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Dynamic stability of shallow arch with elastic supports—application in the dynamic stability analysis of inner winding of transformer during short circuit
International Journal of Non-Linear Mechanics 37 (2002) 909–920
Dynamic stability of shallow arch with elastic supports—application in the dynamic stability analysis of inner winding of transformer during short circuit Jian-Xue Xu ∗ , Hong Huang, Pei-Zhen Zhang, Ji-Qing Zhou Institute of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China
Abstract In this paper, the dynamic stability of a shallow arch with elastic supports subjected to impulsive load is used as a theoretical model to investigate the dynamic stability problem of inner windings of power transformer under short-circuit condition. Firstly, the series solution representing the equilibrium con0gurations of a shallow arch is obtained by solving the corresponding non-linear integration-di1erential equation. The local stability of each equilibrium con0guration is discussed, and the su3cient condition for stability of the shallow arch system as well as the critical load against snap-through is obtained. Secondly, the equivalent relation between short-circuit load and impulsive one, and the electrical forces transferred pattern between the coils of inner windings are assumed. Then the results of the shallow arch model are applied to the case of the inner winding of transformer and the formulas for computing critical electromagnetic force and the dynamic stability criterion of the inner windings are established. Finally, examples are o1ered and the theoretical results are shown to agree well with the experimental ones. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Shallow arch; Dynamic stability; Snap-through; Transformer; Short circuit
1. Introduction The problem of stability of elastic supporting shallow arch is a complicated stability problem of non-linear system, and is a general problem of mechanics with wide application background. In 1954, Ho1 [1] presented the analytical solution of dynamic stability of a shallow arch with two ends simple supporting. In 1966, John and Humphreys [2] provided the analytical analysis on dynamic stability with sti1ness simple supporting and Gereokin ∗
Corresponding author.
numerical analysis by using the Budianshy–Routh stability criterion. From 1966 to 1975, Hsu published a series of papers, [3– 6] which solved the dynamic stability problems with di1erent boundary conditions and the subject of impulsion and step loads analytically. In 1968, Lock [7,8] provided the numerical analysis of sti1ness supporting shallow arch and shallow done based on the Budiansky– Routh stability criterion and corresponding experimental results. In 1976 Lo [9] solved this problem by using the method of integration equation. In 1978 and 1980, Johnson [10,11] studied the response and the e1ect of damping on dynamic snap-through of a shallow circle arch. In 1989, Kounadis et al. [12]
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 1 0 5 - 6
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provided the results for dynamic buckling of an arch model under impact loading. In a chapter of Simitses’s book [13], the dynamic stability of shallow arch of pinned and clamped supports was summarized. In these contributions, the transverse elastic support has not been considered. In the last 90 years, the theoretical and experimental works on non-linear dynamic response, dynamic behaviour, and global dynamic stability of shallow arch [14 –16] and the experimental study on regular and irregular motions of shallow arch with elastic supports [17] have been reported independently. The capability of inner winding to resist short-circuit dynamic instability (buckling) is the weakest ring of the large transformer bearing short-circuit hit or strike. Through analyzing a lot of short-circuit accidents, it has been shown that the buckling of the inner winding is the main reason for the accidents. In the past 0fty years, a series of papers analyzing short-circuit stability by static method have been published. In the transformer sub-sessions of two international large electric network conferences held in 1972 and 1980, the short-circuit mechanical strength of transformer was the central topic. Steel et al. [18] presented the dynamic analysis of a simpli0ed model that is far from the real structure of the inner windings. Since 1983, authors have shown that the shallow arch with elastic supports is an appropriate model for analyzing short-circuit dynamic stability of the inner circle coil of inner winding of transformer. At the same time, the impulsive load is regarded as short-circuit hitting load approximately so that the models of the coils, elasticity of the insulation spacer, and the sudden appearance of short-circuit electric forces approximate the real structure of transformer very well. All of these can be used for practical engineering. In this paper, the non-linear integrationdi1erential motion equations of shallow arch with transversal elastic supports are established. The static buckling characteristic form is solved, and it is used as characteristic mode shape of motion, then the integration-di1erential equation of elastic support shallow arch and the in0nite-dimensional ordinary di1erential equation of series solution form are derived. In in0nite-dimensional gener-
alized displacement and velocity phase space, all static equilibrium positions (singularity points) are found, and the proofs of local stability of each equilibrium point are given. In the view of energy, the global stable region is analyzed. The su3cient condition, criterion formulas, and critical load against snap through instability for elastic support shallow arch subjected to impulsive load are derived. Finally, the relationship between impulsion load and the short-circuit load of inner winding is derived. By considering the transfer from the outer circle coils of inner winding to inner circle coil of inner winding, the formula for calculating critical electric forces and stability criterion of inner winding during short circuit are obtained.
2. Mechanics and mathematics model of elastic support shallow arch Mechanics and mathematics model of elastic support shallow arch is described in Fig. 1. The conditions are: (1) the curvature of arch is small, Ymax =L ¡ 1=10– 1=50; (2) the e1ect of axis force induced by axial strain is considered, and axial strain is assumed to be constant; (3) the deformation is located in elastic range; (4) the initial deKection of arch is the coordinate function. Thus, the integration-di1erential motion equation of transversal shallow arch and boundary with elastic
Fig. 1. Mechanical model of shallow arch.
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
the integration-di1erential motion equation and boundary conditions, (1) – (6) are transferred into
supporting conditions can be described as A
@4 (y − y0 ) @2 y @2 y = − EI − S @t 2 @x4 @x2
@2 y0 − q(x; t); @x2 2 2 @y0 @y EA L=2 − d x; S = S0 + 2L −L=2 @x @x + S0
@3 y − EI 3 @x
x=−L=2
@3 y0 + EI @x3
uN = −(u − u0 ) − (G0 + G)u (1) (2)
x=−L=2
G0 =
S0 L ; 2 EI
1 Q(; ) = 2
=k 1=2
A I
G=
2
=2
−=2
(3) (4) (5) (6)
u = u0
= −
=
; 2 = : 2
; 2
= −
L ; 2 EI
(10) (11) (12) (13)
(14)
where B is the load density impulse (kg=s); (t) is the Dirac function, then formula (8) is simpli0ed as uN = − (u − u0 ) − Gu :
(15)
Assuming that Eqs. (15) and (9) possess the solution having series form: u = u0 +
∞
n ()Yn ();
(16)
n=1
where space function Yn () can adopt the static buckling characteristic form of straight beam with the same boundary conditions of elastic support shallow arch, and is determined by the following equations: Y + kY = 0
−Y = Y
(7)
; 2
(9)
Here, it is agreed that “·” describes derivative against ; and “ ” represents derivative against . Let initial axial force S0 = 0; i.e. G0 = 0; and the load be impulsive load: q(x; t) = B(t);
3
L4 q(x; t); 4 EI
u = u0
(8)
[(u0 )2 − (u )2 ] d;
u − u0 = u
where y is the displacement of arch (m); y0 the initial displacement of arch (m); S the axial force of arch (N); S0 the initial axial force of arch (N); q(x; t) the load density (N=m); A the section area of arch (m2 ); the mass density of arch (kg=m3 ); E the modulus of elasticity (Pa); I the moment of inertia for section area (m4 ); L the span length (m); k1 = k2 = k the sti1ness of transversal elastic support (N=m). Introducing the following non-dimensional quantities: 1=2 1=2 y0 A y A ; u0 = ; u= 2 I 2 I 1=2 EI4 x = t; = ; L AL4 2
+ G0 u0 − Q(; );
−u + u0 = u
L = k1 y − ; t ; 2 @3 y @3 y0 L ;t ; EI 3 − EI = k2 y @x x=L=2 @x3 x=L=2 2 @y0 @y = ; @x x=−L=2 @x x=−L=2 @y0 @y = ; @x x=L=2 @x x=L=2
911
Y = Y
(17)
= −
=
; 2
; 2 (18)
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
Cn =
kn
1 1 + (6kn2 = − 1) cos2
kn 2
;
n = 2; 4; 6; : : : ;
Fig. 2. Functions curves of both sides of Eq. (21).
Y =0 Y =0
; 2 = : 2 = −
Solving Eqs. (17) – (19), the static buckling characteristic form of straight beam can be obtained as (n+1)=2
Yn () = Cn [1 − (−1)
cos(n + 1)];
n = 2; 4; 6; : : : ;
where kn is determined by the following formula: k n kn = − : 2 2
(21)
The curves of left- and right-side functions of this formula are described in Fig. 2, and the values of the intersection points are the roots of Eq. (21). Substituting static buckling characteristic form (20) into the canonical condition 2 =2 Yn Yn d = 1 −=2 the coe3cients Cn can be determined as 1 Cn = ; (n + 1)2
n = 1; 3; 5; : : : ;
2
1
m = n;
0
m = n;
=2
−=2
Yn Yn d =
1 ; n = 1; 3; 5; : : : ; (n+1)2
1 2;
kn
(20)
kn3
mn =
an =
n = 1; 3; 5; : : : ;
kn ; Yn () = Cn sin kn − kn cos 2
tan
Obviously, in formula (20), Y1 ; Y3 ; Y5 are symmetric characteristic form about the center of the beam, and Y2 ; Y4 ; Y6 ; : : : are antisymmetric. It can be proved that Yn () satis0es the following orthogonal relations: 2 =2 Yn Ym d = mn ; (23) −=2 2 =2 2 =2 Yn Ym d = − Yn Ym d = an mn ; −=2 −=2 (24) where
(19)
(22)
n = 2; 4; 6; : : : : (25)
From Fig. 2 and formula (25), once can 0nd that the following relation of coe3cients an holds: a1 ¿ a3 ¿ a5 ¿ · · · ;
(26)
a2 ¿ a4 ¿ a6 ¿ · · · ;
(27)
· · · ¿ a2m−3 ¿ a2m−2 ¿ a2m ¿ a2m−1 ¿ a2m+2
¿ a2m+1 ¿ · · · ;
(28)
where m is an integer related to the non-dimensional sti1ness of elastic support : 2m − 2 ¡ 6 2m: 2 Substituting the series form solution (16) of buckling characteristic mode shapes into integrationdi1erential motion equations of arch (15), and
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
assuming that initial displacement u0 is u0 =
∞
3. Equilibrium con guration and its local stability
bm Ym ()
3.1. Equilibrium con3guration
n=1
one can obtain ∞
∞
Nn ()Yn () =
n=1
2 +
−=2
×
u0
=2
+
By making left-hand side of Eq. (29) equal zero, the equation determining the singular point can be obtained:
n Yn
n=1
(u0 )2 −
∞
u0
+
∞
2 n Yn d
n=1
n Yn
:
n=1
Multiplying both sides of this formula by (2=)Yn (), and integrating it for variable from −=2 to =2, and using orthogonal relations (23) and (24), one obtain ˙m = !m ; ∞
!m = 0;
Cmn !n = − m +G(am m +bm );
n=1
m = 1; 2; 3; : : : ; (29)
(31)
m − G(am m + bm ) = 0;
m = 1; 2; 3 ; : : : :
U0 = b(1 + cos 2); bm = 0;
m = 2; 3; : : : ;
formula (30) and equation (32) become 1 − G(a1 1 + b) = 0;
where
m (1 − Gam ) = 0;
bm = −
The roots Sm of Eq. (33) are as follows:
2 =2 Ym u0 d; −=2 2 =2 Ym Yn d; Cmn = −=2
G=−2
M =1
bm am −
∞
am a2m :
(1)
(29 ) (30)
m=1
In formula (29), m ; !m are generalized displacement and generalized velocity, which span in0nite-dimensional Euclidean phase space. Then the problem on dynamic stability of shallow arch becomes the stability problem of nonlinear dynamical system in in0nite-dimensional phase space, which is described by in0nite ordinary di1erential equations (29). This problem consists of three parts: (1) determination of equilibrium point (singular point); (2) the local stability of equilibrium con0guration; (3) the global stability.
(32)
Formula (31) represents that the velocities are equal to zero; hence, it is just needed to solve Eq. (32) to 0nd the singular point described by generalized displacement, i.e. the equilibrium con0guration. Assuming that the initial displacement is
b1 = b;
∞
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m = 2; 3; : : : :
Sm = 0; m = 1; 2;
(33)
(34)
This solution represents the initial displacement of arch. Denote it by P0 . Substituting it into (16), we obtain u = u0 = b(1 + cos 2): √ (3b± b2 −4) ; Sm = 0; m = 2; 3; : : : : (2) S1 = − 2a1 (35)
The pair of these equilibrium con0gurations just depends on 1 , and only exists while b ¿ 2. It is denoted by P1± . Then √ b ± b2 − 4 (1 + cos 2): u1± = u0 + S1 Y1 () = − 2
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
(3) S1 = −
b aj
1 Sj = ± aj Sm = 0;
a1 −1 ; aj
1=2 (a1 − 2)=aj b2 ; 2 − 1 a1 =aj − 1
m = 1;
m = j:
j = 1; (36)
The pairs of these equilibrium con0gurations just exist while the following formulas hold. It is denoted by Pj± . Then 2 a1 a1 2 −2 b ¿ −1 (37) aj aj and
Introducing non-dimensional quantities (7), the non-dimensional energy is AL3 H = 4 H ∗: (39) EI Under the assumption of zero initial axial force and impulsive load, we have 2 =2 2 2 =2 2 u˙ d + [u˙ − u0 ]2 d H= −=2 −=2 2 2 1 2 + u2 − ; + u2 ; + G : 2 2 2 (40) Considering that the coordinates of equilibrium con0guration in phase space are i = Si ; i = 1; 2; : : : ; !1 = !2 = · · · = 0, we can construct Lyapunov function
uj± = u0 + S1 Y1 () + Sj Yj ():
V (c) = H (c) − H (P);
3.2. Local stability of equilibrium con3guration
where c is a point in the neighboring region of P, its coordinates are
Here, the local stability is the stability in the Lyapunov meaning, i.e. the stability under small perturbation: the total energy of system reaches limit value in equilibrium con0guration, the con0guration is stable when it is maximum, and unstable when it is minimum. Hence, the total energy at any point C, which is located in the small neighboring region of a stable equilibrium con0guration P, is always greater than the energy of P, but for unstable equilibrium con0guration it is not so. The total energy of elastic support shallow arch, (1) – (6), is 2 1 L=2 1 L=2 @y ∗ A dx + EI H = 2 −L=2 @t 2 −L=2 2 2 2 2 @2 y @ y0 @ y @ 2 y0 × − − 2 − @x2 @x2 @x2 @x2 2 S 2L @ y0 S 0 y0 + dx + + 2 @x EI 2EA 1 1 + k1 (y2 − y02 )x=−L=2 + k2 (y2 − y02 )x=L=2 2 2 L=2 + q(y − y0 ) d x: (38) −L=2
i = Si + i ;
(41)
!i = *i ; i = 1; 2; 3; : : : :
Substituting the coordinates of P and C into formula (41), and considering orthogonal relation (24), we can obtain ∞ ∞ Cmn *m *n + J (c) + O(3m ); (42) V (c) = m=1 n=1
where Cmn is determined by (29), and J (c) is the quadratic form ∞ 2 2 2 2 2 a1 am Sm J (c) = 1 1+2b +6ba1 S1 +3a1 S1 + m=2
+ 4(b + a1 S1 )1 +
∞
2n
∞
am Sm m
m=2
1+2an bS1 +2a2n S2n +
n=2
+
∞ ∞
∞
an am S2m
m=1
2an am Sn Sm n m :
(43)
n=2 m=2 m=n
In su3ciently small neighboring region, the nature of V (c) is determined by the quadratic term in formula (42). Obviously, the quadratic form of
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
generalized velocity perturbation *m is positive definite; thus, it is just needed to check the quadratic form J (c) of n . J (c) is written in standard form as J (c) = −
k
aSn 2n
+
n=1
∞
bSn 2n ;
aSn ¿ 0; bSn ¿ 0
n=k+1
(44)
where n = 1; 2; : : : is the linear combination of m ; m = 1; 2. For local stability, it is needed that the quadratic form is positive de0nite, i.e. for formula (44), k = 0 [3,4]. Substituting (34) – (36) into formula (44), the local stability of equilibrium con0guration of arch P0 ; P1± ; Pj± can be determined (1) P0 is always stable, and corresponds to the initial con0guration having non-deformation. (2) For P1± , there exists ∞ 2 2 J (c1± ) = d± d± n n ; 1 1 + n=2
where 2 2 2 d± 1 = 1 + 2b + 6ba1 S1 + 3a1 S1 √ b2 − 4 2 ( b − 4 ± 3b); = 2 1 an (±b b2 − 4 − b2 − 2): d± n =1 + 2 a1
Hence, P1− is always unstable. If the following formula holds, P1+ is stable, otherwise it is unstable. 2 a1 a1 2 −2 b ¿ −1 : (45) a2 a2 (3) For Pj± , there exists
a1 b ± J (cj ) = 1 − 1 21 + 2 aj a1 =aj − 1
(a1 =aj − 2)b2 ∓ −1 (a1 =aj − 1)2 +
∞ n=2 n=j
1−
an aj
2
1=2
2n :
Therefore Pj± is always unstable.
j
915
4. Su!cient condition for stability of shallow arch against snap-through instability If the system (29) possesses stable equilibrium con0gurations more than one, there is an optimal equilibrium con0guration among them, which possesses lowest energy, and is called nature equilibrium con0guration. When the system is subjected to initial perturbation, it comes back to this equilibrium con0guration. But if the perturbation is great enough, the system may jump to another stable equilibrium con0guration, and these jumps between equilibrium con0gurations are called snap-through instability or collapse. In the in0nite-dimensional phase space, the initial perturbation is presented as initial phase point, and the motion of system is presented as phase orbit starting from initial phase point. The stability problem of shallow arch against snap-through instability becomes the problem that is from which phase point the phase orbit approaches the nature equilibrium con0guration. All of these phase points consist of the region of global stability. 4.1. Su5cient condition for stability of shallow arch against snap-through instability Using the Lyapunov function de0ned in Section 3.2 formula (41), V (c) is expressed as V (c) = H (c) − H (P0 );
(46)
where P0 is nature equilibrium con0guration; thus, in the neighboring region of P0 . V (c) is always positive de0nite for all i , *i ; I = 1; 2; : : :, hence, one can draw a series closed equal energy surfaces V (c) = W , and these equal energy surfaces expand further as W increases. The singular point that encounters the equal energy surface for the 0rst time is denoted by P∗ , and this singular point is called critical point. Obviously, P∗ is a local instability singular point, and whole points c inside the equal energy surface passing through P∗ satisfy V (c) ¡ V (P∗ ):
(47)
The region inside the equal energy surface V (c) = V (Pn ) is denoted by S(Pn ); we can see that
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
S(Pn ) is the su3cient region of stability. When initial state phase points locate in S(Pn ), the phase orbit of system motion will always stay in S(Pn ), and cannot pass through the equal energy surface V (c) = V (Pn ) to reach another stable equilibrium con0guration. Hence, the system is stable [3]. Consequently, the formula (47) is regarded as a su3cient condition for stability of shallow arch against snap-through instability. Besides, for the shallow arch system, if the formula (45) does not hold, the system possesses only one local stable equilibrium con0guration P0 ; hence, no matter how great the load is, the snap-through instability does not occur for shallow arch. Specially, when the non-dimensional sti1ness of elastic support 6 27:5, from formula (25) and Fig. 2, a1 =a2 6 2; therefore, formula (45) does always not hold, and it is of no problem of elastic snap-through instability. When formula (45) holds, the non-dimensional arch high satis0es b2 ¿
(a1 =a2 − 1)2 : a1 =a2 − 2
(48)
Besides P0 , P1+ is also locally stable. In this case, the snap-through instability problem still exists, the su3cient condition for stability is determined by formula (47), and the critical point Pn can be found by analyzing the distribution of singular points, and the structure and expanding case of equal energy surfaces V (c), around P0 . Substituting (34) – (36) into energy formula (40), the energy of singular points can be obtained as H (P0 ) = 0;
(49)
H (P1± ) = 12 (−3b ∓
1 a2j
Fig. 4. Section graphs of equal energy surfaces around P0 on plane a1 − a2 .
b2 − 4)2 ×[8 + (b ∓ b2 − 4)2 ];
H (Pj± ) =
Fig. 3. Curves of singular energy via arch high H .
aj b2 1 − a1 − aj 2
(50)
j = 2; 3; 4; : : : : (51)
The curves of energies H via arch high b for the 0rst several equilibrium con0gurations are drawn in Fig. 3. Figs. 4 and 5 show the sections of equal energy surfaces around P0 in planes, 1 –2 , 2 –3 .
By analyzing formula (50), (51), and considering Fig. 2 and formula (28), it is easy to prove that P2± is unstable singular point of lowest energy. From Figs. 4 and 5, it can be derived that P2± is just the critical point Pn . Using formula (47), the su3cient condition for stability of shallow arch against snap-through instability can be obtained. a2 b2 1 1 − : V (c) ¡ V (P∗ ) = V (P2± ) = 2 a2 a1 − a2 2 (52)
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
917
Under the assumption of zero initial axial force, from (49) we can obtain 1 L4 V (c) = T = B2 : 2 4 EI 2 From formula (47), the su3cient condition for stability of shallow arch against snap-through instability subjected to impulsive load can be obtained as 1 L4 B2 ¡ V (P∗ ): 2 4 EI 2 The critical load impulse Bc is determined by the following formula:
4 1=2 2 EI 2 ± V (P∗ ) : (53) Bc = L4
Fig. 5. Section graphs of equal energy surfaces around P0 on plane a2 − a3 .
4.2. Critical load under applying impulsion load The action of impulsive load corresponds to an initial perturbation described by a non-zero phase point of system. Its coordinates are initial displacement and initial velocity. By calculating the Lyapunov function of this initial phase point, V (c), one can determine the stability against snap-through instability under applied impulsive load, i.e. dynamic stability by formula (52). Under applied impulsive load q(xt) = B(t), Fig. 1 shallow arch system gains initial velocity, it is B @y(x; t) = : @t A Initial kinetic energy is 2 1 L=2 1 B2 @y ∗ L: A dx = T = 2 −L=2 @t 2 A Non-dimensional kinetic energy is T=
AL3 ∗ 1 L4 T = B2 : 4 EI 2 2 4 EI 2
5. Analysis of short-circuit dynamic stability of inner winding of transformer The winding of transformer having discus type is described in Fig. 6. The inner circle coil of inner winding is simpli0ed as an elastic thin circle ring supported on the elastic radius supports which are separated by equal distance and composed of elastic insulation stays and elastic insulation cylinder. Thus, the inner circle coil is divided into many spans by these supports. since each span of inner circle is symmetric about the center, and the number of supports is very large, each span can be regarded as a radius direction (transverse) elastic clamped support shallow arch. During short circuit of transformer, the shortcircuit force applied on the inner circle coil is applied in radius direction mainly and with equal distribution along the whole circle. Then, the analysis of short-circuit dynamic stability of inner circle coil can be regarded as the problem of dynamic stability of elastic clamped support shallow arch subjected to short-circuit radius load. This shallow arch is constituted of a span of inner circle coil separated by two insulation stays. Short-circuit load of inner circle coil q(x; t) (Kg=s2 ) is induced by short-circuit electric current: i = I (e−t=Ta − cos !t) where Ta is time constant of short-circuit current damping, $ is circle frequency of steady state
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
electrical force is denoted as q0 , then we obtain q0 = P10 +!(P20 + !(P30 + !(P40 + · · · + !Pm0 )) · · ·); (55) where Pi0 is the coe3cient of single circle of each copper coil, and the electrical force of the ith copper coil Pi = Pi0 F(t). In general, time constant of current Ta is 0.05 – 1.0, and from formula (54), q(t) is decreasing obviously; hence, the 0rst wave of q(t) can be regarded as an impulsion approximately, where the duration time of part of q(t) ¿ 0 is about equal to VT = (3=4)(2=!). When VT is approaching the natural period of elastic support shallow arch, the short-circuit load can approximate impulsion very well: VT VT q(t) dt = B(t) dt = B: (56) 0
Fig. 6. Structural graph of discus-type winding of transformer.
current. The short-circuit load can be regarded as uniform distribution, q(x; t) = q(t), which includes the electrical force applied to inner coil and the force applied to form other circle coils of inner winding. Assume that the forces between coils are transferred according to static form, and the transferring coef0cient is !, then
0
From the above formula, the relation between coe3cient q0 of short-circuit electrical force and impulsive load impulse B can be obtained. Further, from the critical load impulse of shallow arch system against snap-through instability, Bc , the critical short-circuit electrical force coe3cient of dynamic stability of copper coil, q0c can be calculated by using the following formula. Bc q0c = VT : (57) F(t) dt 0 Finally, the short-circuit dynamic stability condition of inner winding of transformer can be obtained:
q(t) = P1 +!(P2 +!(P3 +!(P4 + · · · +!Pm )) · · ·);
q0 ¡ q0c
where Pi represents the electrical force of the ith layer circle coil, and inner circle coil is regarded as the 0rst layer; $ is the total layer number. Considering the simultaneousness of electromagnetic 0eld, we have
6. Example and comparison with experimental results
q(t) = q0 (N1 + N2 e
−2t=Ta
+ N3 e
+ N4 cos 2!t)Vq0 F(t);
−t=Ta
cos !t (54)
where N1 ; N2 ; N3 ; N4 are constants. From electromagnetic calculation, the coe3cient of short-circuit
(58)
The authors have performed short-circuit dynamic stability experiments of inner winding of model transformer [19,20], the diameter, section width high of inner copper coil of inner winding are D = 0:44m, b = 0:0108m, = 0:0032m, respectively. A circular coil is supported by 12 insulation stays. The elastic modulus of coil material is
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
Ee = 1:2 × 1011 Pa. By calculating, the elastic supporting sti1ness is found to be C = 5 × 107 N=m. Short-circuit shock of large current has been done 20 times. The measurement values of strains increase with square of current. Elastic and plastic stages have been reached; for a model, a symmetric buckling instability of inner winding occurred at current I = 15000 A, while another model was destroyed and burned at I = 24000 A. According to electromagnetic calculation, the value of this critical electric force of instability is qc = 4450 N=m. By using the above method and formulas, the theoretical value of critical load of elastic instability is obtained, which is q = 5250 N=m. Taking into consideration the situation that circle coil has entered the elastic-plastic stage before the load reaches the critical value of radius load of elastic instability, we make a simple elastic-plastic correction, the critical load value of elastic-plastic instability will be qr = 3106 N=m which is lower than the theoretical value of the elastic case. Hence, it coincides with the results of the experiments. Indeed, if we adopt the result of analysis of sti1ness supporting shallow arch dynamic stability, the critical load reached qn = 45700 N=m, it is very far from the practical experiment.
7. Conclusions (1) For transverse elastic clamped supporting case, analytical analysis results on dynamic stability of shallow arch subjected to traverse impulse load are provided. They are the series solution representing the equilibrium con0gurations of shallow arch by solving the corresponding non-linear integration-di1erential equation, the local stability of each equilibrium con0guration, and the su3cient condition for stability of the shallow arch system as well as the critical load against snap-through. (2) It is rational that the problem of short-circuit dynamic stability of inner winding of transformer should be investigated by using the theoretical model of the analysis of dynamic stability of transverse elastic clamped support shallow arch subjected to impulsive load.
919
(3) The formulas of critical load of shallow arch against snap-through and the values obtained by calculating these formulas in the examples are veri0ed to be coincident with experimental results. Acknowledgements This work was supported by the National Science Foundation of the People’s Republic of China. References [1] N.J. Ho1, V.G. Bruce, Dynamic analysis of the buckling of laterally loaded Kat archer, J. Math. Phys. 32 (1954) 276–288. [2] J.S. Humphreys, Dynamic snap buckling of shallow arches, AIAA J. 4 (1966) 878–886. [3] C.S. Hsu, On dynamic stability of elastic bodies with prescribed initial conditions, Int. J. Eng. Sci. 4 (1) (1966) 1–21. [4] C.S. Hsu, The e1ects of various parameters on the dynamic stability of a shallow arch, ASME J. Appl. Mech. 34 (1967) 349–358. [5] C.S. Hsu, Stability of shallow arches against snap-through under timeless step loads, ASME J. Appl. Mech. 35 (1) (1968) 31–39. [6] C.S. Hsu, Equilibrium con0gurations of a stability character, Int. J. Non-linear Mech. 3 (1968) 113–136. [7] M.M. Lock, Snapping of a shallow sinusoidal arch under a step pressure load, AIAA J. 4 (7) (1966) 1249–1259. [8] M.M. Lock, S. Okubo, J.S. Whittier, Experiments on snapping of a shallow dome under a step pressure load, AIAA J. 6 (7) (1968) 1320–1326. [9] D.L.C. Lo, Dynamic buckling of shallow arches, ASCE 102 EM5 (1976) 901–916. [10] E.R. Johnson, The e1ect of spatial distribution on dynamic snap through, ASME J. Appl. Mech. 45 (1978) 612–618. [11] E.R. Johnson, E1ect of damping on dynamic snap-through, ASME J. Appl. Mech. 47 (3) (1980) 601– 606. [12] A.N. Kounadis, J. Raftoyiannis, J. Mallis, Dynamic buckling of an arch model under impact loading, J. Sound Vib. 134 (2) (1989) 193–202. [13] G.J. Simitses, Dynamic Stability of Suddenly Loaded Structures, Springer, New York, 1990. [14] K.B. Blair, C.M. Krousrill, T.N. Farris, Non-linear dynamic response of shallow arch to harmonic forcing, J. Sound Vib. 193 (3) (1996) 353–367. [15] P. Patricio, M. Adda-Bedia, M. Ben Amar, Elastica problem: instabilities of an elastic arch, Physica D 124 (1–3) (1998) 285–295.
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[16] J. Levitas, J. Singer, T. Weller, Global dynamic stability of a shallow arch by Poincare-like simple cell mapping, Int. J. Non-linear Mech. 32 (2) (1997) 411–424. [17] Jianxue Xu, Zhenmao Chen, Experiments on regular and chaotic motions of forced shallow arch oscillator, in: Editor in Chief: Du Qinghua, Proceedings of the International Conference on Vibration Problems in Engineering, 17–24 June, 1990, Wuhan-Chongqing, China, International Academic Publishers, 1990, p. 410.
[18] R.B. Steel, W.M. Johnson, J.J. Narbus, M.R. Patel, R.A. Nelson, Dynamic measurements in power transformers under short-circuit conditions, Proceedings of CIGRE 12 (1) (1972), pp. 1–20. [19] Zhang Peizheng, Xu Jianxue, Zhou Jiqing et al., Radius stability of power transformer during short circuit, Transformer 10 (1985) 2–7 (in Chinese). [20] Zhou Jiqing, Xu Jianxue, Zhang Peizheng, Study on short circuit radius stability of inner winding of transformer, J. Xi’an Jiaotong University 20 (4) (1986) 101–112 (in Chinese).
Dynamic stability of shallow arch with elastic supports—application in the dynamic stability analysis of inner winding of transformer during short circuit Jian-Xue Xu ∗ , Hong Huang, Pei-Zhen Zhang, Ji-Qing Zhou Institute of Engineering Mechanics, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China
Abstract In this paper, the dynamic stability of a shallow arch with elastic supports subjected to impulsive load is used as a theoretical model to investigate the dynamic stability problem of inner windings of power transformer under short-circuit condition. Firstly, the series solution representing the equilibrium con0gurations of a shallow arch is obtained by solving the corresponding non-linear integration-di1erential equation. The local stability of each equilibrium con0guration is discussed, and the su3cient condition for stability of the shallow arch system as well as the critical load against snap-through is obtained. Secondly, the equivalent relation between short-circuit load and impulsive one, and the electrical forces transferred pattern between the coils of inner windings are assumed. Then the results of the shallow arch model are applied to the case of the inner winding of transformer and the formulas for computing critical electromagnetic force and the dynamic stability criterion of the inner windings are established. Finally, examples are o1ered and the theoretical results are shown to agree well with the experimental ones. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Shallow arch; Dynamic stability; Snap-through; Transformer; Short circuit
1. Introduction The problem of stability of elastic supporting shallow arch is a complicated stability problem of non-linear system, and is a general problem of mechanics with wide application background. In 1954, Ho1 [1] presented the analytical solution of dynamic stability of a shallow arch with two ends simple supporting. In 1966, John and Humphreys [2] provided the analytical analysis on dynamic stability with sti1ness simple supporting and Gereokin ∗
Corresponding author.
numerical analysis by using the Budianshy–Routh stability criterion. From 1966 to 1975, Hsu published a series of papers, [3– 6] which solved the dynamic stability problems with di1erent boundary conditions and the subject of impulsion and step loads analytically. In 1968, Lock [7,8] provided the numerical analysis of sti1ness supporting shallow arch and shallow done based on the Budiansky– Routh stability criterion and corresponding experimental results. In 1976 Lo [9] solved this problem by using the method of integration equation. In 1978 and 1980, Johnson [10,11] studied the response and the e1ect of damping on dynamic snap-through of a shallow circle arch. In 1989, Kounadis et al. [12]
0020-7462/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 1 ) 0 0 1 0 5 - 6
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
provided the results for dynamic buckling of an arch model under impact loading. In a chapter of Simitses’s book [13], the dynamic stability of shallow arch of pinned and clamped supports was summarized. In these contributions, the transverse elastic support has not been considered. In the last 90 years, the theoretical and experimental works on non-linear dynamic response, dynamic behaviour, and global dynamic stability of shallow arch [14 –16] and the experimental study on regular and irregular motions of shallow arch with elastic supports [17] have been reported independently. The capability of inner winding to resist short-circuit dynamic instability (buckling) is the weakest ring of the large transformer bearing short-circuit hit or strike. Through analyzing a lot of short-circuit accidents, it has been shown that the buckling of the inner winding is the main reason for the accidents. In the past 0fty years, a series of papers analyzing short-circuit stability by static method have been published. In the transformer sub-sessions of two international large electric network conferences held in 1972 and 1980, the short-circuit mechanical strength of transformer was the central topic. Steel et al. [18] presented the dynamic analysis of a simpli0ed model that is far from the real structure of the inner windings. Since 1983, authors have shown that the shallow arch with elastic supports is an appropriate model for analyzing short-circuit dynamic stability of the inner circle coil of inner winding of transformer. At the same time, the impulsive load is regarded as short-circuit hitting load approximately so that the models of the coils, elasticity of the insulation spacer, and the sudden appearance of short-circuit electric forces approximate the real structure of transformer very well. All of these can be used for practical engineering. In this paper, the non-linear integrationdi1erential motion equations of shallow arch with transversal elastic supports are established. The static buckling characteristic form is solved, and it is used as characteristic mode shape of motion, then the integration-di1erential equation of elastic support shallow arch and the in0nite-dimensional ordinary di1erential equation of series solution form are derived. In in0nite-dimensional gener-
alized displacement and velocity phase space, all static equilibrium positions (singularity points) are found, and the proofs of local stability of each equilibrium point are given. In the view of energy, the global stable region is analyzed. The su3cient condition, criterion formulas, and critical load against snap through instability for elastic support shallow arch subjected to impulsive load are derived. Finally, the relationship between impulsion load and the short-circuit load of inner winding is derived. By considering the transfer from the outer circle coils of inner winding to inner circle coil of inner winding, the formula for calculating critical electric forces and stability criterion of inner winding during short circuit are obtained.
2. Mechanics and mathematics model of elastic support shallow arch Mechanics and mathematics model of elastic support shallow arch is described in Fig. 1. The conditions are: (1) the curvature of arch is small, Ymax =L ¡ 1=10– 1=50; (2) the e1ect of axis force induced by axial strain is considered, and axial strain is assumed to be constant; (3) the deformation is located in elastic range; (4) the initial deKection of arch is the coordinate function. Thus, the integration-di1erential motion equation of transversal shallow arch and boundary with elastic
Fig. 1. Mechanical model of shallow arch.
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
the integration-di1erential motion equation and boundary conditions, (1) – (6) are transferred into
supporting conditions can be described as A
@4 (y − y0 ) @2 y @2 y = − EI − S @t 2 @x4 @x2
@2 y0 − q(x; t); @x2 2 2 @y0 @y EA L=2 − d x; S = S0 + 2L −L=2 @x @x + S0
@3 y − EI 3 @x
x=−L=2
@3 y0 + EI @x3
uN = −(u − u0 ) − (G0 + G)u (1) (2)
x=−L=2
G0 =
S0 L ; 2 EI
1 Q(; ) = 2
=k 1=2
A I
G=
2
=2
−=2
(3) (4) (5) (6)
u = u0
= −
=
; 2 = : 2
; 2
= −
L ; 2 EI
(10) (11) (12) (13)
(14)
where B is the load density impulse (kg=s); (t) is the Dirac function, then formula (8) is simpli0ed as uN = − (u − u0 ) − Gu :
(15)
Assuming that Eqs. (15) and (9) possess the solution having series form: u = u0 +
∞
n ()Yn ();
(16)
n=1
where space function Yn () can adopt the static buckling characteristic form of straight beam with the same boundary conditions of elastic support shallow arch, and is determined by the following equations: Y + kY = 0
−Y = Y
(7)
; 2
(9)
Here, it is agreed that “·” describes derivative against ; and “ ” represents derivative against . Let initial axial force S0 = 0; i.e. G0 = 0; and the load be impulsive load: q(x; t) = B(t);
3
L4 q(x; t); 4 EI
u = u0
(8)
[(u0 )2 − (u )2 ] d;
u − u0 = u
where y is the displacement of arch (m); y0 the initial displacement of arch (m); S the axial force of arch (N); S0 the initial axial force of arch (N); q(x; t) the load density (N=m); A the section area of arch (m2 ); the mass density of arch (kg=m3 ); E the modulus of elasticity (Pa); I the moment of inertia for section area (m4 ); L the span length (m); k1 = k2 = k the sti1ness of transversal elastic support (N=m). Introducing the following non-dimensional quantities: 1=2 1=2 y0 A y A ; u0 = ; u= 2 I 2 I 1=2 EI4 x = t; = ; L AL4 2
+ G0 u0 − Q(; );
−u + u0 = u
L = k1 y − ; t ; 2 @3 y @3 y0 L ;t ; EI 3 − EI = k2 y @x x=L=2 @x3 x=L=2 2 @y0 @y = ; @x x=−L=2 @x x=−L=2 @y0 @y = ; @x x=L=2 @x x=L=2
911
Y = Y
(17)
= −
=
; 2
; 2 (18)
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
Cn =
kn
1 1 + (6kn2 = − 1) cos2
kn 2
;
n = 2; 4; 6; : : : ;
Fig. 2. Functions curves of both sides of Eq. (21).
Y =0 Y =0
; 2 = : 2 = −
Solving Eqs. (17) – (19), the static buckling characteristic form of straight beam can be obtained as (n+1)=2
Yn () = Cn [1 − (−1)
cos(n + 1)];
n = 2; 4; 6; : : : ;
where kn is determined by the following formula: k n kn = − : 2 2
(21)
The curves of left- and right-side functions of this formula are described in Fig. 2, and the values of the intersection points are the roots of Eq. (21). Substituting static buckling characteristic form (20) into the canonical condition 2 =2 Yn Yn d = 1 −=2 the coe3cients Cn can be determined as 1 Cn = ; (n + 1)2
n = 1; 3; 5; : : : ;
2
1
m = n;
0
m = n;
=2
−=2
Yn Yn d =
1 ; n = 1; 3; 5; : : : ; (n+1)2
1 2;
kn
(20)
kn3
mn =
an =
n = 1; 3; 5; : : : ;
kn ; Yn () = Cn sin kn − kn cos 2
tan
Obviously, in formula (20), Y1 ; Y3 ; Y5 are symmetric characteristic form about the center of the beam, and Y2 ; Y4 ; Y6 ; : : : are antisymmetric. It can be proved that Yn () satis0es the following orthogonal relations: 2 =2 Yn Ym d = mn ; (23) −=2 2 =2 2 =2 Yn Ym d = − Yn Ym d = an mn ; −=2 −=2 (24) where
(19)
(22)
n = 2; 4; 6; : : : : (25)
From Fig. 2 and formula (25), once can 0nd that the following relation of coe3cients an holds: a1 ¿ a3 ¿ a5 ¿ · · · ;
(26)
a2 ¿ a4 ¿ a6 ¿ · · · ;
(27)
· · · ¿ a2m−3 ¿ a2m−2 ¿ a2m ¿ a2m−1 ¿ a2m+2
¿ a2m+1 ¿ · · · ;
(28)
where m is an integer related to the non-dimensional sti1ness of elastic support : 2m − 2 ¡ 6 2m: 2 Substituting the series form solution (16) of buckling characteristic mode shapes into integrationdi1erential motion equations of arch (15), and
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
assuming that initial displacement u0 is u0 =
∞
3. Equilibrium con guration and its local stability
bm Ym ()
3.1. Equilibrium con3guration
n=1
one can obtain ∞
∞
Nn ()Yn () =
n=1
2 +
−=2
×
u0
=2
+
By making left-hand side of Eq. (29) equal zero, the equation determining the singular point can be obtained:
n Yn
n=1
(u0 )2 −
∞
u0
+
∞
2 n Yn d
n=1
n Yn
:
n=1
Multiplying both sides of this formula by (2=)Yn (), and integrating it for variable from −=2 to =2, and using orthogonal relations (23) and (24), one obtain ˙m = !m ; ∞
!m = 0;
Cmn !n = − m +G(am m +bm );
n=1
m = 1; 2; 3; : : : ; (29)
(31)
m − G(am m + bm ) = 0;
m = 1; 2; 3 ; : : : :
U0 = b(1 + cos 2); bm = 0;
m = 2; 3; : : : ;
formula (30) and equation (32) become 1 − G(a1 1 + b) = 0;
where
m (1 − Gam ) = 0;
bm = −
The roots Sm of Eq. (33) are as follows:
2 =2 Ym u0 d; −=2 2 =2 Ym Yn d; Cmn = −=2
G=−2
M =1
bm am −
∞
am a2m :
(1)
(29 ) (30)
m=1
In formula (29), m ; !m are generalized displacement and generalized velocity, which span in0nite-dimensional Euclidean phase space. Then the problem on dynamic stability of shallow arch becomes the stability problem of nonlinear dynamical system in in0nite-dimensional phase space, which is described by in0nite ordinary di1erential equations (29). This problem consists of three parts: (1) determination of equilibrium point (singular point); (2) the local stability of equilibrium con0guration; (3) the global stability.
(32)
Formula (31) represents that the velocities are equal to zero; hence, it is just needed to solve Eq. (32) to 0nd the singular point described by generalized displacement, i.e. the equilibrium con0guration. Assuming that the initial displacement is
b1 = b;
∞
913
m = 2; 3; : : : :
Sm = 0; m = 1; 2;
(33)
(34)
This solution represents the initial displacement of arch. Denote it by P0 . Substituting it into (16), we obtain u = u0 = b(1 + cos 2): √ (3b± b2 −4) ; Sm = 0; m = 2; 3; : : : : (2) S1 = − 2a1 (35)
The pair of these equilibrium con0gurations just depends on 1 , and only exists while b ¿ 2. It is denoted by P1± . Then √ b ± b2 − 4 (1 + cos 2): u1± = u0 + S1 Y1 () = − 2
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
(3) S1 = −
b aj
1 Sj = ± aj Sm = 0;
a1 −1 ; aj
1=2 (a1 − 2)=aj b2 ; 2 − 1 a1 =aj − 1
m = 1;
m = j:
j = 1; (36)
The pairs of these equilibrium con0gurations just exist while the following formulas hold. It is denoted by Pj± . Then 2 a1 a1 2 −2 b ¿ −1 (37) aj aj and
Introducing non-dimensional quantities (7), the non-dimensional energy is AL3 H = 4 H ∗: (39) EI Under the assumption of zero initial axial force and impulsive load, we have 2 =2 2 2 =2 2 u˙ d + [u˙ − u0 ]2 d H= −=2 −=2 2 2 1 2 + u2 − ; + u2 ; + G : 2 2 2 (40) Considering that the coordinates of equilibrium con0guration in phase space are i = Si ; i = 1; 2; : : : ; !1 = !2 = · · · = 0, we can construct Lyapunov function
uj± = u0 + S1 Y1 () + Sj Yj ():
V (c) = H (c) − H (P);
3.2. Local stability of equilibrium con3guration
where c is a point in the neighboring region of P, its coordinates are
Here, the local stability is the stability in the Lyapunov meaning, i.e. the stability under small perturbation: the total energy of system reaches limit value in equilibrium con0guration, the con0guration is stable when it is maximum, and unstable when it is minimum. Hence, the total energy at any point C, which is located in the small neighboring region of a stable equilibrium con0guration P, is always greater than the energy of P, but for unstable equilibrium con0guration it is not so. The total energy of elastic support shallow arch, (1) – (6), is 2 1 L=2 1 L=2 @y ∗ A dx + EI H = 2 −L=2 @t 2 −L=2 2 2 2 2 @2 y @ y0 @ y @ 2 y0 × − − 2 − @x2 @x2 @x2 @x2 2 S 2L @ y0 S 0 y0 + dx + + 2 @x EI 2EA 1 1 + k1 (y2 − y02 )x=−L=2 + k2 (y2 − y02 )x=L=2 2 2 L=2 + q(y − y0 ) d x: (38) −L=2
i = Si + i ;
(41)
!i = *i ; i = 1; 2; 3; : : : :
Substituting the coordinates of P and C into formula (41), and considering orthogonal relation (24), we can obtain ∞ ∞ Cmn *m *n + J (c) + O(3m ); (42) V (c) = m=1 n=1
where Cmn is determined by (29), and J (c) is the quadratic form ∞ 2 2 2 2 2 a1 am Sm J (c) = 1 1+2b +6ba1 S1 +3a1 S1 + m=2
+ 4(b + a1 S1 )1 +
∞
2n
∞
am Sm m
m=2
1+2an bS1 +2a2n S2n +
n=2
+
∞ ∞
∞
an am S2m
m=1
2an am Sn Sm n m :
(43)
n=2 m=2 m=n
In su3ciently small neighboring region, the nature of V (c) is determined by the quadratic term in formula (42). Obviously, the quadratic form of
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
generalized velocity perturbation *m is positive definite; thus, it is just needed to check the quadratic form J (c) of n . J (c) is written in standard form as J (c) = −
k
aSn 2n
+
n=1
∞
bSn 2n ;
aSn ¿ 0; bSn ¿ 0
n=k+1
(44)
where n = 1; 2; : : : is the linear combination of m ; m = 1; 2. For local stability, it is needed that the quadratic form is positive de0nite, i.e. for formula (44), k = 0 [3,4]. Substituting (34) – (36) into formula (44), the local stability of equilibrium con0guration of arch P0 ; P1± ; Pj± can be determined (1) P0 is always stable, and corresponds to the initial con0guration having non-deformation. (2) For P1± , there exists ∞ 2 2 J (c1± ) = d± d± n n ; 1 1 + n=2
where 2 2 2 d± 1 = 1 + 2b + 6ba1 S1 + 3a1 S1 √ b2 − 4 2 ( b − 4 ± 3b); = 2 1 an (±b b2 − 4 − b2 − 2): d± n =1 + 2 a1
Hence, P1− is always unstable. If the following formula holds, P1+ is stable, otherwise it is unstable. 2 a1 a1 2 −2 b ¿ −1 : (45) a2 a2 (3) For Pj± , there exists
a1 b ± J (cj ) = 1 − 1 21 + 2 aj a1 =aj − 1
(a1 =aj − 2)b2 ∓ −1 (a1 =aj − 1)2 +
∞ n=2 n=j
1−
an aj
2
1=2
2n :
Therefore Pj± is always unstable.
j
915
4. Su!cient condition for stability of shallow arch against snap-through instability If the system (29) possesses stable equilibrium con0gurations more than one, there is an optimal equilibrium con0guration among them, which possesses lowest energy, and is called nature equilibrium con0guration. When the system is subjected to initial perturbation, it comes back to this equilibrium con0guration. But if the perturbation is great enough, the system may jump to another stable equilibrium con0guration, and these jumps between equilibrium con0gurations are called snap-through instability or collapse. In the in0nite-dimensional phase space, the initial perturbation is presented as initial phase point, and the motion of system is presented as phase orbit starting from initial phase point. The stability problem of shallow arch against snap-through instability becomes the problem that is from which phase point the phase orbit approaches the nature equilibrium con0guration. All of these phase points consist of the region of global stability. 4.1. Su5cient condition for stability of shallow arch against snap-through instability Using the Lyapunov function de0ned in Section 3.2 formula (41), V (c) is expressed as V (c) = H (c) − H (P0 );
(46)
where P0 is nature equilibrium con0guration; thus, in the neighboring region of P0 . V (c) is always positive de0nite for all i , *i ; I = 1; 2; : : :, hence, one can draw a series closed equal energy surfaces V (c) = W , and these equal energy surfaces expand further as W increases. The singular point that encounters the equal energy surface for the 0rst time is denoted by P∗ , and this singular point is called critical point. Obviously, P∗ is a local instability singular point, and whole points c inside the equal energy surface passing through P∗ satisfy V (c) ¡ V (P∗ ):
(47)
The region inside the equal energy surface V (c) = V (Pn ) is denoted by S(Pn ); we can see that
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J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
S(Pn ) is the su3cient region of stability. When initial state phase points locate in S(Pn ), the phase orbit of system motion will always stay in S(Pn ), and cannot pass through the equal energy surface V (c) = V (Pn ) to reach another stable equilibrium con0guration. Hence, the system is stable [3]. Consequently, the formula (47) is regarded as a su3cient condition for stability of shallow arch against snap-through instability. Besides, for the shallow arch system, if the formula (45) does not hold, the system possesses only one local stable equilibrium con0guration P0 ; hence, no matter how great the load is, the snap-through instability does not occur for shallow arch. Specially, when the non-dimensional sti1ness of elastic support 6 27:5, from formula (25) and Fig. 2, a1 =a2 6 2; therefore, formula (45) does always not hold, and it is of no problem of elastic snap-through instability. When formula (45) holds, the non-dimensional arch high satis0es b2 ¿
(a1 =a2 − 1)2 : a1 =a2 − 2
(48)
Besides P0 , P1+ is also locally stable. In this case, the snap-through instability problem still exists, the su3cient condition for stability is determined by formula (47), and the critical point Pn can be found by analyzing the distribution of singular points, and the structure and expanding case of equal energy surfaces V (c), around P0 . Substituting (34) – (36) into energy formula (40), the energy of singular points can be obtained as H (P0 ) = 0;
(49)
H (P1± ) = 12 (−3b ∓
1 a2j
Fig. 4. Section graphs of equal energy surfaces around P0 on plane a1 − a2 .
b2 − 4)2 ×[8 + (b ∓ b2 − 4)2 ];
H (Pj± ) =
Fig. 3. Curves of singular energy via arch high H .
aj b2 1 − a1 − aj 2
(50)
j = 2; 3; 4; : : : : (51)
The curves of energies H via arch high b for the 0rst several equilibrium con0gurations are drawn in Fig. 3. Figs. 4 and 5 show the sections of equal energy surfaces around P0 in planes, 1 –2 , 2 –3 .
By analyzing formula (50), (51), and considering Fig. 2 and formula (28), it is easy to prove that P2± is unstable singular point of lowest energy. From Figs. 4 and 5, it can be derived that P2± is just the critical point Pn . Using formula (47), the su3cient condition for stability of shallow arch against snap-through instability can be obtained. a2 b2 1 1 − : V (c) ¡ V (P∗ ) = V (P2± ) = 2 a2 a1 − a2 2 (52)
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
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Under the assumption of zero initial axial force, from (49) we can obtain 1 L4 V (c) = T = B2 : 2 4 EI 2 From formula (47), the su3cient condition for stability of shallow arch against snap-through instability subjected to impulsive load can be obtained as 1 L4 B2 ¡ V (P∗ ): 2 4 EI 2 The critical load impulse Bc is determined by the following formula:
4 1=2 2 EI 2 ± V (P∗ ) : (53) Bc = L4
Fig. 5. Section graphs of equal energy surfaces around P0 on plane a2 − a3 .
4.2. Critical load under applying impulsion load The action of impulsive load corresponds to an initial perturbation described by a non-zero phase point of system. Its coordinates are initial displacement and initial velocity. By calculating the Lyapunov function of this initial phase point, V (c), one can determine the stability against snap-through instability under applied impulsive load, i.e. dynamic stability by formula (52). Under applied impulsive load q(xt) = B(t), Fig. 1 shallow arch system gains initial velocity, it is B @y(x; t) = : @t A Initial kinetic energy is 2 1 L=2 1 B2 @y ∗ L: A dx = T = 2 −L=2 @t 2 A Non-dimensional kinetic energy is T=
AL3 ∗ 1 L4 T = B2 : 4 EI 2 2 4 EI 2
5. Analysis of short-circuit dynamic stability of inner winding of transformer The winding of transformer having discus type is described in Fig. 6. The inner circle coil of inner winding is simpli0ed as an elastic thin circle ring supported on the elastic radius supports which are separated by equal distance and composed of elastic insulation stays and elastic insulation cylinder. Thus, the inner circle coil is divided into many spans by these supports. since each span of inner circle is symmetric about the center, and the number of supports is very large, each span can be regarded as a radius direction (transverse) elastic clamped support shallow arch. During short circuit of transformer, the shortcircuit force applied on the inner circle coil is applied in radius direction mainly and with equal distribution along the whole circle. Then, the analysis of short-circuit dynamic stability of inner circle coil can be regarded as the problem of dynamic stability of elastic clamped support shallow arch subjected to short-circuit radius load. This shallow arch is constituted of a span of inner circle coil separated by two insulation stays. Short-circuit load of inner circle coil q(x; t) (Kg=s2 ) is induced by short-circuit electric current: i = I (e−t=Ta − cos !t) where Ta is time constant of short-circuit current damping, $ is circle frequency of steady state
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electrical force is denoted as q0 , then we obtain q0 = P10 +!(P20 + !(P30 + !(P40 + · · · + !Pm0 )) · · ·); (55) where Pi0 is the coe3cient of single circle of each copper coil, and the electrical force of the ith copper coil Pi = Pi0 F(t). In general, time constant of current Ta is 0.05 – 1.0, and from formula (54), q(t) is decreasing obviously; hence, the 0rst wave of q(t) can be regarded as an impulsion approximately, where the duration time of part of q(t) ¿ 0 is about equal to VT = (3=4)(2=!). When VT is approaching the natural period of elastic support shallow arch, the short-circuit load can approximate impulsion very well: VT VT q(t) dt = B(t) dt = B: (56) 0
Fig. 6. Structural graph of discus-type winding of transformer.
current. The short-circuit load can be regarded as uniform distribution, q(x; t) = q(t), which includes the electrical force applied to inner coil and the force applied to form other circle coils of inner winding. Assume that the forces between coils are transferred according to static form, and the transferring coef0cient is !, then
0
From the above formula, the relation between coe3cient q0 of short-circuit electrical force and impulsive load impulse B can be obtained. Further, from the critical load impulse of shallow arch system against snap-through instability, Bc , the critical short-circuit electrical force coe3cient of dynamic stability of copper coil, q0c can be calculated by using the following formula. Bc q0c = VT : (57) F(t) dt 0 Finally, the short-circuit dynamic stability condition of inner winding of transformer can be obtained:
q(t) = P1 +!(P2 +!(P3 +!(P4 + · · · +!Pm )) · · ·);
q0 ¡ q0c
where Pi represents the electrical force of the ith layer circle coil, and inner circle coil is regarded as the 0rst layer; $ is the total layer number. Considering the simultaneousness of electromagnetic 0eld, we have
6. Example and comparison with experimental results
q(t) = q0 (N1 + N2 e
−2t=Ta
+ N3 e
+ N4 cos 2!t)Vq0 F(t);
−t=Ta
cos !t (54)
where N1 ; N2 ; N3 ; N4 are constants. From electromagnetic calculation, the coe3cient of short-circuit
(58)
The authors have performed short-circuit dynamic stability experiments of inner winding of model transformer [19,20], the diameter, section width high of inner copper coil of inner winding are D = 0:44m, b = 0:0108m, = 0:0032m, respectively. A circular coil is supported by 12 insulation stays. The elastic modulus of coil material is
J.-X. Xu et al. / International Journal of Non-Linear Mechanics 37 (2002) 909–920
Ee = 1:2 × 1011 Pa. By calculating, the elastic supporting sti1ness is found to be C = 5 × 107 N=m. Short-circuit shock of large current has been done 20 times. The measurement values of strains increase with square of current. Elastic and plastic stages have been reached; for a model, a symmetric buckling instability of inner winding occurred at current I = 15000 A, while another model was destroyed and burned at I = 24000 A. According to electromagnetic calculation, the value of this critical electric force of instability is qc = 4450 N=m. By using the above method and formulas, the theoretical value of critical load of elastic instability is obtained, which is q = 5250 N=m. Taking into consideration the situation that circle coil has entered the elastic-plastic stage before the load reaches the critical value of radius load of elastic instability, we make a simple elastic-plastic correction, the critical load value of elastic-plastic instability will be qr = 3106 N=m which is lower than the theoretical value of the elastic case. Hence, it coincides with the results of the experiments. Indeed, if we adopt the result of analysis of sti1ness supporting shallow arch dynamic stability, the critical load reached qn = 45700 N=m, it is very far from the practical experiment.
7. Conclusions (1) For transverse elastic clamped supporting case, analytical analysis results on dynamic stability of shallow arch subjected to traverse impulse load are provided. They are the series solution representing the equilibrium con0gurations of shallow arch by solving the corresponding non-linear integration-di1erential equation, the local stability of each equilibrium con0guration, and the su3cient condition for stability of the shallow arch system as well as the critical load against snap-through. (2) It is rational that the problem of short-circuit dynamic stability of inner winding of transformer should be investigated by using the theoretical model of the analysis of dynamic stability of transverse elastic clamped support shallow arch subjected to impulsive load.
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(3) The formulas of critical load of shallow arch against snap-through and the values obtained by calculating these formulas in the examples are veri0ed to be coincident with experimental results. Acknowledgements This work was supported by the National Science Foundation of the People’s Republic of China. References [1] N.J. Ho1, V.G. Bruce, Dynamic analysis of the buckling of laterally loaded Kat archer, J. Math. Phys. 32 (1954) 276–288. [2] J.S. Humphreys, Dynamic snap buckling of shallow arches, AIAA J. 4 (1966) 878–886. [3] C.S. Hsu, On dynamic stability of elastic bodies with prescribed initial conditions, Int. J. Eng. Sci. 4 (1) (1966) 1–21. [4] C.S. Hsu, The e1ects of various parameters on the dynamic stability of a shallow arch, ASME J. Appl. Mech. 34 (1967) 349–358. [5] C.S. Hsu, Stability of shallow arches against snap-through under timeless step loads, ASME J. Appl. Mech. 35 (1) (1968) 31–39. [6] C.S. Hsu, Equilibrium con0gurations of a stability character, Int. J. Non-linear Mech. 3 (1968) 113–136. [7] M.M. Lock, Snapping of a shallow sinusoidal arch under a step pressure load, AIAA J. 4 (7) (1966) 1249–1259. [8] M.M. Lock, S. Okubo, J.S. Whittier, Experiments on snapping of a shallow dome under a step pressure load, AIAA J. 6 (7) (1968) 1320–1326. [9] D.L.C. Lo, Dynamic buckling of shallow arches, ASCE 102 EM5 (1976) 901–916. [10] E.R. Johnson, The e1ect of spatial distribution on dynamic snap through, ASME J. Appl. Mech. 45 (1978) 612–618. [11] E.R. Johnson, E1ect of damping on dynamic snap-through, ASME J. Appl. Mech. 47 (3) (1980) 601– 606. [12] A.N. Kounadis, J. Raftoyiannis, J. Mallis, Dynamic buckling of an arch model under impact loading, J. Sound Vib. 134 (2) (1989) 193–202. [13] G.J. Simitses, Dynamic Stability of Suddenly Loaded Structures, Springer, New York, 1990. [14] K.B. Blair, C.M. Krousrill, T.N. Farris, Non-linear dynamic response of shallow arch to harmonic forcing, J. Sound Vib. 193 (3) (1996) 353–367. [15] P. Patricio, M. Adda-Bedia, M. Ben Amar, Elastica problem: instabilities of an elastic arch, Physica D 124 (1–3) (1998) 285–295.
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[16] J. Levitas, J. Singer, T. Weller, Global dynamic stability of a shallow arch by Poincare-like simple cell mapping, Int. J. Non-linear Mech. 32 (2) (1997) 411–424. [17] Jianxue Xu, Zhenmao Chen, Experiments on regular and chaotic motions of forced shallow arch oscillator, in: Editor in Chief: Du Qinghua, Proceedings of the International Conference on Vibration Problems in Engineering, 17–24 June, 1990, Wuhan-Chongqing, China, International Academic Publishers, 1990, p. 410.
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