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Stability of a nonuniform cantilever subjected to dissipative and nonconservative forces
compulers B 9rucrurc3 Vol. I I, pp. 175-180 Pergamon Press Ltd., 1980. Printed in Great Britain
STABILITY OF A NONUNIFORM CANTILEVER SUBJECTED TO DISSIPATIVE AND NONCONSERVATIVE FORCES R. C. KAR Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302,India (Received 28 August 1978;receivedfor publication 11 January 1979) Abstmc-The stability of a tapered cantilever beam subjected to a circulatory force at its free end is investigated. The effects of internal and external damping are included in the partial ditIerential equation of motion. An adjoint variational principle has been used to determine approximately the values of the critical flutter load of the system. Graphs which demonstrate the variation of the critical flutter load with taper, damping and tangency coefficient are presented.
1. INTRODumON
EZ(xthtt
In recent years problems in the theory of nonconservative elastic stability have been the subjects of numerous investigators [l. 21. Even though much work has been done to study the stability of uniform beams subjected to follower forces, nonconservative stability of nonuniform beams received less attention in the past. Massey and Van der Meer[3] studied the stability of tapered cantilever columns subjected to a tangential tip load for breadth taper only. Sankaran and Rao[4] used Galerkin finite element method to study the stability of tapered cantilever columns subjected to follower forces with both breadth and thickness tapers. However, the effects of damping and tangency coefficient (also called “nonconservativeness parameter”) were not included in these studies. In this study attention has been focussed upon the stability of a tapered cantilever beam subjected to a nonconservative force at its free end and under the influence of internal and external damping. An adjoint variational principle has been used for the purpose of determining approximately the values of the critical load of the system as it depends upon its damping, taper and nonconservativeness parameters. The variation of the critical flutter load with these parameters is shown in a series of graphs.
+ E*ItXt)W,tt,
= [Ez(xtht,l,t + w*z(xthtt,I,t - P(ac- l)w,, = 0 at xt = L,
(3)
where w(x,, t) is the transverse deflection of the beam at the point xr and at time f, p the density, A(x,) the area of a generic cross section, Z(x,) the moment of inertia of the cross section, E the Young’s modulus, E* the internal damping coefficient, & the external (air) damping coefficient, and (. . .),t = a(. . .)/ax,, (. . At = a(. . .)/at, etc. Introducing the following dimensionless quantities q = w/L, 5 =x,/L, r = cf, q* = E*cjE, /I* = &cL’/EZ,,, Q = PL’/EZv, with c2 = EZv/pA0L4, and defining Z(xt) = Z&l), A(x,),= Avm(l), eqns (l)-(3) become, with the
notation (. . .) = a(. . .)/a~,(. . .)’ = a(. . .)/a& etc. [S(!%“l” + n*[s([)$‘]” t Qis” t m(&j- t /3*?j= 0, O
s(n($‘t
0<7,
(4)
at l=o,
(5)
n*+/“) = [S(J)($ -I-q*$‘)l
- Q(a, - l)$ = 0 at 5 = 1,
(6)
respectively. 2. EQUATION OF MOTION Consider a cantilever beam of length L and variable rectangular cross section subjected to a circulatory force of constant magnitude P at the free end as shown in Fig. 1. The line of action of this force is specified by the tangency coefficient a,. It is assumed that the material of the beam is a Kelvin solid. The apposite partial differential equation of motion and boundary conditions, including the effects of internal and external damping, are
P
[EZ(xt)w,,tI,tt +[E*Z(xthttrl9tt +Pw,tt+pA(xt)w,rt+Btw,,=O, O
o
(1)
at
x,=0,
(2) 175
Fig. 1. Geometry and dimensions of the circulatory system.
176
R.C.KAR
The expression ii(d 7) = n(5):‘*
(7)
where A is, in general, a complex number, is substituted into eqns (4)-(6) yielding (l+ n*A)[s(&r”]“+
essentially the Ritz technique in conjunction with the functional in eqn (14). It is well-known that, in the application of Ritz method, the co-ordinate functions need only satisfy the geometric boundary conditions and not necessarily the natural boundary conditions. It can be easily verified that the following sets of polynomials, namely,
Q$‘t m(&i27j +P*hr/=O,O
q = q’ = 0 at 5 = 0,
4m
(9)
= r+‘(a2
-
2p2!J + y&,
r a
(18)
1,
where
SUM1•t q*Ah” = (I+ q*A)[S(Oq”l -Q(s-l)q'=O
n,(g) = f’+‘(a* - W15 + nl’),
(8)
at l=l,
(lo)
a1=(rt2)(rt3)[(rtl)(r+2)S2(1)tQ(at-l)S(l)], /3* = (rt1Hrt3)[r(r t2)s2(l)t
with the present notation (. . .)’ = d(. . .)/d&
Q(a, - l)S(l)],
y1 = (r t 1Xr t 2)[r(r t l)S2( 1)t Q(ar - l)S( 111, 3. THB ADJOINT PROBLEM AND VARIATIONAL PRINCIF’LE
It can be easily shown that the boundary value problem that is adjoint to the problem stated in eqns (8)-(10) is the following:
a2=(rtl)(rt2)2(rt3)SZ(l)t3(rtl)(rt2)S(l)Qa, -(rt2)(rt3)S(1)Qt2(rt2)S’(l)QartQ2ar, /32=r(r+1)(r+2)(r+3)S2(1)t3(rt1)*S(1)Qal - (rt
(lt q*A)[S(l)t"]"t Qf't m(f)A' .$tfl*A{=O,O<[
(11)
l)(rt 3)S(l)Qt(2rt3)S'(l)Qatt Q2al,
y2= r(r t l)*(r t 2)S2(1)t 3r(r t l)S(l)Qa, -(rtl)(rt2)S(l)Qt2(rtl)S'(1)Qa,tQ2a,,
[=f=O (1 + n*A)S(W+
atl=O,
(12) satisfy all the boundary conditions in eqns (16) and (17). Substitution of eqn (15) into the adjoint variational principle in eqn (14) leads to
Qa& = (1+ n*A) x [S(l)[')'t Qc = 0 at .$= 1.
(13) WB” C”) = 0,
The variational principle associated with the boundary value problems in eqns (8)-(10) and (11)-(13) is s
’ U + ~*AP(lh”t”
(19)
where
- @it’+ A’h3vZ
t A/3*?[] dl+ %n’(l)Al))
= 0.
(14)
t A2Mh&
t V*%&l dl+ &$WB(l)).
(20)
It follows from the variational principle in eqn (19) that 4. TtUX APPROXIMATB SOLUTION the necessary conditions for the functional J to have For the original boundary value problem given in eqns stationary value are (8)-(10) and its adjoint problem in eqns (11)-(13), itis now assumed that the functions q(C) and ((5) can be a.f -0, g=o. (21) approximated by xm (15) where N is a positive integer, B, and C, are constants, and u&) and [,(I) are the co-ordinate functions chosen to satisfy the boundary conditions
which are obtained from eqns (9), (lo), (12) and (13) by deleting the terms in the natural boundary conditions involving the eigenvalue parameter A. This is permissible because of the fact that the method being applied here is
From the second condition in eqn (21), one obtains, for the original problem, in view of eqn (20), ,&LA*+F,A+G,)B,=O,
n=l,2,3
,...,
N,
(22)
where
5. THENUMERICAL REstJLTs Numerical calculations were carried out on IBM 370/M and EC 1030 computers using double precision arithmetic to determine critical loads from eqn (22) for a
Stabilityof a nonuniformcantileversubjectedto nonconservativeforces
linearly tapered cantilever beam with rectangular cross section (Fig. 1) whose breadth b and thickness h were assumed to vary according to the relations
177
2.0
b = bo( 1 - a{),
h = ho(l- /3l). I.5
Consequently, the stiffness distribution S(l) and the mass distribution m(l) are given by W) = (1 - ar)(l - 85Y9 ?
m(l) = (1- G)(l - BS).
0‘ I.C
In Fig. 2 the variation of the critical flutter load parameter Qr versus the breadth taper parameter a is plotted for u*= /3* =O, at = 1 for four values of the thickness taper parameter @. The critical load for these curves, in the complete absence of damping (i.e. q* = /3* = 0), is sometimes called the quasi-critical value [5]. Thevaluesof QforabOand/3=Oaswellasa=Oand /3 > 0 are in excellent agreement with the results reported by Sankaran and Rao[4]. The curves reveal that the value of Qr decreases monotonically as a increases for small values of /3, whereas for a large value of /3, e.g. /3 = 0.6, the value of Qr decreases to a minimum value and thereafter increases as a increases. For a given value of a, the curves depicted in this figure show again that the value of Qf decreases with the increase of /3. The variation of the critical flutter load parameter Qr is plotted against the breadth taper parameter a in Fig. 3 for three values of a, with 0 = 0.2, n* = 0.01 and /3* = 0.1. All curves reveal that the critical load decreases monotonically as the value of a increases over its entire range considered, with the rate of decrease becoming
2.5 l Sankoran
t -
and
Roe
[41
O.!
I
0.2!
I 0.4
0.2
I 0.6
6
a
Fig. 3. Variation of critical flutter load with breadth taper for three values of tangency coefficient.
more pronounced for the super-tangential value of at(a, = 1.5). For given values of a, /3, q* and /3*, the curves shown in the figure indicate that the value of Qt increases with the increase of a, from subtangential to super-tangential values. Figure 4 shows the variation of Qr with a for /3 = 0.2, /3* = 0 and a, = 1 for various values of internal damping parameter n*. The dashed curve corresponds to the case of n* = 0 (representing quasi-critical load in the terminology of [5]), whereas the solid curves refer to
Author 2.5 q=p*=o,
a,=1
8.0.2,
t
p**o,
a,=1
2.0 I-
.
/
0.E
I
I
I
0.2
0.4
0.6
0. 6
a
Fii. 2. Variation of critical flutter load with breadth taper for differentvalues of thiikness taper. CM Vol. Il. No. ?-C
Fii. 4. Variation of critical flutter load with breadth taper for various values of internaldamping.
178
R. C. KAR
specified positive values of v*. It is evident that, for small values of q*, for example, q* = 0.01 and v* = 0.1 as shown in the figure, the critical load in the presence of internal damping is less than the corresponding quasicritical value for a given breadth taper a, whereas for q* sufficiently large, e.g. v* = 0.2, the quasi-critical value of the flutter load is less than the flutter load computed for this value of the damping parameter. Moreover, the quasi-critical and critical flutter load for q* = 0 and q* = 0.01 respectively decreases monotonically as the value of a increases, whereas for q* = 0.1 and 0.2, as the value of a is increased from zero, the critical load decreases to a minimum value after which it grows as a is successively increased, the rate of growth becoming moie rapid for the larger value of the internal damping parameter T* (e.g. q* = 0.2). In Fig. 5 the variation of Qf is plotted against a for /3 =0.2, q* =O.Ol and a, = 1 for four values of the external damping parameter p*. The effect of increasing the value of /I* is to raise the value of the critical flutter load for given values of a, /?, q* and a,. For prescribed values of /3, q*, at and /3*, the curves depicted in this figure show again the monotonic nature of decrease of the value of Q, with the increase in the value of a. The variation of Q, is plotted against the thickness taper parameter b in Fig. 6 with a = 0.2, q* = 0.01 and /I* = 0.1 for three values of at. It can be noted from these graphs that while the curves for a, = 0.7 (subtangential case) and at = 1.5 (super-tangential case) show the typical pattern of decrease in the value of Qr as the value of /3 increases, with the curve for at = 1.5 being concave upward over the entire range of values of /3 shown, the curve for at = 1 (tangential case) indicates a decrease in the value of Q, to a minimum value, attained at a large value of p, after which Qf increases a relatively small amount as the value of /3 is increased further. For given values of a, p, q* and B*, increasing the value of a, increases the value of the critical flutter load Q. In Fig. 7 the curves for o* = 0.01 and 0.1 are below the
2.5
Nt \ 0’
Fig. 6. Variation of critical flutter load with thickness taper for three values of tangency coefficient.
2.0 B ao.2,
I 0.2
r)‘=O.Ol,
I 0.4
o,=I
I
0.6
I 0.6
a Fii.
5.
Variation ..- of critical . flutter _ load. with . breadth taper for dltferent values of external dampmg.
01
0
I
I
I
0.2
0.4
0.6
.I)
B
Fig. 7. Variation of critical flutter load with thickness taper for various values of internal damping.
Stabilityof a nonuniformcantileversubjectedto nonconservativeforces
curve for q* = 0 (elastic case), whereas the curve for q* =0.2 is above that for v* = 0. In essence, for the prescribed values of [I, /3, /3* and al small values of v* (e.g. q* = 0.01 and 0.1) have destabilizing effect, while a large value of V* (e.g. 17* = 0.2) has a stabilizing effect. The curves for q*>O reveal that as ,!? increases from zero, the value of Q, tends to decrease to a minimum value and then increases slowly. It can be observed that the minima of these curves have been shifted to the left, i.e. an increase in the value of q* means that the minimum value of Qr occurs at a lower value of j3. But, the value of the critical flutter load Q for T* = 0 decreases steeply to attain a minimum value at a large value of /3 and then it increases a relatively small amount after which it again decreases with further increase in the value of /3. Plots of the variation of Qr versus /3 for four values of /3* appear in Fig. 8. The curves for fl* = 0 and 0.1 appearing in this figure behave in a manner quite analogous to the curve for q* = 0.01 in Fig. 7. Besides, the curve for /3* = 2 of Fig. 8 and that for q* = 0 of Fig. 7 are qualitatively quite similar. However, the curve for /3* = 4 shows a slightly different behaviour. In this case the value of Qr drops more rapidly over a large range of values of /3 as /3 is increased from zero and then the rate of decrease becomes very small for higher values of B(e.g. p > 0.6). For prescribed values of o, /3, q* and a, increase in the value of fi* increases the value of the critical flutter load Q,. To investigate further the effects of at, o* and /3* on & plots of Q versus ‘I* and Q, versus /3* for three values of a, are shown in Figs. 9 and 10 respectively. Figure 9 depicts that for all curves the value of Qr decreases rapidly to a minimum value as the value of ‘I* is increased from zero and thereafter it increases at a faster rate with the rate of increase being higher for higher values of at. From the curves for ar = 1.0and 1.5 it is also evident that at sufficiently
large values
of
v*,
179
) Fig. 9. Variationof criticalflutterload with internaldampingfor three values of tangencycoefficient.
0.8.0.2,
7)*=0.01
3.0
the value
1
2.0
N \
t
l
0‘
1.0
0.:
7 s0
I
I
I
5
IO
I5
I0
B’
Fig. 10. Variationof critical flutter load with external damping for three values of tangencycoefficient.
0
0.2
04
0.6
0.6
Fii. 8. Variationof critical flutterload with thickness taper for differentvalues of externaldamping.
of Qr exceeds that computed for q* = 0. The effect of increasing ar is to raise the value of Qr for given values internal damping parameter v*, the external damping parameter /3* and the two taper’parameters a and B with the rise being higher for larger values of q*. A similar behaviour is also exhibited by the curves in Fig. 10, i.e.
180
R. C. KAR
an increase in the value of at raises the value of Q for given values of 0, /3, q* and fi*, the increase being higher for higher values of /3*. Moreover, it is also evident that for all curves in Fig. 10 the value of Q, increases monotonically as the value of p* is increased from zero and they are concave downwards with higher rate of growth for smaller values of /3*. 6. CONCLUSIONS
Critical flutter load for an undamped, tangentially loaded, linearly tapered cantilever beam of rectangular cross section decreases monotonically with increasing breadth taper when the thickness taper is either completely absent or has a small value. For sufficiently large thickness taper, the critical load may be either raised or lowered upon increasing the amount of breadth taper of the beam. Reduction in the value of critical flutter load is more pronounced with increase in thickness taper than with increase in breadth taper. For a given damped tapered beam, increase in the value of the tangency coefficient increases the value of the critical flutter load. The effect of an increase in breadth taper is to decrease the critical flutter load of a tangentially loaded cantilever in the absence or presence of small internal damping. When internal damping is very large, the value of the critical clutter load at large values of the breadth taper is higher than the corresponding value of the critical flutter load calculated for the same value of internal damping
but in the absence of breadth taper. Consequently, one may conclude that large values of breadth taper have stabilizing effect for sufficiently large internal damping. The familiar stabilization and destabilization phenomena due to large and small internal damping respectively are observed from the numerical results. On the other hand, external damping has a stabilizing effect only. For the damped system the critical flutter load decreases as thickness taper increases, say up to 0.7, and then increases/decreases by a ralatively small amount with a further increase in the thickness taper.
1. G. Herrmann, Stability of equilibrium of elastic systems subjected to nonconservativeforces. Appl. Mech. Rev. u), 103408 (1%7). 2. C. Sundararajan, The vibration and stability of elastic systerms subjected to follower forces. The Shock and Vibration Digest 7,89-105 (1975). 3. C. Massey and A. T. Van der Meer, Errata on stability of nonprismatic cantilever columns under tangential loading. 1. App. Mech. 269-271(March 1971);Trans. A.S.M.E., L Appl. Mech. 38, 390 (1971). 4. G. V. Sankaran and G. V. Rao, Stability of tapered cantilever columns subjected to follower forces. Comput. Structures 6, 217-220(1976). 5. V. V. Volotin and N. I. Zhinzher, Effects of damping on
stability of elastic systems subjected to nonconservative forces. Int. J. Solids Structures 5, %5-989 (1%9).
STABILITY OF A NONUNIFORM CANTILEVER SUBJECTED TO DISSIPATIVE AND NONCONSERVATIVE FORCES R. C. KAR Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur-721302,India (Received 28 August 1978;receivedfor publication 11 January 1979) Abstmc-The stability of a tapered cantilever beam subjected to a circulatory force at its free end is investigated. The effects of internal and external damping are included in the partial ditIerential equation of motion. An adjoint variational principle has been used to determine approximately the values of the critical flutter load of the system. Graphs which demonstrate the variation of the critical flutter load with taper, damping and tangency coefficient are presented.
1. INTRODumON
EZ(xthtt
In recent years problems in the theory of nonconservative elastic stability have been the subjects of numerous investigators [l. 21. Even though much work has been done to study the stability of uniform beams subjected to follower forces, nonconservative stability of nonuniform beams received less attention in the past. Massey and Van der Meer[3] studied the stability of tapered cantilever columns subjected to a tangential tip load for breadth taper only. Sankaran and Rao[4] used Galerkin finite element method to study the stability of tapered cantilever columns subjected to follower forces with both breadth and thickness tapers. However, the effects of damping and tangency coefficient (also called “nonconservativeness parameter”) were not included in these studies. In this study attention has been focussed upon the stability of a tapered cantilever beam subjected to a nonconservative force at its free end and under the influence of internal and external damping. An adjoint variational principle has been used for the purpose of determining approximately the values of the critical load of the system as it depends upon its damping, taper and nonconservativeness parameters. The variation of the critical flutter load with these parameters is shown in a series of graphs.
+ E*ItXt)W,tt,
= [Ez(xtht,l,t + w*z(xthtt,I,t - P(ac- l)w,, = 0 at xt = L,
(3)
where w(x,, t) is the transverse deflection of the beam at the point xr and at time f, p the density, A(x,) the area of a generic cross section, Z(x,) the moment of inertia of the cross section, E the Young’s modulus, E* the internal damping coefficient, & the external (air) damping coefficient, and (. . .),t = a(. . .)/ax,, (. . At = a(. . .)/at, etc. Introducing the following dimensionless quantities q = w/L, 5 =x,/L, r = cf, q* = E*cjE, /I* = &cL’/EZ,,, Q = PL’/EZv, with c2 = EZv/pA0L4, and defining Z(xt) = Z&l), A(x,),= Avm(l), eqns (l)-(3) become, with the
notation (. . .) = a(. . .)/a~,(. . .)’ = a(. . .)/a& etc. [S(!%“l” + n*[s([)$‘]” t Qis” t m(&j- t /3*?j= 0, O
s(n($‘t
0<7,
(4)
at l=o,
(5)
n*+/“) = [S(J)($ -I-q*$‘)l
- Q(a, - l)$ = 0 at 5 = 1,
(6)
respectively. 2. EQUATION OF MOTION Consider a cantilever beam of length L and variable rectangular cross section subjected to a circulatory force of constant magnitude P at the free end as shown in Fig. 1. The line of action of this force is specified by the tangency coefficient a,. It is assumed that the material of the beam is a Kelvin solid. The apposite partial differential equation of motion and boundary conditions, including the effects of internal and external damping, are
P
[EZ(xt)w,,tI,tt +[E*Z(xthttrl9tt +Pw,tt+pA(xt)w,rt+Btw,,=O, O
o
(1)
at
x,=0,
(2) 175
Fig. 1. Geometry and dimensions of the circulatory system.
176
R.C.KAR
The expression ii(d 7) = n(5):‘*
(7)
where A is, in general, a complex number, is substituted into eqns (4)-(6) yielding (l+ n*A)[s(&r”]“+
essentially the Ritz technique in conjunction with the functional in eqn (14). It is well-known that, in the application of Ritz method, the co-ordinate functions need only satisfy the geometric boundary conditions and not necessarily the natural boundary conditions. It can be easily verified that the following sets of polynomials, namely,
Q$‘t m(&i27j +P*hr/=O,O
q = q’ = 0 at 5 = 0,
4m
(9)
= r+‘(a2
-
2p2!J + y&,
r a
(18)
1,
where
SUM1•t q*Ah” = (I+ q*A)[S(Oq”l -Q(s-l)q'=O
n,(g) = f’+‘(a* - W15 + nl’),
(8)
at l=l,
(lo)
a1=(rt2)(rt3)[(rtl)(r+2)S2(1)tQ(at-l)S(l)], /3* = (rt1Hrt3)[r(r t2)s2(l)t
with the present notation (. . .)’ = d(. . .)/d&
Q(a, - l)S(l)],
y1 = (r t 1Xr t 2)[r(r t l)S2( 1)t Q(ar - l)S( 111, 3. THB ADJOINT PROBLEM AND VARIATIONAL PRINCIF’LE
It can be easily shown that the boundary value problem that is adjoint to the problem stated in eqns (8)-(10) is the following:
a2=(rtl)(rt2)2(rt3)SZ(l)t3(rtl)(rt2)S(l)Qa, -(rt2)(rt3)S(1)Qt2(rt2)S’(l)QartQ2ar, /32=r(r+1)(r+2)(r+3)S2(1)t3(rt1)*S(1)Qal - (rt
(lt q*A)[S(l)t"]"t Qf't m(f)A' .$tfl*A{=O,O<[
(11)
l)(rt 3)S(l)Qt(2rt3)S'(l)Qatt Q2al,
y2= r(r t l)*(r t 2)S2(1)t 3r(r t l)S(l)Qa, -(rtl)(rt2)S(l)Qt2(rtl)S'(1)Qa,tQ2a,,
[=f=O (1 + n*A)S(W+
atl=O,
(12) satisfy all the boundary conditions in eqns (16) and (17). Substitution of eqn (15) into the adjoint variational principle in eqn (14) leads to
Qa& = (1+ n*A) x [S(l)[')'t Qc = 0 at .$= 1.
(13) WB” C”) = 0,
The variational principle associated with the boundary value problems in eqns (8)-(10) and (11)-(13) is s
’ U + ~*AP(lh”t”
(19)
where
- @it’+ A’h3vZ
t A/3*?[] dl+ %n’(l)Al))
= 0.
(14)
t A2Mh&
t V*%&l dl+ &$WB(l)).
(20)
It follows from the variational principle in eqn (19) that 4. TtUX APPROXIMATB SOLUTION the necessary conditions for the functional J to have For the original boundary value problem given in eqns stationary value are (8)-(10) and its adjoint problem in eqns (11)-(13), itis now assumed that the functions q(C) and ((5) can be a.f -0, g=o. (21) approximated by xm (15) where N is a positive integer, B, and C, are constants, and u&) and [,(I) are the co-ordinate functions chosen to satisfy the boundary conditions
which are obtained from eqns (9), (lo), (12) and (13) by deleting the terms in the natural boundary conditions involving the eigenvalue parameter A. This is permissible because of the fact that the method being applied here is
From the second condition in eqn (21), one obtains, for the original problem, in view of eqn (20), ,&LA*+F,A+G,)B,=O,
n=l,2,3
,...,
N,
(22)
where
5. THENUMERICAL REstJLTs Numerical calculations were carried out on IBM 370/M and EC 1030 computers using double precision arithmetic to determine critical loads from eqn (22) for a
Stabilityof a nonuniformcantileversubjectedto nonconservativeforces
linearly tapered cantilever beam with rectangular cross section (Fig. 1) whose breadth b and thickness h were assumed to vary according to the relations
177
2.0
b = bo( 1 - a{),
h = ho(l- /3l). I.5
Consequently, the stiffness distribution S(l) and the mass distribution m(l) are given by W) = (1 - ar)(l - 85Y9 ?
m(l) = (1- G)(l - BS).
0‘ I.C
In Fig. 2 the variation of the critical flutter load parameter Qr versus the breadth taper parameter a is plotted for u*= /3* =O, at = 1 for four values of the thickness taper parameter @. The critical load for these curves, in the complete absence of damping (i.e. q* = /3* = 0), is sometimes called the quasi-critical value [5]. Thevaluesof QforabOand/3=Oaswellasa=Oand /3 > 0 are in excellent agreement with the results reported by Sankaran and Rao[4]. The curves reveal that the value of Qr decreases monotonically as a increases for small values of /3, whereas for a large value of /3, e.g. /3 = 0.6, the value of Qr decreases to a minimum value and thereafter increases as a increases. For a given value of a, the curves depicted in this figure show again that the value of Qf decreases with the increase of /3. The variation of the critical flutter load parameter Qr is plotted against the breadth taper parameter a in Fig. 3 for three values of a, with 0 = 0.2, n* = 0.01 and /3* = 0.1. All curves reveal that the critical load decreases monotonically as the value of a increases over its entire range considered, with the rate of decrease becoming
2.5 l Sankoran
t -
and
Roe
[41
O.!
I
0.2!
I 0.4
0.2
I 0.6
6
a
Fig. 3. Variation of critical flutter load with breadth taper for three values of tangency coefficient.
more pronounced for the super-tangential value of at(a, = 1.5). For given values of a, /3, q* and /3*, the curves shown in the figure indicate that the value of Qt increases with the increase of a, from subtangential to super-tangential values. Figure 4 shows the variation of Qr with a for /3 = 0.2, /3* = 0 and a, = 1 for various values of internal damping parameter n*. The dashed curve corresponds to the case of n* = 0 (representing quasi-critical load in the terminology of [5]), whereas the solid curves refer to
Author 2.5 q=p*=o,
a,=1
8.0.2,
t
p**o,
a,=1
2.0 I-
.
/
0.E
I
I
I
0.2
0.4
0.6
0. 6
a
Fii. 2. Variation of critical flutter load with breadth taper for differentvalues of thiikness taper. CM Vol. Il. No. ?-C
Fii. 4. Variation of critical flutter load with breadth taper for various values of internaldamping.
178
R. C. KAR
specified positive values of v*. It is evident that, for small values of q*, for example, q* = 0.01 and v* = 0.1 as shown in the figure, the critical load in the presence of internal damping is less than the corresponding quasicritical value for a given breadth taper a, whereas for q* sufficiently large, e.g. v* = 0.2, the quasi-critical value of the flutter load is less than the flutter load computed for this value of the damping parameter. Moreover, the quasi-critical and critical flutter load for q* = 0 and q* = 0.01 respectively decreases monotonically as the value of a increases, whereas for q* = 0.1 and 0.2, as the value of a is increased from zero, the critical load decreases to a minimum value after which it grows as a is successively increased, the rate of growth becoming moie rapid for the larger value of the internal damping parameter T* (e.g. q* = 0.2). In Fig. 5 the variation of Qf is plotted against a for /3 =0.2, q* =O.Ol and a, = 1 for four values of the external damping parameter p*. The effect of increasing the value of /I* is to raise the value of the critical flutter load for given values of a, /?, q* and a,. For prescribed values of /3, q*, at and /3*, the curves depicted in this figure show again the monotonic nature of decrease of the value of Q, with the increase in the value of a. The variation of Q, is plotted against the thickness taper parameter b in Fig. 6 with a = 0.2, q* = 0.01 and /I* = 0.1 for three values of at. It can be noted from these graphs that while the curves for a, = 0.7 (subtangential case) and at = 1.5 (super-tangential case) show the typical pattern of decrease in the value of Qr as the value of /3 increases, with the curve for at = 1.5 being concave upward over the entire range of values of /3 shown, the curve for at = 1 (tangential case) indicates a decrease in the value of Q, to a minimum value, attained at a large value of p, after which Qf increases a relatively small amount as the value of /3 is increased further. For given values of a, p, q* and B*, increasing the value of a, increases the value of the critical flutter load Q. In Fig. 7 the curves for o* = 0.01 and 0.1 are below the
2.5
Nt \ 0’
Fig. 6. Variation of critical flutter load with thickness taper for three values of tangency coefficient.
2.0 B ao.2,
I 0.2
r)‘=O.Ol,
I 0.4
o,=I
I
0.6
I 0.6
a Fii.
5.
Variation ..- of critical . flutter _ load. with . breadth taper for dltferent values of external dampmg.
01
0
I
I
I
0.2
0.4
0.6
.I)
B
Fig. 7. Variation of critical flutter load with thickness taper for various values of internal damping.
Stabilityof a nonuniformcantileversubjectedto nonconservativeforces
curve for q* = 0 (elastic case), whereas the curve for q* =0.2 is above that for v* = 0. In essence, for the prescribed values of [I, /3, /3* and al small values of v* (e.g. q* = 0.01 and 0.1) have destabilizing effect, while a large value of V* (e.g. 17* = 0.2) has a stabilizing effect. The curves for q*>O reveal that as ,!? increases from zero, the value of Q, tends to decrease to a minimum value and then increases slowly. It can be observed that the minima of these curves have been shifted to the left, i.e. an increase in the value of q* means that the minimum value of Qr occurs at a lower value of j3. But, the value of the critical flutter load Q for T* = 0 decreases steeply to attain a minimum value at a large value of /3 and then it increases a relatively small amount after which it again decreases with further increase in the value of /3. Plots of the variation of Qr versus /3 for four values of /3* appear in Fig. 8. The curves for fl* = 0 and 0.1 appearing in this figure behave in a manner quite analogous to the curve for q* = 0.01 in Fig. 7. Besides, the curve for /3* = 2 of Fig. 8 and that for q* = 0 of Fig. 7 are qualitatively quite similar. However, the curve for /3* = 4 shows a slightly different behaviour. In this case the value of Qr drops more rapidly over a large range of values of /3 as /3 is increased from zero and then the rate of decrease becomes very small for higher values of B(e.g. p > 0.6). For prescribed values of o, /3, q* and a, increase in the value of fi* increases the value of the critical flutter load Q,. To investigate further the effects of at, o* and /3* on & plots of Q versus ‘I* and Q, versus /3* for three values of a, are shown in Figs. 9 and 10 respectively. Figure 9 depicts that for all curves the value of Qr decreases rapidly to a minimum value as the value of ‘I* is increased from zero and thereafter it increases at a faster rate with the rate of increase being higher for higher values of at. From the curves for ar = 1.0and 1.5 it is also evident that at sufficiently
large values
of
v*,
179
) Fig. 9. Variationof criticalflutterload with internaldampingfor three values of tangencycoefficient.
0.8.0.2,
7)*=0.01
3.0
the value
1
2.0
N \
t
l
0‘
1.0
0.:
7 s0
I
I
I
5
IO
I5
I0
B’
Fig. 10. Variationof critical flutter load with external damping for three values of tangencycoefficient.
0
0.2
04
0.6
0.6
Fii. 8. Variationof critical flutterload with thickness taper for differentvalues of externaldamping.
of Qr exceeds that computed for q* = 0. The effect of increasing ar is to raise the value of Qr for given values internal damping parameter v*, the external damping parameter /3* and the two taper’parameters a and B with the rise being higher for larger values of q*. A similar behaviour is also exhibited by the curves in Fig. 10, i.e.
180
R. C. KAR
an increase in the value of at raises the value of Q for given values of 0, /3, q* and fi*, the increase being higher for higher values of /3*. Moreover, it is also evident that for all curves in Fig. 10 the value of Q, increases monotonically as the value of p* is increased from zero and they are concave downwards with higher rate of growth for smaller values of /3*. 6. CONCLUSIONS
Critical flutter load for an undamped, tangentially loaded, linearly tapered cantilever beam of rectangular cross section decreases monotonically with increasing breadth taper when the thickness taper is either completely absent or has a small value. For sufficiently large thickness taper, the critical load may be either raised or lowered upon increasing the amount of breadth taper of the beam. Reduction in the value of critical flutter load is more pronounced with increase in thickness taper than with increase in breadth taper. For a given damped tapered beam, increase in the value of the tangency coefficient increases the value of the critical flutter load. The effect of an increase in breadth taper is to decrease the critical flutter load of a tangentially loaded cantilever in the absence or presence of small internal damping. When internal damping is very large, the value of the critical clutter load at large values of the breadth taper is higher than the corresponding value of the critical flutter load calculated for the same value of internal damping
but in the absence of breadth taper. Consequently, one may conclude that large values of breadth taper have stabilizing effect for sufficiently large internal damping. The familiar stabilization and destabilization phenomena due to large and small internal damping respectively are observed from the numerical results. On the other hand, external damping has a stabilizing effect only. For the damped system the critical flutter load decreases as thickness taper increases, say up to 0.7, and then increases/decreases by a ralatively small amount with a further increase in the thickness taper.
1. G. Herrmann, Stability of equilibrium of elastic systems subjected to nonconservativeforces. Appl. Mech. Rev. u), 103408 (1%7). 2. C. Sundararajan, The vibration and stability of elastic systerms subjected to follower forces. The Shock and Vibration Digest 7,89-105 (1975). 3. C. Massey and A. T. Van der Meer, Errata on stability of nonprismatic cantilever columns under tangential loading. 1. App. Mech. 269-271(March 1971);Trans. A.S.M.E., L Appl. Mech. 38, 390 (1971). 4. G. V. Sankaran and G. V. Rao, Stability of tapered cantilever columns subjected to follower forces. Comput. Structures 6, 217-220(1976). 5. V. V. Volotin and N. I. Zhinzher, Effects of damping on
stability of elastic systems subjected to nonconservative forces. Int. J. Solids Structures 5, %5-989 (1%9).