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NSE TECHNICAL TRANSACTION VOLAS, NOJ.JULY - SEPTEMBER,.Z010 
0.E . SIMQLOWO A N D  0. A. B-0 
SOFTWARE DEVELPOED FOR POLYNOMIAL 
CAMS DESIGN AND ANALYSIS 
0. E, SIMOLOWO* AND 0. A BAMKRO* 
Mechanical Engineering Department 
University Of Ibadan Nigeria 
Abstract 
Tke n w r i e a f  &sign of c a m  that leads Precisely, the eflect of using $ha 3-4-5 
to their 3rd m w n ~ u r eha s always and @- OApTol yn~mialf unctions .on 
been ussmiated with much tedium and S h e  two parametem while designing 
time consuwu'ng tnathmattcul mrd plate cams with r e c i p ~ s i nfgf at face 
gmphical proeedrrres. This is die to the followers has been andyzed in this 
number of lengthy eqtrutiotrs artd many work Results show that these two 
h i g n  options involved in such &sign parameters are directly influenced by 
processes. Therefore, in this work a the type of cam function used rather than 
signifimt contribution has been ma& the cam angular segments in the 
in cam s y a m  design Md analysis, by, optimum design und'final ~llrmIrf?adure 
developing a sofhvnre package and of cam system. In this work reliable 
applying it to the analysis ofpolynomid software capable of petforming 
cams. o critical pammelers extensive optimum design analyses when 
assmiatcd with t design and properly utilized has been deveIopd 
nan~fmtureso f cum system'hm been 
Key Words; Cmu; S o j b ~ r eA;  na&ses; 
analyzed; the pdme circle radius of the 
0pti11ur1n -Design 
cam und face wath of the Joltower. 
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NSE TECHNICAL TRANSAmION VOL.45, N0.3, JULY - SEPTEMBER, 2010 
0. E. SlMOLOWO AND 0.A , BAMIRO 
R, :: is Prime circle radius i,e. radius of the smallest circle drawn with center at the cam 
rotation axis and tangent to the pitch curve. 
P, :f ollower face width 
fl  :c am angle interval for motion segments 
6 :o ffset of the follower face 
0 :a ngle of cam rotation 
C :t he instanbmus center of curvature. 
p :r adius of curvature corresponding to the current contact point of cam and follower 
: m i n i mr adius of curvature 
{ P, Y] :c omdimtes of the contact point in a coordinate system attached to the carn 
s . :d istance of travel, of the point of contact either side of the cam rotation center 
Y :d isplacement of follower with respect to 8 
Y' :m t derivatives of displacement of follower with respect to 0 
Yw :s econd derivatives of displacement of follower with respect to e 
r :distance of the instantaneous center of curvatwe to the earn center 
L Im aximum displacement of follower (lift) 
conclusions on how they affect the final 
manufacture of such cam systems. The 
The impact of newly developed computer numerical and mathematical procedures of 
aided methud, in form of CAD software and 
cam systems design are tedious, 
packages in the industrial and engineering cumbersome and ti& involving, This work 
w d d  today cannot be over stated. As a 
oveicomes all associated tedium with cam 
follow up to this trend, the objectives of this profile design and simulation analysis for 
work are, 6 ) t o develop a software package optimum design and find manufacture of 
for the design and simulation of plate cams such systems and in s h m r t ime. 
with flat-face follower (ii) perform cam 
design analysis using different polynomial The synthesis and application of relevant 
cam functims and derive inferential design concepts and equations in the 
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NSE TECEINICAL TRANSACTION VOLAS, NO.3, JULY - SE-ER, 2010 
0.E . SIIMOLOWO ,AND 0. A, BAMIRO 
development of software for selected emanated from the p-t'work include; the 
options were based on works done by development of a profile simulation, 
various researchers. Shigley and UicIrer, software , for both trigonometric and 
(1980) expounded on the theory of cam polynomial cams (Simolowo and Barniro, 
mechanisms and dwelt much on different 2007); the development of an anaysis- 
types of follower motions with emphasis on intensive software for improved cams 
the roller and flat face follower cam systems design (Sirnolowo and Barniro, 
mechanisms. This has led to the design and 2008). 
analysis using the flat-face fo~lower as 
carried out in this work Tesar and Mathew, Also, presented in this paper are numerical 
able cam systems detailed'design procedures for cam systems (1976) were to classify 
based on the generalized fundamental 
based on values of the dynamic factor (pd) 
functions p d n t e d  by most researchers 
and made recommendations for cam 
mentioned above. In addition, the result of 
designers b w d  on these values. Andrezej 
analysis brings out valuable inferences on 
Oledzki, (1980) approached cam design T- 
the optimum design of Crlm profiles and 
from dynamic analysis of the mechanism. 
presents chrer pictures in the design trend 
Virgil Moires,(1 965) was able to identify for such system using 3 - 4 -5 and 8" order 
and reckon with all the fbrces acting on the 
plate cam in his analysis of force and torque 
2.0 DESIGN THIEORY 
on cam systems. Concepts and conclusions POLYNOMIAL CAMS 
obtained from all the above-mehtioaed 
Although there are other basic cam motions 
works contributed immensely to software 
that could be us4  in the numerical design 
algorithm &veloped and applied in this 
and cornmetcia1 production of cams 
work. 
generally, the PolynomiaI motion curves 
Particularly, the present computer aided have been chosen in this design analysis. 
design methad employed is av hprovement They 'are one of those motions that address 
on previous works done by Olaniyi, (1977) high-speed applications such as obtains in 
and Udoh, (2001). Ocher works that have automobiles. 
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I 0.E . SIMOLQWO AND 0. A. BAMIRO 
Zl The Polynomial Motton 
The polynomial family of motion curves has their basic equation as present&di n equation (1 ) 
Synthesizing a full- rise curve with boundary conditions 
e=o;  Y=O; Y1=O; Y"=O 
9=$; Y=IL; Y'=O; YA=O 
equation (1) becomes equation (2) with six unknown constants. 
Solving these equations simult8neously (3) and (4) are obtained for the 3 4 5  
gives the results to the six constants polynomial equations. Similar boundary 
Q = O ;  Cl=O;  G=O; c3=10L conditions are also chosen to obtain the 8' 
&=-15L C F ~ L  order plynomid equations presented in 
Substituting these constants in equation (2a) , equations (5) and (6). 
and applying a parallel procedure, equations 
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NSE TECHNICAL TRANSACTION VOLAS, NO.3, JULY. - SEmMBER, 2010 
0. E. SIMOLOWO AND 0. A. BAMIRO 
Return: 
23r CsndFoIIower Design Parameters 
The vectors present at the point of contact The overdl software and necessary complier 
between a plate cam and a flat-face follower, files developed.this work has a considerable 
typical of reciprocating internal combustion memory seize. A large number of cdes of 
engines, rn &M in this work are C1C-w high level language were written in 
shown in figure 1. Using complex polar the development of the cam design software. 
notations (Shigley and Uickl980)  in 
figure 1, the applicable cam profile design Software Fea- 
equations were obtained. The prime circle 
Two of the features among other common 
radius of the cam geometry is determined by 
fa- that enhance proper design analysis 
equation (7) include. 
%>  (pfnin-Y-Y W')m ax ....*.................(.7..) (i) The ability to resah& very large 
The width of the follower face is obtained profiles and hreby make the scald profile 
visible on the computer interface. 
by equation (8) 
(ii) Distinct graphics demarcation of 
different motion segments on all generated 
cam pfde. 
The co&dhks of the cam are also obtained 
using the equations (9a) and (9b) 'Irhese features make the developed software 
u =  &+Y) sine+ rws (0) ...... @a) peculiar. Easy correction and redesign of 
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the affected cam profile segments can easily plotting of cam coordinate u, v for all degree 
identified. They also make the developed intervals using all follower and cam &on 
software more extensive in design analyses, equations. F2:- Calculations of 
synthesis and stimulation based on its ability displacements an8 its derivatives in t h ~  
to give precise segment defects such as corresponding dispIammnt-time profile for 
profile discontinuities (cursps) i d c arry out every cam profile design. F3:- 
faster and more accurate re-design Determination of minimum values of 
procedures based & graphics distinction for velocity, acceleration aid displacement 
each segment. Screen shot 1 shows the values. F4:- Calculation of output design 
graphics distinctions on a generated profile parameters s q h  as Face 'width (I?and -), 
while screen shot 2 presents' tlte graphics prime circle radius a).Th e algorithm for 
distinction and profilmizing -featuresf or a design p d u r e  and methodology we 
generated carn profile. responsible fdr the foIlowing: ~5:-S equence 
of calculatians to be done. F6:- The tmusfer 
3J Functions of Softwnre Algorithms. of result from one part of the program to 
The developed software is based other part of the pn>grani to,& used as input on two 
of dgmithms, nameIy: for further calculations. F7:- The ~ n t r ool f  categoria (a) 
algorithm. Viud display overall program flow as software is operated Arithmetic (b) 
algorithm. Figure depicts the types of by the user. 2 
algorithms employed in the development of V h l D isplay Algorithm 
the software. 
The visual algorithms depicted in fig. 2 are 
Arithmetic algorithm responsible for the following; MI:- All 
The arithmetic algorithm comprises (i) graphics display on the screen. F9Smen 
Design-analysis, ch~Culationsa nd logic (F1- selection. F1O:- Making the program a self . 
F4). (ii) Design p r d u r e a nd Methodology operating program outside complier 
(FS-F'). Making referem fig. 2, the environment. Four basic modes as described to 
functions for below were used in the visual display of alg-o rithms design-analysis, 
calculation and logic in sofhvare are algorithm for the developmg~t of the the 
described. Fk- Calculations and software. They are.the Text mode, Graphics herein 
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NSE TECHNICAL~TRANGACPIONV OL.45, NO.3, JULY - SEPTEFdBER, 2010 
0. E. SrMOLOWO AND 0. A*B AmRO 
mode, Text in graphics modes, and eke (iv) Dwell: - OD-220'; Rise: - 220'-280"; 
,AN51-o r control. Dwell: - 280 O - 300 Return: - 300'-360'; 
(v) Maximum ufr: - 9.57mm. 
MD1:- Text Mode: This mode was used for 
(1) Dwell (OD - 220') or (0 - l l d 9  
all words and texts in the whole program 
radians) 
and dso appearing on the screen. MD: This TI  
mode was used for the following: (i) MI the Since at this stage, the follower dwells for 
color designs appearing on various screen on 1 ld9 radians, i.e. for 0 O c 0 < 220°, 
the VDU. (ii) Displaying strings of text at Y=O;t hus. Y = Y' = Y" = 0. 
the cursor position or at the 'x' 'y' position on (2) Rise (220' - 2804 or (1 1139 - 14d9 
the screen. MD4:- ANSI cmor control: radians) 
This cursor system is used to directly control The values of Y, Y' and Y" me generated 
the position of cursor on the screen. This by the computer dgorithm using the rise 
type'of system generally comes with most equgtions (5) for flh order polynomial 
IBM PC family of computers as well as equations, Where, L is given as 9.57m-m: P 
most MS-DOS compatibles. However, since =roo" 
the system cursor control does not exist on 
(3) Dwell (280' - 3007 or (14m-  
the read-only memory WOW of the 
15Dn radians) 
computers, software algorithms have to be 
written in conn8ction with the existing ANSI At this stage the follower dwells for 7d9 
files located in the ogerating system so as to radians, since for 280 c 0 c 30O0, Y=L 
make the cursor control operative. Y=Yt'=0 
far (4) Return (30Q9-  360') or ( 1 5 M-  2n 33 Deriptlon of Algorithm Cam 
w g n  
A step-by-step -@on of the computer The values of Y, Y, and Y" are also 
algorithm involved in a typical profile generated ' using the 8h order poIynominl 
design is presented in this section using a 
return quation (6) for 3W0-<8  r: 360'. B = 
sample case study design with the following 
W (the cam angle i n w a l  for the return- 
criteria: (i) plate catn; (ii) recipmchg flat- 
motion). 
face follower, (iii) eight- wder Polynomial; 
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NSE TECElNXCAL TRANSACTION VOL.45, NO39 JULY - SEWEFdBER, 201 0 
0. E. SIlMOLOWO AND 0. A. BAMIRO 
(5) 'Ilae value8 of Y, Y' and Y" we Therefore, the development of a software 
generated in stages 1-4. Shown in figure 2 procedure carried out in this work has 
are the motion profdes. enabled faster and more extensive 
6) From equation Q) and (81, Ro andm researches to be carried out in this area. The 
F, are obtained respectively. Y at Y' min is four different aspects af simulations studied 
also obtained from the, values generated in in this parametric qnalysis for optimum 
stages 2 mi4 ; design are: 
(7) Cam geometry and motion Case (I)O: btaining the values of prime 
coordinates of profile W ~ n gde signed are circle radius of the cam &) for a 
g e n d u sing equations 9a and 9b for the combination of carn angular interval using 
four stages of d o n s c onsidered (D-R-D- the 3 4 5 a nd 8" order polynomial functions 
R). The cam profile is obtained by plotting for a RiseReturn (R-R)f ollower motion. 
the generakd values as shown in figure 3 
Case Obtaining the values of follower 
The summary of the design output face width (Fw) for a combination of cam 
parameters, the follower motion' profiles angular intervals using the 3 4 5  and 8" 
and cam pro% as obtained for the sample wder .plynomid functions for n Rise- 
computer-aided h i g n  describe.. in stages khm (R-R)P ollower motion. 
1-7 above is presented below in Table 1 and 
Case 0:Obta ining h values of prim 
figure 3 nq=tiveIy. 
circle radius of .the cam CRo) for a 
3.4 Softwan Applicatfon on Optimum combiiation of cam angular intervds using 
~ w ~ Y *  the 3 4 5 a nd 8' polynomial functions for a 
The developed software was used to carry Rise-Return (R-D-R) Follower d o n .  
out optigmm design analysis from a total of 
Case Obtaining the values of face 
29 stimulations (Simolowo, 2004). 
width (Fw) for a combination of cam angular 
Ordinarily each- of these simulations takes intervals using the 3 4 5  and 8' order 
horn to accomplish using the conventional polynomial functions for a Rise-Dwell- 
numerical procedure described in section 2. Rebum (R-D-R) Pollower motion. 
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WE TECHNICAL TRANSACTION V O U , N O 4 J ULY - SEPTEMBER, 2010 
0.E" SIM0mwo:AIYX)A-0.. BAMIRO 
Far the fwrr different cam angle particularly bflttenced by the type of cam 
cm&inatim w m k ept ahstant for each of fuhctioned used rath* the follower motion 
the followex d m ,w hile the polynomial sepmts amsidered, This is because in 
cammotioas werevaried.The effectsofthe figure 5 themvalueso f Fw using orda 
p b n y n o ~ ~ ~ ~ t h e p~oly~nornf idso arpe d istinctively lower than 
* a g n m ~ t - , & ~ , M  t h w  of 3 4 5  p o l y n ~ d sirx espective of 
tide radius af cam .&) and followm faoe either the D-R-D or R-R follower motion 
wid@ (&).I t should be noted that nmm segments. 
design analysis could also be 
when the 3 4 5  polynomiotl is utd ,  more 
mamiah will be nqnired in the find 
mufactwe of the flat face follower while 
ledssermakrials will be medal fwthe cam 
-on cases (I) aad 0; these sets 
mufactwe. Depending on the design 
of dm show that the values of ptime 
nquiremts, a ptaper combination of these 
circle radius -1 obtain@ using the 8' 
pdynodals motions aad others in cam 
order. polywmia3 are generally higher for design will give the most reliable d 
more cam an@= codinatims thm those 
economics). cam system when the results 
of tfie 3 4 5 p olynomiah as shown in figure 
given M n are  put into consideratian and 
4 and 6. Thus lesser would be 
similar d y s i s a lso extended to other cam 
needed generally in the manufacture of the mtions. 
.cam units when 3 4 5  polynomial is wed 5Al CONCLUSION 
b to the smdh values of &) Mote so, in 
n w k e s pact available is a-critical The objective of developing software with 
enhancing featum for extensive for cam 
wwrMbr: abtkrcbiceoutofth two. deign 4a nalysis has been achieved in 
this work. Alm important inferaces have 
Shown io-egurc5  and 7 am the sirrmldon 
.& ,fmm ,wfication of he 
m d t s f or cases 01) and (Ill)re spectively. 
'&&ware in plynomioil cam design as case 
Haein it k also m e d as  in cases (I) and stndy. The ampurn ai&d simulation 
0t hat the foHowSr faca widths (Fw) are package is highly friendly, requiring 
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NSE TECHNICAL -SACTION VOLA5, NO& JULY - SEPIEMBER, 2030 
. 0. E SmOLOWO AND 0. A. BAMIRO- 
basically the knowledge qf operating the The simulation analysis of the polynomial 
computer by the ca9;1 designer. The Wgn cams performed in, this work opens up to 
out-put param%m are much more reliable further analysis . in cam design and 
than those obtained nuidcdly and can be manufacturing using other cam motions 
used for all @cal p&~uch as the 
manufactming of the emrging a ns ystem. 
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0. E: SIMOLOWO AND O,A. BAMIRO 
F'OLUIwlglp MOTION 
I 220D 28U' 3W 3W 46.00 34.00 -16.11 16.11 45.92 46.00 
I Y' -6o.m  
Cam Angle 
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e mmled sample cam p- 
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NSE TECHNICAL TRANSACTlON VOLAS, N03, JULY - SEFtEMBER, m10 
0.E . SiMOLOWO AND 0. A*B -0 
-RO 8TH -RQ 3 4 4  
100 - E 
80 r 
::- 
:W -  
% a -  
-B  "30 : 
S 20. 
10 - 
 1 - 0 3
2 3 4 5 6 7 8 0 l O t l t 2  
No. of cam angle comMnations 
m g  4: - S b d a k d v alues of Il, for &a nd 3-4-5 P o l p d a l s  for 
R-BR f o R mP dIk 
30 1 
13 :: 
5 1 5  - ; 
8 10 A 
V 
2 5 -  
0 ~ ' m r r r r m I r r r r ~  
1 2 3 4 5 6 7 8 S l O 1 1 1 2 1 3  
No of cam angles combinations 
I 
Fig 5: - Simulated vaIws OFF, for Sm order and 3 4 5 F olymmlahi for 
R-PR.iohwer Profile 
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-PW8TH -W346 
15 y 
gY0  15 w m : ~ /  
5 0 -  
I 
uL  
5 - 
O ? l m r . . . ~ l w w w . a a n ~  
1 2 3 4 5 0 7 8 0 1 0 1 1 1 2 1 3 1 4 1 5 l 6  
No. of cam angle combinations ' 
Fig. 7: Simulated val- d F, for 8h order and 3-4-5 Polpwdals for 
R-R tollorrrer P d U e  
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NSE TECENICAL TRANSACTION VOLAS, NO& JULY - SEFTEMBEK 2010 
0.E . SXMOlLOWO AND 0. A. BAMlRO 
Engineering. Volume 4, No. 4 pgs. 15- 
1. Andrezej GO. Cam Mechanism. 24. 
('The University Preps, Ames IOWA 
6, Tesar D. and Matthew G. K 
State, 1980) Olardyi M 0. Cum Design: lk? Dyrrcunics Sydhesis, Analysis and 
Anao*' of Design of Modeled Cam Syste~n(  D.C 
Science Roject Depwtmmt of 
HeaIth kington,  Mass 1976) 
Mechanical Engindng, Univbity of 
 bada an, Nigeria (Unpublished 1997). 7. Udoh E.M Computer Aiifed 
Design of Cams.  achel lor' of Science 
2' ShideJ~m z 'jcker JJ hjcet. Depmnt of Mechanical 
Theory of Machines and Mechanism Engineering, University of 
(McGraw-Hill Book Company, 1980) 
3. Simolowo OX Computer-Airled Virgil. M F. M e c h i c d  Engiireenng 
Design of C m  Pmfibs. Master of B i i g n .  United State Post Graduate 
badan, Wlgeria. (Unpublished ul04) 
4. .Simolowo 0s.a nd Barairo 0.A 
(2008). 'IXAe Development of An 
Analysis- Jntensive software for 
improved Cams Systems Design. 
JosrntrrL of Science and Tcchrtology 
'(JUST) Ghana, Kwame' lVknunah 
University -of~ cie&e rmd Technology 
(KNUST) VOL 28. No. 1.pgs 103 - I IS' 
5. Sirnolowo 0s. and Bamiro 
0.A (2007) Profile Simulation Software 
fd Tt.igonondc and Polynomial Cam 
Design. Joumd of Research in 
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