Seria: M E C H A N IK A z. 116 N r kol. 1231
W itold G R A B Y S Z , E ugeniusz ŚW ITO Ń SK I K a te d ra M echaniki Technicznej
P olitech n ik a Śląska
S T A B IL N O Ś Ć D Y N A M IC Z N A P R Ę T A C IE N K O Ś C IE N N E G O O B C IĄ Ż O N E G O M IM O Ś R O D O W O
Streszczenie. W referacie przedstaw iono algorytm określania m apy obszarów Stabilności prętów , oparty na m etodzie różnic skończonych. P o k azan o przy
kładow y wynik dla p rę ta cienkościennego obciążonego m im ośrodow o.
T H E D Y N A M IC ST A B IL IT Y O F A T H IN -W A L L E D B A R S U B JE C T E D T O AN E C C E N T R IC L O A D
Sum m ary. A finite-difference b ased algorithm to d eterm in e a stability chart fo r b a rs is p re se n te d in the p ap er. A n exam ple result fo r a th in -w alled b a r su b jected to an eccentric load is shown.
H H H A M H ilE C K A S I y C T O H B H B O C T b T O H K O C T E H H O F O C T E P X 1 1 S I I1 P H B H E U .E H T P E H H O M jJ E H C T B H H H A T P y 3 K H
P e 3 P n e . B pa6o're npeitCTaBJien anroptiTM on p en en en n fi xapTH ycTOHUHBOCTM CTepxtnen, ocHOBan ua Meroxte K oiiem iux p a3H oc- T eii. IloKa3aH npHkepitUH pe3yjibTaT üjiíi ToiiKOCTeHiioro CTepxatst HarpyxteHHoro BireueiiTpeiiHO.
1. IN T R O D U C T IO N
T h e so called technical theory o f thin-w alled bars o f o p en cross-section has b een p ro p o s e d in th e w ork [4]. This theory describes dynamics o f th e b a r subjected to the load show n in Fig. 1 by fo u r partial differential equations w ith variable coefficients. T h ree o f th ese eq u atio n s are coupled (cf. [4, 2, 1]). T h e uncoupled one defines longitudinal vibrations an d is identical w ith th e vibration equation o f a prism atic bar. This eq u atio n is not to be analyzed here.
A g en eral, analytical solution o f th e above m entioned system o f eq u atio n s is not know n, how ever, fo r p articu lar cases o f a b a r’s geom etry, a rran g em en t o f th e load and b o u n d ary conditions, th e system could be substantially simplified. In th e ex trem e case, th e stability investigation leads to an analysis o f a system o f M athieu eq u atio n s (cf. [1, 3]).
If th e conditions o f th e p roblem d o n ’t allow to uncouple the system an d th e exciting force
is given by th e form ula
P = Pm( 1 + a s in u i) ( 1)
it is possible to apply an algorithm o f finding stability regions o f th e th in -w alled bar, taking ad v an ta g e o f p ro p e rtie s o f ordinary differential eq u atio n s w ith periodic coefficients. A disadvantage o f such an a p p ro ach is a call fo r a deco m p o sitio n o f th e exciting fo rce in to harm onics. T h e transition from th e system o f P D E s to th e system of O D E s could b e d o n e by th e G alerk in m eth o d (cf. [2, 3]) o r applying th e fin ite-d ifferen ce m e th o d (cf. [9])1. It should b e em phasized, th a t practical exam ples involved in [2, 3, 9]
co n cern only th e principal region o f instability.
A differen t a p p ro a c h is p ro p o u n d e d in the p re se n t p a p e r. T h e direct stability analysis gives a w ay to solving eq u atio n s o f dynam ics for varying p a ra m e te rs o f th e load an d judging (according to o b tain ed responses o f chosen variables) w h eth er a given p o in t of th e in p u t p a ra m e te rs state, e.g. (Pm, u , a ), is an elem en t o f th e stability region o r is not.
R e p e titio n o f this o p e ra tio n fo r changed p a ram eters com es up w ith a stability c h art w ith re s p e c t to th e se p a ra m e te rs.
T h e re a re a few ways to find approxim ate solutions o f th e eq u atio n s o f th e th in -w alled b a r’s dynam ics. A m eth o d b ased on pow er series is p re s e n te d in p a p e rs [5, 6]. A ssu m p tio n a b o u t displacem ent functions to b e in form o f p o w er series though, is ac c e p ta b le only fo r a h arm o n ic load. Such a lim itation d o esn ’t exist in th e fi
n ite -d iffe re n c e m eth o d . T his m eth o d w as em ployed in p a p e rs [7, 8] to solve an eig en p ro b lem o f a th in -w alled b a r o f v ariable cross-section. O n th e o th e r hand, the
1 T h e p a p e r [9] is n o t d evoted to thin-w alled bars, how ever, th e p ro b lem touched th e re is also d escrib ed by a system o f coupled partial differential equations.
a u th o rs o f [9] show an exam ple o f a num erically o btained displacem ent function fo r th e m id p o in t o f th e shaft. D erivatives w ith respect to tim e have b e e n ap p ro x im ated by the p re d ic to r-c o rre c to r m ethod.
N u m ero u s exem plifications on the subject for statics an d dynamics, b u t first o f all the sim ple id e a o f th e fin ite-d ifferen ce m eth o d have p ersu ad ed th e a u th o r to b e b ased on th e m e th o d in th e p re se n t p a p e r as well as in the dissertation [1], T h e m ost com plicated case co n sid ered in [1] concerns an I-b e a m subjected to an eccentric load, b u t it seem s to b e adm issible to p u t a hypothesis about effectiveness o f th e m ethod in th e g eneral case. M uch a tte n tio n w as p aid to th e com parison o f results o b tain ed for a b a r subjected to a c en tral load w ith corresponding analytical ones.
2. A B IS Y M M E T R IC T H IN -W A L L E D B A R S U B JE C T E D T O A N E C C E N T R IC L O A D
T h e dynam ic stability o f a thin-w alled b a r which cross-section has tw o axes of sym m etry, lo a d e d as show n in Fig. 2 is described by tw o2 equations (cf. [2, 4, 1])
Fig. 2. A n I-b e a m subjected to an eccentric load Rys. 2. D w uteow nik obciążony m im ośrodow o
- e A = o
d x 2 r dx 2
(
2)
2 D isregarding longitudinal vibrations as well as vibrations in direction o f z-axis
w here
ri - d isp lacem en t o f th e sh ear cen tre in direction o f y-axis,
<p - angle o f torsion a b o u t x-axis, F - cro ss-sectio n al area,
I = f y 2 dF, I u =f<->2 dF, I0= a /3 z V ? (cf. [4]),
«a - secto ria l coo rd in ate, E - Y o u n g ’s m odulus, G - sh e a r m odulus, p - density o f m aterial,
b - d am p in g coefficient fo r longitudinal vibration, b T - d am p in g coefficient for torsional vibration, r = ( I y + I j / F + y l + z l ,
y D , z D - c o o rd in ates o f th e sh ear centre.
3. T H E F IN IT E -D IF F E R E N C E M E T H O D
L e t us d eco m p o se th e set o f equations (2) into a system o f firs t-o rd e r eq u atio n s v = — - Or)
dt t = i n dx M = —
dx f = M
dx
E l — + p F — + b v + PM - e P B = 0
z dx dt * (3)
a = dt e = ^
dx
dx H = dl
dx
EIa — + p F r2— + bTQ + (P r2 - G IJ B - e,P M = 0
dx dt
A s fa r as derivatives w ith resp ect to x are concerned, let us ap p ro x im ate th em by central differences.
Fig. 3. A n exam ple o f th e displacem ent function Rys. 3. Przykładowy przebieg przem ieszczenia
Fig. 4. A n exam ple o f the. displacem ent function (loss o f stability) Rys. 4. Przykładowy przebieg przem ieszczenia (u tra ta stabilności)
F o r exam ple, th e angle o f deflection could be approxim ated as follows3
*?.! = - tiM1) (4)
w h ere h is th e length o f th e segm ent (k, k + 1 ).
A s fo r derivatives w ith resp ect to tim e w e’ll use the im plicit m eth o d (the C ran k -N ich o lso n m ethod, cf. [10]).
3 T h e su p erscrip t depicts a p o in t along th e bar, th e subscript — a m o m en t o f time.
Fig. 5. A stability ch art with respect to o> and a Rys. 5 M ap a stabilności ze w zględu na o i a T h u s fo r th e first o f eq u atio n s (3) w e have
. i(v,* * v,:,)
T 2
w h e re r depicts th e tim e ste p length.
As th e effect o f th e above described approxim ation, instead o f th e system (3) we get a system o f linear, algebraic equations. A fter having p resu m ed bou n d ary conditions in th e form
V rj? = 11^ = 0 ( N = max(&)) (
V <P? = <pf = 0
' (6)
V M,1 = M * = 0 i
V B. = B'N =0 i
as well as n o n -triv ia l initial conditions (v * 0, fl * 0) an d a fte r having substituted n u m erical values o f geom etric an d m aterial constants, solving th e system step by step, we com e into possession o f o u tp u t functions o f desirable variables.
E x am p les o f th e se functions are shown in Fig. 3 an d Fig. 4. B eing b ased on such o u tp u t g rap h s4 w e decide a b o u t stability (o r ab o u t lack o f stability).
C hanging som e p a ra m e te rs and rep eatin g the re p o rte d p ro ced u re, it is possible to g e n e ra te a stability ch art w ith resp ect to these p aram eters.
A stability ch art w ith resp ect to to and a, in the case w hen th e exciting force is given by th e fo rm u la (1) is show n in Fig. 5. T h e thick line depicts th e b o rd e r b etw een th e stability region an d th e instability region for a b a r subjected to an eccentric load. T h e thin line is th e b o rd e r in th e case o f a central load. T h e influence o f th e shift o f th e force ap p licatio n p o in t on stability is clearly visible.
B IB L IO G R A P H Y
[1] G rabysz W .: T h e D ynam ic Stability o f T h in -W a lled Bars. M.Sc. d issertatio n (in Polish). Pol. Śl., Gliwice 1993.
[2] B olotin V .V.: T h e D ynam ic Stability o f E lastic System. H o ld en -D ay , Inc., San F rancisco 1964.
[3] L eipholz H .: Stability Theory. John Willey & Sons, C hichester 1987.
[4] V lasov V. Z.: T h in -W a lled E lastic Bars (in R ussian). G IF -M L , M oscow 1959.
[5j Św itoński E.: P roblem s o f D ynam ics of T h in -W alled B ars (in Polish). Z.N . Pol.Śl., M ech an ik a 67, Gliwice 1974.
4 T h e re a re a g re a t n u m b er o f th em in th e w ork [1].
[6] Św itoński E .: Stability o f T h in -W a lle d B ars W ith V ariab le C ro ss-S ectio n . J. T h e o re tic a l a n d A p p lied M echanics, 4, 31, W arszaw a 1993.
[7] Cywiński Z.: T h e T echnical T heory o f T h in -W a lled B ars o f V ariable, O p e n and C om plex C ro ss-S ectio n s (in Polish). Z.N . Pol. G dańskiej, 134, G d ań sk 1968.
[8] Cywiński Z.: T h e P arad o x o f T orsional Buckling. J. T h eo retical an d A pplied M echanics, 4, 30, W arszaw a 1992.
[9] Suwaj S., W ojciech S.: P rincipal R egions o f Instability o f V ib ratio n s for R o tatin g -O scillatin g Shafts (in Polish). J. T h eo retical an d A p p lied M echanics, 1, 21, W arszaw a 1983.
[10] B jo rck A., D ah lq u ist G.: N um erical M ethods. P re n tic e -H a ll, E nglew ood Cliffs 1974.
R ecen zen t: Prof. d r hab. inż. Jerzy W róbel W płynęło do R ed ak cji w grudniu 1993 r.
Streszczenie
D yn am ik a p rę ta cienkościennego o profilu otw artym w ogólnym p rzy p ad k u opisana je s t p rz e z u k ład czte rech rów nań różniczkowych cząstkowych, z których trzy są ze sobą sp rzężo n e. W przy p ad k u gdy przekrój poprzeczny rozpatryw anego p rę ta cienkościennego p o siad a dw ie osie sym etrii, a siła osiowa nie je s t przyłożona centralnie, to drgania g ię tn o -sk rę tn e o p isan e są u k ład em dwu rów nań sprzężonych (2). W odniesieniu do p rę ta o w ym iarach takich ja k założono w niniejszej pracy, rów nania te o k reślają możliw ość w ystąpienia w yboczenia. W p racach [2, 3, 9] podany je s t algorytm o k reślan ia m apy obszarów stabilności, w ykorzystujący w łasności rów nań różniczkowych o okresow ych w spółczynnikach. W niniejszej pracy przedstaw iona je s t o d m ien n a m e to d a b ad an ia stabilności. O p ie ra się o n a na rozwiązywaniu układu rów nań (2) m e to d ą różnic skończonych. O trzym uje się w tedy wykresy przebiegów wybranych zm iennych, na przykład przem ieszczenia śro d k a ścinania przek ro ju leżącego w połow ie długości p ręta.
P rzykładow e przebiegi p o k azan e są na rys. 3 o raz 4. P ow tarzając obliczenia dla różnych w ielkości p ara m e tró w , m ożna utw orzyć m apę obszarów stabilności ze w zględu na te p aram etry . N a rys. 5 p o k azan a je s t m a p a stabilności ze w zględu n a p a ra m e try siły w ym uszającej (porów naj rów nanie (1)) &> i a. N a rysunku tym m o żn a zauw ażyć wpływ p rzesu n ięcia p u n k tu przyłożenia siły n a stabilność.
