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On the equations of motion of thin rings
Journal of Sotmd and Vibration (1973) 26 (4), 465-488
ON THE EQUATIONS OF MOTION OF THIN RINGS H. E. WILLIAMS Department of Engineering, Ilarrey M ttdd College, Claremont, California 91711, U.S.A. (Receiced 12 June 1972, attd ht revisedform 14 A'ot'ember 1972) The equations of motion governing small, elastic displacements, including warping effects, are derived by using Hamilton's Theorem. The stress-resultant displacement equations are identified through the natural boundary conditions. It is shown that the governing equations obtained are consistent with those obtained by using momentum principles. Of particular importance is the appearance of a "twist" equation of motion relating the twisting moment and the St. Venant torsion moment. Numerical results are presented for the frequencies and mode shapes of both complete rings and ring segments. 1. INTRODUCTION In discussing the equations governing small, elastic displacements of thin circular rings, it is important to emphasize the role of the shape of the transverse cross section in establishing the relative importance of warping effects. As the technical theory ofrings may be considered to be a generalization of the theory of straight bars, one might expect a similar qualitative behaviour in that (i) warping vanishes identically for circular cross sections, (ii) warping effects are confined to the neighborhood of the supports for solid (dosed) cross sections, and are negligible outside this region, and (iii) warping effects are important over the full length of the bar for thin walled, open cross sections. Thus, the form of the appropriate governing equations for rings can be expected to he strongly influenced by the shape of the transverse cross section. It should also be noted that, if the developed length of the ring is not large in comparison with the transverse dimensions, transverse shearing strain may be important. For transverse cross sections that are circular, the governing equations for rings are a straightforward generalization of those for straight bars. Probably the most well known work in this area is that by Love [1] who determined the frequency of free radial and transverse vibrations of complete rings. If one assumes that warping effects are negligible in rings with solid cross sections, the equations developed by Love may be applied to other than circular cross sections. The frequency of the lowest mode of free, in-plane vibration of ring segments was obtained by Den Hartog [2] using a Rayleigh-Ritz procedure and equations which omitted warping effects. The frequency of the lowest out-of-plane mode was also found [3] in the same way. This work was later extended by Volterra and Morell [4, 5, 6]. More general studies of the equations of motion and the dynamic behavior of rings have also been made. Notable among these are the contributions of Ojalvo [7], Hammoud and Archer [8], and Rao [9]. In none of these papers were warping effects included, though the effect of rotarY, inertia was considered in both references [8] and [9]. Warping effects were first studied by Culver [10] using equations developed by Vlasov [1 I] and modified for dynamic effects by including a distributed inertia force and twisting moment. (Culver also verified the equations by deriving them from Hamilton's Principle.) A frequency equation was presented for the free, transverse vibration of simply supported 465
466
rl. E. WILLIAMS
ring segments of symmetrical cross sections (I,~ = 0)t for which the shear center and centroid coincide. A Rayleigh-Ritz solution was also given for non-simply supported boundary conditions for the same class of cross section. The purpose of this paper is to explore in detail the equations of motion of thin rings including both warping and rotary inertia effects. The equations of motion are first derived from Hamilton's Theorem upon assuming a set of generalized displacements derived earlier [12]. It is then shown that, through Hamilton's Theorem, one can establish both the stress-resultant, displacement equations and the equations of motion. Furthermore, the equations of motion are consistent with those obtained by using the classical approach: i.e., by applying momentum principles to a volume element of ring, and with the momentcurvature relations inferred from Hamilton's Theorem. A most interesting result is the natural appearance of a "twist" equation of motion describing the relationship between the St. Venant torsion moment and the warping moment. Finally, numerical results are presented for the frequencies and mode shapes of free transverse vibration of complete rings and ring segments for a variety of cross-sectional shapes. 2. EQUATIONS OF MOTION As is well known, the equations of motion can be identified as the Euler equations associated with Hamilton's Principle expressed in the form 5
(S - K) dr = o
dr (Work done by external forces and moments in a virtual displacement),
(1)
where S = total strain energy of ring segment (01 -<<0 ~< 02), K = total kinetic energy of ring segment (01 ~< 0 ~< 02) and 3ui = 0 at z = Zo, zl. The natural boundary conditions define the choices from which actual boundary conditions may be selected. In order to use equation (1) to develop an approximate theory for thin rings, the displacement components must be expressed in terms of generalized displacements. These generalized displacements must be sufficiently general to adequately describe the expected motion and also simple enough to lead to a manageable set of equations. The generalized displacements used here are those developed earlier [12] for static loading, and given by
v = v(o) - ~,(o) + ~(o)
+ ~(~, 0 ~o(o),
where to, = W ' / R ,
tc, = ( V - - U ' ) / R ,
too = (fl -4- W / R ) ' / R
and
~( ) (7=
80
•
The coordinates (~, () are centroidal (see Figure 1), and satisfy the requirements that
fa
d A ( ~ , ~) = 0
"j" A list of nomenclature is given in Appendix II.
(2)
EQUATIONS OF MOTION OF THIN RINGS
467
over the transverse cross section. >X
u,,~ Y
,
Z~
,0
¢_.
w,Maz
>( .
r
R
t
*;
z
Figure 1. Sign convention.
The function ~b(~,~) is the St. Venant Torsion Function which is harmonic everywhere in the cross section and satisfies the condition that
1~+
m~=m~-l~
on the bounding contour. The quantities (/, m) are the direction cosines of the outward pointing unit normal vector. As q5 is known only to within an additive constant, it is convenient also to require that
a dAq~ = O . This assumption fixes V(O) to be the area average of the circumferential displacement: i.e., da v(0) = fa v(~, ~, 0)-2-
As is noted in reference [12], the generalized displacements can be interpreted as producing a rigid body motion of a transverse cross section (in which the components of the transverse shear strain vanish at the centroid) plus a warping displ~icement analogous to that of St. Venant Torsion. Thus, the model of the ring studied here includes the same basic degree of freedom as in the torsion-bending theory of straight bars. In the paragraphs to follow, the quantities appearing in equation (1) are defined in terms of the generalized displacement variables (equation (2)) and the equations resulting from the indicated variational process are determined. The total strain energy in the rings segment (01 ~< 0 ~< 02) is defined in terms of the strain energy density (Vo), where G Vo = 2e2/2 + G(e,,2 + too2 + e~:2) + -~- (y,o2 + 7~..2 + ~,0~2) for a linearly elastic material. The strain components (e,,, etc.) follow from the displacement
468
H.E. WILLIAMS
field (equations (2)), and are given by ~oo =
1
--[(u
+ v ' ) + ~(p -
r
Yo, ="
+
_
~'o: =
04
~__ +
I"
~,, = 0,
•
W"/R) + ~(V' - U")/R + ¢Ko'],
R Rico,
R O~ ] Rt,"o '
~:: = 0,
y,: = 0.
Thus, the total strain energy stored in a ring segment (0t ~ 0 ~< 02) can be computed, and is given by
f2 oL
S=
rdAV o=
dO E A R
+ i~
1
(3) where the area properties (i:, etc.) are defined in Appendix I. The total kinetic energy of the ring segment (0t ~< 0 ~< 02) is K
r dA ~p,,,2, dOl
JA
where (v), the velocity o f a general point, is v = e,(0 + (/~) + e0(~' - (£', + ~g': + ¢~o) + ez(W - ~.fl)
(4)
and (') = O( )lOt. In terms of area properties (i:', etc.) defined in Appendix I, it follows that the total kinetic energy stored in a ring segement (0~ ~< 0 < 02). is p A R a f°'d0 {02 + 1;"2 + i+"2 K~-
"2
,]o,
.
R2
+ [~2(i" + i=' + i : ' / + iz:S)
+ ~',2(1,,' + i..,/) + ,:: 2(i:, + i : : , ) + (RK.o)2F, _ 2~,~':(i,,' + i,,:') + 2R~0[Kz(~
+ ~ 2 ) _ I;',(~' + ~. 2)] + 2i,~'
+ 2~jP~.o + 2l:.'- ( (/~:~ • R
"~) }
"
R '
(5)
The work done on the ring segment (01 ~< 0 ~< 0z) in a variation of displacements, due to both distributed forces (Po Po, P,) and a twisting moment (t#) applied along the line of
EQUATIONS OF MOTION OF THIN RINGS
469
centroids, and normal (Zoo)and shear stresses (%. %z) distributed over the end cross sections (0 = 01, 02), is given by
6(Work) =
R dO(ta6fl + po6V + p,6U + p:6W)
)O
I
+
dA(zo,&¢ + Too6V + Zo=&O
•
(6)
0=0,
On substituting the assumed form for the displacements (equation (2)), and identifying the following stress resultants, (Fr,Fo,F..) =
f A(zo , zoo,To:),
Mor = - fa dA~%o,
Al°" = f a dA~roo ,
Moo = fa dA((%, - ~Zo:),
AI~ = [ dA
q~r00 R '
dA
the secon d integral in equation (6), the work done by stresses distributed over the end cross sections, becomes
[Fr*3U + Fo*6V + Fz*3W + Moo'611 + M0r*6h'r + Moz*3h"= + M~,*S(RXo)]
, O=Oi
where ( )* denotes boundary data. At this stage, with all the quantities appearing in equation (1) defined, it is possible to complete the derivation of the governing equations. It should be noted, however, that the requirements in Hamilton's Theorem that
t~tlI =
0
at
t = to, ti
are equivalent to requiring 6U, 6U', etc. = 0
at
t = to, t t .
Thus, on substituting equations (3), (5) and (6) into equation (1), and carrying out independent variations of the variables U, V, IV and fl, one obtains the following differential equations of motion:
mu+
..,
-
_ ,
(.3 -- -
~tzl~_
" i (I=- I + ~-=
-
i==')
9,}
I2 S - -
= R2p,,
(7)
470
H. E. WILLIAMS
-- pAR 3 {-- -~ - g',(l=' + 1==') + n,(l,, + I,= ) -- R~o(~' + Fez) + 1,/f,, i ' V" _ i = % }
= R%,
(8)
U + U" R +-~d2 [PRtt°' + ( ' ( f l if,
]
J
W-~"R')- ~ s U +RU'' ] _ "-ff GJ"Rico ,~J
_ ,
- p A R 3 - .-~ + I,, II +
- ~= (I. + i . / -
(sRKo'
- ,...,
(/,,'
+ 1=, )K r + F'RF,-o'
~' - ~:1 - ,¢, (~, + e~ ~)
- ((~' + ~,~)R~o' - ( i , / -
.~/)
(9)
= R~e,,
U + U" ] R + ~Rt¢o' U + U'" ] + - ~a2 [PRh-o' + (, (fl - W.-~--R')- ~, - R
G] -~" m:o ] J -
0 -- pAR 3 - fl(i,; + l~.: + L,/ + I-=,') - l,='--~
if,
~) -
+ i=' ~ + r'Rr,-o' + i V ( ~ / +
r¢,'(¢/+
~)
.°
0o) and the natural boundary conditions that either ,W=o,
or
EAR
-~
a[
L
U+U" n
- p,4R 3 {,~(i=' + i = ' ) - ,~,(Lz' + L,z') + R,~o(~,' + V2)
(11)
471
EQUATIONS OF MOTION OF THIN RINGS
5V = O,
or
6 W = O,
or
U + V'
EAR--
R
(12)
= RFo* + Mo~*,
U + U"
R
000 [ F Rtc°' + ~s ( fl - W-VR) - ~ U +
-- pAR 3 -- (i,/ + 1,~, )t¢,
+ ~RKo'
]
] +---E -Rtc°}od
F'Rfc o + fq(i,~'
+ i.:' - ~' - ~ ) + r~,(~; + ~2) + g~o(~f + ~2)
6fl = 0,
or
+~
Rt¢o - p A R 3 - F'R~o
- ~ . e : + ~ ) + , . ¢ : + ~ ' ) - ~.'-~} = Moo*, 6U' = O,
or
EAR [i~ U + U" R t =
5IV'=0,
(14)
(15)
- Mo..*,
or
. 6fl' = O,
]}
= Mo,* + Me*,
(16)
= Me*,
(17)
or
on both 0 = 0i, 02. 3. ALTERNATE FORM OF THE EQUATION OF MOTION The equations of motion (7)-(10) and the natural boundary conditions (11)-(17) are
472
tl. E. WILLIAMS
sufficient to define a solution for U(O, z), etc., given the loading, and the initial and boundary conditions o f a particular problem. As the equations are linear and possess constant coefficients, solutions are obtainable without analytical difficulty. However, before undertaking any solution, let us formulate the equations in terms of stress resultants in an attempt to understand their significance. As was done previously [12] for the equations ofequilibrium, the force and moment stress resultants are identified through their appearance in the natural boundary conditions. In these equations, one observes that a force or moment stress resultant is equated to some function of the displacements at the bounding cross sections (0 = 0~, 02). However, as these bounding cross sections can be chosen arbitrarily without affecting the form of the displacement function, it follows that the natural boundary conditions essentially define the form for the stress resultant, displacement equations for all 0. However, before elaborating on this point, it is convenient first to evaluate the linear and moment of momentum of a differential volume element as these expressions can be identified in the governing equations. Let the basic differential volume element be that bounded by transverse planes of cross section A located at 0 + AO]2. Within this element, there exist doubly differential volume elements of volume rA0 dA. Associating a velocity (see equation (4)) with these elements, it follows that the linear momentum (AM) and the moment of linear momentum (AL) about the centroid of the basic differential element are AM = pAO~ardAv i.
AL
= pAOJa r dA[(~e¢ + (eg) x v'].
With sectional properties given in Appendix I, and with the definitions AM lim ~ = m c e , + m,e{ + moeo, ,0~o A0 AL lira ~ = lceg + l~ec + loeo, A0-.o A0 it lbllows that
"'~ = oAR2(O/R + M,/),
,,,~ = pAR20~'/R - - / ~ i J ) ,
mo= pAR2(~'/R - t?fl-,.' + K-z1**'+ RK-o~s'), -
-
i=,,')-- ~(L~' + 1,r=) -- R~'O(~; + ~ )
R
r: + ~,(Lr' +
R
:~ + / ~ ( L r ' + L z ' + i : , / + 1 z : : ) + - ~ i , z '
,
,
[~ ~ pAR3 I ~ ]zz' -- hr(]rz' + ],r,') + l~zjzz' + ]zzz') + Rh'O(~s' "l" r~2) ] • AS will be seen later, it is convenient to define a quantity (/+), without immediate physical
473
EQUATIONS OF MOTION OF THIN RINGS
significance, given by
lee = paRa [F'Rg:o + t~,=(*s'+ P,2) - ~,,((,' + P,~2) + ~s"f-l~] • This quantity appears to be a momentum associated with warping. In terms of these momentum variables, the natural boundary conditions (11)-(17) suggest that the force and moment stress resultants are related to the displacements in the following way:
(18) RFo = EAR {,U + V'
[i~ U + U''
(19)
U + U"
] + (sRh'o'
J
O00[rR'c°'+(s(F--W-i~--R)
-~'u+U''R-] U + U"
+ GJRtc°} + lee + l~, "l
Mo, = - E A R [L(fl - W"[R) - Lz - -R
+ (sR~co'],
Mee = EAR rlrR~:o' +
IR+v,' ~j
(s~ -
W"/R) - ~s U
I_
Mo: = - E A R []: U +R U''
(21)
(22)
/r--(fl -- W"/R) - ~sRt¢o'] ,
Moo = EAR {-- ~ [PRr.o'+ (s~ -- W"/R) -- ~s -U- ~
(20)
(23)
] + GJR,:o} + lee. (24)
With equations (18)-(24) as definitions, it is then sufficient to prescribe either U or F,, V or Fo, tV or Fz, fl or Moo, U' or Moz, IV' or Mo, and fl' or M , on both 0 = 01, 02. Further interesting consequences follow from the definitions (18)-(24) when the force stress resultants are rewritten in terms of the moment stress resultants. With Mo,, Mee and Ma, defined in equations (21), (22) and (23), respectively, it follows that equations (18), (19) and (20) are equivalent to
RF, = Mo:' - l~,
Rro EAR( U ~ V') =
-
-
(18a) -
Mo:,
RF= = --Mo; + (--M O' + GJl¢o + lee)+ l¢.
(19a) (20a)
Furthermore, equation (24) requires that
Moo = --Mee' + GJtco + lee. 3
(25)
474
n.E. WILLIAMS
It appears from equation (25) that the twisting moment is defined in terms of the St. Venant Torsion moment (GJh'o) and the warping moment (--M4,') by an equation having the form of a "twist equation of motion". The appearance of the quantity lo is not readily justified on physical grounds nor can the term be shown in general to be important. It will be shown later however, that l0 is insignificant in comparison with M0' for the transverse bending modes of free vibration, but is comparable with Mo' for the torsion modes. When one now returns to equation (20a), it follows from the definition of Moo (25) that
Mo/ - Moo + R G = 1¢.
(20b)
It is significant to observe that equations (18a) and (20b) are the axial and radial components, respectively, of the Principle of Moment of Momentum. Looking now at the Euler equations (7)-(10), and rewriting them in terms of the moment stress resultants, we obtain
M o z " - - I ~ ' - - ( F , AR U +R V"
-
Mo~ ) = Rd~ -- R2pr,
(7a)
=
(Sa)
R,
,o -
R%,
--Mo/' + Moo' + l¢' = Rdl~ -- R2p,,
(9a)
Moo' + Mo, = lo -- Rrp.
(lOa)
However, upon taking equations (18a), (19a) and (20b) into account, it follows that equations (7a), (Sa), and (9a) are equivalent to
F ; - Fo = - R p r + the,
(7b)
Fo'+G=
(8b)
--Rpo+lho,
F=' -------Rp~ + lhg.
(9b)
Equations (7b), (8b) and (9b) can be identified as the radial, circumferential and axial components of the Principle of Linear Momentum, while equation (10a) is the circumferential component of the Principle of Moment of Momentum. In summary then, it has been shown that (i) the natural boundary conditions can be interpreted as identifying the moment and force stress-resultant displacement equations, and, (ii), these stress resultants are consistent with the classical equations of motion obtained by applying the Principles of Linear and Moment of Momentum to a differential volume element. It should be noted that the equations developed in this section contain no new information, but are simply a more convenient form of the previous equations and lead to useful physical interpretation. 4. APPLICATION--FREE VIBRATION In the sections which follow, the frequencies and mode shapes of free vibration of complete rings and ring segments are derived. The results for the complete ring are particularly useful in establishing the order of magnitude of warping and rotary inertia effects. The results for the ring segment may be compared with similar results for straight beams in an attempt to assess the effect of coupling between rotation and deflection in the transverse bending modes.
EQUATIONS OF blOTION OF THIN RINGS
475
4.1. EXAMPLE I--COMPLETE RINGS
The equations for the frequencies and corresponding mode shapes of free vibration of complete rings may be obtained by substituting (O/R, W/R, fl) = ( C, A, B) cos nO cos coz, V/R = B sin nO cos ogr
0* >/2),
into the differential equations of motion (7)-(10). This assumed form is a solution provided that Zn2{(L= - ¢,) O*2 - 1) -- ~2[L :' + L,~' -- (~,' + p¢2)-]} + B{(i.~ - . ~ . ) (,,2 - 1) - ~=[i.:' - , : ( ~ : + ~¢~)-1} + C{L(,: - 1y + 1 -~[I
+ nXi=' + L=')]}
+ .DE1 - ~ X 2 i = ' + L = ' ) ] = o,
(26)
-,,tn~212/,J + l , f f - (2~,' + F¢2)] + ~2nB(2~, + F¢2)
+ .C[1 - ~2(2i=" + L = ' ) ] + ~{,,~ - ~211 + 3i=' + L = ' ] } = o, . . ~ { , : ( i . - 2(, + F) + .~C,J - ~ [ 1 - 2(,' - 2 ~ = ) 3 } + ~ { , : ( r
+ nXi: + L,/+
(27)
F'
_ L) + ,,2~7 + ,,2(i. - L)
+ ~2[izz, -- n2p, + n2¢s, ._}_~g 2)']} + C,,2{Q, = - 1) (L, - *~) - ~2(L=' + L,=' - *,' - Fez)} - B,,~=(21.; + i . / -
2~,' - ~=) = 0,
(28)
and .~o:(r
- L) + ,,2(i. - L + ~J) + ~=Eizz' + n2(¢:
+ e~: - F,)]} + B { F , : + (~J - 2(~),: + i, - ~ 2 [ i . ' + L ; + L,,' + L = ' + , : r , ] }
+ C((n 2 - 1) (i,, - nZ~s) - ~ 2 [ i , , _ n2(~, + ~¢2)]} + Bn~a(2~s, + ~¢2) = 0,
(29)
are satisfied, where G = G/E,
~2 = pR2o2/E.
As is well known, these equations possess a non-trivial solution provided that the determinant of the coefficient matrix vanishes identically. This requirement leads to a frequency equation which is fourth-order in ~2, and can be expected to lead to four distinct roots for ~2 in general. Without simplifying assumptions, it is not possible to obtain general results--only results for specific numerical values of the cross-sectional properties. In order to determine some properties of the motion, let us restrict our attention to symmetric
476
H. E. ~,VILLIAMS
cross sections for which
C~: =o,
i , : , i , / = o,
- I ~,I2, ~,:,2, i-z z z I ' lrrz ,~ 1 •
Let us further take ~[R ,~ I so that we may ignore the distinction between the primed and unprimed area properties: i.e., i , / ~ i,,, etc. Consistent with this restriction is the consequence that i, and i= are small in comparison with unity. For this restricted class of cross sections, equations (26)-(29) simplify considerably and lead to an uncoupling of the in- and out-of-plane motion. The frequency equation for in-plane motion only is given by ~4(1 + n2i=)
_
~21112 d~ 1 .~
211215012
--
3)]
+ l z l l 2 ( l l 2 - - 1) 2 = 0 ,
(30)
where D/C-" = 11(1 -- 2lzU2)[(~ 2 - n2).
(31)
An approximate solution of equation (30) valid for n4i~ <~ 1 is given by ~2
= ~ 2 e x t = i12 .~_ l _ 112012 - - 1) 2
=~°~=~
,--:-Ff
'
where ~ t , ~ben~ refer to the extensional, bending (inextensional) modes, respectively. This distinction is readily understood, following the observation
(Die)e=
~ ,1,
(~lC)~e,~ ~
-1/,,,
when it is noted that U + V' ~00(~, ~ = 0) = ~ = (C + 11/7)cos 110cos co~. R
The frequency equation for out-of-plane motion only is given by t~4{/,[1 + n2(l, + i~ - 2(~)] + L(I - 4n2(s) + n2/~(1 + n2l,) - n'(~ z} -
~={i.[1 + n~(,l ~ + 1) ( i ,
2~) + .r,l~(2,14 - ,2 + l)]
+ GJ -
+ LnZEi, Ol2 + 2) + 3112F -Jr 3G7 - 2(,(2n 2 + 1)] + Fn 4 + G J n 2 - 2(sn2[1 +
!12(!l 2 - -
..~ !12(112 - - 1)2 CL(~,7 .dr. n2(i,p
l)(s] }
_ (2)] = 0 ,
(32)
where B
,14(i, - 2(~ + P) + n2t~J - ~21-1 + ,12(i, + P - 2(~)3
-- --~ = n4(/~ -- (s) + n2( ~rJ + L -- (s) + ~ 2 ( 1 r - - n 2 p + n2(s) "
(33)
For the case of circular cross sections, the warping function (~b) vanishes, and equations (32) and (33) reduce to ~412 Or (2/l 2 -- 1)i3 - ~2{1*21~',]1i Or 1 --I-112[(2112 Or 3)i
+ G.7(n 2 + 4)]} + n=(n = -
I)2G,] =
0,
(34)
EQUATIONS OF MOTION OF THIN RINGS
B
!l4 + n 2 G j ] i -- ~2(1 + n2i)]l
,4
n2(1 + GJ/i) + ~2
,
477
(35)
where J = 2i. An approximate solution of equation (34) valid for n4i ,~ 1 is given by ~2
(1 "1- 21125)]2
= ~2to r =
n2(n 2 = ~2bend =
1 +
1)
Sj
2n2G
'
where ~2to,. ~2be,a refer to the torsional and bending modes, respectively. This distinction follows from the fact that 1 1 + 2n2t~ i 1 +2nz(1 + 3 5 ) '
=
=
-
,,2
-
1 -+ 2 5
1 + 2n2G" It should be noted that the approximate solutions of equations (30) and (34) agree with results given by Den Hartog [13] if we write E]E = 1. This point will be discussed later. Upon returning now to equations (32) and (33), it is of interest to obtain numerical results for particular cross sections in order to observe the effects of the parameters ir, (s, r and J. This is accomplished by comparing results for a thin walled circular cross section, an equal legged I-section and an equal legged channel section. The section properties used in the calculations are tabulated below.
TABLE I
Section properties Property
Circular
Section I (equal legs) Channel (equal legs)
]r iz j p
a218R 2 a2[8R 2 a214R 2 0
7a2[36R 2
7a2136R 2
a2]18R 2 ,~2/3R2
a2[gR 2 ,r2]3R 2
a4[72R 4
43ag[324R 4
(s
0
0
-- 4a3/27.R 3
The reduced frequencies for the torsional and bending modes are presented in Figure 2 for 5 = ½. The corresponding mode shapes are presented in Figure 3. It is interesting to note that whereas the frequencies of the torsional mode are of comparable order of magnitude for all sections, the frequencies of the bending mode for the channel and I-section are an order of magnitude smaller than those of a circular cross
478
It. E. W I L L I A M S I
I'0
I
(o)
I
i
I
I0
j.._K_.~L'~" I120
g 08 i
x
x
I
I
I
I
I
-~/
'
I
13
~
i
1
2
.~ 0-4
O
o~ 0"1
~.~
0
2
4
l
I
6
8
I0
I
I
I
I
I
1
1
(f)l
I
l
i
I
I
/viol
I ,lo,y?.~ /"~,':/IO, I 0 2 4. 6
n
l/a. 11.5//"
..J ~-~
o3 om.-I~ o 2 I~
~la'lllOiR.q __. v.Z..II
I-
0"I
ja/l?III0
~04
/ (°; o21- ~, I '~.,io.115
1
I
05
031-
03
l
I
0
4
2
1
I
6
0-5:
~"
I
4 /R I
I
I
6
,o
~" 0"4 I~1 02
I
8
12
06
- (c) I
,
, ~uI
8
10
0"I
O
!
I
2
l
4
I
6
I I0
B
n
Figure 2. Natural frequencies of complete rings. (a) Circular section thin-walled, (b) I-section, (c) channel section, (d) circular section thin-walled, (e) I-section, (f) channel section. I
I
(o)
'~
I
I
I
/ ,,,'R- ] //,IZO j
"°''°J/
/
sso
'
I
(b:
i
l
I
T Z
~-oo
_k
--
'
:3so - (¢~
J
J ' ! ~
~oo
250
250
lla.lllO - I ~ 200 i ~,..~,o.~. I~_. 150
200 150
•"
'
K-a-~
14 12
II II
f- / T
,,°
8
\\
I00 80
leO
~/a'VS~".~
,~.~~,,,o -~
o
a/R= f/20 O/A~•
0
~
~
"°
~
M'~
'
'
o/R
1/20
'
, ~;~o"
, ,~r ,,,°,
O
'
"
50
4
700
,
'
(,)'
'
'
~j/~/~
2 0
I
I
!
|
!
GO0
601
500
,
o,..4
/,,,°/~
~
/
~
aiR-
T
I / c/ i Ri/io. t/a.vzo~_~,/
501
s//.
400
a/R.'-]~ 300 1120
~
I air. //lllO"
1/20
30
I O0
o/R-
I/5
IO
1/10 0
l
I
l
l
l
2
4
6
8
10
n
J
0
l
I
1
l
I
2
4
6
8
10
/'?
0
I
I
|
l
2
4
G
8
l 10
/7
Figure 3. Mode shapes of complete rings. (a) Channel section, (b) I-section, (c) channel section, (d) and (e) circular section thin walled, (f) I-section. section of equal i,. Further, the frequencies for the channel section are less than those for the I-section for bending modes, while the opposite is true for the torsion modes. It has also been observed that the static deflections of a channel section are greater than those of a similarly loaded I-section. This, at least, makes plausible the observation concerning the results for the bending mode. In comparing the mode shapes, it is apparent that there is considerably more rotation
EQUATIONS OF MOTION OF TInN RINGS
479
measured by (B/~) in the bending mode for the channel and I-section than for the circular cross section. The opposite seems to be the case in the torsion modes. Further, on comparing the channel and I-section, it appears that there is generally less rotation in the torsion mode for the I-section; however, there is more rotation in the bending mode for the I-section. With the orders of magnitude of the frequencies and mode shapes established, it is now possible to estimate the relative importance of warping and rotary inertia for the transverse mode of vibration. The relative importance of warping can be established by comparing the warping moment ( - M , ' ) and the St. Venant Torsion moment in the "twist" equation of motion (25). For the uncoupled motion, it follows that M ; = E A R [ f l ( $ + B) - (s(A + B/Iz2)]n 3 sin nO cos o r , Gd~o = -- E A R G 3 (71 + B)n sin nO cos o r . N o w , as
Crj[P = 8"~2R2/a4 (I-section) and 324 "I:2R2 -
387 a* (I---section),
and 1 ~< zZR2/a'*<.< 16 for the sections studied here, it is apparent that the twisting contribution from the F-term is certainly important for such thin walled open sections. However, as ~,J _
3 z2R (V-section)
(~
4
a3
and 1/I0 ~< "c2R/a3 <~ 4/5 for the sections studied here, the effect of twist-bending coupling may even be more important than twisting itself in providing torsional stiffness for sections for which the shear center and centroid do not coincide ((s # 0). The question of the relative importance of rotary inertia can be generalized to the question of when it is justifiable to write the net "inertia" transverse force and twisting moment for the out-of-plane motion as
,:I¢IR,~ pX if',
lolR ~ pp(l,,+ I,,).
Essentially, this question can be answered by looking at the coefficients of the frequency terms in the simplified versions of equations (28) and (29). It follows that, subject to the restrictions made earlier on the cross sections for which the calculations were made, the kinetic energy contributions to equations (28) and (29) are ~2{.g[1 +
n2(i,+ / ~
-- 2(,)3 - B[i~ - n2(/TM- (s)])
and ~(,~[r
_ ,:(r
- (s)] - B [ i , + L + , : c ] } ,
respectively. The dominant terms are * x 1 in the first expression and B x (i, + ]~) in the second expression. From the results for the mode shapes (Figure 3), it is apparent that higher order effects (rotary inertia, "warping momentum") are negligible provided that n2i, ,~ I for the bending mode (as B / ~ ,~ 1/i~), and
,:r ~ (I,+ L )
480
H.E.
WILLIAMS
for the torsion mode (as B[~ ~ I). These restrictions imply that ll2a2/R 2 ,~ 1
for the cross sections studied above. Hence, the higher order kinetic energy effects can be appreciable, even for the lower modes, for the values of air studied here. Finally, it is of interest to evaluate the relative importance of the warping momentum (/¢) and the warping moment (M'¢) in the "twist" equation of motion (25). As
M~' = EARna[F(B + A) - (s(B/n 2 + ,4)] sin nO cos co~, and
l~ = E A R n ~ 2 [/~(B + .~) - (s -~] sin nO cos tot, it is apparent that l¢ is negligible in comparison with M~' for the bending modes but is definitely comparable to M~' for the torsion modes. Thus, the customary assumption that the twisting moment (Moo) is equal to the warping moment ( - M ~ ' ) plus the St. Venant Torsion Moment (GJh'o) is shown to be in error for an important class of motion. 4.2. EXAMPLE I I - - R I N G SEGMENTS The equation for the frequencies of free vibration and the corresponding mode shapes for ring segments with general boundary conditions may be obtained by substituting
(U/R, V]R, W/R, fl) = (C, D, A, 13) e ~° cos tot into the differential equations of motion (7)-(10), and seeking values of 2 and to that satisfy both the resulting equations and the appropriate boundary conditions. This assumed form is a solution provided that .A{2z( 1 dr 22) (l,z -- 4s) -- ~ 2 ) 2 [ _ ( ] , , dr lrr, ) dr (4s' dr ~2)]} + J g { - O + ;.') (l,~ + ;.if3 - ~2[i,~' + ,t2(4, ' + ~¢~)]}
+ O{1 + i=(1 + 3.2)2 -
~2[1
-
).2(i~=' + iz,_.')] }
+ D{I -- ~2(21~.' + 1::~')} 2 = 0,
(36)
~2;.[_ (2i# + i.D + (24,' + F¢2)] + B~22(24, ' + ?2) dr C'2[1 -- ~z(2[~' + L~z')] dr DD .2 dr ~2( 1 dr 31~.' + L:.-')] = o, /~{,~.4(i t - - 2 ( s dr /~) - - ,~28, ~ -- ~ 2 F 1 -
(37)
,~2(irr t dr i z r , t) - r t Z 2
+ 2 ~ ( ( ~ ' + ~:~)]} + B { a f f a e ( r - L) + L - i, - GJ] + me[;:(r"
-
L' -
~2) + L.-']} + C(;:(1 + z ~) (L~ - 4,)
dr e 2 2 2 ( / J dr L , J - 4,' -- f2)} _ ~ 2 2 ( 2 i , _ ., dr L,,' - 24,' - F2) = 0, (38)
+ B{;:E;?P + 28. - 332 + i. - ~ f f i , ; + L / +
L: + i='
-- 22F']} + G'{-(1 + 22) (i,, + 224,) -- ~2[i,., dr 22(4,, dr ~¢2)]} dr ~2~2(24s, dr ?z) = 0.
(39)
EQUATIONS OF MOTION OF THIN RINGS
481
As is well known, these equations possess a non-trivial solution provided that the determinant of the coefficient matrix vanishes identically. The resulting characteristic equation is a polynomial of the seventh order in 22, containing ~2 as a parameter. If ~2 were known, it would be possible, by numerical means, to determine the seven generallycomplex roots for 22 . In what follows, the characteristics of the motion are determined for the restricted class of cross section that was studied above for the complete ring. For this class of cross section, equations (36)-(39) simplify considerably and lead to an uncoupling of the in- and out-of-plane motion. The out-of-plane motion only will be studied here as the in-plane motion has been extensively reported on by Lang [14, 15]. Thus, subject to the restrictions noted earlier, the characteristic equation associated with the simplified forms of equations (38) and (39) becomes : : ( F i , - ~,2) + ;.6E2r L - 2~,'- - GJ L + 2 ~ 2 ( r i , - ~:)]
+ ) : { [ / , ( r - 2GJ) - U ] + ~ ( r i , + ~'i-r
- U)
- 6 J i , - 2 t & - 2 U + (L + L) (4¢, - i3]}
+ ;?{-GSl,
+ ~4[-r
- i,(/, - 2¢,) + i=0¢, - 133
+ ~q-OJ - 2¢~ + L(2i: + / , + r - 2~,)3}
+ ~ 2 E - L + ~ ' ( L + i 3 ] = o.
(40)
The corresponding mode shape for each solution (2) is given by B
J.4(/r - - 2¢, + F ) - - ,2.2GJ -
= - ):[~(r
- ¢3 + (¢, - i , -
N2[1 -
22(L + P -
GJ)] + ~[,~=(r
2¢s)]
- ¢3 + L ] "
(41)
Note that, for circular cross sections, the coefficient of the first term in equation (40) vanishes, yielding a cubic in 22. This is consistent with the fact that only three conditions can be prescribed at each end of a ring of circular cross section. Let the roots )z of equation (40) be given in the form )(1)" = C l + id, .J.(2)2 ~-- C 1 - -
id,
2 (a)2 = - C 2 , /].(4)2 = C 3 ,
where C1, C2, Ca and d are real and d, Ca, Ca > 0. The corresponding roots 2 follow as
&(~)
2¢ ~) =
= ~ + #,
22(1) =
_ (~ + # ) ,
;..Y)
-(~
-
= ~ -
#,
;.c~) = i , , / ~ ,
; 4 4~ = ~ / ~
~oca, = _ ~ , / ~ ,
: 4 , ) = _4-6;3,
iv),
where = dl2v > 0,
V = x/(~/C-i 2 + d 2 - C012 > 0.
With the roots (2) known, the ratio B ] ~ can be obtained from equation (41) and the
482
ti. E, WILLIAMS
solution for the displacement (W) and rotation (fl) written out. If we define % = _ C22(L - 2(~ + F) + C2CJ - ~2[1 + C2(i, + r - 2(~)] c ~ [ c ~ ( r - 0 - (L - L - o J ) ] + ~ [ i ~ - c ~ ( r - ~ , ) ] ' 2~ + ;) -
c3[c~(r
- (,) + (~ - i , - a J ] + ~ [ i ~
.~t cos fi =
c3G3 - ~[1
G~G3 + G2G4 G32 + G42 ,
-
c ~ ( i , + r - 2L)]
c3~(i, -
+ c ~ ( r - (s)] '
R1 sin 6 = G t G , - GzG3 G32 + G42
where G, -- (C~ - d 2) (i, - 2(s +/~)( - C I G j ) - ~2[1 - C,(i, - 2( s + F ) ] , 02 = d[(2C, + ~2) (i, - 2(s + F) - ~ J ] , 63 = ( c , 2 - d 2) ( r - (~) - c 1 ( i , - (~ + G i ) + ~ 2 [ c , ( r
- (,) + i ~ ] ,
G4 = d[(2C1 + ~2) (F - (~) - (i, - ~ + a J ) ] , it can be shown that W]R = cos ¢o'¢{e~°(aI cos 70 +/12 sin 70) + e-~°(Aa cos 70 + A4 sin 70) + A5 cos x/~2-20 + A6 sin ~/~20 + ATe "/-c'7° + Ase ='/c--~}
(42)
and fl = cos ~or{Rxe=°[Ax(eos 6 cos 70 -- sin 6 sin 70) + A2(cos 6 sin 70 + sin c5cos 70)] + Rae-~°[A3(cos ~ cos 70 + sin 6 sin 70) + A4(cos 6 sin 70 - sin 6 cos 70)] + -R2(A5 cos x/~220 + A 6 sin x/~20) + -~3(Ave"/~;° + Ase-'/cc--~)} •
(43)
As an example of the above analysis, numerical results were obtainedt for a ring segment of channel cross section, clamped at 0 = __+rr/2. A number of cases were studied to obtain the dependency of the solution on the parameters a i r and z/a. The mode shapes are presented in Figures 4--11 for the first three modes. The corresponding roots (2) and the frequencies are presented in Tables 2 and 3. A solution is also presented for comparison for the lowest bending mode for a circular cross section. The natural frequencies and the corresponding mode shapes must satisfy the characteristic equation (40) and the boundary conditions that W, fl, fl', W' = 0
at
0 = _ n/2.
The method of solution is basically that proposed by Lang [15]--a value of ~ is assumed and corrected until the determinant of the coefficient matrix changes sign and then approaches sufficiently close to zero. The process is readily automated and was carried out on a digital computer. "I"The author is indebted to S. G. Groff and R. T. Weimer for performing the calculations, and to Harvey Mudd Collegefor supporting this phase of the work.
EQUATIONS OF MOTION OF TIIIN RINGS I-0
I'0
0-8
0"8
0-6
0"6
0.4
04
0"2
0"2 I'0 - "":-:--~-~-~-~-~1.5
0"5
483
I
0
I
0"5
I'0
1"5
d
B
Figure 5
Figure 4-
Figures 4 and 5. First even transverse bending modes. Figure 4, channel section. Figure 5, circular section thin-walled.
I
I'0
0
I
Z
~
I'0
~
0"8
//
I
-
~'~ o/R-I/10,1/20
O'G
~ 0-4
-I.0
0"2 a/t?=l/lO
I
I
0'5
1.0
1'5
0"5
I'0
@
d
Figure 6. Second even transverse bending mode. Channel section.
I
I ~,t/a
1'5
Figure 7. First odd transverse bending mode. Channel section.
,MIO ~
i
V~=~.,/,,-J/5:.'V" \
I
38! ~2 ]-- ./~#~r'~aAg'Ll20,tla'I/5
]'0
I ~ / ~ -~- a/R.lll O,tlo.l15 ./ ~'- o/R,I/10,
r/o-l/lo
t
I
I
0
0.5
I'0
1.2'-"
1"5
0
I
t
0-5
I-0
d
d
Figure 8
Figure 9
1'5
Figures 8 and 9. First even transverse bonding mode. Figure 8, channel section. Figure 9, circular section thin-waiTed.
484
it. E. WILLIAMS |
I
2 I t
,o
I
0
I
-I
om.1/lff
~
~-~
/\o,R.,,2o
-I0
/
-4
l/a',l/lO
-15
0
I
I
0"5
I0
0
1"5
8 Figure 10. Second even transverse bending mode. Channel section.
I 05
I I-0
1.5
8 Figure 11. First odd transverse bending mode. Channel section.
TABLE 2
Natural frequencies of a channel cross section Mode
First even
First odd Second even
air
z/a
]r x 103
~2 X 103
~2/Jr
kit
1/I0 1/20 1/10 1/20 1110 1/20 1110 1/20 1[10 1/20
1]5 1/5 1/10 I]10 1/5 1/5 1/I0 1110 1/10 1/10
1"944 0.4861 1.944 0.4861 1.944 0.4861 1.944 0-4861 1-944 0.4861
2"635 0.540 1-812 0.264 14.81 4"071 10.97 2-224 34.20 8"719
1"355 1.I I 0.9321 0.543 7.619 8.375 5.643 4"575 17.59 17.94
3"55 (4"73) 3-36 3-23 2"82 5"46 (7"85) 5-59 5"06 4"80 6"72 (I 1"0) 6-75
Natttral frequencies of a circtdar cross section Mode
i x 103
~2× 103
~2/i
kit
First even
2.500 0.6250
8.127 2.040
3.251 3.264
4.40 (4.73) 4.43
t Numbers in parenthesis are for straight beams.
It is interesting to note (from Figures 4 and 5) that the transverse deflection of the lowest bending mode is generally larger for the circular cross section and decreases for the channel cross sections as the section gets thinner (r/a decreasing). Alternatively (from Figures 8 and 9), the curves for the corresponding rotation show little similarity, with those for the channel cross section being practically double the amplitude of those of the circular cross section and having a definite positive region for 0 > 0.6 (say). As can be seen from Table 2, the reduced frequency for the channel section depends
485
EQUATIONS OF MOTION OF THIN RINGS TABLE 3
Characteristic roots for a channel cross section--equal legs Mode
First even
First odd Second even
aiR
z/a
ct
]~
~[C 2
~]C 3
1/I0 1/20 1/10 1/20 1/10 1/20 1/10 1/20 1/10 1/20
1/5 1/5 1/I0 1/I0 1/5 1/5 1/10 1/10 1/10 1/10
1.562 1.508 1-788 1-714 2.113 2"199 2.341 2-515 2"720 3.129
0.9096 0.9452 1.209 1.087 0.9894 1-131 1.392 1.402 1-349 1"582
2-339 2.199 2.533 2.363 3.218 3-111 3.441 3.364 4-264 4"326
4.758 9.470 2.558 4.741 4.873 9"495 2.887 4"795 3.248 4"953
significantly on the thickness parameter T/a as well as on the curvature parameter aiR. This behavior was also observed in the results for the complete ring, and illustrates the role of the torsional rigidity (J)--the only area property depending on x--in affecting the motion. It also follows from the two examples presented that the reduced frequency of the first even mode for the circular section is very nearly proportional to the reduced area moment (i). As a final note, there is presented in the last column of Table 2 the values of the parameter kl, where kl is the reduced frequency for a straight beam of length I, clamped at both ends, and given by co = ( k l ) 2 x / t ~ p A l 4 . The length I was chosen as the developed length of the ring: i.e., 1 = hR. It is apparent that the reduced frequency of a circular cross section correlates very closely with that for a straight beam (given in parenthesis in the last column), but the reduced frequencies for the channel cross section are quite different. Further, whereas the reduced frequencies for the first three modes for the straight beams are quite far apart, those for the channel cross section are relatively close together. 5. DISCUSSION The equations governing the motion of thin, circular rings for a general cross section have been derived by using Hamilton's Variational Principle. These equations have also been shown to be consistent with the Principles of Linear and Angular Momentum by using the definitions of the stress resultants and the components of momentum together with assumed forms for the displacement field. Of particular importance is the natural occurence of the "twist" equation of motion (25) which relates the twisting moment to the warping moment and the St. Venant Torsion Moment. This relation is in marked contrast to the customary procedure of constructing the twisting moment as the superposition of the two contributions in a purely statical sense. It is significant that the dynamic contribution can be important in the higher frequency (torsion) modes. It has been shown from the analysis of complete rings that the relative importance of torsional stiffness and warping restraint is measured by the parameter z2R2/a4 for the thin-walled sections studied here. For solid sections, one would expect that the relative importance would be measured by the parameter R2/a 2, where a represents a characteristic transverse dimension of the section. Thus, as a i r ,~ I for the present theory to be applicable,
486
rl. E. WILLIAMS
it is apparent that warping restraint is important only for thin-walled sections for which z/a <~ 1. Further, as the ratio of the warping moment to the St. Venant Torsion Moment increases as n 2, it is increasingly important to consider warping restraint in higher order modes. The relative importance of dynamic terms besides the leading terms--the "rotary inertia" and "twist momentum" terms--has been shown to be measured by the parameter n2a2/R 2. Upon assuming n2a2/R 2 = O(a[R) as defining the order of error, and replacing the wave number n by 2foR[l, where I is the wavelength of the deformation, it follows that the higher order dynamic terms are appreciable for l/a = ( a / R ) - ½ 0 ( 1 ) .
Thus, these terms may be appreciable for aiR = 1/20 (say) for wave numbers in excess of five. Note that, as l/a >> 1, the present theory should still be expected to apply. As a final note, the appearance of the material constant E in place of the customary E in equations (30) and (34) suggests that a comparison be made between this and the classical derivation wherein E is the parameter. The comparions can actually be made at the stage where the strain energy density was formulated. In this paper, no assumptions were made as to the nature of the stress distribution--only the form of the displacement field was assumed. Hence, the expression for the strain energy density Vo used here is without prior restriction. In comparsion, it is customarily assumed, as in beam theory, that the transverse normal stress components vanish identically. This has the effect of dropping the strain variables e,, and ez~ from the analysis, as err = ezz = -- Ve,oo ,
and leads to the following expression for the strain energy density (Vo): E Vo = 5-
G + T
2 +
This expression could have been used to evaluate the total strain energy (3) instead of the general expression that was actually used. However, as the form of the generalized displacements used in this paper (2) leads to a strain field with e00, ~'0,, )'0z as the only non-vanishing components, it follows that the general expression for Vo reduces to that given above if E is replaced by E. Thus, the results obtained in this paper can alternatively be interpreted as following from a restricted form of the strain energy density if/~ is replaced by E. In this way the classical results are recovered, but at the expense of making inconsistent stress and strain assumptions. REFERENCES 1. A. E. H. Love 1952,4 Treatise on the AIathematical Theory o f Elasticity. Cambridge University Press Fourth Edition. 2. J. P. DEN HARTOO1928 Philosophical Magazine 5, 400--408. The lowest natural frequency of circular arcs. 3. J. P. DEN HARTOG 1947 Mechanical Vibration. New York: McGraw-Hill Book Co. Third edition. 4. E. VoL'rERRAand J. D. MORELL1960 Transactions o f the American Society o f ltlechanical Engineers, Journal o f Applied Mechanics 27, 744-746. A note on the lowest natural frequency of elastic arcs.
EQUATIONS OF MOTION OF THIN RINGS
487
5. E. VOLTERRAand J. D. MORELL 1961 Journal of the Acoustical Society of America 33, 17871790. Lowest natural frequencies of elastic hinged arcs. 6. E. VOLTERRAand J. D. MORELL 1961 Transactions of the American Society o f AIechanical Engineers, Journal of Applied Alechanics 28, 624-627. Lowest natural frequency of elastic arc for vibrations outside the plane of initial curvature. 7. I. U. OJALVO 1962 International Jourttal of Mechanical Sciences 4, 53-72. Coupled twistbending vibrations of incomplete elastic rings. 8. A. S. HAMMONDand R. R. ARCHER 1963 in Developments in l~Iechanics, Vohtme 2, Part 2: Solid Mechanics, 489-524. On the free vibration of complete and incomplete rings. 9. S. S. RAO 1971 Journal of Sound and Vibration 16, 551-566. Effects of transverse shear and rotary inertia on the coupled twist-bending vibration of circular rings. 10. C. G. CULVER 1967 Proceedings of the American Society of Civil Engineers, Journal of the Structural Division 93, 189-203. Natural frequencies of horizontaly curved beams. 11. V. S. VLASOV 1961 Thht-walled elastic beams. Israel Program for Scientific Translation, for National Science Foundation, U.S. Department of Commerce. 12. H. E. WILLIAMSTo appear in Journal of Engineering Alechanics Division, Proceedings of the American Society of Civil Engineers. On the linear theory of thin rings. 13. J. P. DEN HARTOa 1962 in Handbook of Enghwerhlg Mechanics (Editor: W. Flugge). New York: McGraw-Hill Book Co. 14. T. E. LANa 1962 (July) Jet Propulsion Laboratory Technical Report No. 32-261. Vibration of thin circular rings--Part I. Solutions for modal characteristics and forced excitation. 15. T. E. LANG 1963 (March) Jet Propttlsion Laboratory Technical Report No. 32-261. Vibration of thin circular rings--Part II. Modal functions and eigenvalues of constrained semi-circular rings.
APPENDIX I THE AREA PROPERTIES
The cross-sectional area properties appearing in the expressions for the strain energy (3) and kinetic energy (5) are defined below: (I., I,,, I,) = f d A ,) A J
(~, ~¢, ¢2) 1 + ~/R
O~
r
'
r
$
0~
1 ['dA R
(~' ~) = ~ - ~ J a
9- ~ x
(~, O,
F' = ~R4 f dAR¢2,
1 3 dA ¢ (~;, ~;) = -x~f,
× (¢, O,
(I..,,', I , j , I.~') = fadA(~2~, ~3, ~2~),
1
(~g, ~ ) = -A-~ f adA¢(¢L ¢¢).
tI. E. WILLIAMS
488
A P P E N D I X II NOMENCLATURE transverse dimension o f cross section (see Figures 2 and 3) cross-sectional area E 2+2G F.. components o f force stress resultants G shear modulus o f material a
A
F,,Fo,
G C/E
t,,Ir:,I= area moments o f inertia of cross section it, etc. lr/AR 2, etc. J
Y
lvto, l~ 1, m~, too, m, Mo,, 3Ioo, Mo=
M, II
Pr, Po,Pz R T
7:t~ ll~ U~ |9
U,V,W
P r
P CO
torsional stiffness J/AR 2
components of angular momentum per radian warping momentum per radian components of linear momentum per radian components o f moment stress resultant bimoment wave number components of distributed force per length o f centroidal curve radius o f centroidal curve thickness dimension of cross section (see Figures 2 and 3) distributed twisting moment per length o f centroidal curve components o f displacement o f a general point components o f displacement o f the centroid circumferential component o f rotation o f cross section warping constant axial (centroidal) coordinate o f cross section moment o f torsion function circumferential coordinate rate of twist (Rt¢o = fl' + W'/R) radial (centroidai) coordinate o f cross section moment o f torsion function density o f material torsion function frequency of free vibration
coR~/p/E
()' a()/~O (9 O()/&
ON THE EQUATIONS OF MOTION OF THIN RINGS H. E. WILLIAMS Department of Engineering, Ilarrey M ttdd College, Claremont, California 91711, U.S.A. (Receiced 12 June 1972, attd ht revisedform 14 A'ot'ember 1972) The equations of motion governing small, elastic displacements, including warping effects, are derived by using Hamilton's Theorem. The stress-resultant displacement equations are identified through the natural boundary conditions. It is shown that the governing equations obtained are consistent with those obtained by using momentum principles. Of particular importance is the appearance of a "twist" equation of motion relating the twisting moment and the St. Venant torsion moment. Numerical results are presented for the frequencies and mode shapes of both complete rings and ring segments. 1. INTRODUCTION In discussing the equations governing small, elastic displacements of thin circular rings, it is important to emphasize the role of the shape of the transverse cross section in establishing the relative importance of warping effects. As the technical theory ofrings may be considered to be a generalization of the theory of straight bars, one might expect a similar qualitative behaviour in that (i) warping vanishes identically for circular cross sections, (ii) warping effects are confined to the neighborhood of the supports for solid (dosed) cross sections, and are negligible outside this region, and (iii) warping effects are important over the full length of the bar for thin walled, open cross sections. Thus, the form of the appropriate governing equations for rings can be expected to he strongly influenced by the shape of the transverse cross section. It should also be noted that, if the developed length of the ring is not large in comparison with the transverse dimensions, transverse shearing strain may be important. For transverse cross sections that are circular, the governing equations for rings are a straightforward generalization of those for straight bars. Probably the most well known work in this area is that by Love [1] who determined the frequency of free radial and transverse vibrations of complete rings. If one assumes that warping effects are negligible in rings with solid cross sections, the equations developed by Love may be applied to other than circular cross sections. The frequency of the lowest mode of free, in-plane vibration of ring segments was obtained by Den Hartog [2] using a Rayleigh-Ritz procedure and equations which omitted warping effects. The frequency of the lowest out-of-plane mode was also found [3] in the same way. This work was later extended by Volterra and Morell [4, 5, 6]. More general studies of the equations of motion and the dynamic behavior of rings have also been made. Notable among these are the contributions of Ojalvo [7], Hammoud and Archer [8], and Rao [9]. In none of these papers were warping effects included, though the effect of rotarY, inertia was considered in both references [8] and [9]. Warping effects were first studied by Culver [10] using equations developed by Vlasov [1 I] and modified for dynamic effects by including a distributed inertia force and twisting moment. (Culver also verified the equations by deriving them from Hamilton's Principle.) A frequency equation was presented for the free, transverse vibration of simply supported 465
466
rl. E. WILLIAMS
ring segments of symmetrical cross sections (I,~ = 0)t for which the shear center and centroid coincide. A Rayleigh-Ritz solution was also given for non-simply supported boundary conditions for the same class of cross section. The purpose of this paper is to explore in detail the equations of motion of thin rings including both warping and rotary inertia effects. The equations of motion are first derived from Hamilton's Theorem upon assuming a set of generalized displacements derived earlier [12]. It is then shown that, through Hamilton's Theorem, one can establish both the stress-resultant, displacement equations and the equations of motion. Furthermore, the equations of motion are consistent with those obtained by using the classical approach: i.e., by applying momentum principles to a volume element of ring, and with the momentcurvature relations inferred from Hamilton's Theorem. A most interesting result is the natural appearance of a "twist" equation of motion describing the relationship between the St. Venant torsion moment and the warping moment. Finally, numerical results are presented for the frequencies and mode shapes of free transverse vibration of complete rings and ring segments for a variety of cross-sectional shapes. 2. EQUATIONS OF MOTION As is well known, the equations of motion can be identified as the Euler equations associated with Hamilton's Principle expressed in the form 5
(S - K) dr = o
dr (Work done by external forces and moments in a virtual displacement),
(1)
where S = total strain energy of ring segment (01 -<<0 ~< 02), K = total kinetic energy of ring segment (01 ~< 0 ~< 02) and 3ui = 0 at z = Zo, zl. The natural boundary conditions define the choices from which actual boundary conditions may be selected. In order to use equation (1) to develop an approximate theory for thin rings, the displacement components must be expressed in terms of generalized displacements. These generalized displacements must be sufficiently general to adequately describe the expected motion and also simple enough to lead to a manageable set of equations. The generalized displacements used here are those developed earlier [12] for static loading, and given by
v = v(o) - ~,(o) + ~(o)
+ ~(~, 0 ~o(o),
where to, = W ' / R ,
tc, = ( V - - U ' ) / R ,
too = (fl -4- W / R ) ' / R
and
~( ) (7=
80
•
The coordinates (~, () are centroidal (see Figure 1), and satisfy the requirements that
fa
d A ( ~ , ~) = 0
"j" A list of nomenclature is given in Appendix II.
(2)
EQUATIONS OF MOTION OF THIN RINGS
467
over the transverse cross section. >X
u,,~ Y
,
Z~
,0
¢_.
w,Maz
>( .
r
R
t
*;
z
Figure 1. Sign convention.
The function ~b(~,~) is the St. Venant Torsion Function which is harmonic everywhere in the cross section and satisfies the condition that
1~+
m~=m~-l~
on the bounding contour. The quantities (/, m) are the direction cosines of the outward pointing unit normal vector. As q5 is known only to within an additive constant, it is convenient also to require that
a dAq~ = O . This assumption fixes V(O) to be the area average of the circumferential displacement: i.e., da v(0) = fa v(~, ~, 0)-2-
As is noted in reference [12], the generalized displacements can be interpreted as producing a rigid body motion of a transverse cross section (in which the components of the transverse shear strain vanish at the centroid) plus a warping displ~icement analogous to that of St. Venant Torsion. Thus, the model of the ring studied here includes the same basic degree of freedom as in the torsion-bending theory of straight bars. In the paragraphs to follow, the quantities appearing in equation (1) are defined in terms of the generalized displacement variables (equation (2)) and the equations resulting from the indicated variational process are determined. The total strain energy in the rings segment (01 ~< 0 ~< 02) is defined in terms of the strain energy density (Vo), where G Vo = 2e2/2 + G(e,,2 + too2 + e~:2) + -~- (y,o2 + 7~..2 + ~,0~2) for a linearly elastic material. The strain components (e,,, etc.) follow from the displacement
468
H.E. WILLIAMS
field (equations (2)), and are given by ~oo =
1
--[(u
+ v ' ) + ~(p -
r
Yo, ="
+
_
~'o: =
04
~__ +
I"
~,, = 0,
•
W"/R) + ~(V' - U")/R + ¢Ko'],
R Rico,
R O~ ] Rt,"o '
~:: = 0,
y,: = 0.
Thus, the total strain energy stored in a ring segment (0t ~ 0 ~< 02) can be computed, and is given by
f2 oL
S=
rdAV o=
dO E A R
+ i~
1
(3) where the area properties (i:, etc.) are defined in Appendix I. The total kinetic energy of the ring segment (0t ~< 0 ~< 02) is K
r dA ~p,,,2, dOl
JA
where (v), the velocity o f a general point, is v = e,(0 + (/~) + e0(~' - (£', + ~g': + ¢~o) + ez(W - ~.fl)
(4)
and (') = O( )lOt. In terms of area properties (i:', etc.) defined in Appendix I, it follows that the total kinetic energy stored in a ring segement (0~ ~< 0 < 02). is p A R a f°'d0 {02 + 1;"2 + i+"2 K~-
"2
,]o,
.
R2
+ [~2(i" + i=' + i : ' / + iz:S)
+ ~',2(1,,' + i..,/) + ,:: 2(i:, + i : : , ) + (RK.o)2F, _ 2~,~':(i,,' + i,,:') + 2R~0[Kz(~
+ ~ 2 ) _ I;',(~' + ~. 2)] + 2i,~'
+ 2~jP~.o + 2l:.'- ( (/~:~ • R
"~) }
"
R '
(5)
The work done on the ring segment (01 ~< 0 ~< 0z) in a variation of displacements, due to both distributed forces (Po Po, P,) and a twisting moment (t#) applied along the line of
EQUATIONS OF MOTION OF THIN RINGS
469
centroids, and normal (Zoo)and shear stresses (%. %z) distributed over the end cross sections (0 = 01, 02), is given by
6(Work) =
R dO(ta6fl + po6V + p,6U + p:6W)
)O
I
+
dA(zo,&¢ + Too6V + Zo=&O
•
(6)
0=0,
On substituting the assumed form for the displacements (equation (2)), and identifying the following stress resultants, (Fr,Fo,F..) =
f A(zo , zoo,To:),
Mor = - fa dA~%o,
Al°" = f a dA~roo ,
Moo = fa dA((%, - ~Zo:),
AI~ = [ dA
q~r00 R '
dA
the secon d integral in equation (6), the work done by stresses distributed over the end cross sections, becomes
[Fr*3U + Fo*6V + Fz*3W + Moo'611 + M0r*6h'r + Moz*3h"= + M~,*S(RXo)]
, O=Oi
where ( )* denotes boundary data. At this stage, with all the quantities appearing in equation (1) defined, it is possible to complete the derivation of the governing equations. It should be noted, however, that the requirements in Hamilton's Theorem that
t~tlI =
0
at
t = to, ti
are equivalent to requiring 6U, 6U', etc. = 0
at
t = to, t t .
Thus, on substituting equations (3), (5) and (6) into equation (1), and carrying out independent variations of the variables U, V, IV and fl, one obtains the following differential equations of motion:
mu+
..,
-
_ ,
(.3 -- -
~tzl~_
" i (I=- I + ~-=
-
i==')
9,}
I2 S - -
= R2p,,
(7)
470
H. E. WILLIAMS
-- pAR 3 {-- -~ - g',(l=' + 1==') + n,(l,, + I,= ) -- R~o(~' + Fez) + 1,/f,, i ' V" _ i = % }
= R%,
(8)
U + U" R +-~d2 [PRtt°' + ( ' ( f l if,
]
J
W-~"R')- ~ s U +RU'' ] _ "-ff GJ"Rico ,~J
_ ,
- p A R 3 - .-~ + I,, II +
- ~= (I. + i . / -
(sRKo'
- ,...,
(/,,'
+ 1=, )K r + F'RF,-o'
~' - ~:1 - ,¢, (~, + e~ ~)
- ((~' + ~,~)R~o' - ( i , / -
.~/)
(9)
= R~e,,
U + U" ] R + ~Rt¢o' U + U'" ] + - ~a2 [PRh-o' + (, (fl - W.-~--R')- ~, - R
G] -~" m:o ] J -
0 -- pAR 3 - fl(i,; + l~.: + L,/ + I-=,') - l,='--~
if,
~) -
+ i=' ~ + r'Rr,-o' + i V ( ~ / +
r¢,'(¢/+
~)
.°
0o) and the natural boundary conditions that either ,W=o,
or
EAR
-~
a[
L
U+U" n
- p,4R 3 {,~(i=' + i = ' ) - ,~,(Lz' + L,z') + R,~o(~,' + V2)
(11)
471
EQUATIONS OF MOTION OF THIN RINGS
5V = O,
or
6 W = O,
or
U + V'
EAR--
R
(12)
= RFo* + Mo~*,
U + U"
R
000 [ F Rtc°' + ~s ( fl - W-VR) - ~ U +
-- pAR 3 -- (i,/ + 1,~, )t¢,
+ ~RKo'
]
] +---E -Rtc°}od
F'Rfc o + fq(i,~'
+ i.:' - ~' - ~ ) + r~,(~; + ~2) + g~o(~f + ~2)
6fl = 0,
or
+~
Rt¢o - p A R 3 - F'R~o
- ~ . e : + ~ ) + , . ¢ : + ~ ' ) - ~.'-~} = Moo*, 6U' = O,
or
EAR [i~ U + U" R t =
5IV'=0,
(14)
(15)
- Mo..*,
or
. 6fl' = O,
]}
= Mo,* + Me*,
(16)
= Me*,
(17)
or
on both 0 = 0i, 02. 3. ALTERNATE FORM OF THE EQUATION OF MOTION The equations of motion (7)-(10) and the natural boundary conditions (11)-(17) are
472
tl. E. WILLIAMS
sufficient to define a solution for U(O, z), etc., given the loading, and the initial and boundary conditions o f a particular problem. As the equations are linear and possess constant coefficients, solutions are obtainable without analytical difficulty. However, before undertaking any solution, let us formulate the equations in terms of stress resultants in an attempt to understand their significance. As was done previously [12] for the equations ofequilibrium, the force and moment stress resultants are identified through their appearance in the natural boundary conditions. In these equations, one observes that a force or moment stress resultant is equated to some function of the displacements at the bounding cross sections (0 = 0~, 02). However, as these bounding cross sections can be chosen arbitrarily without affecting the form of the displacement function, it follows that the natural boundary conditions essentially define the form for the stress resultant, displacement equations for all 0. However, before elaborating on this point, it is convenient first to evaluate the linear and moment of momentum of a differential volume element as these expressions can be identified in the governing equations. Let the basic differential volume element be that bounded by transverse planes of cross section A located at 0 + AO]2. Within this element, there exist doubly differential volume elements of volume rA0 dA. Associating a velocity (see equation (4)) with these elements, it follows that the linear momentum (AM) and the moment of linear momentum (AL) about the centroid of the basic differential element are AM = pAO~ardAv i.
AL
= pAOJa r dA[(~e¢ + (eg) x v'].
With sectional properties given in Appendix I, and with the definitions AM lim ~ = m c e , + m,e{ + moeo, ,0~o A0 AL lira ~ = lceg + l~ec + loeo, A0-.o A0 it lbllows that
"'~ = oAR2(O/R + M,/),
,,,~ = pAR20~'/R - - / ~ i J ) ,
mo= pAR2(~'/R - t?fl-,.' + K-z1**'+ RK-o~s'), -
-
i=,,')-- ~(L~' + 1,r=) -- R~'O(~; + ~ )
R
r: + ~,(Lr' +
R
:~ + / ~ ( L r ' + L z ' + i : , / + 1 z : : ) + - ~ i , z '
,
,
[~ ~ pAR3 I ~ ]zz' -- hr(]rz' + ],r,') + l~zjzz' + ]zzz') + Rh'O(~s' "l" r~2) ] • AS will be seen later, it is convenient to define a quantity (/+), without immediate physical
473
EQUATIONS OF MOTION OF THIN RINGS
significance, given by
lee = paRa [F'Rg:o + t~,=(*s'+ P,2) - ~,,((,' + P,~2) + ~s"f-l~] • This quantity appears to be a momentum associated with warping. In terms of these momentum variables, the natural boundary conditions (11)-(17) suggest that the force and moment stress resultants are related to the displacements in the following way:
(18) RFo = EAR {,U + V'
[i~ U + U''
(19)
U + U"
] + (sRh'o'
J
O00[rR'c°'+(s(F--W-i~--R)
-~'u+U''R-] U + U"
+ GJRtc°} + lee + l~, "l
Mo, = - E A R [L(fl - W"[R) - Lz - -R
+ (sR~co'],
Mee = EAR rlrR~:o' +
IR+v,' ~j
(s~ -
W"/R) - ~s U
I_
Mo: = - E A R []: U +R U''
(21)
(22)
/r--(fl -- W"/R) - ~sRt¢o'] ,
Moo = EAR {-- ~ [PRr.o'+ (s~ -- W"/R) -- ~s -U- ~
(20)
(23)
] + GJR,:o} + lee. (24)
With equations (18)-(24) as definitions, it is then sufficient to prescribe either U or F,, V or Fo, tV or Fz, fl or Moo, U' or Moz, IV' or Mo, and fl' or M , on both 0 = 01, 02. Further interesting consequences follow from the definitions (18)-(24) when the force stress resultants are rewritten in terms of the moment stress resultants. With Mo,, Mee and Ma, defined in equations (21), (22) and (23), respectively, it follows that equations (18), (19) and (20) are equivalent to
RF, = Mo:' - l~,
Rro EAR( U ~ V') =
-
-
(18a) -
Mo:,
RF= = --Mo; + (--M O' + GJl¢o + lee)+ l¢.
(19a) (20a)
Furthermore, equation (24) requires that
Moo = --Mee' + GJtco + lee. 3
(25)
474
n.E. WILLIAMS
It appears from equation (25) that the twisting moment is defined in terms of the St. Venant Torsion moment (GJh'o) and the warping moment (--M4,') by an equation having the form of a "twist equation of motion". The appearance of the quantity lo is not readily justified on physical grounds nor can the term be shown in general to be important. It will be shown later however, that l0 is insignificant in comparison with M0' for the transverse bending modes of free vibration, but is comparable with Mo' for the torsion modes. When one now returns to equation (20a), it follows from the definition of Moo (25) that
Mo/ - Moo + R G = 1¢.
(20b)
It is significant to observe that equations (18a) and (20b) are the axial and radial components, respectively, of the Principle of Moment of Momentum. Looking now at the Euler equations (7)-(10), and rewriting them in terms of the moment stress resultants, we obtain
M o z " - - I ~ ' - - ( F , AR U +R V"
-
Mo~ ) = Rd~ -- R2pr,
(7a)
=
(Sa)
R,
,o -
R%,
--Mo/' + Moo' + l¢' = Rdl~ -- R2p,,
(9a)
Moo' + Mo, = lo -- Rrp.
(lOa)
However, upon taking equations (18a), (19a) and (20b) into account, it follows that equations (7a), (Sa), and (9a) are equivalent to
F ; - Fo = - R p r + the,
(7b)
Fo'+G=
(8b)
--Rpo+lho,
F=' -------Rp~ + lhg.
(9b)
Equations (7b), (8b) and (9b) can be identified as the radial, circumferential and axial components of the Principle of Linear Momentum, while equation (10a) is the circumferential component of the Principle of Moment of Momentum. In summary then, it has been shown that (i) the natural boundary conditions can be interpreted as identifying the moment and force stress-resultant displacement equations, and, (ii), these stress resultants are consistent with the classical equations of motion obtained by applying the Principles of Linear and Moment of Momentum to a differential volume element. It should be noted that the equations developed in this section contain no new information, but are simply a more convenient form of the previous equations and lead to useful physical interpretation. 4. APPLICATION--FREE VIBRATION In the sections which follow, the frequencies and mode shapes of free vibration of complete rings and ring segments are derived. The results for the complete ring are particularly useful in establishing the order of magnitude of warping and rotary inertia effects. The results for the ring segment may be compared with similar results for straight beams in an attempt to assess the effect of coupling between rotation and deflection in the transverse bending modes.
EQUATIONS OF blOTION OF THIN RINGS
475
4.1. EXAMPLE I--COMPLETE RINGS
The equations for the frequencies and corresponding mode shapes of free vibration of complete rings may be obtained by substituting (O/R, W/R, fl) = ( C, A, B) cos nO cos coz, V/R = B sin nO cos ogr
0* >/2),
into the differential equations of motion (7)-(10). This assumed form is a solution provided that Zn2{(L= - ¢,) O*2 - 1) -- ~2[L :' + L,~' -- (~,' + p¢2)-]} + B{(i.~ - . ~ . ) (,,2 - 1) - ~=[i.:' - , : ( ~ : + ~¢~)-1} + C{L(,: - 1y + 1 -~[I
+ nXi=' + L=')]}
+ .DE1 - ~ X 2 i = ' + L = ' ) ] = o,
(26)
-,,tn~212/,J + l , f f - (2~,' + F¢2)] + ~2nB(2~, + F¢2)
+ .C[1 - ~2(2i=" + L = ' ) ] + ~{,,~ - ~211 + 3i=' + L = ' ] } = o, . . ~ { , : ( i . - 2(, + F) + .~C,J - ~ [ 1 - 2(,' - 2 ~ = ) 3 } + ~ { , : ( r
+ nXi: + L,/+
(27)
F'
_ L) + ,,2~7 + ,,2(i. - L)
+ ~2[izz, -- n2p, + n2¢s, ._}_~g 2)']} + C,,2{Q, = - 1) (L, - *~) - ~2(L=' + L,=' - *,' - Fez)} - B,,~=(21.; + i . / -
2~,' - ~=) = 0,
(28)
and .~o:(r
- L) + ,,2(i. - L + ~J) + ~=Eizz' + n2(¢:
+ e~: - F,)]} + B { F , : + (~J - 2(~),: + i, - ~ 2 [ i . ' + L ; + L,,' + L = ' + , : r , ] }
+ C((n 2 - 1) (i,, - nZ~s) - ~ 2 [ i , , _ n2(~, + ~¢2)]} + Bn~a(2~s, + ~¢2) = 0,
(29)
are satisfied, where G = G/E,
~2 = pR2o2/E.
As is well known, these equations possess a non-trivial solution provided that the determinant of the coefficient matrix vanishes identically. This requirement leads to a frequency equation which is fourth-order in ~2, and can be expected to lead to four distinct roots for ~2 in general. Without simplifying assumptions, it is not possible to obtain general results--only results for specific numerical values of the cross-sectional properties. In order to determine some properties of the motion, let us restrict our attention to symmetric
476
H. E. ~,VILLIAMS
cross sections for which
C~: =o,
i , : , i , / = o,
- I ~,I2, ~,:,2, i-z z z I ' lrrz ,~ 1 •
Let us further take ~[R ,~ I so that we may ignore the distinction between the primed and unprimed area properties: i.e., i , / ~ i,,, etc. Consistent with this restriction is the consequence that i, and i= are small in comparison with unity. For this restricted class of cross sections, equations (26)-(29) simplify considerably and lead to an uncoupling of the in- and out-of-plane motion. The frequency equation for in-plane motion only is given by ~4(1 + n2i=)
_
~21112 d~ 1 .~
211215012
--
3)]
+ l z l l 2 ( l l 2 - - 1) 2 = 0 ,
(30)
where D/C-" = 11(1 -- 2lzU2)[(~ 2 - n2).
(31)
An approximate solution of equation (30) valid for n4i~ <~ 1 is given by ~2
= ~ 2 e x t = i12 .~_ l _ 112012 - - 1) 2
=~°~=~
,--:-Ff
'
where ~ t , ~ben~ refer to the extensional, bending (inextensional) modes, respectively. This distinction is readily understood, following the observation
(Die)e=
~ ,1,
(~lC)~e,~ ~
-1/,,,
when it is noted that U + V' ~00(~, ~ = 0) = ~ = (C + 11/7)cos 110cos co~. R
The frequency equation for out-of-plane motion only is given by t~4{/,[1 + n2(l, + i~ - 2(~)] + L(I - 4n2(s) + n2/~(1 + n2l,) - n'(~ z} -
~={i.[1 + n~(,l ~ + 1) ( i ,
2~) + .r,l~(2,14 - ,2 + l)]
+ GJ -
+ LnZEi, Ol2 + 2) + 3112F -Jr 3G7 - 2(,(2n 2 + 1)] + Fn 4 + G J n 2 - 2(sn2[1 +
!12(!l 2 - -
..~ !12(112 - - 1)2 CL(~,7 .dr. n2(i,p
l)(s] }
_ (2)] = 0 ,
(32)
where B
,14(i, - 2(~ + P) + n2t~J - ~21-1 + ,12(i, + P - 2(~)3
-- --~ = n4(/~ -- (s) + n2( ~rJ + L -- (s) + ~ 2 ( 1 r - - n 2 p + n2(s) "
(33)
For the case of circular cross sections, the warping function (~b) vanishes, and equations (32) and (33) reduce to ~412 Or (2/l 2 -- 1)i3 - ~2{1*21~',]1i Or 1 --I-112[(2112 Or 3)i
+ G.7(n 2 + 4)]} + n=(n = -
I)2G,] =
0,
(34)
EQUATIONS OF MOTION OF THIN RINGS
B
!l4 + n 2 G j ] i -- ~2(1 + n2i)]l
,4
n2(1 + GJ/i) + ~2
,
477
(35)
where J = 2i. An approximate solution of equation (34) valid for n4i ,~ 1 is given by ~2
(1 "1- 21125)]2
= ~2to r =
n2(n 2 = ~2bend =
1 +
1)
Sj
2n2G
'
where ~2to,. ~2be,a refer to the torsional and bending modes, respectively. This distinction follows from the fact that 1 1 + 2n2t~ i 1 +2nz(1 + 3 5 ) '
=
=
-
,,2
-
1 -+ 2 5
1 + 2n2G" It should be noted that the approximate solutions of equations (30) and (34) agree with results given by Den Hartog [13] if we write E]E = 1. This point will be discussed later. Upon returning now to equations (32) and (33), it is of interest to obtain numerical results for particular cross sections in order to observe the effects of the parameters ir, (s, r and J. This is accomplished by comparing results for a thin walled circular cross section, an equal legged I-section and an equal legged channel section. The section properties used in the calculations are tabulated below.
TABLE I
Section properties Property
Circular
Section I (equal legs) Channel (equal legs)
]r iz j p
a218R 2 a2[8R 2 a214R 2 0
7a2[36R 2
7a2136R 2
a2]18R 2 ,~2/3R2
a2[gR 2 ,r2]3R 2
a4[72R 4
43ag[324R 4
(s
0
0
-- 4a3/27.R 3
The reduced frequencies for the torsional and bending modes are presented in Figure 2 for 5 = ½. The corresponding mode shapes are presented in Figure 3. It is interesting to note that whereas the frequencies of the torsional mode are of comparable order of magnitude for all sections, the frequencies of the bending mode for the channel and I-section are an order of magnitude smaller than those of a circular cross
478
It. E. W I L L I A M S I
I'0
I
(o)
I
i
I
I0
j.._K_.~L'~" I120
g 08 i
x
x
I
I
I
I
I
-~/
'
I
13
~
i
1
2
.~ 0-4
O
o~ 0"1
~.~
0
2
4
l
I
6
8
I0
I
I
I
I
I
1
1
(f)l
I
l
i
I
I
/viol
I ,lo,y?.~ /"~,':/IO, I 0 2 4. 6
n
l/a. 11.5//"
..J ~-~
o3 om.-I~ o 2 I~
~la'lllOiR.q __. v.Z..II
I-
0"I
ja/l?III0
~04
/ (°; o21- ~, I '~.,io.115
1
I
05
031-
03
l
I
0
4
2
1
I
6
0-5:
~"
I
4 /R I
I
I
6
,o
~" 0"4 I~1 02
I
8
12
06
- (c) I
,
, ~uI
8
10
0"I
O
!
I
2
l
4
I
6
I I0
B
n
Figure 2. Natural frequencies of complete rings. (a) Circular section thin-walled, (b) I-section, (c) channel section, (d) circular section thin-walled, (e) I-section, (f) channel section. I
I
(o)
'~
I
I
I
/ ,,,'R- ] //,IZO j
"°''°J/
/
sso
'
I
(b:
i
l
I
T Z
~-oo
_k
--
'
:3so - (¢~
J
J ' ! ~
~oo
250
250
lla.lllO - I ~ 200 i ~,..~,o.~. I~_. 150
200 150
•"
'
K-a-~
14 12
II II
f- / T
,,°
8
\\
I00 80
leO
~/a'VS~".~
,~.~~,,,o -~
o
a/R= f/20 O/A~•
0
~
~
"°
~
M'~
'
'
o/R
1/20
'
, ~;~o"
, ,~r ,,,°,
O
'
"
50
4
700
,
'
(,)'
'
'
~j/~/~
2 0
I
I
!
|
!
GO0
601
500
,
o,..4
/,,,°/~
~
/
~
aiR-
T
I / c/ i Ri/io. t/a.vzo~_~,/
501
s//.
400
a/R.'-]~ 300 1120
~
I air. //lllO"
1/20
30
I O0
o/R-
I/5
IO
1/10 0
l
I
l
l
l
2
4
6
8
10
n
J
0
l
I
1
l
I
2
4
6
8
10
/'?
0
I
I
|
l
2
4
G
8
l 10
/7
Figure 3. Mode shapes of complete rings. (a) Channel section, (b) I-section, (c) channel section, (d) and (e) circular section thin walled, (f) I-section. section of equal i,. Further, the frequencies for the channel section are less than those for the I-section for bending modes, while the opposite is true for the torsion modes. It has also been observed that the static deflections of a channel section are greater than those of a similarly loaded I-section. This, at least, makes plausible the observation concerning the results for the bending mode. In comparing the mode shapes, it is apparent that there is considerably more rotation
EQUATIONS OF MOTION OF TInN RINGS
479
measured by (B/~) in the bending mode for the channel and I-section than for the circular cross section. The opposite seems to be the case in the torsion modes. Further, on comparing the channel and I-section, it appears that there is generally less rotation in the torsion mode for the I-section; however, there is more rotation in the bending mode for the I-section. With the orders of magnitude of the frequencies and mode shapes established, it is now possible to estimate the relative importance of warping and rotary inertia for the transverse mode of vibration. The relative importance of warping can be established by comparing the warping moment ( - M , ' ) and the St. Venant Torsion moment in the "twist" equation of motion (25). For the uncoupled motion, it follows that M ; = E A R [ f l ( $ + B) - (s(A + B/Iz2)]n 3 sin nO cos o r , Gd~o = -- E A R G 3 (71 + B)n sin nO cos o r . N o w , as
Crj[P = 8"~2R2/a4 (I-section) and 324 "I:2R2 -
387 a* (I---section),
and 1 ~< zZR2/a'*<.< 16 for the sections studied here, it is apparent that the twisting contribution from the F-term is certainly important for such thin walled open sections. However, as ~,J _
3 z2R (V-section)
(~
4
a3
and 1/I0 ~< "c2R/a3 <~ 4/5 for the sections studied here, the effect of twist-bending coupling may even be more important than twisting itself in providing torsional stiffness for sections for which the shear center and centroid do not coincide ((s # 0). The question of the relative importance of rotary inertia can be generalized to the question of when it is justifiable to write the net "inertia" transverse force and twisting moment for the out-of-plane motion as
,:I¢IR,~ pX if',
lolR ~ pp(l,,+ I,,).
Essentially, this question can be answered by looking at the coefficients of the frequency terms in the simplified versions of equations (28) and (29). It follows that, subject to the restrictions made earlier on the cross sections for which the calculations were made, the kinetic energy contributions to equations (28) and (29) are ~2{.g[1 +
n2(i,+ / ~
-- 2(,)3 - B[i~ - n2(/TM- (s)])
and ~(,~[r
_ ,:(r
- (s)] - B [ i , + L + , : c ] } ,
respectively. The dominant terms are * x 1 in the first expression and B x (i, + ]~) in the second expression. From the results for the mode shapes (Figure 3), it is apparent that higher order effects (rotary inertia, "warping momentum") are negligible provided that n2i, ,~ I for the bending mode (as B / ~ ,~ 1/i~), and
,:r ~ (I,+ L )
480
H.E.
WILLIAMS
for the torsion mode (as B[~ ~ I). These restrictions imply that ll2a2/R 2 ,~ 1
for the cross sections studied above. Hence, the higher order kinetic energy effects can be appreciable, even for the lower modes, for the values of air studied here. Finally, it is of interest to evaluate the relative importance of the warping momentum (/¢) and the warping moment (M'¢) in the "twist" equation of motion (25). As
M~' = EARna[F(B + A) - (s(B/n 2 + ,4)] sin nO cos co~, and
l~ = E A R n ~ 2 [/~(B + .~) - (s -~] sin nO cos tot, it is apparent that l¢ is negligible in comparison with M~' for the bending modes but is definitely comparable to M~' for the torsion modes. Thus, the customary assumption that the twisting moment (Moo) is equal to the warping moment ( - M ~ ' ) plus the St. Venant Torsion Moment (GJh'o) is shown to be in error for an important class of motion. 4.2. EXAMPLE I I - - R I N G SEGMENTS The equation for the frequencies of free vibration and the corresponding mode shapes for ring segments with general boundary conditions may be obtained by substituting
(U/R, V]R, W/R, fl) = (C, D, A, 13) e ~° cos tot into the differential equations of motion (7)-(10), and seeking values of 2 and to that satisfy both the resulting equations and the appropriate boundary conditions. This assumed form is a solution provided that .A{2z( 1 dr 22) (l,z -- 4s) -- ~ 2 ) 2 [ _ ( ] , , dr lrr, ) dr (4s' dr ~2)]} + J g { - O + ;.') (l,~ + ;.if3 - ~2[i,~' + ,t2(4, ' + ~¢~)]}
+ O{1 + i=(1 + 3.2)2 -
~2[1
-
).2(i~=' + iz,_.')] }
+ D{I -- ~2(21~.' + 1::~')} 2 = 0,
(36)
~2;.[_ (2i# + i.D + (24,' + F¢2)] + B~22(24, ' + ?2) dr C'2[1 -- ~z(2[~' + L~z')] dr DD .2 dr ~2( 1 dr 31~.' + L:.-')] = o, /~{,~.4(i t - - 2 ( s dr /~) - - ,~28, ~ -- ~ 2 F 1 -
(37)
,~2(irr t dr i z r , t) - r t Z 2
+ 2 ~ ( ( ~ ' + ~:~)]} + B { a f f a e ( r - L) + L - i, - GJ] + me[;:(r"
-
L' -
~2) + L.-']} + C(;:(1 + z ~) (L~ - 4,)
dr e 2 2 2 ( / J dr L , J - 4,' -- f2)} _ ~ 2 2 ( 2 i , _ ., dr L,,' - 24,' - F2) = 0, (38)
+ B{;:E;?P + 28. - 332 + i. - ~ f f i , ; + L / +
L: + i='
-- 22F']} + G'{-(1 + 22) (i,, + 224,) -- ~2[i,., dr 22(4,, dr ~¢2)]} dr ~2~2(24s, dr ?z) = 0.
(39)
EQUATIONS OF MOTION OF THIN RINGS
481
As is well known, these equations possess a non-trivial solution provided that the determinant of the coefficient matrix vanishes identically. The resulting characteristic equation is a polynomial of the seventh order in 22, containing ~2 as a parameter. If ~2 were known, it would be possible, by numerical means, to determine the seven generallycomplex roots for 22 . In what follows, the characteristics of the motion are determined for the restricted class of cross section that was studied above for the complete ring. For this class of cross section, equations (36)-(39) simplify considerably and lead to an uncoupling of the in- and out-of-plane motion. The out-of-plane motion only will be studied here as the in-plane motion has been extensively reported on by Lang [14, 15]. Thus, subject to the restrictions noted earlier, the characteristic equation associated with the simplified forms of equations (38) and (39) becomes : : ( F i , - ~,2) + ;.6E2r L - 2~,'- - GJ L + 2 ~ 2 ( r i , - ~:)]
+ ) : { [ / , ( r - 2GJ) - U ] + ~ ( r i , + ~'i-r
- U)
- 6 J i , - 2 t & - 2 U + (L + L) (4¢, - i3]}
+ ;?{-GSl,
+ ~4[-r
- i,(/, - 2¢,) + i=0¢, - 133
+ ~q-OJ - 2¢~ + L(2i: + / , + r - 2~,)3}
+ ~ 2 E - L + ~ ' ( L + i 3 ] = o.
(40)
The corresponding mode shape for each solution (2) is given by B
J.4(/r - - 2¢, + F ) - - ,2.2GJ -
= - ):[~(r
- ¢3 + (¢, - i , -
N2[1 -
22(L + P -
GJ)] + ~[,~=(r
2¢s)]
- ¢3 + L ] "
(41)
Note that, for circular cross sections, the coefficient of the first term in equation (40) vanishes, yielding a cubic in 22. This is consistent with the fact that only three conditions can be prescribed at each end of a ring of circular cross section. Let the roots )z of equation (40) be given in the form )(1)" = C l + id, .J.(2)2 ~-- C 1 - -
id,
2 (a)2 = - C 2 , /].(4)2 = C 3 ,
where C1, C2, Ca and d are real and d, Ca, Ca > 0. The corresponding roots 2 follow as
&(~)
2¢ ~) =
= ~ + #,
22(1) =
_ (~ + # ) ,
;..Y)
-(~
-
= ~ -
#,
;.c~) = i , , / ~ ,
; 4 4~ = ~ / ~
~oca, = _ ~ , / ~ ,
: 4 , ) = _4-6;3,
iv),
where = dl2v > 0,
V = x/(~/C-i 2 + d 2 - C012 > 0.
With the roots (2) known, the ratio B ] ~ can be obtained from equation (41) and the
482
ti. E, WILLIAMS
solution for the displacement (W) and rotation (fl) written out. If we define % = _ C22(L - 2(~ + F) + C2CJ - ~2[1 + C2(i, + r - 2(~)] c ~ [ c ~ ( r - 0 - (L - L - o J ) ] + ~ [ i ~ - c ~ ( r - ~ , ) ] ' 2~ + ;) -
c3[c~(r
- (,) + (~ - i , - a J ] + ~ [ i ~
.~t cos fi =
c3G3 - ~[1
G~G3 + G2G4 G32 + G42 ,
-
c ~ ( i , + r - 2L)]
c3~(i, -
+ c ~ ( r - (s)] '
R1 sin 6 = G t G , - GzG3 G32 + G42
where G, -- (C~ - d 2) (i, - 2(s +/~)( - C I G j ) - ~2[1 - C,(i, - 2( s + F ) ] , 02 = d[(2C, + ~2) (i, - 2(s + F) - ~ J ] , 63 = ( c , 2 - d 2) ( r - (~) - c 1 ( i , - (~ + G i ) + ~ 2 [ c , ( r
- (,) + i ~ ] ,
G4 = d[(2C1 + ~2) (F - (~) - (i, - ~ + a J ) ] , it can be shown that W]R = cos ¢o'¢{e~°(aI cos 70 +/12 sin 70) + e-~°(Aa cos 70 + A4 sin 70) + A5 cos x/~2-20 + A6 sin ~/~20 + ATe "/-c'7° + Ase ='/c--~}
(42)
and fl = cos ~or{Rxe=°[Ax(eos 6 cos 70 -- sin 6 sin 70) + A2(cos 6 sin 70 + sin c5cos 70)] + Rae-~°[A3(cos ~ cos 70 + sin 6 sin 70) + A4(cos 6 sin 70 - sin 6 cos 70)] + -R2(A5 cos x/~220 + A 6 sin x/~20) + -~3(Ave"/~;° + Ase-'/cc--~)} •
(43)
As an example of the above analysis, numerical results were obtainedt for a ring segment of channel cross section, clamped at 0 = __+rr/2. A number of cases were studied to obtain the dependency of the solution on the parameters a i r and z/a. The mode shapes are presented in Figures 4--11 for the first three modes. The corresponding roots (2) and the frequencies are presented in Tables 2 and 3. A solution is also presented for comparison for the lowest bending mode for a circular cross section. The natural frequencies and the corresponding mode shapes must satisfy the characteristic equation (40) and the boundary conditions that W, fl, fl', W' = 0
at
0 = _ n/2.
The method of solution is basically that proposed by Lang [15]--a value of ~ is assumed and corrected until the determinant of the coefficient matrix changes sign and then approaches sufficiently close to zero. The process is readily automated and was carried out on a digital computer. "I"The author is indebted to S. G. Groff and R. T. Weimer for performing the calculations, and to Harvey Mudd Collegefor supporting this phase of the work.
EQUATIONS OF MOTION OF TIIIN RINGS I-0
I'0
0-8
0"8
0-6
0"6
0.4
04
0"2
0"2 I'0 - "":-:--~-~-~-~-~1.5
0"5
483
I
0
I
0"5
I'0
1"5
d
B
Figure 5
Figure 4-
Figures 4 and 5. First even transverse bending modes. Figure 4, channel section. Figure 5, circular section thin-walled.
I
I'0
0
I
Z
~
I'0
~
0"8
//
I
-
~'~ o/R-I/10,1/20
O'G
~ 0-4
-I.0
0"2 a/t?=l/lO
I
I
0'5
1.0
1'5
0"5
I'0
@
d
Figure 6. Second even transverse bending mode. Channel section.
I
I ~,t/a
1'5
Figure 7. First odd transverse bending mode. Channel section.
,MIO ~
i
V~=~.,/,,-J/5:.'V" \
I
38! ~2 ]-- ./~#~r'~aAg'Ll20,tla'I/5
]'0
I ~ / ~ -~- a/R.lll O,tlo.l15 ./ ~'- o/R,I/10,
r/o-l/lo
t
I
I
0
0.5
I'0
1.2'-"
1"5
0
I
t
0-5
I-0
d
d
Figure 8
Figure 9
1'5
Figures 8 and 9. First even transverse bonding mode. Figure 8, channel section. Figure 9, circular section thin-waiTed.
484
it. E. WILLIAMS |
I
2 I t
,o
I
0
I
-I
om.1/lff
~
~-~
/\o,R.,,2o
-I0
/
-4
l/a',l/lO
-15
0
I
I
0"5
I0
0
1"5
8 Figure 10. Second even transverse bending mode. Channel section.
I 05
I I-0
1.5
8 Figure 11. First odd transverse bending mode. Channel section.
TABLE 2
Natural frequencies of a channel cross section Mode
First even
First odd Second even
air
z/a
]r x 103
~2 X 103
~2/Jr
kit
1/I0 1/20 1/10 1/20 1110 1/20 1110 1/20 1[10 1/20
1]5 1/5 1/10 I]10 1/5 1/5 1/I0 1110 1/10 1/10
1"944 0.4861 1.944 0.4861 1.944 0.4861 1.944 0-4861 1-944 0.4861
2"635 0.540 1-812 0.264 14.81 4"071 10.97 2-224 34.20 8"719
1"355 1.I I 0.9321 0.543 7.619 8.375 5.643 4"575 17.59 17.94
3"55 (4"73) 3-36 3-23 2"82 5"46 (7"85) 5-59 5"06 4"80 6"72 (I 1"0) 6-75
Natttral frequencies of a circtdar cross section Mode
i x 103
~2× 103
~2/i
kit
First even
2.500 0.6250
8.127 2.040
3.251 3.264
4.40 (4.73) 4.43
t Numbers in parenthesis are for straight beams.
It is interesting to note (from Figures 4 and 5) that the transverse deflection of the lowest bending mode is generally larger for the circular cross section and decreases for the channel cross sections as the section gets thinner (r/a decreasing). Alternatively (from Figures 8 and 9), the curves for the corresponding rotation show little similarity, with those for the channel cross section being practically double the amplitude of those of the circular cross section and having a definite positive region for 0 > 0.6 (say). As can be seen from Table 2, the reduced frequency for the channel section depends
485
EQUATIONS OF MOTION OF THIN RINGS TABLE 3
Characteristic roots for a channel cross section--equal legs Mode
First even
First odd Second even
aiR
z/a
ct
]~
~[C 2
~]C 3
1/I0 1/20 1/10 1/20 1/10 1/20 1/10 1/20 1/10 1/20
1/5 1/5 1/I0 1/I0 1/5 1/5 1/10 1/10 1/10 1/10
1.562 1.508 1-788 1-714 2.113 2"199 2.341 2-515 2"720 3.129
0.9096 0.9452 1.209 1.087 0.9894 1-131 1.392 1.402 1-349 1"582
2-339 2.199 2.533 2.363 3.218 3-111 3.441 3.364 4-264 4"326
4.758 9.470 2.558 4.741 4.873 9"495 2.887 4"795 3.248 4"953
significantly on the thickness parameter T/a as well as on the curvature parameter aiR. This behavior was also observed in the results for the complete ring, and illustrates the role of the torsional rigidity (J)--the only area property depending on x--in affecting the motion. It also follows from the two examples presented that the reduced frequency of the first even mode for the circular section is very nearly proportional to the reduced area moment (i). As a final note, there is presented in the last column of Table 2 the values of the parameter kl, where kl is the reduced frequency for a straight beam of length I, clamped at both ends, and given by co = ( k l ) 2 x / t ~ p A l 4 . The length I was chosen as the developed length of the ring: i.e., 1 = hR. It is apparent that the reduced frequency of a circular cross section correlates very closely with that for a straight beam (given in parenthesis in the last column), but the reduced frequencies for the channel cross section are quite different. Further, whereas the reduced frequencies for the first three modes for the straight beams are quite far apart, those for the channel cross section are relatively close together. 5. DISCUSSION The equations governing the motion of thin, circular rings for a general cross section have been derived by using Hamilton's Variational Principle. These equations have also been shown to be consistent with the Principles of Linear and Angular Momentum by using the definitions of the stress resultants and the components of momentum together with assumed forms for the displacement field. Of particular importance is the natural occurence of the "twist" equation of motion (25) which relates the twisting moment to the warping moment and the St. Venant Torsion Moment. This relation is in marked contrast to the customary procedure of constructing the twisting moment as the superposition of the two contributions in a purely statical sense. It is significant that the dynamic contribution can be important in the higher frequency (torsion) modes. It has been shown from the analysis of complete rings that the relative importance of torsional stiffness and warping restraint is measured by the parameter z2R2/a4 for the thin-walled sections studied here. For solid sections, one would expect that the relative importance would be measured by the parameter R2/a 2, where a represents a characteristic transverse dimension of the section. Thus, as a i r ,~ I for the present theory to be applicable,
486
rl. E. WILLIAMS
it is apparent that warping restraint is important only for thin-walled sections for which z/a <~ 1. Further, as the ratio of the warping moment to the St. Venant Torsion Moment increases as n 2, it is increasingly important to consider warping restraint in higher order modes. The relative importance of dynamic terms besides the leading terms--the "rotary inertia" and "twist momentum" terms--has been shown to be measured by the parameter n2a2/R 2. Upon assuming n2a2/R 2 = O(a[R) as defining the order of error, and replacing the wave number n by 2foR[l, where I is the wavelength of the deformation, it follows that the higher order dynamic terms are appreciable for l/a = ( a / R ) - ½ 0 ( 1 ) .
Thus, these terms may be appreciable for aiR = 1/20 (say) for wave numbers in excess of five. Note that, as l/a >> 1, the present theory should still be expected to apply. As a final note, the appearance of the material constant E in place of the customary E in equations (30) and (34) suggests that a comparison be made between this and the classical derivation wherein E is the parameter. The comparions can actually be made at the stage where the strain energy density was formulated. In this paper, no assumptions were made as to the nature of the stress distribution--only the form of the displacement field was assumed. Hence, the expression for the strain energy density Vo used here is without prior restriction. In comparsion, it is customarily assumed, as in beam theory, that the transverse normal stress components vanish identically. This has the effect of dropping the strain variables e,, and ez~ from the analysis, as err = ezz = -- Ve,oo ,
and leads to the following expression for the strain energy density (Vo): E Vo = 5-
G + T
2 +
This expression could have been used to evaluate the total strain energy (3) instead of the general expression that was actually used. However, as the form of the generalized displacements used in this paper (2) leads to a strain field with e00, ~'0,, )'0z as the only non-vanishing components, it follows that the general expression for Vo reduces to that given above if E is replaced by E. Thus, the results obtained in this paper can alternatively be interpreted as following from a restricted form of the strain energy density if/~ is replaced by E. In this way the classical results are recovered, but at the expense of making inconsistent stress and strain assumptions. REFERENCES 1. A. E. H. Love 1952,4 Treatise on the AIathematical Theory o f Elasticity. Cambridge University Press Fourth Edition. 2. J. P. DEN HARTOO1928 Philosophical Magazine 5, 400--408. The lowest natural frequency of circular arcs. 3. J. P. DEN HARTOG 1947 Mechanical Vibration. New York: McGraw-Hill Book Co. Third edition. 4. E. VoL'rERRAand J. D. MORELL1960 Transactions o f the American Society o f ltlechanical Engineers, Journal o f Applied Mechanics 27, 744-746. A note on the lowest natural frequency of elastic arcs.
EQUATIONS OF MOTION OF THIN RINGS
487
5. E. VOLTERRAand J. D. MORELL 1961 Journal of the Acoustical Society of America 33, 17871790. Lowest natural frequencies of elastic hinged arcs. 6. E. VOLTERRAand J. D. MORELL 1961 Transactions of the American Society o f AIechanical Engineers, Journal of Applied Alechanics 28, 624-627. Lowest natural frequency of elastic arc for vibrations outside the plane of initial curvature. 7. I. U. OJALVO 1962 International Jourttal of Mechanical Sciences 4, 53-72. Coupled twistbending vibrations of incomplete elastic rings. 8. A. S. HAMMONDand R. R. ARCHER 1963 in Developments in l~Iechanics, Vohtme 2, Part 2: Solid Mechanics, 489-524. On the free vibration of complete and incomplete rings. 9. S. S. RAO 1971 Journal of Sound and Vibration 16, 551-566. Effects of transverse shear and rotary inertia on the coupled twist-bending vibration of circular rings. 10. C. G. CULVER 1967 Proceedings of the American Society of Civil Engineers, Journal of the Structural Division 93, 189-203. Natural frequencies of horizontaly curved beams. 11. V. S. VLASOV 1961 Thht-walled elastic beams. Israel Program for Scientific Translation, for National Science Foundation, U.S. Department of Commerce. 12. H. E. WILLIAMSTo appear in Journal of Engineering Alechanics Division, Proceedings of the American Society of Civil Engineers. On the linear theory of thin rings. 13. J. P. DEN HARTOa 1962 in Handbook of Enghwerhlg Mechanics (Editor: W. Flugge). New York: McGraw-Hill Book Co. 14. T. E. LANa 1962 (July) Jet Propulsion Laboratory Technical Report No. 32-261. Vibration of thin circular rings--Part I. Solutions for modal characteristics and forced excitation. 15. T. E. LANG 1963 (March) Jet Propttlsion Laboratory Technical Report No. 32-261. Vibration of thin circular rings--Part II. Modal functions and eigenvalues of constrained semi-circular rings.
APPENDIX I THE AREA PROPERTIES
The cross-sectional area properties appearing in the expressions for the strain energy (3) and kinetic energy (5) are defined below: (I., I,,, I,) = f d A ,) A J
(~, ~¢, ¢2) 1 + ~/R
O~
r
'
r
$
0~
1 ['dA R
(~' ~) = ~ - ~ J a
9- ~ x
(~, O,
F' = ~R4 f dAR¢2,
1 3 dA ¢ (~;, ~;) = -x~f,
× (¢, O,
(I..,,', I , j , I.~') = fadA(~2~, ~3, ~2~),
1
(~g, ~ ) = -A-~ f adA¢(¢L ¢¢).
tI. E. WILLIAMS
488
A P P E N D I X II NOMENCLATURE transverse dimension o f cross section (see Figures 2 and 3) cross-sectional area E 2+2G F.. components o f force stress resultants G shear modulus o f material a
A
F,,Fo,
G C/E
t,,Ir:,I= area moments o f inertia of cross section it, etc. lr/AR 2, etc. J
Y
lvto, l~ 1, m~, too, m, Mo,, 3Ioo, Mo=
M, II
Pr, Po,Pz R T
7:t~ ll~ U~ |9
U,V,W
P r
P CO
torsional stiffness J/AR 2
components of angular momentum per radian warping momentum per radian components of linear momentum per radian components o f moment stress resultant bimoment wave number components of distributed force per length o f centroidal curve radius o f centroidal curve thickness dimension of cross section (see Figures 2 and 3) distributed twisting moment per length o f centroidal curve components o f displacement o f a general point components o f displacement o f the centroid circumferential component o f rotation o f cross section warping constant axial (centroidal) coordinate o f cross section moment o f torsion function circumferential coordinate rate of twist (Rt¢o = fl' + W'/R) radial (centroidai) coordinate o f cross section moment o f torsion function density o f material torsion function frequency of free vibration
coR~/p/E
()' a()/~O (9 O()/&