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Z E S Z Y T Y N A U K O W E P O L IT E C H N IK I ŚLĄ SK IEJ 1995

Seria MECHANIKA z. 122 Nr kol. 1267

Vladimir ZEMAN, Josef

n

£MEÖEK Department o f Mechanics

University of West Bohemia in Pilsen

MODELOWANIE DRGAŃ UKŁADÓW SZYBKOOBROTOWYCH WALÓW

Streszczenie. Przedstawiony jest sposób modelowania drgań sprzężonych wałów, wykorzystujący syntezę modalną oraz metodę elementów skończonych. Model bierze pod uwagę zjawiska giroskopowe, bezwładność rotacyjną, odkształcenia związane ze zginaniem, skręcaniem i ścinaniem oraz przemieszczeniami osiowymi, jak również sprężystości i tłumienie łożysk.

MODELLING OF HIGH SPEED SHAFT SYSTEMS VIBRATION

Summary The modal synthesis method and FEM based modelling o f vibration o f coupled shafts including the effects o f gyroscopic forces, rotational inertia, bending torsional, longitudinal and shearing deformation o f shafts, elasticity and damping of bearings is presented.

M OZIEJIHPOBAHHE K O JIEEA H H ÏÏ BH CO KOOBOPOTH HX BAJIOB

P e 3 D M e . B CTSTbe pacM BTpHBaeTca npHMSHSHHe u e T o a a M o a a n b H o ro cHtrre

3

a m MoaenHpoBBHHH roneßaHHfi cbh3q h h x

BaaoB c yveTOM rHapocKonmiecrHx m o m b h t o b, po t b ii,h o hho0 H H ep iI,H H t H 3 rH Ö H X , K p y T H I b H H X H ItpO Æ O U bH H X A e $ O p M a U ,H fi,

ae$ûpMau,Hft caBHra, noaaTiHBocTH h 3BTyxaHHH onop.

(2)

306 V. Zem an, J. Nfcmećek

1. INTRODUCTION

We consider the high speed shaft system composed o f several shafts with circular cross-sections with rigid discs supported by rolling-element bearings. The shafts are joined by flexible couplings or gears. Each shaft will be assumed to rotate stationary.

The presented mathematical modelling of spatial vibration of the shaft systems is based on the 1D-FEM shaft models [1] and modal synthesis method [2], In comparison with [1] the shaft models are described in the fixed coordinate systems and include the effect o f shear deformation caused by bending phenomenon.

2. MATRICES OF A SHAFT ELEMENT

The shaft will be assumed to rotate about its theoretical axis X with constant angular velocity

(ot

and to execute combined bending, torsional and longitudinal vibration.

The vibration o f the shaft element is expressed by displacements u(x), v(x), w (x) of neutral axis points and by small turn angles $(x), \|/(x), <p(x) (Fig. 1).

7

X

I

Fig. 1. Scheme of the shaft element in the fixed system Rys. 1. Schemat elementu wału w układzie nieruchomym Kinetic energy of the shaft element "e" of length 1 is

E(k*’ = ^ 0 A ( x ) v T(x )-v (x )+ c o T(x)-J(x)co (x)]pdx

(

1

)

(3)

M o d ellin g o f high sp eed shaft system s vibrations 307

w h ere A (x) and r are the cross-sectional area and density, respectively. T he speed o f th e neutral axis points

v ( x ) = [ u (x ), v ( x ) ,w ( x ) ] T , (2)

th e v ec to r cd(x) o f th e infinitesimal mass elem ent angle velocity in cartesian coordinate system tj, q

co(x) = [©„ +<p(x), S ( x ) - r o {)v ( x ) , vj/(x) +<o0S ( x ) ] T (3) a t the diagonal m atrix o f cross-sectional area m om ents o f inertia

J ( x ) = diag (2 J (x ), J (x ), J (x )) (4)

(4 ) w ere introduced in the expression (1).

Potential energy o f th e shaft elem ent is

E';> = - i j o1{E£ ^ (x ) + G [ y ^ ( x ) + Y^ ( x ) ] } A ( x ) d x , (5)

w h ere E and G are Y ou n g m odulus and shear m odulus, respectively. C o m ponents o f the strain vecto r in (5) are

W

d u , d u , du 9 u d u x

= — y „ = — - + — - , y „ ( x ) = — *-3--(6)

3x dx

<3y d x d z

T h e displacem ents o f an arbitrary point (x,y,z) o f the shaft elem ent in direction o f th e fixed axis x,y,z are

u x = u ( x ) - y \ j / ( x ) + z S ( x ) , u y = v (x ) - zc p (x ), u , = w (x ) + ycp(x).

T h e transversal displacem ents can be approxim ated by cubic polynom ials

(7)

v (x ) = < $ (x )c ,, w (x ) = C>(x)c2 , ® ( x ) = [ 1 x x 2 x 3 ] , (8) longitudinal and torsional displacem ents by linear polynom ials

v ( x ) = T ( x ) c 3 , p ( x ) = Ÿ ( x ) c 4 , ¥ ( x ) = [ 1 x ] ( 9 ) A ccording to th e M indlin beam theory including the effect o f shear, deform ation plane sections o f th e shaft rem ain plane after the déform ation but are not perpendicular to th e deform ed neutral axis. H ence turn angles j(x), y(x) can b e approxim ated by quadratic polynom ials

y ( x ) = © (x )b , , S ( x ) = 0 ( x ) b 2 , 0 ( x ) = [ l x x 2 ]. (10) V ectors b j, b 2 o f unknow n coefficients will be determ ined from the conditions [3]

(4)

308 V. Z em an, J. Ném eóek

5 s v c?3 w

k G ÿ iy A = Q I = k G ÿ „ A = - E J ^ r ,

(H)

w h ere Q y , Q z are internal shear forces transfered by th e cross section in corresponding directions and

k = - J 2 A j

S (y) b 2( y ) '

dA

d \ d w .

“ V , ï ^ — + S

dx 5x

(

12

)

H ere S(y) denotes th e statical m om ent about z axis o f the part o f the cross-sectional area cut o f f by a section o f the w idth b(y) in distance y from axis z.

F ro m the relations (8), (10), (11) and (12) w e get k A G ^ ' ^ - e b . ^ - E J <D'"Cl

k A G ( i> 'c 2 - 0 b 2) = -E J < D " ,c i and further

w here

b , = H e, , b 2 = - H c2 , (13)

H =

0 1 0 K l2

0 0 2 0

0 0 0 3

and k = 6E J k A G l2

Substituting (13) into (10) w e get

y ( x ) = 0 ( x ) H e, , 9 ( x ) = - 0 ( x ) H c 2 , (14) T he configuration o f th e shaft elem ent "e" o f length 1 in th e fixed coordinate system x,y,z can be expressed by th e vector o f displacem ents o f nodes 1 (x = 0) and 2 (x = l)

„<0 _ r -T _T _T _T 1 (15)

*1 ~ [ ftl > Qz > fts > *1« J w here

q t = [ v ( 0 ) v ( o ) v (i) v ( i ) ] T . q 2 = [ w ( ° ) 9 ( ° ) w d ) 9 0 ) ] T q3 = [ u (0 ) 0(1) f , q 4 = [ (p(0) <p(l) f .

T h ese nodal displacem ents satisfy th e relations (8), (9), (14) and hence

q¡ =

c ¡ , c¡ =

S7'

q ¡ , i = 1,2 , 3 ,4 , (16)

(5)

M o d ellin g o f high speed shaft system s vibrations 309

w here

S ,=

s , =

1 0 0 0

0 1 0 k12

1 1 l 2 l3 ’

0 1 21 (3 + k)12

1 0 0 0 -

0 - 1 0 - kI 2 ' 1 0

1 1 l 2 l 3 3 “ 4 “

1 1

0 - 1 -21 - ( 3 + k)12_

U sing relations (8), (9), (14) and (16), the displacem ents o f internal points th e shaft elem ent are given by expressions

v ( x ) = < b (x )S '1q , , H/(x) = 0 ( x ) H S ; ' q , ,

w ( x ) = <t>(x)S2' q 2 , 9 ( x ) = - 0 ( x ) H S !,q I , (17) u ( x ) = 4 '( x ) S - ‘ q 3 , <p(x) = 0 ( x ) H S 3'q 3

T he m atrices o f a shaft elem ent satisfy th e L agrange condition

d i æ ^ M dE[e) [ dE(;> _

d t [ ô q (0)

J

oq<e) + (5q<c)

M (e) q(,) +o»0 G (e) q<5> + (k<*> - (o2K '3e))q (e)

(18)

S ubstituting (1), (2) into (18) and considering all above presented relations w e get th e sym m etrical m ass m atrix M (e), th e antisym m etrical matrix o f gyroscopic effects G w , the sym m etrical static stiffness m atrix K (se) and th e sym m etrical dynam ic stiffness m atrix K D (m ultiplied by m j) o f the shaft elem ent All m atrices are o f o rd er 12 and have th e form

M (e) =

0 0 0

0 s

2

t

(

i

, +

i

4)

s

2'

0 0

0 0

S3

t

I5S3’

0

0 0 0 s ; TW

G (s) =

1 o -2S1

t

I,S2 -' 0 0

2S2

t

I,S,1 0 0 0

0 0 0 0

f o 0 0 0

(

19

)

(6)

312

. V. Zem an, J. NSmeiek

S (T( l 7 + I 12)S1I

0 0 0

0

s2t(i7 +iI2)s-'

0 0

0 0

S“TI3 10 3

0

0 0 0 s;Ti„ s r J

0 0 o

---

1

0 s - TI 1s ¡ , 0 0

0 0 0 0

1 o 0 0

---

1

o

w h ere

i i i

I, = J j H r 0 T 0 H p d x , I 4 = j A d > TO p d x , I, = J A y T \\i p d x ,

0 0 0

1 I 1

I 6 = 2 j j v / > p d x , I 7 = | E J Ht© ,T0 'H d x , I „ = Je Ai|/,t V 'd x ,

0 0 0

1 1

I „ = 2 j G J i | / ' T V ' d x , I 12= j G A ( d ) ' T - H T 0 T) ( 0 ’- 0 H ) d x .

0 0

T he internal dam ping matrix B w o f the shaft elem ent can be considered proportional in fo rm B"> = p ('> K ^ .

3. M A T H E M A T IC A L M O D E L O F TH E SHA FT SY STEM

L e t the shaft system (Fig. 2) be com posed o f N shafts (here N = 5) with rigid discs.

T he shafts are supported by viscous-elastic bearings and joined by discrete couplings (here flexible couplings s = 1,2 and gear m eshings z = 1,2).

(7)

Modelling o f high speed shaft systems vibrations 311

ij- W « —

bV> 5=1

1 I

\ X ^

r r

^ J j

v

J . O

H c

T7T*

■ v * c m ? n g

"irTT T tT • n r

¿ 2=2

(

20

)

Fig. 2. S chem e o f th e shaft system Rys.2. S chem at u kładu napędow ego

T he equation o f m otion o f th e dism em bered shaft "j" is described in m atrix form co rresponding to (18)

M j q (t) + (B + coOJG j)q j(t) + ( K sj-C );j K ra)q j (t) =

f f ( t ) + f f ( t ) , j =

1 , 2 ,. .. ., N ,

w h ere M j, Bj, G j, K ^ , are square mass, dam ping, gyroscopic, static stiffness and dynam ic stiffness m atrices o f th e isolated shaft "j" and w is its given co nstant angular velocity. T h e generalized coordinate vector qj(t) o f dim ension n, in this form (i is index o f th e n ode)

qj = [ . . . , u i , v 1,\tii , w i , B 1,<p1, . . . J

expresses nodal displacem ents o f the shaft "j". The structure o f the m atrices M j, Bj, Gj, K Sj and Kpj results from th e schem e

Ks j

DJ

(8)

m V. Zem an, J. N em ecek

T he cross-hatched subm atrices o f the order 12 in th e first m em ber represent the tran sfo rm ed shaft elem ent m atrices (19) in form Tt X T .where X e { M (e), G ( , ) , , K p }and th e transform ational m atrix T exchanges only row s and colum ns o f the original elem ent matrices. The cross-hatched subm atrices o f the o rd er 6 in th e second m em ber correspond to discrete param eters (rigid discs, supports) External excitation o f the shaft by nodal forces is expressed by th e vector f f ( t ) . T he internal force effect o f th e neighbouring shaft connected to the "j" shaft is described by the coupling fo rc e vecto r f E( t) .

T he configuration o f th e w hole shaft system is described by th e generalized

N

coo rd in ate vector q (t) = [qj(t)] o f dim ension n = £ ^ . The influence o f th e elastic j=i

viscous properties and

kinem atic errors in couplings (in case o f gear meshings) is expressed in m athem atical m odel o f th e system by stiffness and dam ping coupling m atrices K c, B c and by th e internal excitation force vector f ( t) satisfying

fC ( ‘) = = ~ K c q ( t ) ) - B c q ( t ) + f ' ( t ) , (21)

oq 3q

w h ere f ° ( t ) = [f,c ( t ) ' | is the potential energy and Ed is th e dissipative function o f the discrete couplings u n dergoing the vibration

E very isolated shaft is characterised by spectral and m odal m atrix - h V) o f the conservative part. These m atrices satisfy the orthonorm ality conditions

X j M j Vj = I j , V / ( k , - « * k J v j = A j , j = 1 , 2 , . .. ,N ,

and can b e devided into subm atrices o f m, m aster m ode shapes (index m ), Sj slave m ode shapes (index s) and the o ther shapes (index o), resp.,

V ^ - V j ‘Vj » v j , A j = d i a g ( - A J> 'A j > *AJ ,).

T he m athem atical m odel (20) can be transform ed into the condensed m athem atical m odel

N

w ith th e relatively sm aller num ber m = ^ m ^ n o f degrees o f freedom in m aster m odal j=!

co ordinates o f th e isolated shafts [2]

rax ( t ) + ( mD + m G + m VtC Bc mv ) mi ( t ) + ( n,A + “ V TC K c rav ) mx ( t) = mV T[ C f ' ( t ) + ( l - C K c H ) f E( t ) ] ,

(9)

M odelling o f high speed shaft systems vibrations 313

w here

-D = diag ( mV / Bj “ V j), raG = d.ag (o 0J m\ J G j “ V, ) , H = diagi’Vj ‘A-/ ‘\ J ) , “A = d i a g ( - A j , "V = diag(-Vj),

f E(t) = [ f / ( t ) ] , C = ( l + K e H ) _l

The matrix C expresses approximately an influence o f frequentionally higher slave mode shapes o f uncoupled subsystems. The transformational relations

q (t) = mV mx (t) + H [ f c (t) + f ‘(t)]

f c ( t) = - C [ K c “ V ” x (t) + B c “ V “ i ( t ) - f * ( t ) + K c H f E(t)]

enable after integration o f the condensed model (22) to determine the vector o f coupling forces and generalized coordinates

The conservative part o f the condensed model (22) in form

mx ( t ) + ( "A + mV T C K c "V ) “ x ( t) = 0 (23) can be used for calculation o f the natural frequencies Q 0, u = 1, 2 ,.., m o f the w hole shaft system. The corresponding eigenvectors “ x 0 o f the condensed conservative model (23) have to be transformed by means o f the relation

q 0 = ( l + H K c ) '' “ V “ x u into the space o f the generalized coordinates.

U sage o f the model (23) for spectral tuning is presented in [4],

BIBLIOGRA PH Y

[1] D upal J., Zem an, V.: FEM dynamic analysis o f the helical rotors. "Zeszyty N aukow e Politechniki Śląskiej M ECHANIKA", z. 116, Gliwice 1994, p 119 -1 2 6 .

[2] Zeman, V.: Vibration o f Mechanical Systems by the Modal Synthesis M ethod.

"Z.angew. Math. Mech." 74, Nr. 4, 1994, T99 - T101.

[3] Bittnar Z.,Śęjnoha J.: Numericke metody mechaniky 1. Vyd. ĆVUT, Praha 1992.

[4] Zem an, V., Nćmećek, J.: Spectral tuning o f shaft systems with gears. Journal o f Czech and Slovak Mechanical Engineering 44, 1993, p. 21 - 30.

Recenzent: prof. dr hab. inż. A. Olędzki W płynęło do Redakcji w grudniu 1994 r.

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