- Email: [email protected]
Coupling effects on a cantilever subjected to a follower force
Journal
ofSound and
COUPLING
Vibration (1979) 62(l), 13l-l 39
EFFECTS
A. N.
ON A CANTILEVER FOLLOWER FORCE
National (Received
J.T.
KOUNADISAND Technical
University,
SUBJECTED
TO A
KATSIKADELIS Athens,
8 May 1978, and in revised.form
Greece 25 July 1978)
In this investigation a solution methodology is presented for studying the stability of a uniform cantilever having a translational and rotational spring at its support, carrying two concentrated masses, one at the support and the other at its tip, and subjected to a follower compressive force at its free end. The analysis is based on Timoshenko’s beam theory by considering the cantilever as a continuous elastic system. The coupling effects on the flutter load are fully assessed for a variety of parameters such as translational and rotational springs at the support, translational and rotational inertia of the concentrated masses, and cross-sectional shape, as well as transverse shear deformation and rotatory Inertia of the mass of the column.
1. INTRODUCTION
In the recent past, some technological demands have brought into focus the importance of stability of elastic systems subjected to non-conservative forces, i.e., non-potential forces: tangential or follower forces are examples of non-dissipative forces which are non-conservative; such forces have also been termed circulatory by Ziegler [l]. Lanczos [2] preferred the term polygenic rather than non-conservative, and monogenic rather than conservative. Pfliiger [3] investigated the stability of a fixed cantilever subjected to a follower compressive force at its tip. From this analysis one can derive the paradoxical result that no critical load exists. contrary to intuitive expectation. Later on, Ziegler [4], using a two-degree of freedom model, resolved this paradox; he showed that such a problem can be treated meaningfully only by employing the kinetic stability criterion. Using this criterion Beck [5] actually solved the foregoing problem of the fixed cantilever under a follower compressive force at its tip (Beck’s column). The failure of the equilibrium (Euler) and energy methods for establishing the critical load in this case, was attributed to the fact that the follower force is not derivable from a potential. However, the failure of the foregoing (static) methods is not a necessary consequence of the non-conservativeness of the loading. This was shown also by Pfltiger on the basis of the static method and verified by Leipholz [6] by virtue of the kinetic criterion. This led to the conclusion that the failure of the static methods is due not only to the nature of the loading but also to the existing constraints (boundary conditions). Leipholz [7] has also shown which non-conservative systems can be analyzed by virtue of static methods. The type of instability considered herein is flutter. The non-conservative loading, being a function of displacements, does not depend on time explicitly (autonomous system) and it is non-dissipative; when such a non-conservative loading is present the governing equation of motion is non-self-adjoint. Therefore, the eigenfrequencies of the flexural vibrations may be complex. The frequency equation is derived by using linearized equations of motion and the flexural eigenfrequencies are established as functions of the non-conservative loading. 131 0022-460X/79/01
013 1 +09 SOZ.OO/O
@ 1979 Academic
Press Inc. (London)
Limited
132
A. N. KOIJNADIS
AND J. T. KATSIKADELIS
The purpose of the present study is to investigate the stability of a uniform cantilever (Beck’s column) having a translational and a rotational spring at its support, carrying concentrated masses at its tip and support, and subjected to a follower compressive force at the free end. The effect of the moment of inertia of the concentrated masses is also included. The analysis is based on Timoshenko’s beam theory by considering the cantilever as a continuous elastic system. The shear coefficient is established by using Cowper’s [IS] formulae, which give satisfactory results for lower modes of vibration, even for short wave lengths. In the present investigation stubby columns (for which the transverse shear effect is significant) with a low critical slenderness ratio are considered. The individual effect of several of the foregoing parameters upon the flutter load such as elasticity of the support, position of a concentrated mass and transverse shear and rotatory inertia have been examined by Simkins and Anderson [9], Kapoor and Leipholz [lo], Nemat-Nasser [11] and Walter and Levinson [12]. Pertinent to the present work are those by Kounadis and Katsikadelis [ 131, Kounadis [14] and Celep [lS]. In this investigation the coupling effects upon the flutter load of two concentrated masses (one placed at the support and the other at the tip) of a uniform elastically supported Timoshenko cantilever under a follower force at its free end are examined.
2.
EQUATION
OF
MOTION:
FREQUENCY
EQUATION
Consider the elastically restrained uniform cantilever with length I, flexural rigidity area A, mass per unit length m = pA, and translational and rotational spring constants C, and C,, respectively, shown in Figure 1. The beam carries at its support and free end the concentrated masses M, and M, with rotational inertias J, and J,, respectively, and it is subjected to a follower compressive force S at its end. El, cross-sectional
Figure 1. Elastically sive force S.
restrained
cantilever
with concentrated
masses M, and M, subjected
to a follower compres-
According to the kinetic criterion, sufficiently small lateral displacements from the equilibrium position of the cantilever are imposed and the disturbed motion, which consists of small oscillations about the original position, is discussed. According to Timoshenko’s beam theory, the points of the cantilever undergo total lateral deflections y(x, t) and angles of rotation of the cross-sections J/(x, t). In this case for the bending moment M(x, t) and
CANTILEVER
133
SUBJECTED TO A FOLLOWER FORCE
shearing force Q(x, t) normal to the undeformed expressions are valid: M = -EZ$‘,
axis of the cantilever, the following
Q = K’AG(ey’
-
I&
(111
where e = 1 - S/(K’AG). K’is the Timoshenko shear coefficient and G is the shear modulus,, Use of the principle of virtual work due to the aforementioned displacements yields + J; pZlJiS$ dx + 6U + Sy’(Z,t) 6y(Z,t) - S ’y’dj’ dx s0
jj6ydx
+
M&l,
M,jV, t) dy(O,t) + J,$(O, t) 6+(0,t) +
+ J&L
t) 6y(l, t)
t) WV, t) + C,y(O, t) 6Y(O, t) + C,$(O, t) &qo, t) = 0,
(2)
where the functional of the strain energy U is given by U = f
’ [EZ+” + K’AG(y’ -
t+b)“]dx.
(3)
s0
In the foregoing relations the prime denotes partial differentiation with respect to the axial co-ordinate x, while the dot denotes partial differentiation with respect to the time t. By introducing the variation of equation (3) into equation (1) and integrating by parts, after some rearrangements and use of equation (1) the following variational equations and boundary conditions are obtained [14] : EZf”
-
pZt./’+ mj + Sy” = 0,
K’AG(y”
-
$‘) -
pAj
-
Sy” = 0;
(4)
atx = 0: -Q(O, t) + M&O, t) + C,y(O, t) = 0, atx
EZ+‘(O, t) -
.Z,$(O, t) -
C&O,
t) = 0;
= 1: EZl+qZ,t) + .Z,$b(l, t) = 0,
Q(l, t) + Sy”(l, t) + M,j;(l, t) = 0.
(5)
By virtue of the second of equations (4) the first of these equations becomes e_Y’ - p{(l/K’G) + e/E}jj,’ + (m/EZ)ji + (S/EZ)y” + (p2/EK’G) Y = 0.
(6)
It is worth noticing that the shearing force due to S is taken normal to the deformed axis
of the cantilever. A modified expression [16] can be obtained if the shearing force is calculated from the angle of rotation of the cross-section, which is due only to bending. If this is so, Sy’ should be replaced by S+ and the second of equations (4) becomes K’AG(y” - I,/?‘-) pAj - S$’ = 0.
(7)
By using this expression, the first of equations (4) can be transformed into ,111,
4
- p{(l/K’G) + l/E)j” + (m/EZ*)j; + (S/EZ*)y” + (p2/EK’G) y = 0,
(8)
where Z* = Z[l + S/(K’AG)]. Notice that in this case Z in equations (5) should be replaced by I*. Equation (8) is identical to equation (1) of reference [17]. In the present analysis equation (8) will be used for the sake of obtaining comparable results with those in other investigations. Any slight difference will be investigated in a future paper. For free vibrations with circular frequency w equation (8) has a solution of the form y(x, t) = u(x) e’“‘.
(9)
134
A. N. KOUNADIS
AND J. T. KATSIKADELIS
Substitution of the above equation into equation (8) and introduction independent variable 5 = x/l yield
of the dimensionless
d4u(t)/dt4 + /I” d%(<)/d<’ - yv(t) = 0
(10)
where B’ = pw212/K’G + po212/E + Sl’JEI*,
y = (mo212/EI*)(1 - /I~w~I*/~K’G).
(1 la, b) Similarly the boundary conditions (5) become, at l = 0, El*@(O) - C&(O) + c&,$(O)
Q(0) - C&O) + 02M,u(0) = 0,
= 0,
at i” = 1 EI*$‘(l) - 02.J2t&l) = 0,
Q(1) + (S/l)u’(l) - o2 M,u(l) = 0.
(12)
The shear force, Q(c), the bending moment, M(t), and the rotation of the cross-section, ll/(<),due only to bending, are given by [ 131 Q(t) = - [(EZ*/13)/(1 - p2c021*/mK’G)][u”‘(<) M(t) = -(EI*/12)[u”(5) VW) = C(W(1
+
+ fi”u’(<)],
+ (po212/K’G) u(t)],
,NmK’G)l[~‘(~) - bQ(NmK’Gl.
(13)
The general solution of equation (10) is u(t) = C, cash 1 + C, sinh Lg + C, cos x{ + C, sin It,
(14)
where 2 = J-
(P2/2) + JIIVVX
1 = J + (P2/2) + J@V4%.
(15)
Notice that I and x satisfy the following relations: p’ = x2 - AZ,
y = x2/12.
(16)
Application of the boundary conditions (12) leads to a system of four homogeneous linear algebraic equations in C,, C,, C, and C,. For a non-trivial solution the determinant (B,I of the coefficients of the unknowns must vanish. This condition leads to the frequency equation, which is a transcendental equation of the form -- -- - -_ (‘ijJ = ‘(“, E,K’, s, S, C,, C,, ‘~,, J,, M,, J,) = 0, (17) where all the arguments are dimensionless variables, defined as follows: Sz = pAo214/E1 is the dimensionless
frequency;
F = E/G = 2(1 + v), v being
Poisson’s
ratio;
s = 21/r is the
slenderness of the column, r being the radius of gyration; S = S/S, is the dimensionless axial force, S, = 7c2EZ/412being the Euler buckling load; C, = CJEI is the dimensionless rotational spring constant; C, = C,13/EI is the dimensionless translational spring constant; and Mi = Mjml, Ji = J/ml3 (i = 1,2) are the dimensionless translational and rotational inertias, respectively. Hence, the eigenfrequencies of the free lateral vibrations are established as functions of the axial compressive force. For a sufficient small magnitude of the follower force, the other parameters being kept constant, all frequencies obtained from equation (17) are real. As S increases, then for a certain finite value S = S, at least two consecutive eigenfrequencies become equal. Beyond this value, the frequency equation yields a pair of complex conjugate
CANTILEVER
SUBJECTED
TO A FOLLOWER
13s
FORCE
eigenfrequencies and the resulting motion is an oscillation with exponentially increasing amplitude. The value S = S, is the critical load and it is referred to as the flutter load 01 the column. The frequency equation is obtained in the form of a determinant lBijl = 0, the elements of which are obtained by application of the boundary conditions (12). By virtue of equations (13) and (14) the elements B, (i, j = 1,2,3,4) are given as follows: B,,
B,, B,, B,,
B13 = 1 - B’&?,/C,,
= 1 - Q2&?i/CT,
B,,
= 0;l~2/~,, B,,
= - [L’ + 4Q2s/(K’s2)]/C,, = - [ - A2 + 4Q2s/(K’s2)]/C,,
B,,
= -tU2;2/c,,
= (1 - &Q2/c,)[1
+ 4~81~/(K’s~)]A,
= (1 - &Q2/c,)[1
- 4s&J2/(K’s2)]&
B,, = [P + 4Q2&/(K’?)] cash 1 - Q2.Z21[l + ~EB~~/(K’s~)]sinh il, B,,
= [i’ +
B,,
= [ - A2 + 4Q2s/(K’s2)] cos 1 + Q2.Z2A[1- 4a8A2/(K’s2)] sin A,
B,,
= [-A”
4Q2E/(K’S2)]
sinh i - Q2J2L[1 + 4sQ;Z2/(K’s2)]cash i,
+ 4Q2s/(K’s2)] sin 2 - Q’.Z,;i[l - 4s8A2/(K’s2)] cos 2,
B,,
= A(- 0A2 + 7c2s/4) sinh j. - Q2R2 cash II,
B,,
= A(- 8J2 + 7~~514) cash 2 - Q2M2 sinh 2,
B,,
= -i1(di2
B,,
+ 7r25/4) sin i - Q2R2 cos ;i,
= x(8i2 + 7-c2s/4)cos 1 - Q2R2 sin 1,
(18)
where 8 = l/[r - 16Q%/(K’s4)],
r = [l +
n2S~/(K’s2)].
Notice that for C, = cc, M, = .I, = 0; equation (17) then becomes identical to equation (17) of reference [13].
3. NUMERICAL RESULTS AND DISCUSSION In all cases considered in this investigation the flutter load was obtained by numerical evaluation of the frequency equation (17) by step-increasing the magnitude of the follower force S on a CDC/CYBER 72 digital computer. It should be noticed that in cases where the effect of transverse shear deformation is taken into account the chosen cross-section is a thin Z-section (t,/d = l/10, tw/d = l/20, b = h = d, v = 0.30 [8]) with K’ = 0.186, for which this effect is more pronounced. From Figure 2 one can see the dependence of the flutter load S, upon the magnitude of a concentrated mass %i, at the tip of a partially fixed cantilever subjected to a compressive follower force s with (curve I) and without (curve II) the effect of shear deformation and rotatory inertia of the mass of the column. It is clear that in both cases the translational inertia of the concentrated mass has an appreciable destabilizing effect. Note that for &i, + cc the critical load of a beam with one end partially fixed and the other hinged is obtained. This is a divergence buckling load (w’ + 0) because the horizontal non-conservative component of the load has no effect on the response of the beam. The critical load. equal to 11.598, may be also obtained from the buckling equation k (cot k - k/C,) = 1,
k2 = S12/EZ.
(19)
136
A. N. KOUNADIS
AND J. T. KATSIKADELIS
Asymptote
___-___-___------_---~---~_---__
Asymptote K’=0486+
8007.00
1 00
1 0.2
1
l 0.4
1
1 0.6
0
rotary l 0.6
1
l I.0
inertia,
s=2
L/r=60
fi
’ 50.0
I IOQO
./Jl es.0
I 150~
I 200.0
.$,=I b598.$=10~018
I 2500
1 500.0
K
Figure 1, c,
2. Flutter
= 03, R,
load s, versus concentrated
arbitrary,
mass m2 at the tip of an elastically
supported
cantilever.
e,
=
J, = J, = 0.
Figure 3 shows, in the dashed line, the dependence of the flutter load S, on the moment of inertia 1, of a concentrated mass R, at the support of a partially fixed (C, = 1) cantilever with (curve I) and without (curve II) the effect of shear deformation and rotatory inertia of the mass of the column reported in references [13] and [14}. If in addition to these parameters a concentrated mass H2 = 0.5 with a moment of inertia J, = 1 is placed at the tip of the cantilever the previousIy mentioned curves are transformed into curves III and IV, drawn with solid lines; the resulting considerable destabilizing effect is evident. Moreover, two significant observations are worth mentioning. (a) In both curves III and IV the flutter load is obtained by the coalescing of the first and second eigenfrequencies when J, varies from zero up to the corresponding peak values (points A’, D’), while beyond these peak values the flutter load is obtained by the coalescing of the second and third eigenfrequencies. The same phenomenon was observed in curves I and II. In contradistinction to the case of absence of a tip mass M,, where the effect of shear deformation is appreciable, the presence of this mass almost cancels this effect. (b) In all the curves, 1, II, III and IV, before reaching the peak points A, D, A’and D’there is a very small interval of values of the varying parameter .I1 in which the following phenomenon occurs: the first and second eigenfrequencies, by the coalescing of which the first
6.00
Figure 3. Flutter load S, verslis moment of inertia 3, of a concentrated mass R, at the support of a partially fixed cantilever carrying also a concentrated mass H, at the tip. Curves I, II: c, = 1. c, = 00, R, = 0, J, = 0. Curwes III, IV: C, = 1, C, = 00, !iZz = 0,5,J, = 1.
CANTILEVER
SUBJECTED
TO A FOLLOWER
--------__________________________
Figure 4. Flutter with a concentrated
137
FORCE
hsysyrrptote
=
3+01.....
load S, versus translational spring constant cr of an elastically supported mass R, at its tip. si, = 1, R, = J, = 0, R, = J, = O-5. K’ = nfi.
cantilever
(c,
= 1)
flutter load is obtained, appear again as the follower force increases beyond this critical value. The new presence of all the eigenfrequencies indicates the existence of a post-critical stability region. This stability region ends up at that value of the flutter load which is obtained by the coalescing of the second and third eigenfrequencies. This small interval of values of the varying parameter j1 may be characterized as a transition zone between the range of values of 1, for which the flutter load is obtained by the coalescing of the first and second eigenfrequencies and the range of values of this parameter for which the flutter load is obtained by the coalescing of the second and third eigenfrequencies. This phenomenon has also been observed during other investigations [13, 141 but it has not been previously reported. Figure 4 shows the dependence of the flutter load S, upon the translational spring constant C_r of an elastically supported cantilever (C, = 1) carrying a concentrated mass R, = J_2 = O-5at its tip. The considerable stabilizing effect is evident as the varying parameter C, increases. On the contrary, from Figure 5 one can see that the analogous effect due to the rotational spring constant C, is very small.
Figure 5. Flutter with a concentrated
load 3, versus rotational spring constant c, of an elastically supported mass Mi, at its tip. C, = 1. RI = .?, = 0, R, = f, = 0.5, K’ = co.
cantilever
(i5, = 1)
Figure 6 shows the plot of the flutter load S, versus the rotatory inertia .I, of a concentrated mass R, = 0.5 at the tip of an elastically restrained cantilever for the two cases C, = cx), C, = 1 (curve I) and C, = 1, C, = co (curve II). Curve I shows an abrupt destabilizing effect for very small values of .I,, while for the other values of this parameter the flutter load is always less than 4.10. Curve II shows a destabilizing effect approximately up to the value J, = 2. while beyond this value a considerable stabilizing effect appears.
138
A. N. KOUNADJS AND J. T. KATSIKADELIS
, ,
I
,
,
,
( ( ,
I, I
I,
-------------__---_-___---__----___-~__~.~
1
,I?
,I,
1,,
,
,
,
,
Asymptote .? =9*87
Figure 6. Flutter loads s, versus rotator j, of a concentrated mass M, at the tip of an elastically supported cantilever. R, = .7, = 0, M;I, = 0.5. K’ = “3.
4. CONCLUSIONS
Among the most important conclusions of this investigation one may list the following. (a) The effect of the translational spring as well as the effect of the moment of inertia of a concentrated mass placed either at the support or at the tip may be considerable. (b) The shear deformation for thin cross-sections with small values of K’ may have a considerable destabilizing effect. (c) In all cases where the flutter load for some range of values of the varying parameter is obtained by the coalescing of the second and third eigenfrequencies, while for another range of values of this parameter it is obtained by the coalescing of the first and second eigenfrequencies, there exists a stability region. This region corresponding to a small interval of values of the varying parameter is the transition zone which connects two ranges of values of the foregoing parameter: namely the first range, preceding the transition zone, where the flutter load is obtained by the coalescing of the first and second eigenfrequencies, and the second range, where the flutter load is obtained by the coalescing of the second and third eigenfrequencies.
REFERENCES 1. H. ZIEGLER 1953 Zeitschrift ftir angewandte Mathematik und Physik 4, 89-121. Linear elastic stability. 2. C. LANCZOS 1970 The Variational Principle of Mechanics. University of Toronto Press, fourth edition. 3. A. PILLAGER1950 Stabilitiitsproblem der Elastostatik. Berlin: Springer Verlag. See p. 217. der Elastomechanik. 4. H. ZIEGLER 1952 Ingenieur-Archiu 20, 49-56. Die Stabilitatskriterien 5. M. BECK 1952 Zeitschrift ftir anyewandte Mathematik und Physik 3, 225-228. Die Knicklast des einseitig eingespannten, tangential gedrtickten Stabes. 6. H. LEIPHOLZ 1962 Zeitschrif ftir angewandte Mathemutik und Physik 13, 359-372. Anwendung des Galerkinschen Verfahrens an nichtkonservative Stabilitltsprobleme des elastichen Stabes. 7. H. LEIPHOLZ 1970 Stability Theory. New York and London : Academic Press. See pp. 204-2 12. 8. G. R. COWPER 1966 Journal ofApplied Mathematics 33,335-340. The shear coefficient in Timoshenko beam theory. 9. T. E. SIMKINS and G. L. ANDERSON 1975 Journal of Sound and Vibration 39, 359-369. Stability of Beck’s column considering support characteristics.
CANTILEVER SUBJECTED TO A FOLLOWER FORCE
139
10. R. N. KAPOOR and H. H. LEIPHOLZ 1974 Zngenieur-Archiu. 43, 233-239. Stability analysis of damped polygenic systems with relocatable mass along its length. Il. S. NEMAT-NASSER 1967 .Iournal of AppIied Mechanics 34,484485. Instability of a cantilever under a follower force according to Timoshenko’s beam theory. 12. W. W. WALTERAND M. LEVINSON1968 Canadian Aeronautical and Space Institute 1, 91-93. Destabilization of a nonconservatively loaded elastic system due to rotatory inertia. 13. A. N. KOUNADISand J. T. KATSIKADELIS1976 Journal of Sound and Vibration 49, 171L178. Shear and rotatory inertia effect on Beck’s column. 14. A. N. KOUNADIS1977 Journal of Applied Mechanics 44,731-736. Stability of elastically restrained Timoshenko cantilevers with attached masses subjected to a follower force. 15. Z. CELEP1977 Zeitschrijtftir angewandte Mathematik und Mechanik 57,555-557. On the vibration and stability of Beck’s column subjected to vertical and follower forces. 16. S. P. TIMOSHENKOand J. M. GERE 1961 Theory of Elastic Stability New York: McGraw-Hill Book Company, Inc. See pp. 132-135. 17. A. N. KOUNADIS1975 Zngenieur-Archiu SS,43-5 1. Beam columns of varying cross-sections under lateral harmonic loads.
ofSound and
COUPLING
Vibration (1979) 62(l), 13l-l 39
EFFECTS
A. N.
ON A CANTILEVER FOLLOWER FORCE
National (Received
J.T.
KOUNADISAND Technical
University,
SUBJECTED
TO A
KATSIKADELIS Athens,
8 May 1978, and in revised.form
Greece 25 July 1978)
In this investigation a solution methodology is presented for studying the stability of a uniform cantilever having a translational and rotational spring at its support, carrying two concentrated masses, one at the support and the other at its tip, and subjected to a follower compressive force at its free end. The analysis is based on Timoshenko’s beam theory by considering the cantilever as a continuous elastic system. The coupling effects on the flutter load are fully assessed for a variety of parameters such as translational and rotational springs at the support, translational and rotational inertia of the concentrated masses, and cross-sectional shape, as well as transverse shear deformation and rotatory Inertia of the mass of the column.
1. INTRODUCTION
In the recent past, some technological demands have brought into focus the importance of stability of elastic systems subjected to non-conservative forces, i.e., non-potential forces: tangential or follower forces are examples of non-dissipative forces which are non-conservative; such forces have also been termed circulatory by Ziegler [l]. Lanczos [2] preferred the term polygenic rather than non-conservative, and monogenic rather than conservative. Pfliiger [3] investigated the stability of a fixed cantilever subjected to a follower compressive force at its tip. From this analysis one can derive the paradoxical result that no critical load exists. contrary to intuitive expectation. Later on, Ziegler [4], using a two-degree of freedom model, resolved this paradox; he showed that such a problem can be treated meaningfully only by employing the kinetic stability criterion. Using this criterion Beck [5] actually solved the foregoing problem of the fixed cantilever under a follower compressive force at its tip (Beck’s column). The failure of the equilibrium (Euler) and energy methods for establishing the critical load in this case, was attributed to the fact that the follower force is not derivable from a potential. However, the failure of the foregoing (static) methods is not a necessary consequence of the non-conservativeness of the loading. This was shown also by Pfltiger on the basis of the static method and verified by Leipholz [6] by virtue of the kinetic criterion. This led to the conclusion that the failure of the static methods is due not only to the nature of the loading but also to the existing constraints (boundary conditions). Leipholz [7] has also shown which non-conservative systems can be analyzed by virtue of static methods. The type of instability considered herein is flutter. The non-conservative loading, being a function of displacements, does not depend on time explicitly (autonomous system) and it is non-dissipative; when such a non-conservative loading is present the governing equation of motion is non-self-adjoint. Therefore, the eigenfrequencies of the flexural vibrations may be complex. The frequency equation is derived by using linearized equations of motion and the flexural eigenfrequencies are established as functions of the non-conservative loading. 131 0022-460X/79/01
013 1 +09 SOZ.OO/O
@ 1979 Academic
Press Inc. (London)
Limited
132
A. N. KOIJNADIS
AND J. T. KATSIKADELIS
The purpose of the present study is to investigate the stability of a uniform cantilever (Beck’s column) having a translational and a rotational spring at its support, carrying concentrated masses at its tip and support, and subjected to a follower compressive force at the free end. The effect of the moment of inertia of the concentrated masses is also included. The analysis is based on Timoshenko’s beam theory by considering the cantilever as a continuous elastic system. The shear coefficient is established by using Cowper’s [IS] formulae, which give satisfactory results for lower modes of vibration, even for short wave lengths. In the present investigation stubby columns (for which the transverse shear effect is significant) with a low critical slenderness ratio are considered. The individual effect of several of the foregoing parameters upon the flutter load such as elasticity of the support, position of a concentrated mass and transverse shear and rotatory inertia have been examined by Simkins and Anderson [9], Kapoor and Leipholz [lo], Nemat-Nasser [11] and Walter and Levinson [12]. Pertinent to the present work are those by Kounadis and Katsikadelis [ 131, Kounadis [14] and Celep [lS]. In this investigation the coupling effects upon the flutter load of two concentrated masses (one placed at the support and the other at the tip) of a uniform elastically supported Timoshenko cantilever under a follower force at its free end are examined.
2.
EQUATION
OF
MOTION:
FREQUENCY
EQUATION
Consider the elastically restrained uniform cantilever with length I, flexural rigidity area A, mass per unit length m = pA, and translational and rotational spring constants C, and C,, respectively, shown in Figure 1. The beam carries at its support and free end the concentrated masses M, and M, with rotational inertias J, and J,, respectively, and it is subjected to a follower compressive force S at its end. El, cross-sectional
Figure 1. Elastically sive force S.
restrained
cantilever
with concentrated
masses M, and M, subjected
to a follower compres-
According to the kinetic criterion, sufficiently small lateral displacements from the equilibrium position of the cantilever are imposed and the disturbed motion, which consists of small oscillations about the original position, is discussed. According to Timoshenko’s beam theory, the points of the cantilever undergo total lateral deflections y(x, t) and angles of rotation of the cross-sections J/(x, t). In this case for the bending moment M(x, t) and
CANTILEVER
133
SUBJECTED TO A FOLLOWER FORCE
shearing force Q(x, t) normal to the undeformed expressions are valid: M = -EZ$‘,
axis of the cantilever, the following
Q = K’AG(ey’
-
I&
(111
where e = 1 - S/(K’AG). K’is the Timoshenko shear coefficient and G is the shear modulus,, Use of the principle of virtual work due to the aforementioned displacements yields + J; pZlJiS$ dx + 6U + Sy’(Z,t) 6y(Z,t) - S ’y’dj’ dx s0
jj6ydx
+
M&l,
M,jV, t) dy(O,t) + J,$(O, t) 6+(0,t) +
+ J&L
t) 6y(l, t)
t) WV, t) + C,y(O, t) 6Y(O, t) + C,$(O, t) &qo, t) = 0,
(2)
where the functional of the strain energy U is given by U = f
’ [EZ+” + K’AG(y’ -
t+b)“]dx.
(3)
s0
In the foregoing relations the prime denotes partial differentiation with respect to the axial co-ordinate x, while the dot denotes partial differentiation with respect to the time t. By introducing the variation of equation (3) into equation (1) and integrating by parts, after some rearrangements and use of equation (1) the following variational equations and boundary conditions are obtained [14] : EZf”
-
pZt./’+ mj + Sy” = 0,
K’AG(y”
-
$‘) -
pAj
-
Sy” = 0;
(4)
atx = 0: -Q(O, t) + M&O, t) + C,y(O, t) = 0, atx
EZ+‘(O, t) -
.Z,$(O, t) -
C&O,
t) = 0;
= 1: EZl+qZ,t) + .Z,$b(l, t) = 0,
Q(l, t) + Sy”(l, t) + M,j;(l, t) = 0.
(5)
By virtue of the second of equations (4) the first of these equations becomes e_Y’ - p{(l/K’G) + e/E}jj,’ + (m/EZ)ji + (S/EZ)y” + (p2/EK’G) Y = 0.
(6)
It is worth noticing that the shearing force due to S is taken normal to the deformed axis
of the cantilever. A modified expression [16] can be obtained if the shearing force is calculated from the angle of rotation of the cross-section, which is due only to bending. If this is so, Sy’ should be replaced by S+ and the second of equations (4) becomes K’AG(y” - I,/?‘-) pAj - S$’ = 0.
(7)
By using this expression, the first of equations (4) can be transformed into ,111,
4
- p{(l/K’G) + l/E)j” + (m/EZ*)j; + (S/EZ*)y” + (p2/EK’G) y = 0,
(8)
where Z* = Z[l + S/(K’AG)]. Notice that in this case Z in equations (5) should be replaced by I*. Equation (8) is identical to equation (1) of reference [17]. In the present analysis equation (8) will be used for the sake of obtaining comparable results with those in other investigations. Any slight difference will be investigated in a future paper. For free vibrations with circular frequency w equation (8) has a solution of the form y(x, t) = u(x) e’“‘.
(9)
134
A. N. KOUNADIS
AND J. T. KATSIKADELIS
Substitution of the above equation into equation (8) and introduction independent variable 5 = x/l yield
of the dimensionless
d4u(t)/dt4 + /I” d%(<)/d<’ - yv(t) = 0
(10)
where B’ = pw212/K’G + po212/E + Sl’JEI*,
y = (mo212/EI*)(1 - /I~w~I*/~K’G).
(1 la, b) Similarly the boundary conditions (5) become, at l = 0, El*@(O) - C&(O) + c&,$(O)
Q(0) - C&O) + 02M,u(0) = 0,
= 0,
at i” = 1 EI*$‘(l) - 02.J2t&l) = 0,
Q(1) + (S/l)u’(l) - o2 M,u(l) = 0.
(12)
The shear force, Q(c), the bending moment, M(t), and the rotation of the cross-section, ll/(<),due only to bending, are given by [ 131 Q(t) = - [(EZ*/13)/(1 - p2c021*/mK’G)][u”‘(<) M(t) = -(EI*/12)[u”(5) VW) = C(W(1
+
+ fi”u’(<)],
+ (po212/K’G) u(t)],
,NmK’G)l[~‘(~) - bQ(NmK’Gl.
(13)
The general solution of equation (10) is u(t) = C, cash 1 + C, sinh Lg + C, cos x{ + C, sin It,
(14)
where 2 = J-
(P2/2) + JIIVVX
1 = J + (P2/2) + J@V4%.
(15)
Notice that I and x satisfy the following relations: p’ = x2 - AZ,
y = x2/12.
(16)
Application of the boundary conditions (12) leads to a system of four homogeneous linear algebraic equations in C,, C,, C, and C,. For a non-trivial solution the determinant (B,I of the coefficients of the unknowns must vanish. This condition leads to the frequency equation, which is a transcendental equation of the form -- -- - -_ (‘ijJ = ‘(“, E,K’, s, S, C,, C,, ‘~,, J,, M,, J,) = 0, (17) where all the arguments are dimensionless variables, defined as follows: Sz = pAo214/E1 is the dimensionless
frequency;
F = E/G = 2(1 + v), v being
Poisson’s
ratio;
s = 21/r is the
slenderness of the column, r being the radius of gyration; S = S/S, is the dimensionless axial force, S, = 7c2EZ/412being the Euler buckling load; C, = CJEI is the dimensionless rotational spring constant; C, = C,13/EI is the dimensionless translational spring constant; and Mi = Mjml, Ji = J/ml3 (i = 1,2) are the dimensionless translational and rotational inertias, respectively. Hence, the eigenfrequencies of the free lateral vibrations are established as functions of the axial compressive force. For a sufficient small magnitude of the follower force, the other parameters being kept constant, all frequencies obtained from equation (17) are real. As S increases, then for a certain finite value S = S, at least two consecutive eigenfrequencies become equal. Beyond this value, the frequency equation yields a pair of complex conjugate
CANTILEVER
SUBJECTED
TO A FOLLOWER
13s
FORCE
eigenfrequencies and the resulting motion is an oscillation with exponentially increasing amplitude. The value S = S, is the critical load and it is referred to as the flutter load 01 the column. The frequency equation is obtained in the form of a determinant lBijl = 0, the elements of which are obtained by application of the boundary conditions (12). By virtue of equations (13) and (14) the elements B, (i, j = 1,2,3,4) are given as follows: B,,
B,, B,, B,,
B13 = 1 - B’&?,/C,,
= 1 - Q2&?i/CT,
B,,
= 0;l~2/~,, B,,
= - [L’ + 4Q2s/(K’s2)]/C,, = - [ - A2 + 4Q2s/(K’s2)]/C,,
B,,
= -tU2;2/c,,
= (1 - &Q2/c,)[1
+ 4~81~/(K’s~)]A,
= (1 - &Q2/c,)[1
- 4s&J2/(K’s2)]&
B,, = [P + 4Q2&/(K’?)] cash 1 - Q2.Z21[l + ~EB~~/(K’s~)]sinh il, B,,
= [i’ +
B,,
= [ - A2 + 4Q2s/(K’s2)] cos 1 + Q2.Z2A[1- 4a8A2/(K’s2)] sin A,
B,,
= [-A”
4Q2E/(K’S2)]
sinh i - Q2J2L[1 + 4sQ;Z2/(K’s2)]cash i,
+ 4Q2s/(K’s2)] sin 2 - Q’.Z,;i[l - 4s8A2/(K’s2)] cos 2,
B,,
= A(- 0A2 + 7c2s/4) sinh j. - Q2R2 cash II,
B,,
= A(- 8J2 + 7~~514) cash 2 - Q2M2 sinh 2,
B,,
= -i1(di2
B,,
+ 7r25/4) sin i - Q2R2 cos ;i,
= x(8i2 + 7-c2s/4)cos 1 - Q2R2 sin 1,
(18)
where 8 = l/[r - 16Q%/(K’s4)],
r = [l +
n2S~/(K’s2)].
Notice that for C, = cc, M, = .I, = 0; equation (17) then becomes identical to equation (17) of reference [13].
3. NUMERICAL RESULTS AND DISCUSSION In all cases considered in this investigation the flutter load was obtained by numerical evaluation of the frequency equation (17) by step-increasing the magnitude of the follower force S on a CDC/CYBER 72 digital computer. It should be noticed that in cases where the effect of transverse shear deformation is taken into account the chosen cross-section is a thin Z-section (t,/d = l/10, tw/d = l/20, b = h = d, v = 0.30 [8]) with K’ = 0.186, for which this effect is more pronounced. From Figure 2 one can see the dependence of the flutter load S, upon the magnitude of a concentrated mass %i, at the tip of a partially fixed cantilever subjected to a compressive follower force s with (curve I) and without (curve II) the effect of shear deformation and rotatory inertia of the mass of the column. It is clear that in both cases the translational inertia of the concentrated mass has an appreciable destabilizing effect. Note that for &i, + cc the critical load of a beam with one end partially fixed and the other hinged is obtained. This is a divergence buckling load (w’ + 0) because the horizontal non-conservative component of the load has no effect on the response of the beam. The critical load. equal to 11.598, may be also obtained from the buckling equation k (cot k - k/C,) = 1,
k2 = S12/EZ.
(19)
136
A. N. KOUNADIS
AND J. T. KATSIKADELIS
Asymptote
___-___-___------_---~---~_---__
Asymptote K’=0486+
8007.00
1 00
1 0.2
1
l 0.4
1
1 0.6
0
rotary l 0.6
1
l I.0
inertia,
s=2
L/r=60
fi
’ 50.0
I IOQO
./Jl es.0
I 150~
I 200.0
.$,=I b598.$=10~018
I 2500
1 500.0
K
Figure 1, c,
2. Flutter
= 03, R,
load s, versus concentrated
arbitrary,
mass m2 at the tip of an elastically
supported
cantilever.
e,
=
J, = J, = 0.
Figure 3 shows, in the dashed line, the dependence of the flutter load S, on the moment of inertia 1, of a concentrated mass R, at the support of a partially fixed (C, = 1) cantilever with (curve I) and without (curve II) the effect of shear deformation and rotatory inertia of the mass of the column reported in references [13] and [14}. If in addition to these parameters a concentrated mass H2 = 0.5 with a moment of inertia J, = 1 is placed at the tip of the cantilever the previousIy mentioned curves are transformed into curves III and IV, drawn with solid lines; the resulting considerable destabilizing effect is evident. Moreover, two significant observations are worth mentioning. (a) In both curves III and IV the flutter load is obtained by the coalescing of the first and second eigenfrequencies when J, varies from zero up to the corresponding peak values (points A’, D’), while beyond these peak values the flutter load is obtained by the coalescing of the second and third eigenfrequencies. The same phenomenon was observed in curves I and II. In contradistinction to the case of absence of a tip mass M,, where the effect of shear deformation is appreciable, the presence of this mass almost cancels this effect. (b) In all the curves, 1, II, III and IV, before reaching the peak points A, D, A’and D’there is a very small interval of values of the varying parameter .I1 in which the following phenomenon occurs: the first and second eigenfrequencies, by the coalescing of which the first
6.00
Figure 3. Flutter load S, verslis moment of inertia 3, of a concentrated mass R, at the support of a partially fixed cantilever carrying also a concentrated mass H, at the tip. Curves I, II: c, = 1. c, = 00, R, = 0, J, = 0. Curwes III, IV: C, = 1, C, = 00, !iZz = 0,5,J, = 1.
CANTILEVER
SUBJECTED
TO A FOLLOWER
--------__________________________
Figure 4. Flutter with a concentrated
137
FORCE
hsysyrrptote
=
3+01.....
load S, versus translational spring constant cr of an elastically supported mass R, at its tip. si, = 1, R, = J, = 0, R, = J, = O-5. K’ = nfi.
cantilever
(c,
= 1)
flutter load is obtained, appear again as the follower force increases beyond this critical value. The new presence of all the eigenfrequencies indicates the existence of a post-critical stability region. This stability region ends up at that value of the flutter load which is obtained by the coalescing of the second and third eigenfrequencies. This small interval of values of the varying parameter j1 may be characterized as a transition zone between the range of values of 1, for which the flutter load is obtained by the coalescing of the first and second eigenfrequencies and the range of values of this parameter for which the flutter load is obtained by the coalescing of the second and third eigenfrequencies. This phenomenon has also been observed during other investigations [13, 141 but it has not been previously reported. Figure 4 shows the dependence of the flutter load S, upon the translational spring constant C_r of an elastically supported cantilever (C, = 1) carrying a concentrated mass R, = J_2 = O-5at its tip. The considerable stabilizing effect is evident as the varying parameter C, increases. On the contrary, from Figure 5 one can see that the analogous effect due to the rotational spring constant C, is very small.
Figure 5. Flutter with a concentrated
load 3, versus rotational spring constant c, of an elastically supported mass Mi, at its tip. C, = 1. RI = .?, = 0, R, = f, = 0.5, K’ = co.
cantilever
(i5, = 1)
Figure 6 shows the plot of the flutter load S, versus the rotatory inertia .I, of a concentrated mass R, = 0.5 at the tip of an elastically restrained cantilever for the two cases C, = cx), C, = 1 (curve I) and C, = 1, C, = co (curve II). Curve I shows an abrupt destabilizing effect for very small values of .I,, while for the other values of this parameter the flutter load is always less than 4.10. Curve II shows a destabilizing effect approximately up to the value J, = 2. while beyond this value a considerable stabilizing effect appears.
138
A. N. KOUNADJS AND J. T. KATSIKADELIS
, ,
I
,
,
,
( ( ,
I, I
I,
-------------__---_-___---__----___-~__~.~
1
,I?
,I,
1,,
,
,
,
,
Asymptote .? =9*87
Figure 6. Flutter loads s, versus rotator j, of a concentrated mass M, at the tip of an elastically supported cantilever. R, = .7, = 0, M;I, = 0.5. K’ = “3.
4. CONCLUSIONS
Among the most important conclusions of this investigation one may list the following. (a) The effect of the translational spring as well as the effect of the moment of inertia of a concentrated mass placed either at the support or at the tip may be considerable. (b) The shear deformation for thin cross-sections with small values of K’ may have a considerable destabilizing effect. (c) In all cases where the flutter load for some range of values of the varying parameter is obtained by the coalescing of the second and third eigenfrequencies, while for another range of values of this parameter it is obtained by the coalescing of the first and second eigenfrequencies, there exists a stability region. This region corresponding to a small interval of values of the varying parameter is the transition zone which connects two ranges of values of the foregoing parameter: namely the first range, preceding the transition zone, where the flutter load is obtained by the coalescing of the first and second eigenfrequencies, and the second range, where the flutter load is obtained by the coalescing of the second and third eigenfrequencies.
REFERENCES 1. H. ZIEGLER 1953 Zeitschrift ftir angewandte Mathematik und Physik 4, 89-121. Linear elastic stability. 2. C. LANCZOS 1970 The Variational Principle of Mechanics. University of Toronto Press, fourth edition. 3. A. PILLAGER1950 Stabilitiitsproblem der Elastostatik. Berlin: Springer Verlag. See p. 217. der Elastomechanik. 4. H. ZIEGLER 1952 Ingenieur-Archiu 20, 49-56. Die Stabilitatskriterien 5. M. BECK 1952 Zeitschrift ftir anyewandte Mathematik und Physik 3, 225-228. Die Knicklast des einseitig eingespannten, tangential gedrtickten Stabes. 6. H. LEIPHOLZ 1962 Zeitschrif ftir angewandte Mathemutik und Physik 13, 359-372. Anwendung des Galerkinschen Verfahrens an nichtkonservative Stabilitltsprobleme des elastichen Stabes. 7. H. LEIPHOLZ 1970 Stability Theory. New York and London : Academic Press. See pp. 204-2 12. 8. G. R. COWPER 1966 Journal ofApplied Mathematics 33,335-340. The shear coefficient in Timoshenko beam theory. 9. T. E. SIMKINS and G. L. ANDERSON 1975 Journal of Sound and Vibration 39, 359-369. Stability of Beck’s column considering support characteristics.
CANTILEVER SUBJECTED TO A FOLLOWER FORCE
139
10. R. N. KAPOOR and H. H. LEIPHOLZ 1974 Zngenieur-Archiu. 43, 233-239. Stability analysis of damped polygenic systems with relocatable mass along its length. Il. S. NEMAT-NASSER 1967 .Iournal of AppIied Mechanics 34,484485. Instability of a cantilever under a follower force according to Timoshenko’s beam theory. 12. W. W. WALTERAND M. LEVINSON1968 Canadian Aeronautical and Space Institute 1, 91-93. Destabilization of a nonconservatively loaded elastic system due to rotatory inertia. 13. A. N. KOUNADISand J. T. KATSIKADELIS1976 Journal of Sound and Vibration 49, 171L178. Shear and rotatory inertia effect on Beck’s column. 14. A. N. KOUNADIS1977 Journal of Applied Mechanics 44,731-736. Stability of elastically restrained Timoshenko cantilevers with attached masses subjected to a follower force. 15. Z. CELEP1977 Zeitschrijtftir angewandte Mathematik und Mechanik 57,555-557. On the vibration and stability of Beck’s column subjected to vertical and follower forces. 16. S. P. TIMOSHENKOand J. M. GERE 1961 Theory of Elastic Stability New York: McGraw-Hill Book Company, Inc. See pp. 132-135. 17. A. N. KOUNADIS1975 Zngenieur-Archiu SS,43-5 1. Beam columns of varying cross-sections under lateral harmonic loads.