam axis is not fixed, Fig. 9(a),

1. Draw AB equal to Cr:b<f>/{3 perpendicular to QA in direction of follower motion.

2. Draw AD, indefinite in length, at desired angle

'Y with AB, so that BD will be in direction of cam rotation.

3. Draw BC, indefinite in length, perpendicular..,to AD.

4. With A as center, inscribe any radius, intersect-ing BC at C. (Suggested radius approximately equal to QA.)

5. Draw AO equal and perpendicular to AC, thus locating cam axis 0.

F-4

Cam Size Determination

Radius of Curvatur~Nomenclature total linear displacement of follower (in.) length of swinging arm (in.)

coefficient of velocity at reference point. coefficient of acceleration at reference point velocity of follower at one radian per second cam shaft speed (in./sec.)

a, acceleration of follower at one radian per second cam shaft speed (in./sec. 2)

x

normal acceleration of swinging arm at one radian per second cam shaft speed (in./sec. 2)

offset of follower center from cam axis (in.) total angular displacement of swinging a.rm (deg.)

total angular displacement of cam (deg.)

On-Center Translating Follower. See Fig. F-IO(a) and (b).

(

(a)

E

FIG. F-10

To determine radius of curvature at any point F,

I. Draw OA equal to v1

=

57.3C )&/{3, turned 90°

in direction of cam rotation, directed toward 0. 2. Draw OB equal to a₁ = Cah(57.3/{3)2, directed

to-ward 0.

3. Draw BC perpendicular to AF.

4. Draw CD, indefinite in length, parallel to OA. 5. Dra.w AD, parallel to OF, interesecting CD at D. 6. Draw OD, intersecting AF at E.

7. Scale EF, which is the radius of curvature.

Off'set Translating Follower. See Fig. F-11 (a) and (b).

To determine radius of curvature at any point F, pro-ceed exactly as in the case of the on-center follower.

(a)

E

FIG. F-11

Swinging Arm Follower. See Fig. F-12.

To determine the radiu~ of curvature at any point F,

I. Draw OA equal to v1

=

Cib<l>//3, turned 90° in

direction of cam rotation directed toward

0.

2. Draw FH,. equal to OA, intersecting circle QHF

at H.

3. Draw HG perpendicular to QF.

4. Draw OJ, equal to a,.

=

GF, parallel to QF, di-rected in sense of F to Q.

5. Draw JB equal to a1

=

57.3C,,b4>/f32, directed

toward J.

6. Draw BC perpendicular to AF.

7. Draw CD, indefinite in length, parallel to OA. 8. Draw AD, parallel to OF, intersecting CD at D. 9. Draw OD, intersecting AF at E.

10. Scale EF, which is the radius of cu'rvature.

/\

I I '

FIG. F-12

References

1. Cams-Design, Dynamics and Accuracy. H. A. Rothbart. John Wiley & Rons, Inc. 1956.

2. Cam Pressure Angles. R. T. Hinkle. Machine Design. July

1955.

3. Disc Cam Curvature. J. Hirachom. Fifth Mechanism Con-ference Tr&DBactions. 1958.