A mechanical model of the vibration conveyor
Abstract
V různých odvětvích technické praxe, zejména pak v automobilovém průmyslu, se používají jako součásti montážních linek vibrační dopravníky. Umožňují transportovat nejen díly různých tvarů a velikostí, ale i sypké materiály. Princip dopravy dílů je založen na kmitavém pohybu nosného členu ve směru, který uděluje dopravovanému předmětu současně vertikální i horizontální rychlost. Pohon vibračních dopravníků se provádí pomocí mechanických nebo elektromagnetických budičů. Mechanické budiče založené na rotaci nevyvážené hmoty jsou připojeny buď přímo k nosnému členu dopravníku nebo ke členu, který je s ním spojen pružnou vazbou. Elektromagnetické budiče vytvářejí periodické sily nebo momenty mezi nosičem a rámem. V zásadě je snaha naladit systém tak, aby vlastní frekvence příslušná hlavnímu transportnímu pohybu byla blízká nebo shodná s frekvencí budící. V rezonanční oblasti je dopravní výkon nejvyšší ve vztahu k energetickým nárokům pohonu.
Full text
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A MECHANICAL MODEL OF THE VIBRATION CONVEYOR
Ma ek Pešík
Technical Uni e si y o Libe ec
Facul y o Mechanical Enginee ing
S uden ská 2, 461 17, Libe ec 1, Czech Republic
[email p o ec ed]
Abs ac
Vib a ing con eyo s as componen s o assembly lines a e used in a ious echnical b anches,
especially in he au omo i e indus y. They do no only p o ide mechanical handling
o di e en ly shaped and sized pa s, bu hey also allow he anspo o loose ma e ials.
He eby he main p inciple o he anspo a ion o pa s is based on he oscilla ing
mo emen s o he ca ying elemen which, a he same ime, impa s he e ical and
ho izon al eloci y o he anspo ed i em. The ib a ion con eyo s a e d i en by means o
mechanical exci e s o elec omagne ic componen s. The mechanical exci e s, based on he
o a ion o an unbalanced objec , a e ei he di ec ly a ached o he ca ying elemen o he
con eye o a e connec ed o he elemen ha is joined o he con eyo by a lexible linkage.
The elec omagne ic componen s induce pe iodic powe o momen s be ween he ca ying
elemen and he ame. Gene ally, i is aimed o une he sys em in a way o ha e he na u al
equency acco ding o he main anspo a ion mo emen ma ching o nea ly ma ching wi h
he exci e equency. The anspo pe o mance in ela ion o he ene ge ic equi emen s o
he d i e is highes in he esonance zone.
In oduc ion
Vib a ing con eyo s ha e he ca ying elemen o en di ec ly connec ed o he ame by ei he
lea sp ings o h ough an ine ia elemen aiming on eac i e dynamic powe minimiza ion.
The lea sp ings o m a lexible linkage be ween he ca ying elemen and ei he he ame o
he ine ia mass, and hey pe o m a lead mechanism unc ion simul aneously. F om he
mechanical poin o iew, i is basically a wo mass model wi h one o he bodies (ei he he
ame o he ine ia mass) connec ed lexibly o he base o o he ame o easons o
ib oisola ion. The second objec is connec ed by a mo able and lexible link-up. To p o ide
an op imal ib a ion con eyo pe o mance, i is necessa y o know he in luence o di e en
dynamic sys em pa ame e s on he na u al equency acco ding o he main anspo a ion
mo emen . This is only possible by assembling and analysing i s mechanical model. The
ollowing a icle deals wi h he se up o such a mechanical ib a ion con eyo model wi h
ei he a ansla ional mo ion o a sc ew mo ion.
1 Cons uc ion o Vib a ion Con eyo s
In gene al, ib a ing con eyo s ha e hei ca ying elemen s adjus ed o he shape and size o
he objec s o be handled. Addi ionally, he ca ying su ace is modi ied owa ds ab asion
esis ance and, commonly, o achie e a la ge ic ion coe icien . Fu he , he su ace has a
signi ican in luence on he emi ed con eyo noise. The ca ying elemen is di ec ly
connec ed ei he o he ame o o he ine ia elemen by use o he sp ings. Gene ally, he
lexible connec ion consis s o he lea sp ings whose s i ness is ela i ely easy o adap ;
72
hence, hey pe o m simul aneously he conduc ing mechanism unc ion. He eby, he c oss
s i ness o he sp ing is signi ican ly lowe han he longi udinal s i ness.
The lea sp ings a e o be posi ioned as a pa allelog am sys em ha de e mines he ela ion
be ween he ho izon al and e ical ampli udes o he oscilla ing mo emen o he ca ying
elemen . This way i consequen ly es ima es he eloci y o he componen o ma e ial
ans e . Vib a ion con eye s a e cons uc ed o p o ide he ansla ional o sc ew mo ions.
1.1 Con eyo s wi h he ansla ional mo ion
T ansla ionally mo ed con eyo s (Fig. 1) ha e hei ubing o lume-shaped ca ying
elemen s connec ed wi h he ame by elas ic elemen s. Thus, he anspo is handled by bo h
e ically and ho izon ally di ec ed oscilla ion mo emen s ha pe o m a he same ime. Bo h
mo emen s a e bonded wi h each o he by means o a lead mechanism caused by a
pa allelog am o analogue wi h he lea sp ings. The dynamically o ced powe is gene a ed
ia one o wo mechanical exci es ha a e connec ed o he ca ying elemen .
F
ig. 1 Vib a ion con eyo wi h ansla ional mo ion
1.2 Con eyo wi h he sc ew mo ion
Con eyo s wi h he sc ew mo ion (Fig. 2) ha e a cylinde -shaped ca ying elemen on whose
cylinde he e a e placed g ound componen pa s ha mo e ia spi al lanes along he inne
cylinde wall su ace. Thei anspo is ealized simul aneously wi h bo h he oscilla o y
ansla ion mo emen and he oscilla o y o a ion mo emen o he ca ying elemen which is
usually connec ed o he ame by help o he lea sp ings. These lea sp ings a e ei he placed
symme ically owa ds he cylinde base and he ame o wi h espec o he ine ia mass. The
handled componen s mo e hen ia he spi al lane. The o ce powe will be ini ia ed by one o
mo e cen al elec omagne s placed on he longi udinal axis o he ca ying elemen . I is
u he possible o place he magne s in a symme ic ela ion o he pe ime e o he ca ying
elemen .
73
F
ig. 2 Vib a ion con eyo wi h sc ew mo ion
2 Mechanical ib a ing con eyo model
Summing up, each ib a ion con eyo has a ca ying elemen ha is connec ed o ame o
ine ia mass by a mo able and elas ic linkage. The mo able connec ion is ealized wi h a lead
mechanism ha de ines he mu ual ela ion be ween he cinema ic quan i ies o ho izon al and
e ical mo emen . The elas ic connec ion hen consis s o e.g. pi ched sp ings o lea sp ings
ha can also ca y ou a lead mechanism unc ion. To educe he ansmission o dynamic
o ces, he con eye ame is elas ically connec ed wi h he unde lay.
Basically, om he mechanical poin o iew, we analyse a wo mass model wi h one o he
bodies pe o ming a gene al mo ion, whe eas he second body is connec ed h ough a lead
mechanism ha ing speci ied i s cinema ic connec ion.
Wi h espec o he na u e o he con eyo , we can, in he dynamic model, assume one o wo
symme ical planes. In he i s case, we conside he gene al plane mo ion o he ame and
he ela i e mo emen o he ca ying elemen ha is connec ed ia he lead mechanism. In
he second case, he mo emen is aceable in one up igh axis and he o a ion o bo h he
ame and he ca ying elemen a ound ha axis.
2.1 Con eyo s wi h he ansla ional mo ion
The con eyo ame is de e mined by he mass 1
m and i s momen o ine ia abou he axis
passing e ically h ough i s cen e o g a i y owa ds he symme ical axis. To connec wi h
he unde lay we apply s i ness o he elas ic connec ion in bo h he longi udinal di ec ion y
k1
and he c oss di ec ion x
k1 as well as o he damping coe icien s y
b1 and x
b1. The
posi ioning o he sp ing ix u es a he ame and hei dis ances o he ho izon al axis in line
wi h he s i ness coe icien s x
k1 and y
k1 de e mine he o sion s i ness coe icien z
k
1
(Fig. 3).
74
F
ig. 3 Mechanical con eyo model wi h ansla ional mo ion
The ca ying elemen will be de e mined by he mass 1
m and he momen o ine ia z
J2 abou
he ho izon al axis passing h ough he cen e o g a i y. The mo able connec ion wi h
espec o he ame is ealized by help o a pa allelog am wi h le e s o he leng h l and he
basic angle
o he le e s o he e ical di ec ion. The elas ic connec ion wi h he ame
apply he sp ings o s i ness x
k1 and y
k1. Fu he he ix u e dis ances om he cen e o he
ca ying elemen g a i y also de e mine he o sion s i ness z
k
1 (Fig. 3).
The Lag ange equa ion p o ides an equa ions-o -mo ion sys em
.0
dq
dE
qd
dE
dq
dE
qd
dE
d
dp
dkk
(1)
Fi s ly, i is possible o exp ess he kine ic ene gy o he sys em
,
2
1
2
1
2
1
2
1
2
1
2
12
2
2
2
2
2
2
2
2
2
1
1
2
1
1
2
1
1z
z
z
zk JymxmJymxmE
(2)
ollowed by he po en ial ene gy o he sys em is
.
2
1
2
1
2
1
2
1
2
1
2
1
2
122
2
122
2
122
2
1
1
2
1
1
2
1
1
zzssyssx
z
yxp
z
z
kyykxxk
kykxkE
(3)
As he po en ial ene gy o he o mally applied sp ings wi h s i ness coe icien s x
k2, y
k2
and
z
k
2 and connec ion poin s ( s
x1, s
y1) and ( s
x2, s
y2) complies wi h he lea sp ing
ene gy o coe icien 2
k, his ela ion (3) is ew i able as ollows:
b
g
l
T
2
1
T
2
m
2z
J
m
1
J
1z
y
2
y
1
x
2
x
1
2z
1z
k
2
1
k
y
b
1
1
k
z
75
.
2
1
2
1
2
1
2
12
2
2
1
1
2
1
1
2
1
1lkkykxkE z
yxp z
(4)
I is possible o exp ess simila ly he damping ene gy d
E dissipa ed by he dampe s in an
equi alen way. Howe e , as i s applica ion in he ollowing analysis o na u al equencies
would ha e only o mal e ec , i is o e idden in he ollowing ela ions.
Fo he conside ed coo dina e da a 1
x, 1
y, z1
, 2
x, 2
y and z2
desc ibing he kinema ic
condi ion o he ib a ion con eyo sys em, he ollowing equi emen s a e o be e ec i e:
,coscos 112
lxx
z
(5)
,sinsin 112
lyy
z
(6)
and
.
21
z
z
(7)
A e hei subs i u ion in o he simpli ied Lag ange equa ion:
0
dq
dE
qd
dE
d
dp
k
(8)
including ime and he gene alized coo dina e q in o which i is possible o pu 1
x, 1
y, z1
and
g adually, we ecei e ou mo ion equa ions o he sys em.
As a as o he adjus men he e esul s he ollowing:
,0coscos 11122121
xk mlmxmm xz
(9)
,0sinsin 11122121
yk mlmymm yz
(10)
,0sinsincoscos
sincos
112
1121
2
221
z
zzz
z
kl m
yx m mJJ
(11)
.0sinsincoscos
sincos
212
1122
lk m
yxmlm
z
(12)
The men ioned di e en ial equa ions sys em is possible o apply in espec o na u al
equencies and o ensu e he in luence o he sys em’s pa ame e s on hei alues. A co ec
uning o he sys em wi h espec o he esonance zone enables high ib a ion pe o mances
a ela i ely low ene gy exposu es.
2.2 Con eyo s wi h he sc ew mo ion
The con eyo ame is again de e mined wi h he mass 1
m and he momen o ine ia y
J1
abou he e ical axis passing h ough i s cen e o g a i y. To connec i wi h he unde lay,
we apply he s i ness o he elas ic suppo s in he longi udinal di ec ion y
k1 as well as in he
o sion di ec ion y
k
1. The damping coe icien s y
b1 and y
b
1 (Fig. 4) a e possible o be
induc ed in an equi alen way.
76
F
ig. 4 Mechanical con eyo model wi h sc ew mo ion
The ca ying elemen will ha e de e mined he mass 1
m and he momen o ine ia y
J2 abou
he e ical axis passing h ough i s cen e o g a i y. The mo able connec ion in espec o
he ame is ealized by help o he lea sp ings o he leng h l and he basic angle
o he
le e s. The elas ic and damping connec ion wi h he ame is o med by he sp ings o
s i ness y
k2and y
k
2 as well as by he dampe s wi h he damping coe icien s y
b2 and y
b
2
.
Again, an equa ions-o -mo ion sys em can be gained using he Lag ange equa ion. Fi s ly, i
is possible o exp ess he kine ic ene gy o he sys em p o iding he conside ed mo emen s
in o a di ec ion and a ound he e ical axis passing h ough he cen e o g a i y o bo h he
ca ying elemen and he ame.
,
2
1
2
1
2
1
2
12
22
2
22
2
11
2
11 yyyyk JymJymE
(13)
Subsequen ly, i is possible o exp ess he sys em’s po en ial ene gy.
.
2
1
2
1
2
1
2
12
122
2
122
2
1
1
2
1
1yyyy
y
yp kyykkykE y
(14)
I is possible o exp ess he damping ene gy d
E dissipa ed by he dampe s in an equi alen
way. Howe e , o he same easons as men ioned be o e, i will no be conside ed in he
ollowing na u al equencies calcula ion.
Due o he ac ha he po en ial ene gy o he o mally applied sp ings wi h he igidi y
coe icien s y
k2 and y
k2
is iden ical wi h he lea sp ings ene gy wi h s i ness
coe icien 2
k, he ela ion (14) is ew i able as ollows:
y
2y
1y
k
1
k
y
y
k
2
2y
k
l
b
y
2
1
y
1y
m
2
J
2y
1
m
1y
J
2
k
77
.
2
1
2
1
2
12
2
2
1
1
2
1
1lkkykE y
yp y
(15)
Fo he conside ed coo dina es 1
y, y1
, 2
y and y2
desc ibing he kinema ic condi ion o
he ib a ion con eyo sys em wi h a combined mo ion, he ollowing equi emen s a e o be
e ec i e:
sin
12 lyy (16)
and
.cos
12
l
yy (17)
A e hei subs i u ion in o he equa ion (8) i is possible o gain h ee mo ion equa ions o he
sys em o he coo dina es 1
y, y1
and
. The esul o he adjus men will be as ollows:
,0sin 112121 yklmymm y
(18)
,0cos 112121 yzyzy y
k
l
JJJ
(19)
.0
coscos
sinsin
2
2
2
212
2
212
lk
lJ
Jlmym yyy
(20)
The e e ed di e en ial equa ions sys em is, simila ly o he example men ioned be o e,
possible o be applied in espec o na u al equencies as well as o ensu e he in luence o he
sys em’s pa ame e s on hei alues. Again, he co ec uning o sys em wi h espec o he
esonan zone will enable high ib a ion pe o mances a ela i ely low ene gy exposu es.
Conclusion
The analysed mechanical ib a ion con eyo models wi h bo h he ansla ional mo ion and
he sc ew mo ion oge he wi h hei ela ed mo ion equa ions p o ide he sys em uning ha
conside s he na u al equency o he main ib a ion mo emen . I is he only possible
app oach in o de o ensu e anspo a ion e iciency pe o ming a he smalles possible
mechanical esis ance jus in he esonan zone. In e ms o ib a ing con eyo s wi h he
ansla ional mo ion i is possible o ob ain he co ec uning h ough he adjus men o he
s i ness o he ca ying elemen o he ame elas ic connec ion. In case o con eyo s wi h he
sc ew mo emen he men ioned esul is possible o be achie ed by adjus ing he mass
pa ame e s o he ca ying elemen .
Acknowledgemen s:
The pape has been elabo a ed in he ame o he solu ion o he g an p ojec : 1M0553
78
Li e a u e
[1] KOŽEŠNÍK, J.: Kmi ání mechanických sous a . P aha, 2008.
[2] HARRIS, C.M.: Shock and Vib a ion Handbook. Fi h edi ion. McG aw-Hill.
NewYo k, 2005.
[3] BRADSKÝ, Z.; VRZALA, R.: Mechanika III. Dynamika. Libe ec, 1986.
[4] DRESIG, H.; HOLZWEIßIG, F.: Maschinendynamik. 8. Au lage.Sp inge Ve lag.
Be lin, 2008.
[5] NĚMEČEK, P.; PEŠÍK, M.: PUV 2011-24384 K uho ý ib ační dop a ník.
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Ing. Ma ek Pešík
79
MECHANICKÝ MODEL VIBRAČNÍHO DOPRAVNÍKU
V ůzných od ě ích echnické p axe, zejména pak au omobilo ém p ůmyslu, se použí ají
jako součás i mon ážních linek ib ační dop a níky. Umožňují anspo o a nejen díly
ůzných a ů a elikos í, ale i sypké ma e iály. P incip dop a y dílů je založen na kmi a ém
pohybu nosného členu e smě u, k e ý uděluje dop a o anému předmě u současně e ikální
i ho izon ální ychlos . Pohon ib ačních dop a níků se p o ádí pomocí mechanických nebo
elek omagne ických budičů. Mechanické budiče založené na o aci ne y ážené hmo y jsou
připojeny buď přímo k nosnému členu dop a níku nebo ke členu, k e ý je s ním spojen
p užnou azbou. Elek omagne ické budiče y ářejí pe iodické sily nebo momen y mezi
nosičem a ámem. V zásadě je snaha naladi sys ém ak, aby las ní ek ence příslušná
hla nímu anspo nímu pohybu byla blízká nebo shodná s ek encí budící. V ezonanční
oblas i je dop a ní ýkon nej yšší e z ahu k ene ge ickým ná okům pohonu.
MECHANISCHES MODEL EINES VIBRATIONSTRANSPORTERS
In e schiedenen Gebie en de echnischen P axis, o allem in de Au omobilindus ie,
we den Vib a ions anspo e als Teile on Mon agelinien e wende . Sie e möglichen nich
nu Bau eilen on e schiedenen Fo men und G ößen, abe auch Schü ma e iale zu
anspo ie en. Das P inzip des T anspo s de Bau eile wi d au de Schwingungsbewegung
de T äge geg ünde , de dem anspo ie en Bau eil gleichzei ig ho izon ale und e ikale
Geschwindigkei gib . De An ieb on Vib a ions anspo e n wi d du ch mechanische ode
elek omagne ische E ege du chge üh . Die mechanischen E ege geg ünde e au de
Ro a ion de nich ausgewuch e e Masse sind mi dem T äge unmi elba e bunden ode
we den zum Bau eil, de mi dem T äge du ch elas isch e bunden is , be es ig . Die
elek omagne ischen E ege bilden pe iodische K ä e ode Momen e zwischen dem T äge
und dem Rahmen. G undsä zlich gib es Mühe das Sys em so abzus immen, dass die
Eigen equenz en sp echende de Haup anspo bewegung seh nah ode ganz gleiche mi de
E egungs equenz wü de. Im Resonanzbe eich is die höchs e T anspo leis ung in de
Beziehung zum Ene gie e b auch des An iebs.
MECHANICZNY MODEL PRZENOŚNIKA WIBRACYJNEGO
W óżnych dziedzinach p ak yki echnicznej, a zwłaszcza w p zemyśle mo o yzacyjnym,
p zenośniki wib acyjne używane są jako elemen y linii mon ażowych. Pozwalają one nie
ylko na anspo części o óżnym ksz ałcie i wielkości, lecz akże ma e iałów sypkich.
Zasada anspo u części polega na uchu wahadłowym elemen u nośnego w kie unku, k ó y
nadaje anspo owanemu elemen owi jednocześnie nie ylko p ędkość poziomą, ale eż
pionową. P zenośniki wib acyjne napędzane są za pomocą mechanicznych wzbudnic lub
czujników elek omagne ycznych. Mechaniczne wzbudnice wyko zys ują o ację
niewyważonej masy i są podłączone bezpoś ednio do elemen u nośnego p zenośnika lub do
elemen u, k ó y jes z nim połączony elas ycznie. Elek omagne yczne elemen y gene ują siły
ok esowe albo momen y między nośnikiem a amą. W zasadzie podejmowane są s a ania, aby
sys em us awić ak, aby częs o liwość głównego uchu anspo owego była zbliżona lub
zgodna z częs o liwością wzbudnicy. W p zes zeni ezonansowej jes e ek anspo owy
największy w s osunku do zapo zebowania ene ge ycznego napędu.