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Parametric instability of a non-uniform beam with thermal gradient resting on a Pasternak foundation
Compum & Strucrures Vol.29, No. 4, pp. 591-599, 1988 Printed in Great Britain.
0045-7949/88 $3.00 + 0.00 0 1988 Pergamon Pressplc
PARAMETRIC INSTABILITY OF A NON-UNIFORM BEAM WITH THERMAL GRADIENT RESTING ON A PASTERNAK FOUNDATION R. C. KAR and T. SUJATA Department of Mechanicaf Engineering, Indian Institute of Technology, Kharagpur 721302, India (Received 17 April 1987) Abstract-The dynamic stability behaviour of a tapered cantilever beam on a Pastemak foundation under the action of a puhating axia1 force and a steady, one-dimensional temperature gradient is studied. The effects of taper, elastic foundation, shear layer and thermal gradient on the natural frequencies, static buckling loads and principal regions of instability are investigated. The results reveal that increasing taper and stiffening elastic foundation have stabilizing effects, whereas increasing thermal gradient and stiffening of the shear layer have d~~biiizing effects on the beam.
1. INTRODUCFION The practical significant of ins~bility under pulsating loads has become increasingly noted in recent years with the advancement of modern technology.
Such loads appear as variable coefficients in the governing equations and are parametric with respect to certain forms of the deformations. The main objective of investigations of parametric instability is the determination of the boundaries of the regions of instability. A wide-ranging discussion of studies on this subject is given by Evan-Iwanowski [I] and Ibrahim and Barr [2-61. The books by Bolotin[7] and Schmidt [8] present excellent and well-rounded accounts of background knowledge. Noteworthy theoretical and experimental investigations have also been carried out on the dynamic stability of elastic systems, Brown et al. [9] studied the stability of uniform bars with various end conditions using the finite-element method. Ahuja and Duffield [IO] presented experimental and theoretical results on the effects of taper and foundation elasticity on the dynamic stability of a simple supported beam. Abbas and Thomas [I I] extended this work to include the effects of rotary inertia and shear deformation on uniform beams with various end conditions. Takahashi [ 121determined the regions of instability of a uniform cantilever beam acted upon by conservative and non-conservative forces. Datta and Chakraborty [ 131 studied the stability of a simply supported, tapered bar under parametric excitation using the finiteelement method. The effect of temperature on the modulus of elasticity is far from negligible, especially in high-speed atmospheric flights and nuclear engineering applications in which certain parts have to operate under elevated temperatures. Most engineering materials are found to have a linear relationship between Young’s modulus and temperature. Tomar and Jain studied the effect of thermal gradient on the frequencies of rotating beams with [14] and without [15] pre-twist.
Although some studies have been carried out in the past on the parametric instability of beams on Winkler foundations, as well as on the effect of temperature gradient on the frequencies of vibration of beams, it appears, to the authors’ knowledge, that no work exists concerning the effects of Pasternak foundation and thermal gradient on the dynamic stability of beams under pulsating axial loads. Thus, the purpose of this paper is to present the results of an investigation of the dynamic instability of a tapered cantilever beam with parametric excitation lying on a Pasternak foundation and subjected to a steady, onedimensional temperature gradient along its length. The boundary value problem for flexural deformation of the beam relative to its undistur~d equilibrium configuration is derived with the aid of a conservation law. Bolotin’s method [7] is used to determine the natural frequencies, static buckling loads and the principal regions of instability for prescribed values of the geometric and material parameters of the system, namely taper, thermal gradient, shearlayer stiffness and elastic foundation; the results are presented graphically.
2. FORMULATION OF THE PROBLEM
The problem under consideration, as shown in Fig. 1, is a linearly tapered cantilever beam of length L having a rectangular cross-section resting on a Pasternak foundation (i.e. an elastic foundation with a rate-inde~ndent shear layer interposed between the beam and the foundation) and subjected to a periodic time-dependent force, P(t) = P,, + P, cos wr acting along the undeformed axis at its free end. A steady one-dimensional temperature gradient is assumed to exist along the length of the beam. The formulation of the problem is carried out under the following assumptions: (i) the shear layer deforms only in the X, direction; (ii) the shear layer maintains continuous contact with the beam on one
591
R. C. KAR and T.
592
SUJATA
NONUNIFORM
BEAM
P(t)
SHEAR
LAYER
&e L
TAPERED
BEAM
Fig. 1. Geometry and configuration of the system.
side and the elastic foundation on the other; and (iii) there are no tangential shearing forces at the interface between the beam and the shear layer or that between the shear layer and the elastic foundation. Using the Euler-Bernoulli theory and a conservation law, the equation of motion and boundary conditions of the system shown in Fig. 1 can be written as
0 < x, < L, W(0, t) = 0,
0 < t.
(1)
W,, (0, I) = 0,
[E(X,)Z(X,)W,,,llx,=,
p is the density, K, and K, are the stiffnesses of the elastic foundation and shear layer, respectively, and w,, = a w/ax,, w,, = awlat, etc. Introducing the following dimensionless parameters q=W/L,i;=X,/L,~=ct(wherec*=E,Z,/pA,L~), p(z) = P(t)L*/E,Z,, P* = P+L*/E,Z,, k,= K,L4/E,Z,, k, = K,L*/E,Z,, a = P,-,IP*, fl = P,/P*, 9 = w/c and defining A(5) = A,m(c), E(5),= E, iY5), Z(5) = Z,S({) where m(C), S(c), T(t) represent the distribution of mass, moment of inertia and elasticity modulus, respectively, A,, E, and Z, are the area, elasticity modulus and moment of inertia at the end X, = L and P* is the critical buckling load of a uniform beam in the absence of thermal gradient and foundation. Equations (1) and (2) can be written in non-dimensional form as
LTW(tW’l”
=0
and
+ [P(T) - klrl” + ktl + 46% = 0, 0<5<1,
[E(X,)Z(X,)W,,l.,+P(t)W,l,,=,=O
OCT.
(2) tl(O, 7) = 0,
where W(X,, t) is the transverse deflection of the beam at any point X, and at time t, A(X,), E(X,), Z(X,), respectively, represent the area, elasticity modulus and moment of inertia of the beam at X,,
v ‘(0, z ) = 0,
(3)
Parametricinstability of a non-uniform beam and {[s(r)T(5)n”c~,r>~tp
=0
(4)
593
For finding the boundaries of the principal regions of instability, the disturbing frequency 0 is represented as 8 = (~/wo)oo. Equation (9) then reduces to
where ( )’ = d( )/d& (‘) = d( )/dr, etc. A series solution of eqn (3) is sought in the form (5)
rl(L z) = ,$,f,(rk(s),
where thef,(z)s are unknown functions of time and the r,(c)s represent coordinate functions satisfying as many or much of the boundary conditions given in eqn (4) as possible. Choosing q,(c) = 1 -cos(2r and using Galerkin’s matrix form as [Ml IYT)‘Z>> + WI
- l)ne/2,
r = 1,2,. . .
(6)
where y * = (8/0,)~. The dimensionl~s fundamental natural frequency o, and critical buckling load p* for a uniform beam in the absence of foundation and thermal gradient are obtained by solving the eigenvalue problems in eqns (10) and (11) with taper, foundation and thermal gradient parameters set to zero. The boundaries of the principal regions of instability are then determined from eqn (12).
method eqn (3) is written in 4. RESULTS AND DISCUSSION
Numerical calculations have been carried out on a HP-1000 computer using double-precision arithmetic with a five-term approximation for the series solution in eqn (5). The linearly tapered cantilever beam with a rectangular cross-section is assumed to have a width b and depth h varying according to the relations
- aP*WI
where
b = 6,[1 + a*(1 - 5%
-<)I,
h =h,[l+B*(l
where b, , h, are, respectively, the width and depth of the beam at the end { = 1 and a*, B* are the width and depth taper parameters. Consequently, the mass dist~bution m(r) and the moment of inertia distribution S(5) are given by the relations Kl
= VI - MHI + kF1.
(8)
are determined
(
of the principal regions of instability from the equation [7l
[x?-(a-t!Blp*[~l-~tMl){~~=~O}.
(9)
The non-dimensional natural frequencies of free vibration or are computed from the equation (Kl
-~:21~l)Lf~ = fO>,
w
which is obtained from eqn (9) by setting a = 0, /J = 0 and 0/2=0:. With /I = 0, 8 = 0 the non~imen~onal static buckling loads pf are determined from the equation
(WI -P:wlw-} where p: = ap*.
= IO},
Olfl +B*Q - t)l,
S(t) = [l + a*(1 - c)][l + jT*(l - t)]‘.
3. DETERMINATION OF THE INSTABILMV REGIONS The boundaries
MC) = I1 + a*(1 -
(11)
The temperature above the reference temperature at any point e from the fixed end of the beam is assumed to be Y = Yo( 1 - {). Choosing U. = Y, , the temperature at the end < = 1 as the reference temperature, the variation of modulus of elasticity of the beam 1161is denoted by &0=&V
-YY,(l-01,
OG’ylp,< 1
= 4 WEI where y is the coefficient of thermal expansion of the beam materiat 6 = y’Y, is the thermal gradient parameter and T(c) = [l - 6(1 - t)]. The effects of various geometric and material parameters on the natural frequencies o:, static buckling loads p: and the principal regions of instability of the beam have been studied. in order to check the accuracy of the numerical technique used,
R. C. KAR and T. SUJATA
594
the results obtained for a uniform cantilever beam on a Winkler foundation, which is a special case of the present problem, were compared with those of Abbas and Thomas [i I]; the agreement was found to be good. Figures 2 and 3 show the variations of natural frequency OF and static buckling load pf of the first three modes as a function of the thermal gradient parameter 6 for three different combinations of the taper parameters a* and /?* with elastic foundation constant kj’(kt = k,/n’) = 2, shear-layer constant k:(k,* = &/a*) = 2. At fixed values of 6 and /?*, the values of wr and pr increase in all the modes with increase in the value of the width taper parameter a *. A similar effect is also observed on keeping the value of a* fixed and increasing the value of the depth taper parameter jS*. However, this effect is more pronounced with the change in the value of fi* than with that of a*. For a given tapered beam, the values of both CD: and p: decrease monotoni~lly with increase in the value of 6, the rate of decrease being greater for the higher modes. Whereas, at any value of 6, the static buckling loads record larger changes in higher modes with an increase in the value of either of the taper parameters, the natural frequencies show a similar trend with increase in depth taper only. Figures 4 and 5 depict the effect of the elastic foundation constant k: and the shear-layer constant k: on the natural frequencies and static buckling loads of a given tapered beam (e.g. a * = 2, p * = 1) as a function of thermal gradient parameter 6. At given values of 6 and k!, the natural frequency and
buckling load of all the three modes increase as the value of k$ increases, the effect being greater on the natural frequencies of the lower modes. However, for fixed values of 6 and k:, an increase in shear-layer constant causes a reduction in the values of OF and p: of all the three modes. For given foundation constants the variations of wf and p: with increase in the value of 6 are analogous to those in Figs 2 and 3. The influence of various system parameters on the principal regions of instability is shown in Figs 6-9. Figure 6 shows the effect of the taper parameters a* and j?* on the instability regions for k: = kj? = 2, 6 = 0.2 and static load parameter a = 0.5. For a given value of the width taper parameter, as the depth taper increases all the instability regions become narrower and shift towards higher excitation frequencies, thus making the beam less sensitive to periodic forces. A similar effect is also observed on keeping the value of /?* constant and increasing the value of a*. This effect is, however, more pronounced with the change in value of /I* than with that of a*. With variations in either of the taper parameters, the higher regions depict greater shifts than the lower ones. The effect of variation of elastic foundation constant kf on the instability regions for CX*= 2, /?* = 1, k: = 2, 6 = 0.2 and a = 0.5 is shown in Fig. 7. With increase in the value of k$, the instability regions also decrease in width and shift towards higher excitation frequencies. But these effects are less predominant in the higher regions. Thus, stiffening of the elastic foundation makes the beam less susceptible to the
80.0*3
60.02nd
K
:>4.-...-
1st
mode
__
-__
I I I I I I I 0.01 0.00.1 0.2
0.3
Ok&
0.5
0.8
0.7
-
0.8
Fig. 2. Variation of w : versus 6 for three combinations of a * and /I* with k: = 2, kf = 2 (----a* ~*=1;-_--a*=2,~*=l;-a*=2,Pt=2),
= 1,
*c
0.1
;. .
0.2
- ._
0.3
.-..
0.r 6
PL
_-_.
1st
-__._ mode
L
---_ -----
Fig. 3. Variation ofp: verm 6 for three cotnbinations of a* ad 8* with kf = 2, kf = 2 (---_*5~(~*~I;---a*~2,~*=1;-u’=2,~’=2).
YMJ
I
---~-L--_ ,oo_o-_---------
200.0 -
300.0 -
600.0-
500.0-
600.0 -
3
CL
I
0.2
I
0.3
I 0.~ 6
I 0.5
----
I 0.6
---
I 0.7
Fig. 4. Variation of w : versus 6 for three combinations of k$ and k: with a* = 2, @* = I (_---k~=2, k:- 1;---~~=2,b~=2;~k~-l,k~=2).
0.1
f
---------------_-_
0.0
0.O-
111.0-
2(SO-
3[LO -
LI
6C
701.0-
80
90
100,
0.1
---__------_ I
I 0.2
0.4
0.3
6
I
I
--Pm_
1st mode
I
0.6
I
0.5
---___ /I ---_-
A0.7
I
C
Fig. 5 Variation o. r, : versus 6 for three combinations of k: and k: with a l = 2, j * = 1 -_k:=2,k:=2;--k:=l,kf=2). (-___ kT=2, k:=l;
0.0. 0.0
160.01
4oo*o*
I 0.1
I 0.2
I 0.3
I 0.4
P
I 0.5
I 0.6
-----__ I 0.7
I 0.8 Fig. 6. Principal regions of instability for three combinations ameterswithk~=2,k~=2,6=0.2,a=0.5(----a*=1,~*=1;-aa+=2, ^_ .
5.01 0.0
------_
------__
------mm___
___------_
1.0
of taper par-
I 0.9
;’
s
F
;
&
F
F cl
%
---___
-_-----__.-__
___------
::_ -1
___e----
F__
---
US -_-
--
------_--P__
Y
US: Unstable ,__--
I 0.1
** ------__
I
0.3
I
0.2
---
0.4
P
I
0.6
I
0.5
*-
-*
I 0.6
I 0.7
us
__Y--
I
0.9
3
**
1.0
7
Fig. 7.Principal regions of instability for three values of k: with a * = 2, b * = 1, k~=2,S=0.2,a=0.5(----k~=1;-kkf=2;---k~=3).
a.0
3.0
Ii.0
r
7 3" . 8.0 m
-____--------__ --__ 4 F-
18.0-
I
f
---_ 50.0 -
dl.”
I 19.0-
2o.ok
I
-----,___
I
I
I
I
---*
I
us
us
I
I
___-----------_.__
us
----__ us
I
/;
-4
0.0
3.01
0.1
0.2
I
0.3
0.1
0.5
0.6
0.7
0.6
0.9
1.0
__--_---3 I= 4.0
6.0 -
7.0 -
,46.0
---us
P Fig. 8. Principal regions of instability for three values of k: with a* = 2, /?* = 1, ,k: = 1; - /C: = 2; --kf = 3). k: = 2, 6 = 0.2, a = 0.5 (----
9
a
us
-4 -*_--_____-----3 I- ------______
l-
48.0
__-------------_-___
_-- _------------_ =l
=: -,
598
R. C. KAR and T. SIJJATA
US :Unstable
--
---__
--w__
:
-__ .-----__v___ _---w
-~-r-==-----_-------__
5.0
0.0
I 0.1
0.2
0.3
I 0.4
0.5 P
__ ---_ 0.6
us
--
us
I 0.7
I 0.6
0.9
1.0
Fig. 9. Principal regions of instability for three values of 6 with a* = 2, b* = 1,kb = 2, k: 6 = 0.4;--6 = 0.6). (- 6 = 0.2;----
periodic force and has a stabilizing effect, which agrees with the findings of Abbas and Thomas [l 11. Plots of the principal regions of instability for three values of shear-layer constant kf with a * = 2, /I * = 1, k,* = 2, 6 = 0.2 and a = 0.5 appear in Fig. 8. It may be observed that an increase in the value of k: increases the width of all the regions and relocates them at lower forcing frequencies. The lower regions are more affected by changes in the values of k: than the higher ones. Thus, the beam becomes more sensitive to the periodic force with increase in shear-layer stiffness, indicating destabilization. Figure 9 shows the effect of the thermal gradient parameter 6 on the principal regions of instability for a*=2, b*=l, /ct=k:=2 and a=05 All the regions experience a slight increase in width and shift towards lower excitation frequencies with an increase in the value of 6, the shift being more for the second and third regions than for the first one.
= 2, a = 0.5
5. CONCLUSION
Increase in taper parameters and stiffening of the elastic foundation not only increases the natural frequencies and static buckling loads, but also reduces the widths of the principal regions of instability and shifts them towards higher excitation frequencies, thus making the beam less sensitive to periodic forces. However, stiffening of the shear-layer and increase in thermal gradient reduces the natural frequencies and static buckling loads as well as widening the principal regions of instability and shifting them towards lower excitation frequencies, thereby making the beam more sensitive to periodic forces. Thus, it may be inferred that increasing taper and stiffening of elastic foundation have stabilizing effects on the beam, whereas increasing temperature gradient and stiffening of shear-layer have a destabilizing effect.
Parametric instability of a non-uniform beam REFERENCES
1. R. M. Evan-Iwanowski, On the parametric response of structures. Appl. Mech. Rev. 18, 699-702 (1965). 2. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part I: Mechanics of linear problems. Shock Vibr. Dig. 10, 15-29 (1978). 3. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part II: Mechanics of nonlinear problems. Shock Vibr. Dig. 10, 9-24 (1978). 4. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part III: Current problems (1). Shock Vibr. Dig. 10,
41-57 (1978). 5. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part IV: Current problems (2). Shock Vibr. Dig. 10, 19-47 (1978). 6. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part V: Stochastic oroblems. Shock Vibr. Dig. 10,17-38 (1978). 7. V. V. Bolotin, Dynamic Stability of Elastic Systems.
Holden Day, San Francisco (1964). 8. G. Schmidt, Parametererregre Schwingungen. VEB Deutscher Verlag Wissenachaften, Berlin (1975).
599
9. J. E. Brown, J. M. Hutt and A. E. Salama, Finite element solution to dynamic stability of bars. AIAA JI 6, 1423-1428 (1968).
10. R. Ahuja and R. C. Duffield, Parametric instability of variable cross-section beams resting on an elastic foundation. J. Sound Vibr. 39, 159-174 (1975). 11. B. A. H. Abbas and J. Thomas, Dynamic’ stability of Timoshenko beams resting on an elastic foundation. J. Sound Vibr. 60, 33-44 (1978). 12. K. Takahashi, An approach to investigate the instability
of multiple-degree-of-freedom parametric dynamic systems. J. Sound Vibr. 78, 519-529 (1981). 13. P. K. Datta and S. Chakraborty, Parametric instability of tapered beams by finite element method. J. mech. Engng. Sci., Inst. mech. Engrs 24, 205-208 (1982). 14. J. S. Tomar and R. Jain, Thermal effect on frequencies
of coupled vibrations AIAA Jl23,
of pretwisted rotating beams.
1293-1296 (1985).
15. J. S. Tomar and R. Jain, Effect of thermal gradient on frequencies of wedge-shaped rotating beams. AIAA JI 22, 848-850 (1984). 16. M. S. Dhotarad and N. Ganesan, Vibration analysis
of a rectangular plate subjected to a thermal gradient. J. Sound Vibr. 60, 481-497
(1978).
0045-7949/88 $3.00 + 0.00 0 1988 Pergamon Pressplc
PARAMETRIC INSTABILITY OF A NON-UNIFORM BEAM WITH THERMAL GRADIENT RESTING ON A PASTERNAK FOUNDATION R. C. KAR and T. SUJATA Department of Mechanicaf Engineering, Indian Institute of Technology, Kharagpur 721302, India (Received 17 April 1987) Abstract-The dynamic stability behaviour of a tapered cantilever beam on a Pastemak foundation under the action of a puhating axia1 force and a steady, one-dimensional temperature gradient is studied. The effects of taper, elastic foundation, shear layer and thermal gradient on the natural frequencies, static buckling loads and principal regions of instability are investigated. The results reveal that increasing taper and stiffening elastic foundation have stabilizing effects, whereas increasing thermal gradient and stiffening of the shear layer have d~~biiizing effects on the beam.
1. INTRODUCFION The practical significant of ins~bility under pulsating loads has become increasingly noted in recent years with the advancement of modern technology.
Such loads appear as variable coefficients in the governing equations and are parametric with respect to certain forms of the deformations. The main objective of investigations of parametric instability is the determination of the boundaries of the regions of instability. A wide-ranging discussion of studies on this subject is given by Evan-Iwanowski [I] and Ibrahim and Barr [2-61. The books by Bolotin[7] and Schmidt [8] present excellent and well-rounded accounts of background knowledge. Noteworthy theoretical and experimental investigations have also been carried out on the dynamic stability of elastic systems, Brown et al. [9] studied the stability of uniform bars with various end conditions using the finite-element method. Ahuja and Duffield [IO] presented experimental and theoretical results on the effects of taper and foundation elasticity on the dynamic stability of a simple supported beam. Abbas and Thomas [I I] extended this work to include the effects of rotary inertia and shear deformation on uniform beams with various end conditions. Takahashi [ 121determined the regions of instability of a uniform cantilever beam acted upon by conservative and non-conservative forces. Datta and Chakraborty [ 131 studied the stability of a simply supported, tapered bar under parametric excitation using the finiteelement method. The effect of temperature on the modulus of elasticity is far from negligible, especially in high-speed atmospheric flights and nuclear engineering applications in which certain parts have to operate under elevated temperatures. Most engineering materials are found to have a linear relationship between Young’s modulus and temperature. Tomar and Jain studied the effect of thermal gradient on the frequencies of rotating beams with [14] and without [15] pre-twist.
Although some studies have been carried out in the past on the parametric instability of beams on Winkler foundations, as well as on the effect of temperature gradient on the frequencies of vibration of beams, it appears, to the authors’ knowledge, that no work exists concerning the effects of Pasternak foundation and thermal gradient on the dynamic stability of beams under pulsating axial loads. Thus, the purpose of this paper is to present the results of an investigation of the dynamic instability of a tapered cantilever beam with parametric excitation lying on a Pasternak foundation and subjected to a steady, onedimensional temperature gradient along its length. The boundary value problem for flexural deformation of the beam relative to its undistur~d equilibrium configuration is derived with the aid of a conservation law. Bolotin’s method [7] is used to determine the natural frequencies, static buckling loads and the principal regions of instability for prescribed values of the geometric and material parameters of the system, namely taper, thermal gradient, shearlayer stiffness and elastic foundation; the results are presented graphically.
2. FORMULATION OF THE PROBLEM
The problem under consideration, as shown in Fig. 1, is a linearly tapered cantilever beam of length L having a rectangular cross-section resting on a Pasternak foundation (i.e. an elastic foundation with a rate-inde~ndent shear layer interposed between the beam and the foundation) and subjected to a periodic time-dependent force, P(t) = P,, + P, cos wr acting along the undeformed axis at its free end. A steady one-dimensional temperature gradient is assumed to exist along the length of the beam. The formulation of the problem is carried out under the following assumptions: (i) the shear layer deforms only in the X, direction; (ii) the shear layer maintains continuous contact with the beam on one
591
R. C. KAR and T.
592
SUJATA
NONUNIFORM
BEAM
P(t)
SHEAR
LAYER
&e L
TAPERED
BEAM
Fig. 1. Geometry and configuration of the system.
side and the elastic foundation on the other; and (iii) there are no tangential shearing forces at the interface between the beam and the shear layer or that between the shear layer and the elastic foundation. Using the Euler-Bernoulli theory and a conservation law, the equation of motion and boundary conditions of the system shown in Fig. 1 can be written as
0 < x, < L, W(0, t) = 0,
0 < t.
(1)
W,, (0, I) = 0,
[E(X,)Z(X,)W,,,llx,=,
p is the density, K, and K, are the stiffnesses of the elastic foundation and shear layer, respectively, and w,, = a w/ax,, w,, = awlat, etc. Introducing the following dimensionless parameters q=W/L,i;=X,/L,~=ct(wherec*=E,Z,/pA,L~), p(z) = P(t)L*/E,Z,, P* = P+L*/E,Z,, k,= K,L4/E,Z,, k, = K,L*/E,Z,, a = P,-,IP*, fl = P,/P*, 9 = w/c and defining A(5) = A,m(c), E(5),= E, iY5), Z(5) = Z,S({) where m(C), S(c), T(t) represent the distribution of mass, moment of inertia and elasticity modulus, respectively, A,, E, and Z, are the area, elasticity modulus and moment of inertia at the end X, = L and P* is the critical buckling load of a uniform beam in the absence of thermal gradient and foundation. Equations (1) and (2) can be written in non-dimensional form as
LTW(tW’l”
=0
and
+ [P(T) - klrl” + ktl + 46% = 0, 0<5<1,
[E(X,)Z(X,)W,,l.,+P(t)W,l,,=,=O
OCT.
(2) tl(O, 7) = 0,
where W(X,, t) is the transverse deflection of the beam at any point X, and at time t, A(X,), E(X,), Z(X,), respectively, represent the area, elasticity modulus and moment of inertia of the beam at X,,
v ‘(0, z ) = 0,
(3)
Parametricinstability of a non-uniform beam and {[s(r)T(5)n”c~,r>~tp
=0
(4)
593
For finding the boundaries of the principal regions of instability, the disturbing frequency 0 is represented as 8 = (~/wo)oo. Equation (9) then reduces to
where ( )’ = d( )/d& (‘) = d( )/dr, etc. A series solution of eqn (3) is sought in the form (5)
rl(L z) = ,$,f,(rk(s),
where thef,(z)s are unknown functions of time and the r,(c)s represent coordinate functions satisfying as many or much of the boundary conditions given in eqn (4) as possible. Choosing q,(c) = 1 -cos(2r and using Galerkin’s matrix form as [Ml IYT)‘Z>> + WI
- l)ne/2,
r = 1,2,. . .
(6)
where y * = (8/0,)~. The dimensionl~s fundamental natural frequency o, and critical buckling load p* for a uniform beam in the absence of foundation and thermal gradient are obtained by solving the eigenvalue problems in eqns (10) and (11) with taper, foundation and thermal gradient parameters set to zero. The boundaries of the principal regions of instability are then determined from eqn (12).
method eqn (3) is written in 4. RESULTS AND DISCUSSION
Numerical calculations have been carried out on a HP-1000 computer using double-precision arithmetic with a five-term approximation for the series solution in eqn (5). The linearly tapered cantilever beam with a rectangular cross-section is assumed to have a width b and depth h varying according to the relations
- aP*WI
where
b = 6,[1 + a*(1 - 5%
-<)I,
h =h,[l+B*(l
where b, , h, are, respectively, the width and depth of the beam at the end { = 1 and a*, B* are the width and depth taper parameters. Consequently, the mass dist~bution m(r) and the moment of inertia distribution S(5) are given by the relations Kl
= VI - MHI + kF1.
(8)
are determined
(
of the principal regions of instability from the equation [7l
[x?-(a-t!Blp*[~l-~tMl){~~=~O}.
(9)
The non-dimensional natural frequencies of free vibration or are computed from the equation (Kl
-~:21~l)Lf~ = fO>,
w
which is obtained from eqn (9) by setting a = 0, /J = 0 and 0/2=0:. With /I = 0, 8 = 0 the non~imen~onal static buckling loads pf are determined from the equation
(WI -P:wlw-} where p: = ap*.
= IO},
Olfl +B*Q - t)l,
S(t) = [l + a*(1 - c)][l + jT*(l - t)]‘.
3. DETERMINATION OF THE INSTABILMV REGIONS The boundaries
MC) = I1 + a*(1 -
(11)
The temperature above the reference temperature at any point e from the fixed end of the beam is assumed to be Y = Yo( 1 - {). Choosing U. = Y, , the temperature at the end < = 1 as the reference temperature, the variation of modulus of elasticity of the beam 1161is denoted by &0=&V
-YY,(l-01,
OG’ylp,< 1
= 4 WEI where y is the coefficient of thermal expansion of the beam materiat 6 = y’Y, is the thermal gradient parameter and T(c) = [l - 6(1 - t)]. The effects of various geometric and material parameters on the natural frequencies o:, static buckling loads p: and the principal regions of instability of the beam have been studied. in order to check the accuracy of the numerical technique used,
R. C. KAR and T. SUJATA
594
the results obtained for a uniform cantilever beam on a Winkler foundation, which is a special case of the present problem, were compared with those of Abbas and Thomas [i I]; the agreement was found to be good. Figures 2 and 3 show the variations of natural frequency OF and static buckling load pf of the first three modes as a function of the thermal gradient parameter 6 for three different combinations of the taper parameters a* and /?* with elastic foundation constant kj’(kt = k,/n’) = 2, shear-layer constant k:(k,* = &/a*) = 2. At fixed values of 6 and /?*, the values of wr and pr increase in all the modes with increase in the value of the width taper parameter a *. A similar effect is also observed on keeping the value of a* fixed and increasing the value of the depth taper parameter jS*. However, this effect is more pronounced with the change in the value of fi* than with that of a*. For a given tapered beam, the values of both CD: and p: decrease monotoni~lly with increase in the value of 6, the rate of decrease being greater for the higher modes. Whereas, at any value of 6, the static buckling loads record larger changes in higher modes with an increase in the value of either of the taper parameters, the natural frequencies show a similar trend with increase in depth taper only. Figures 4 and 5 depict the effect of the elastic foundation constant k: and the shear-layer constant k: on the natural frequencies and static buckling loads of a given tapered beam (e.g. a * = 2, p * = 1) as a function of thermal gradient parameter 6. At given values of 6 and k!, the natural frequency and
buckling load of all the three modes increase as the value of k$ increases, the effect being greater on the natural frequencies of the lower modes. However, for fixed values of 6 and k:, an increase in shear-layer constant causes a reduction in the values of OF and p: of all the three modes. For given foundation constants the variations of wf and p: with increase in the value of 6 are analogous to those in Figs 2 and 3. The influence of various system parameters on the principal regions of instability is shown in Figs 6-9. Figure 6 shows the effect of the taper parameters a* and j?* on the instability regions for k: = kj? = 2, 6 = 0.2 and static load parameter a = 0.5. For a given value of the width taper parameter, as the depth taper increases all the instability regions become narrower and shift towards higher excitation frequencies, thus making the beam less sensitive to periodic forces. A similar effect is also observed on keeping the value of /?* constant and increasing the value of a*. This effect is, however, more pronounced with the change in value of /I* than with that of a*. With variations in either of the taper parameters, the higher regions depict greater shifts than the lower ones. The effect of variation of elastic foundation constant kf on the instability regions for CX*= 2, /?* = 1, k: = 2, 6 = 0.2 and a = 0.5 is shown in Fig. 7. With increase in the value of k$, the instability regions also decrease in width and shift towards higher excitation frequencies. But these effects are less predominant in the higher regions. Thus, stiffening of the elastic foundation makes the beam less susceptible to the
80.0*3
60.02nd
K
:>4.-...-
1st
mode
__
-__
I I I I I I I 0.01 0.00.1 0.2
0.3
Ok&
0.5
0.8
0.7
-
0.8
Fig. 2. Variation of w : versus 6 for three combinations of a * and /I* with k: = 2, kf = 2 (----a* ~*=1;-_--a*=2,~*=l;-a*=2,Pt=2),
= 1,
*c
0.1
;. .
0.2
- ._
0.3
.-..
0.r 6
PL
_-_.
1st
-__._ mode
L
---_ -----
Fig. 3. Variation ofp: verm 6 for three cotnbinations of a* ad 8* with kf = 2, kf = 2 (---_*5~(~*~I;---a*~2,~*=1;-u’=2,~’=2).
YMJ
I
---~-L--_ ,oo_o-_---------
200.0 -
300.0 -
600.0-
500.0-
600.0 -
3
CL
I
0.2
I
0.3
I 0.~ 6
I 0.5
----
I 0.6
---
I 0.7
Fig. 4. Variation of w : versus 6 for three combinations of k$ and k: with a* = 2, @* = I (_---k~=2, k:- 1;---~~=2,b~=2;~k~-l,k~=2).
0.1
f
---------------_-_
0.0
0.O-
111.0-
2(SO-
3[LO -
LI
6C
701.0-
80
90
100,
0.1
---__------_ I
I 0.2
0.4
0.3
6
I
I
--Pm_
1st mode
I
0.6
I
0.5
---___ /I ---_-
A0.7
I
C
Fig. 5 Variation o. r, : versus 6 for three combinations of k: and k: with a l = 2, j * = 1 -_k:=2,k:=2;--k:=l,kf=2). (-___ kT=2, k:=l;
0.0. 0.0
160.01
4oo*o*
I 0.1
I 0.2
I 0.3
I 0.4
P
I 0.5
I 0.6
-----__ I 0.7
I 0.8 Fig. 6. Principal regions of instability for three combinations ameterswithk~=2,k~=2,6=0.2,a=0.5(----a*=1,~*=1;-aa+=2, ^_ .
5.01 0.0
------_
------__
------mm___
___------_
1.0
of taper par-
I 0.9
;’
s
F
;
&
F
F cl
%
---___
-_-----__.-__
___------
::_ -1
___e----
F__
---
US -_-
--
------_--P__
Y
US: Unstable ,__--
I 0.1
** ------__
I
0.3
I
0.2
---
0.4
P
I
0.6
I
0.5
*-
-*
I 0.6
I 0.7
us
__Y--
I
0.9
3
**
1.0
7
Fig. 7.Principal regions of instability for three values of k: with a * = 2, b * = 1, k~=2,S=0.2,a=0.5(----k~=1;-kkf=2;---k~=3).
a.0
3.0
Ii.0
r
7 3" . 8.0 m
-____--------__ --__ 4 F-
18.0-
I
f
---_ 50.0 -
dl.”
I 19.0-
2o.ok
I
-----,___
I
I
I
I
---*
I
us
us
I
I
___-----------_.__
us
----__ us
I
/;
-4
0.0
3.01
0.1
0.2
I
0.3
0.1
0.5
0.6
0.7
0.6
0.9
1.0
__--_---3 I= 4.0
6.0 -
7.0 -
,46.0
---us
P Fig. 8. Principal regions of instability for three values of k: with a* = 2, /?* = 1, ,k: = 1; - /C: = 2; --kf = 3). k: = 2, 6 = 0.2, a = 0.5 (----
9
a
us
-4 -*_--_____-----3 I- ------______
l-
48.0
__-------------_-___
_-- _------------_ =l
=: -,
598
R. C. KAR and T. SIJJATA
US :Unstable
--
---__
--w__
:
-__ .-----__v___ _---w
-~-r-==-----_-------__
5.0
0.0
I 0.1
0.2
0.3
I 0.4
0.5 P
__ ---_ 0.6
us
--
us
I 0.7
I 0.6
0.9
1.0
Fig. 9. Principal regions of instability for three values of 6 with a* = 2, b* = 1,kb = 2, k: 6 = 0.4;--6 = 0.6). (- 6 = 0.2;----
periodic force and has a stabilizing effect, which agrees with the findings of Abbas and Thomas [l 11. Plots of the principal regions of instability for three values of shear-layer constant kf with a * = 2, /I * = 1, k,* = 2, 6 = 0.2 and a = 0.5 appear in Fig. 8. It may be observed that an increase in the value of k: increases the width of all the regions and relocates them at lower forcing frequencies. The lower regions are more affected by changes in the values of k: than the higher ones. Thus, the beam becomes more sensitive to the periodic force with increase in shear-layer stiffness, indicating destabilization. Figure 9 shows the effect of the thermal gradient parameter 6 on the principal regions of instability for a*=2, b*=l, /ct=k:=2 and a=05 All the regions experience a slight increase in width and shift towards lower excitation frequencies with an increase in the value of 6, the shift being more for the second and third regions than for the first one.
= 2, a = 0.5
5. CONCLUSION
Increase in taper parameters and stiffening of the elastic foundation not only increases the natural frequencies and static buckling loads, but also reduces the widths of the principal regions of instability and shifts them towards higher excitation frequencies, thus making the beam less sensitive to periodic forces. However, stiffening of the shear-layer and increase in thermal gradient reduces the natural frequencies and static buckling loads as well as widening the principal regions of instability and shifting them towards lower excitation frequencies, thereby making the beam more sensitive to periodic forces. Thus, it may be inferred that increasing taper and stiffening of elastic foundation have stabilizing effects on the beam, whereas increasing temperature gradient and stiffening of shear-layer have a destabilizing effect.
Parametric instability of a non-uniform beam REFERENCES
1. R. M. Evan-Iwanowski, On the parametric response of structures. Appl. Mech. Rev. 18, 699-702 (1965). 2. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part I: Mechanics of linear problems. Shock Vibr. Dig. 10, 15-29 (1978). 3. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part II: Mechanics of nonlinear problems. Shock Vibr. Dig. 10, 9-24 (1978). 4. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part III: Current problems (1). Shock Vibr. Dig. 10,
41-57 (1978). 5. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part IV: Current problems (2). Shock Vibr. Dig. 10, 19-47 (1978). 6. R. A. Ibrahim and A. D. S. Barr, Parametric vibration. Part V: Stochastic oroblems. Shock Vibr. Dig. 10,17-38 (1978). 7. V. V. Bolotin, Dynamic Stability of Elastic Systems.
Holden Day, San Francisco (1964). 8. G. Schmidt, Parametererregre Schwingungen. VEB Deutscher Verlag Wissenachaften, Berlin (1975).
599
9. J. E. Brown, J. M. Hutt and A. E. Salama, Finite element solution to dynamic stability of bars. AIAA JI 6, 1423-1428 (1968).
10. R. Ahuja and R. C. Duffield, Parametric instability of variable cross-section beams resting on an elastic foundation. J. Sound Vibr. 39, 159-174 (1975). 11. B. A. H. Abbas and J. Thomas, Dynamic’ stability of Timoshenko beams resting on an elastic foundation. J. Sound Vibr. 60, 33-44 (1978). 12. K. Takahashi, An approach to investigate the instability
of multiple-degree-of-freedom parametric dynamic systems. J. Sound Vibr. 78, 519-529 (1981). 13. P. K. Datta and S. Chakraborty, Parametric instability of tapered beams by finite element method. J. mech. Engng. Sci., Inst. mech. Engrs 24, 205-208 (1982). 14. J. S. Tomar and R. Jain, Thermal effect on frequencies
of coupled vibrations AIAA Jl23,
of pretwisted rotating beams.
1293-1296 (1985).
15. J. S. Tomar and R. Jain, Effect of thermal gradient on frequencies of wedge-shaped rotating beams. AIAA JI 22, 848-850 (1984). 16. M. S. Dhotarad and N. Ganesan, Vibration analysis
of a rectangular plate subjected to a thermal gradient. J. Sound Vibr. 60, 481-497
(1978).