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On non-linear free vibrations of an elastic cable
In! J \
Vol. 15. pp 333-340. Printed in Great Britain
ON NON-LINEAR FREE VIBRATIONS AN ELASTIC CABLE
OF
P. HAGEDCJRN and B. SCHAFER lnstitut fur Mechanik. TH Darmstadt, West Germany (Received 3 April 1980) Abstract-The effect of non-linear terms in the equations of motion on the first normal modes of the oscillations of an elastic flexible cable under the action of gravity is studied. The non-linear equations are derived and approximate solutions are found by the Ritz-Galerkin method. A numerical example is given and the significance of the results is discussed with regard to the galloping oscillations of overheadtransmission lines.
1. INTRODUCTION
Oscillations of heavy cables with negligible bending stiffness but with a finite longitudinal elasticity are of importance in a number of technical applications. One of these is given by the oscillations of overhead transmission lines, discussed for example in reference [l]. The differential equations governing the planar motions of an elastic catenary do not admit an solution in elementary functions. Only the equilibrium curves, i.e. the equations of the ‘elastic catenary’ could be given analytically by Schell [2] in 1880, in a publication little known today. Solutions to the linearized equations of motion were first given by Simpson [l, 31, who also examined the problem oftranslating cables [4] which corresponds to the conveyor belt or similar technical applications. Independently the oscillations of shallow catenary were also studied by Irvine and Caughey [5]. Further literature on the linear oscillations of cables is given in references [3-6]. In all this previous work only the linearized equations of motion were treated. In several technical applications, e.g. in the galloping type oscillations of overhead transmission lines, it is the case that the amplitudes may be very large, sometimes of the same order of magnitude as the sag. It is clear that these oscillations can only be described by non-linear differential equations. The galloping oscillations of transmission lines as described in reference [7] are of the type of self-excited vibrations, usually with an extremely small excitation term. Galloping is then observed in usually one of the first modes of the system. Clearly the knowledge of the non-linear effects on these modes in the free vibrations is then also of importance in the discussion of galloping, and may, for example, be useful in the design of appropriate dampers or vibration absorbers. It is the aim of this paper to discuss the non-linear free oscillations of suspended cables in the case of small sag. 2.
DEVELOPMENT
OF
THE
EQUATIONS
OF
MOTION
The full non-linear equations of the planar motion of a cable with zero bending stiffness were given in reference [3], and the equilibrium position x(y), y(v) as well as the corresponding tension T(y) are found as the solutions of
(1) (2) (3)
334
P. HAGEDORN and B. SCH&R
where y is the arc length of the cable in the unstretched state, EA the longitudinal stiffness, p the mass per unit length and g the gravity acceleration (Fig. 1). Equations (l-3) give -=-
(1’) (2’)
and T2(y)=H2-tp2g2(1/2-y)*,
(3’)
where H is the constant horizontal component of cable tension and 1 the length of the unstretched cable. An exact solution to (1’-3’), which, of course, tends to a catenary for H/EA+O, was given by Schell[2]. In what follows we wish to study the oscillations of the cable about equilibrium for the case of small sag, i.e. d/u = v 4 1, a being the span and d the sag. In overhead transmission lines usually d/u is of the order v = d/a ~0.05, while H/EA is usually much smaller, typically H/EA ~0.002. Therefore from (l’-3’) one obtains T(ykH,
(4) (5)
(6) and if the arc length l+& y ( > measured in the equilibrium position is introduced instead of y, the expressions s=
(7)
x = s, y=4vs
l(
s a>
or y=4vx
1-z
(9)
(
1 give a sufficiently good approximation to the equilibrium. The linear oscillations about this approximate equilibrium position were studied in references [1] and [S] in slightly different ways but with almost identical results. In the equations of motion given in reference [1], linearized about the approximate equilibrium in the displacements u and u (horizontal and vertical respectively) relative to equilibrium as well as in their derivatives .. =‘*’
(10)
(11)
335
On non-linear free vibrations of an elastic cable
additional simplifications were made. In reference [l] terms of the order v2 and of order (H/EA)2 as well as of higher order were discarded. In reference [5] it was noted that frequently H/EA=O(v2) and terms of order v3 and higher were neglected in the linear equations (10) and (i 1). Also in reference [5] the longitudinal inertia terms were set equal to zero, so that equations (10) and (11) reduce to EA(u’ + y’o’)’= 0, (12) [Hu’ + EA y'(u'+ y’u’)]’ = pi;, with CC-,
a at
I=--.
(13)
a ax
The normal modes correspond to solutions of the type u(x, t)= U(x) sin 0% u(x, t)= V(x) sin ~0t 1
(14)
and analytic expressions for the modal functions U(x), V(x) can be obtained. The symmetric and antisymmetric modes are given by different expressions. In reference [5] the antisymmetric modes correspond to V,(X)= -4dV K
l-2: >
sinZnn~+~
(
l-cos2nxX
(15) a>}’
V,(x) = d sin 2nn 9,
(16)
with n= 1, 2, 3, . . . being the order at the antisymmetric eigenmode under consideration. The modal functions are normalized in such a way that the maximal amplitude of the vertical displacement V(x) (occuring at x= a/4) is equal to the sag d. The corresponding circular frequencies are 2nlr H. a”=(17) a /- c1’ they are the same as in a taut string. The linear free vibrations in the antisymmetric modes do not produce any change in the cable tension up to order v2. For the symmetric modes the functions
-&
&x-tan-$
B
(1 --cos &x)-sin
[ V”(x)=d
BP 2 sin &,x-cos
1 -tan
Bp
fis
II,
(18) (19)
are obtained, with the abbreviation fl:l’ H “-664~~ EA
(20)
and where the frequencies (21) follow from tan !a/2 -= Pai2
1
H /3a ’ . -16EAv2 0 z
(22)
The modal functions (18) and (19) are normalized in such a way that the ratio between the maximum slope in the symmetric and in the antisymmetric case is (l/nx)(&a/2) tan &u/2; for the numerical example given later this corresponds to 0.81 so that the slope in both cases
336
P. HAGEDORN and B. SC&&R
is of the same order of magnitude. In contrast to the antisymmetric modes, variations in the cable tension do occur in the symmetric modes. In general it cannot be said which of the two modes, the tist symmetric or the fist antisymmetric, corresponds to the lowest frequency since this depends on the cable parameters. From equations (15), (16) and (18), (19) it follows that the amplitudes of the horizontal motion are small of order v relative to the amplitudes of the vertical motion. We shall derive the non-linear equations of motion about equilibrium, taking into account terms up to the order v3. In the derivation it will be assumed that H/EA=O(v2). Instead of using the non-linear equation given in the literature, we will obtain them directly from Hamilton’s principle. The kinetic and potential energies are EK=$ j&c-
a j&(x) I0
‘(ri2(x, t)+ti2(x, t))dx, 50
+ u(x, t)) dx +
(23)
(H&(x,t) ++EAe2(x, t)) dx,
(24)
where
is the strain with respect to the equilibrium position. The power series development using (8), gives
of E,
+5(x,t) = 11’+ yV + $y’u’ - u’)2 i-f{ -
y’2d3-yV
+ yr(2 -~~)&u’
+i5(y’2(4-y’2)u’4+(4~z
+ (2y’2 - I)&}
-1)t!4+2[2(1
+y4)-1
+4y’[(3y’2 -2)d2 +(3 -2y2)u’2]u’u’} +qds,
lJ+V
V’S).
(26)
From Hamilton’s principle s
” (E,-E,)dt=O I 10
(27)
retaining terms up to order v3 we then obtain after a straightforward linear equation of motion in the form
calculation the non-
jfii =EA[u’ +y’u’ ++uQ]‘,
(28)
pti=EA
, $ u’+ yqu’+ y’v’)+ d(u’ + Jfu’)+ $fu’2 +y (29) [ 1 if we recall that vertical amplitudes of the order ofthe sag we have y’=O(v), u’= O(v),u’= 0(v2).
3. INFLUENCE
OF THE NON-LINEARITIES
ON THE NORMAL
MODES
The influence of the non-linear terms on the first symmetric and first antisymmetric modes will be examined. Solutions to equations (28) and (29) will be sought in the form u(x, t) = U(x)q(t) U(X, r)=
V(x)q(r)
(30)
I’
V(x), V(x) being the eigenfunctions of the simplified linear problem and time function. Substituting equation (30) into (28) and (29) gives
q(r)
an unknown
(31)
On non-linear free vibrations of an elastic cable
337
Applying the Ritz-Galerkin method, i.e. multiplying equations (31) and (32) respectively by V(x), V(x), integrating and adding gives tj+a,q+a,q2+a,q3=0.
(33)
This same equation would of course also have been obtained from the single Lagrange equation for the ‘generalized coordinate’ q if the potential and kinetic energies were calculated from equation (30). The coefficients a,, a2, a, are given by (34) (35) (36) with D=&
r
(U2+ V2) dx.
(37)
0 The quadratic non-linearity in equation (33) is not unexpected since it is clear that the cable is harder’ in the downward than in the upward direction, where it may possibly even become slack for sufficiently large amplitudes. It is therefore expected that a2 is positive. The coefficient a3 is also always positive so that it corresponds to a hardening spring. Equation (33) has been studied recently by Mahaffey [8] in connection with plasma oscillations. Since it describes a conservative system, the energy
h=+fj2++ap,q2 +ia2q3 +&q4
(38)
\ is a
first integral, and the solution can therefore be given by quadratures. Nevertheless an approximate but simple solution is frequently quite useful. The coefficients a,, a2, a, can be calculated from equations (34)-(37) for the symmetric and for the antisymmetric case using the expressions of the modal functions (18), (19) and (15), (16) respectively. This is done in the Appendix; it turns out that, for all practical purposes, a, coincides with the square of the circular frequency computed earlier in equation (22) for the linearized and simplified problem. In the antisymmetric case u2 vanishes, i.e. the phase curves in the q, i-plane are symmetric about the g-axis, as expected. In the first symmetric mode, u2 is positive. In order to judge the influence of the non-linear terms on the galloping oscillations of a real transmission line the coefficients were computed for the Elbe-crossing described in reference [9] with (I= 1200 m, d=18 m, H=40.5 kN, EA=43 214 kN and c(=2.131 kg/m. The fundamental frequencyf, =0.073 Hz corresponds to the first symmetric mode, while the second lowest frequencyf, =O.l 1 Hz is the one of the first antisymmetric mode. Using a normalized time T= ,/h;t and indicating differentiation with respect to T again by a dot, we obtain the equations of motion i+q+0.19q2+2.86q3=0
(39a)
for the fundamental (symmetric) mode and ij+q+3.89q3=0
(39b)
for the first antisymmetric mode. It can easily be checked that q ~0 is the only equilibrium position for equations (39a) and (39b). Both equations show that the non-linearities may be quite important, since with amplitude q= 1 the maximum acceleration is 3-5 times the acceleration of the linearized oscillator.
P. HAGEDORN and B. SCHKFER
338
4.
PERTURBATIONAL
We now wish to find an approximate
SOLUTION
solution to
tj+q+‘l(fi,q2+ci,q3)=0
(W
using the Lindstedt method, q being a small parameter. From q(T)=qg(T.)+tfqt(~)+~*q*(T)+
* * *
8*=1+~el+tj2e2+
...
(41)
>
we obtain from equation (40) li;Ot ciPq, = 0,
(42)
4r +Q*q, =e,q,-ti,qi-t&q&
(43)
42 +a2q2=e2q0+elql
-z2qoq1
-3a3qihl,
w
etc. The zero order solution qO(r)= C sin ii)r
(45)
substituted into equation (43) gives 4”1+c@q, = +,C2+(elC-$C3)
sin Qr++a2C2 cos 5.Trs+&C3
sin 3iir
(46)
and secular terms are avoided by setting e, =&C*.
(47)
The general solution of equation (46) is then qr(r)=C,
sin (ii)r+yJ-
ks
C*- i$
c*cos2ti-
$2
C3 sin 3ii)r,
(48)
with integration constants C, and yr. Setting Cr equal to zero and substituting into equation (44) gives
Again, secular terms are avoided for e2=gs
5 c*
-iif
+z
9
-2 2 a,C
(50)
r
( and the general solution of equation (49) is then __
q*(r)=:%
19i 5 C*+%-$$ C4 cos2GX-
9_ &(~:+,u~C*)~~
sin 3ii)r
1 G*h3 1 6; + 9-6X C4 cos 48~ + -(32)2 ._4 C5 sin 5f37.
(51)
Up to terms of order O(q’) the solution of equation (40) is therefore given by (41) with (45), (48) and (51), and the frequency described by (52) This perturbational solution was compared with the ‘exact’ solution of equations (39a) and (39b) obtained by an analogue computer. Computations were carried out for amplitudes up to d/2 and it was found that the difference between the exact and the perturbational solu-
On non-linear free vibrations of an elastic cable
339
tion was negligible. The frequency shift due to the non-linearities is 22% for equation (39a) and 26% for equation (39b), both for the exact solution and for the perturbational solution. 5. CONCLUSIONS
The effect of non-linearities on the first normal modes in the free vibrations of an elastic cable was studied in this paper. A perturbation analysis showed that the non-linearities may produce a considerable change in the frequency, and also the higher harmonics may become quite important. This may be of importance for the galloping oscillations of overhead transmission lines, for which so far no successful dampers or vibration absorbers could be developed due to the low frequencies involved and small amount of space available. Since primarily only the large amplitude vibrations are to be avoided, the damper could possibly be tuned to one of the higher harmonics. Since the difference between the first frequencies is of the same order of magnitude as the frequency shift due to non-linearities in the case of large amplitudes, it is not at all clear that non-linear coupling between these modes can be disregarded. However, this far more complex problem is beyond the scope of the present paper.
APPENDIX In the first orrri\rmmerric mode one obtains from (15), (16) in (34)-(373 that a,=O, and a, reduces to
(AlI with
642) Since
J;U’ dx/j;
I” dx = O(v’),for practical purposes D&’
D
can be approximated by d2
EA2
’
(A3)
Also. with the same approximation. a, is equal to or: as given by (17). In the first symmetric mode a2 reduces to a*=$
r
$(U’+y’F’)V’2dx,
(A4)
0
where however u’ + f F’ is constant as follows from (12). The value of this constant, c= u’+yv: (AS) corresponds to the amplitude of the variation in tension during the oscillation in this mode, divided by EA. For the mode normalized as above, one obtains
REFERENCES 1. A. Simpson, Oscillations of catenaries and systems of overhead transmission lines, Ph.D. Thesis, University of Bristol (1963). 2. W. Schell, Theorie der Bewegungen und der KrHfte, 2. Band, B. G. Teubner (1880). 3. A. Simpson, Determination of the inplane natural frequencies of multispan transmission lines by a transfermatrix method. Proc. IEEE 113,870-878(1966). 4. A. Simpson, On the oscillatory motions of translating elastic cables. J. Sound Vibration 20, 177-189 (1972). 5. H. M. Irvine and T. K. Caughey, The linear theory of free vibrations of a suspended cable. Proc. Roy. Sot. London A 341,2P!-315 (1974). 6. P. Hagedom, Ein einfacha Rechenmodell zur Berechnung winderregter Schwingungen an Hochspannungsleitungen mit DPmpfem. InpArch. (in press). 7. P. Hagedorn, Nichtlineare Schwingungen, Akademische Verlagsgesellschaft, Wiesbaden (1978). 8. R. A. Mahaffey, A harmonic oscillator description of plasma oscillations. Whys. Fluids 19, 1387-1391 (1976). 9. W. Ann, F. Kiessling and D. Schnakenberg, Die Leiter der 380-kV-Elbckreuxung der Nordwestdeutsche Kraftwerke AG und ihre Verlegung. EIektriziriitswirrscha/t 78,245-256 (1979).
340
P. HAGEDORE; and B
SCHAFER
and Ve2 dx
Substituting into (19) yields
Similarly, one has :Vf4 dx
V4dx =@d”
d 1 +tan’ [(
(A8)
Ba Ba tan61+5tan4i+tan
“a1+1
”
(A9) _I
The constant D is given by
and can be approximated in L (All) and a, corresponds to the squs Resume
-cular frequency as gilen by equation< (21) and (22).
:
On etudie
l’e.-2t de termes non lineaires dans les equations du mouvement stir les premiers modes normaux des oscillations ,;’ tin cable elastique flexible sous l’action de la pesanteur. On etablit les equations non lineaires et on trouve des so!utions approchees par la methode de RitzGalerkin. On donne un exemple numerique et on discute la signification des resultats en ce qui concerne les oscillations galopantes des lignes aeriennes.
Vol. 15. pp 333-340. Printed in Great Britain
ON NON-LINEAR FREE VIBRATIONS AN ELASTIC CABLE
OF
P. HAGEDCJRN and B. SCHAFER lnstitut fur Mechanik. TH Darmstadt, West Germany (Received 3 April 1980) Abstract-The effect of non-linear terms in the equations of motion on the first normal modes of the oscillations of an elastic flexible cable under the action of gravity is studied. The non-linear equations are derived and approximate solutions are found by the Ritz-Galerkin method. A numerical example is given and the significance of the results is discussed with regard to the galloping oscillations of overheadtransmission lines.
1. INTRODUCTION
Oscillations of heavy cables with negligible bending stiffness but with a finite longitudinal elasticity are of importance in a number of technical applications. One of these is given by the oscillations of overhead transmission lines, discussed for example in reference [l]. The differential equations governing the planar motions of an elastic catenary do not admit an solution in elementary functions. Only the equilibrium curves, i.e. the equations of the ‘elastic catenary’ could be given analytically by Schell [2] in 1880, in a publication little known today. Solutions to the linearized equations of motion were first given by Simpson [l, 31, who also examined the problem oftranslating cables [4] which corresponds to the conveyor belt or similar technical applications. Independently the oscillations of shallow catenary were also studied by Irvine and Caughey [5]. Further literature on the linear oscillations of cables is given in references [3-6]. In all this previous work only the linearized equations of motion were treated. In several technical applications, e.g. in the galloping type oscillations of overhead transmission lines, it is the case that the amplitudes may be very large, sometimes of the same order of magnitude as the sag. It is clear that these oscillations can only be described by non-linear differential equations. The galloping oscillations of transmission lines as described in reference [7] are of the type of self-excited vibrations, usually with an extremely small excitation term. Galloping is then observed in usually one of the first modes of the system. Clearly the knowledge of the non-linear effects on these modes in the free vibrations is then also of importance in the discussion of galloping, and may, for example, be useful in the design of appropriate dampers or vibration absorbers. It is the aim of this paper to discuss the non-linear free oscillations of suspended cables in the case of small sag. 2.
DEVELOPMENT
OF
THE
EQUATIONS
OF
MOTION
The full non-linear equations of the planar motion of a cable with zero bending stiffness were given in reference [3], and the equilibrium position x(y), y(v) as well as the corresponding tension T(y) are found as the solutions of
(1) (2) (3)
334
P. HAGEDORN and B. SCH&R
where y is the arc length of the cable in the unstretched state, EA the longitudinal stiffness, p the mass per unit length and g the gravity acceleration (Fig. 1). Equations (l-3) give -=-
(1’) (2’)
and T2(y)=H2-tp2g2(1/2-y)*,
(3’)
where H is the constant horizontal component of cable tension and 1 the length of the unstretched cable. An exact solution to (1’-3’), which, of course, tends to a catenary for H/EA+O, was given by Schell[2]. In what follows we wish to study the oscillations of the cable about equilibrium for the case of small sag, i.e. d/u = v 4 1, a being the span and d the sag. In overhead transmission lines usually d/u is of the order v = d/a ~0.05, while H/EA is usually much smaller, typically H/EA ~0.002. Therefore from (l’-3’) one obtains T(ykH,
(4) (5)
(6) and if the arc length l+& y ( > measured in the equilibrium position is introduced instead of y, the expressions s=
(7)
x = s, y=4vs
l(
s a>
or y=4vx
1-z
(9)
(
1 give a sufficiently good approximation to the equilibrium. The linear oscillations about this approximate equilibrium position were studied in references [1] and [S] in slightly different ways but with almost identical results. In the equations of motion given in reference [1], linearized about the approximate equilibrium in the displacements u and u (horizontal and vertical respectively) relative to equilibrium as well as in their derivatives .. =‘*’
(10)
(11)
335
On non-linear free vibrations of an elastic cable
additional simplifications were made. In reference [l] terms of the order v2 and of order (H/EA)2 as well as of higher order were discarded. In reference [5] it was noted that frequently H/EA=O(v2) and terms of order v3 and higher were neglected in the linear equations (10) and (i 1). Also in reference [5] the longitudinal inertia terms were set equal to zero, so that equations (10) and (11) reduce to EA(u’ + y’o’)’= 0, (12) [Hu’ + EA y'(u'+ y’u’)]’ = pi;, with CC-,
a at
I=--.
(13)
a ax
The normal modes correspond to solutions of the type u(x, t)= U(x) sin 0% u(x, t)= V(x) sin ~0t 1
(14)
and analytic expressions for the modal functions U(x), V(x) can be obtained. The symmetric and antisymmetric modes are given by different expressions. In reference [5] the antisymmetric modes correspond to V,(X)= -4dV K
l-2: >
sinZnn~+~
(
l-cos2nxX
(15) a>}’
V,(x) = d sin 2nn 9,
(16)
with n= 1, 2, 3, . . . being the order at the antisymmetric eigenmode under consideration. The modal functions are normalized in such a way that the maximal amplitude of the vertical displacement V(x) (occuring at x= a/4) is equal to the sag d. The corresponding circular frequencies are 2nlr H. a”=(17) a /- c1’ they are the same as in a taut string. The linear free vibrations in the antisymmetric modes do not produce any change in the cable tension up to order v2. For the symmetric modes the functions
-&
&x-tan-$
B
(1 --cos &x)-sin
[ V”(x)=d
BP 2 sin &,x-cos
1 -tan
Bp
fis
II,
(18) (19)
are obtained, with the abbreviation fl:l’ H “-664~~ EA
(20)
and where the frequencies (21) follow from tan !a/2 -= Pai2
1
H /3a ’ . -16EAv2 0 z
(22)
The modal functions (18) and (19) are normalized in such a way that the ratio between the maximum slope in the symmetric and in the antisymmetric case is (l/nx)(&a/2) tan &u/2; for the numerical example given later this corresponds to 0.81 so that the slope in both cases
336
P. HAGEDORN and B. SC&&R
is of the same order of magnitude. In contrast to the antisymmetric modes, variations in the cable tension do occur in the symmetric modes. In general it cannot be said which of the two modes, the tist symmetric or the fist antisymmetric, corresponds to the lowest frequency since this depends on the cable parameters. From equations (15), (16) and (18), (19) it follows that the amplitudes of the horizontal motion are small of order v relative to the amplitudes of the vertical motion. We shall derive the non-linear equations of motion about equilibrium, taking into account terms up to the order v3. In the derivation it will be assumed that H/EA=O(v2). Instead of using the non-linear equation given in the literature, we will obtain them directly from Hamilton’s principle. The kinetic and potential energies are EK=$ j&c-
a j&(x) I0
‘(ri2(x, t)+ti2(x, t))dx, 50
+ u(x, t)) dx +
(23)
(H&(x,t) ++EAe2(x, t)) dx,
(24)
where
is the strain with respect to the equilibrium position. The power series development using (8), gives
of E,
+5(x,t) = 11’+ yV + $y’u’ - u’)2 i-f{ -
y’2d3-yV
+ yr(2 -~~)&u’
+i5(y’2(4-y’2)u’4+(4~z
+ (2y’2 - I)&}
-1)t!4+2[2(1
+y4)-1
+4y’[(3y’2 -2)d2 +(3 -2y2)u’2]u’u’} +qds,
lJ+V
V’S).
(26)
From Hamilton’s principle s
” (E,-E,)dt=O I 10
(27)
retaining terms up to order v3 we then obtain after a straightforward linear equation of motion in the form
calculation the non-
jfii =EA[u’ +y’u’ ++uQ]‘,
(28)
pti=EA
, $ u’+ yqu’+ y’v’)+ d(u’ + Jfu’)+ $fu’2 +y (29) [ 1 if we recall that vertical amplitudes of the order ofthe sag we have y’=O(v), u’= O(v),u’= 0(v2).
3. INFLUENCE
OF THE NON-LINEARITIES
ON THE NORMAL
MODES
The influence of the non-linear terms on the first symmetric and first antisymmetric modes will be examined. Solutions to equations (28) and (29) will be sought in the form u(x, t) = U(x)q(t) U(X, r)=
V(x)q(r)
(30)
I’
V(x), V(x) being the eigenfunctions of the simplified linear problem and time function. Substituting equation (30) into (28) and (29) gives
q(r)
an unknown
(31)
On non-linear free vibrations of an elastic cable
337
Applying the Ritz-Galerkin method, i.e. multiplying equations (31) and (32) respectively by V(x), V(x), integrating and adding gives tj+a,q+a,q2+a,q3=0.
(33)
This same equation would of course also have been obtained from the single Lagrange equation for the ‘generalized coordinate’ q if the potential and kinetic energies were calculated from equation (30). The coefficients a,, a2, a, are given by (34) (35) (36) with D=&
r
(U2+ V2) dx.
(37)
0 The quadratic non-linearity in equation (33) is not unexpected since it is clear that the cable is harder’ in the downward than in the upward direction, where it may possibly even become slack for sufficiently large amplitudes. It is therefore expected that a2 is positive. The coefficient a3 is also always positive so that it corresponds to a hardening spring. Equation (33) has been studied recently by Mahaffey [8] in connection with plasma oscillations. Since it describes a conservative system, the energy
h=+fj2++ap,q2 +ia2q3 +&q4
(38)
\ is a
first integral, and the solution can therefore be given by quadratures. Nevertheless an approximate but simple solution is frequently quite useful. The coefficients a,, a2, a, can be calculated from equations (34)-(37) for the symmetric and for the antisymmetric case using the expressions of the modal functions (18), (19) and (15), (16) respectively. This is done in the Appendix; it turns out that, for all practical purposes, a, coincides with the square of the circular frequency computed earlier in equation (22) for the linearized and simplified problem. In the antisymmetric case u2 vanishes, i.e. the phase curves in the q, i-plane are symmetric about the g-axis, as expected. In the first symmetric mode, u2 is positive. In order to judge the influence of the non-linear terms on the galloping oscillations of a real transmission line the coefficients were computed for the Elbe-crossing described in reference [9] with (I= 1200 m, d=18 m, H=40.5 kN, EA=43 214 kN and c(=2.131 kg/m. The fundamental frequencyf, =0.073 Hz corresponds to the first symmetric mode, while the second lowest frequencyf, =O.l 1 Hz is the one of the first antisymmetric mode. Using a normalized time T= ,/h;t and indicating differentiation with respect to T again by a dot, we obtain the equations of motion i+q+0.19q2+2.86q3=0
(39a)
for the fundamental (symmetric) mode and ij+q+3.89q3=0
(39b)
for the first antisymmetric mode. It can easily be checked that q ~0 is the only equilibrium position for equations (39a) and (39b). Both equations show that the non-linearities may be quite important, since with amplitude q= 1 the maximum acceleration is 3-5 times the acceleration of the linearized oscillator.
P. HAGEDORN and B. SCHKFER
338
4.
PERTURBATIONAL
We now wish to find an approximate
SOLUTION
solution to
tj+q+‘l(fi,q2+ci,q3)=0
(W
using the Lindstedt method, q being a small parameter. From q(T)=qg(T.)+tfqt(~)+~*q*(T)+
* * *
8*=1+~el+tj2e2+
...
(41)
>
we obtain from equation (40) li;Ot ciPq, = 0,
(42)
4r +Q*q, =e,q,-ti,qi-t&q&
(43)
42 +a2q2=e2q0+elql
-z2qoq1
-3a3qihl,
w
etc. The zero order solution qO(r)= C sin ii)r
(45)
substituted into equation (43) gives 4”1+c@q, = +,C2+(elC-$C3)
sin Qr++a2C2 cos 5.Trs+&C3
sin 3iir
(46)
and secular terms are avoided by setting e, =&C*.
(47)
The general solution of equation (46) is then qr(r)=C,
sin (ii)r+yJ-
ks
C*- i$
c*cos2ti-
$2
C3 sin 3ii)r,
(48)
with integration constants C, and yr. Setting Cr equal to zero and substituting into equation (44) gives
Again, secular terms are avoided for e2=gs
5 c*
-iif
+z
9
-2 2 a,C
(50)
r
( and the general solution of equation (49) is then __
q*(r)=:%
19i 5 C*+%-$$ C4 cos2GX-
9_ &(~:+,u~C*)~~
sin 3ii)r
1 G*h3 1 6; + 9-6X C4 cos 48~ + -(32)2 ._4 C5 sin 5f37.
(51)
Up to terms of order O(q’) the solution of equation (40) is therefore given by (41) with (45), (48) and (51), and the frequency described by (52) This perturbational solution was compared with the ‘exact’ solution of equations (39a) and (39b) obtained by an analogue computer. Computations were carried out for amplitudes up to d/2 and it was found that the difference between the exact and the perturbational solu-
On non-linear free vibrations of an elastic cable
339
tion was negligible. The frequency shift due to the non-linearities is 22% for equation (39a) and 26% for equation (39b), both for the exact solution and for the perturbational solution. 5. CONCLUSIONS
The effect of non-linearities on the first normal modes in the free vibrations of an elastic cable was studied in this paper. A perturbation analysis showed that the non-linearities may produce a considerable change in the frequency, and also the higher harmonics may become quite important. This may be of importance for the galloping oscillations of overhead transmission lines, for which so far no successful dampers or vibration absorbers could be developed due to the low frequencies involved and small amount of space available. Since primarily only the large amplitude vibrations are to be avoided, the damper could possibly be tuned to one of the higher harmonics. Since the difference between the first frequencies is of the same order of magnitude as the frequency shift due to non-linearities in the case of large amplitudes, it is not at all clear that non-linear coupling between these modes can be disregarded. However, this far more complex problem is beyond the scope of the present paper.
APPENDIX In the first orrri\rmmerric mode one obtains from (15), (16) in (34)-(373 that a,=O, and a, reduces to
(AlI with
642) Since
J;U’ dx/j;
I” dx = O(v’),for practical purposes D&’
D
can be approximated by d2
EA2
’
(A3)
Also. with the same approximation. a, is equal to or: as given by (17). In the first symmetric mode a2 reduces to a*=$
r
$(U’+y’F’)V’2dx,
(A4)
0
where however u’ + f F’ is constant as follows from (12). The value of this constant, c= u’+yv: (AS) corresponds to the amplitude of the variation in tension during the oscillation in this mode, divided by EA. For the mode normalized as above, one obtains
REFERENCES 1. A. Simpson, Oscillations of catenaries and systems of overhead transmission lines, Ph.D. Thesis, University of Bristol (1963). 2. W. Schell, Theorie der Bewegungen und der KrHfte, 2. Band, B. G. Teubner (1880). 3. A. Simpson, Determination of the inplane natural frequencies of multispan transmission lines by a transfermatrix method. Proc. IEEE 113,870-878(1966). 4. A. Simpson, On the oscillatory motions of translating elastic cables. J. Sound Vibration 20, 177-189 (1972). 5. H. M. Irvine and T. K. Caughey, The linear theory of free vibrations of a suspended cable. Proc. Roy. Sot. London A 341,2P!-315 (1974). 6. P. Hagedom, Ein einfacha Rechenmodell zur Berechnung winderregter Schwingungen an Hochspannungsleitungen mit DPmpfem. InpArch. (in press). 7. P. Hagedorn, Nichtlineare Schwingungen, Akademische Verlagsgesellschaft, Wiesbaden (1978). 8. R. A. Mahaffey, A harmonic oscillator description of plasma oscillations. Whys. Fluids 19, 1387-1391 (1976). 9. W. Ann, F. Kiessling and D. Schnakenberg, Die Leiter der 380-kV-Elbckreuxung der Nordwestdeutsche Kraftwerke AG und ihre Verlegung. EIektriziriitswirrscha/t 78,245-256 (1979).
340
P. HAGEDORE; and B
SCHAFER
and Ve2 dx
Substituting into (19) yields
Similarly, one has :Vf4 dx
V4dx =@d”
d 1 +tan’ [(
(A8)
Ba Ba tan61+5tan4i+tan
“a1+1
”
(A9) _I
The constant D is given by
and can be approximated in L (All) and a, corresponds to the squs Resume
-cular frequency as gilen by equation< (21) and (22).
:
On etudie
l’e.-2t de termes non lineaires dans les equations du mouvement stir les premiers modes normaux des oscillations ,;’ tin cable elastique flexible sous l’action de la pesanteur. On etablit les equations non lineaires et on trouve des so!utions approchees par la methode de RitzGalerkin. On donne un exemple numerique et on discute la signification des resultats en ce qui concerne les oscillations galopantes des lignes aeriennes.