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Parametric instability of an elastically restrained cantilever beam
0
004E7949/90 s3.00 + 0.00 1990 Pqmma pm. plc
PARAMETRIC INSTABILITY OF AN ELASTICALLY RESTRAINED CANTILEVER BEAM R. C. KAR and T. SUJATA Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721 302, India (Received 10 January 1989)
Abstract-The dynamic stability of a beam elastically restrained at one end and free at the other subjected to pulsating uniaxial and follower forces has been studied. The effects of the tangency coefficient of the applied force, and the rotational and translational end-flexibilities of the beam on the regions of parametric instability of simple and combination resonances have been investigated. The results reveal that these parameters have significant influence on the dynamic stability of the system.
1. INTRODUCTION Vibration and stability of elastically restrained beams have been studied by many researchers. Abbas [l] investigated the effects of translational and rotational end-flexibilities on the natural frequencies of free vibration of Timoshenko beams using a finite element method. Cortinez and Laura [2] dealt with the vibration and buckling of non-uniform beams with a rotational restraint at one end and a concentrated mass at the other. Maurizi et al. [3] studied the free and forced vibrations of elastically restrained beams. They observed the effects of both translational and rotational flexibilities on the displacement, bending moments and shear force amplitudes. In a recent work, Liu [4] presented approximate expressions for the fundamental natural frequency of a restrained, slender cantilever beam of uniform cross-section for various combinations of rotational and translational end-flexibilities. GiirgGze [5] considered a simplysupported vertical beam carrying a concentrated mass, restrained at one end and subjected to a periodic axial displacement excitation at the other. He investigated the effects of various parameters such as mass-ratio, spring-stiffness and damping on the boundaries of the main parametric instability zones as well as on the dynamic response. In an earlier paper, the present authors [6] reported the regions of instability for simple parametric resonance of a nonuniform beam with an elastic end support and thermal gradient under a pulsating axial load. The purpose of the present work is to investigate the dynamic stability of a uniform beam elastically restrained at one end and free at the other, subjected to a directionally controlled pulsating longitudinal force. The first-, second- and third-order regions of parametric instability of simple and combination resonance are determined simultaneously by an eigensolution approach. The effects of end-flexibilities and tangency coefficient of the applied force on the 469
instability regions have been studied through the use of graphs.
2. FORMULATIONOF THE PROBLEM Figure 1 shows a uniform beam of length L elastically restrained against translation and rotation at the end X, = 0 by springs of stiffness K, and K. respectively. A pulsating force p(t) = PO+ P, cos wt with a tangency coefficient a acts on the beam at the end X, = L. Euler-Bernoulli theory, with the aid of a conservation law, yields the equation of motion as
0 < X, < L, and the boundary
conditions
t > 0,
(1)
as
K, WO, t) + EZW,,,,(O, r) + Z’(r)W’,,(O, t) = 0 & w,,(t), 1) - EZW,,, (0, t) = 0 EZW,,,(L,f)=O (a - l)P(t)W,,(L,
t) - EZW,,,,(L,
1) = 0,
(2)
where W(X,, t) is the transverse deflection, E is the Young’s modulus, p is the density, A is the area of cross-section, Z is the moment of inertia, D, is the external damping coefficient, and W,, = 8 W/at, w,, = aw/ax,, etc. Introducing the dimensionless parameters ?l = W/L, { = X,/L, T = ct (where c2 = EZ/pAL4), 8 =w/c,p,=P,L2/EZ,po = P,, L2/EZ, d, = D, L4c/EZ, 8, = EZ/K,L’ and PO= EZ/K,L, eqns (1) and (2) re-
duce to tf”‘+p(r)q”+d,i+tj’=O,
O<<
r>O
(3)
R. C. KAR and T. SUJATA
470
P(t)
*
L
x,
w
Fig. 1. Configuration and
of the system.
where rl(O,7)lBI+rl”‘(O,7)+p(7)?‘(0,7)=0 VW
7)//30- r1”(0,7) s”(l,
(a - I)p(z)rl’(l, respectively,
where
etc. A series solution
(
= 0
7) = 0
7) - ~“‘(1,7)
. ) = a(
= 0,
)/at, (
S’0
Vi(thj(5)
d5
(4)
)’ = c?( )/at,
of eqn (3) is sought
rl(r, 7) = i
D,=d,
in the form
f,(T)rl,(t;)>
Ku=
‘111’(5hj’(t)dt I0
+~:‘(O)~~(O)-~~(O)~j(O)
(9
i,j=l,2
r=l
where the
f,(r)s
the ~(5)s
are the co-ordinate
to
satisfy
conditions
are unknown
as many
functions
in eqn (4) as possible.
chosen
of
n.
(9)
of time and
functions
or as much
,...,
the
so as
boundary
It is further
assumed
3. DETERMINATION OF THE INSTABILITY REGIONS The solution
of eqn (8) is sought
in the form
that q,(t) can be written as %(5)=sin1,5
+A,sinh1,5
(f(7))
+B,cos&<
= eW/2{bo)
+ $,
(&Jsin
+ {b,}cos
+ C, cash A,<, (6) where
where A,, A,, B, and C, are determined boundary
using
the
conditions
tl,(O)/S,+rt:“(O)=O,
k@7
{b,},
{bk}
(10)
. .) are vectors in(10) in eqn (8) and application of harmonic balance method yields the quadratic eigenvalue problem in 1
dependent
{a,},
ker)],
(k = 1,2,.
of time. Substitution
of eqn
rl;(Wso-~:(o)=o (cupful+
q:(l)=o,
r]:“(l)=0
W,l
+ PfoWJ
= v%
(11)
(7)
where which, in turn, are obtained (4) by deleting
from those given in eqns
the terms containing
the load parame-
{X} = [{bo}, {b, 1, . . . , {a, 1, . . Jr.
ter p(7). Substituting the extended
eqns (5) and (6) in eqn (3) and using Galerkin
-PI
method,
Wlcos
one obtains
@7){f(7)1
= P>?
(8)
By introducing the new variable {Y} =1(X}, eqn (11) can be reduced to a standard eigenvalue problem of the double-size matrix
Instability of an elastically restrained cantilever beam In eqn (1 l), if all the eigenvalues have negative real parts, the corresponding basic solution is bounded as z -+ 00 and the solution is stable. On the other hand, if any of the eigenvalues has a positive real part, the solution is unstable. Thus the problem of determination of the stability of the non-trivial solutions becomes that of finding the signs of the eigenvaiues of a real non-symmet~c matrix. 4. RESULTS
AND
4
DISCUSSION
The numerical calculations have been carried out on a CYBER 180~8~A computer system for a beam with static load parameter pa = 0.5 and external damping coefficient d,= 0.01.The natural frequen6 cies obtained have been compared with those given Fig. 3. Regions of instability for /3,= 0.0, be = 0.1, u = 0.5. by Abbas [l] and Liu [4] and the static buckling loads Key as for Fig. 2. with those obtained analytically. In both cases, the results were found to be in good agreement. Moreover, the instability regions for the special case of a neighbourhoods of 2w,, wj and 20~ J3. The only cantilever beam (i.e. 8, = 0, & = 0) agree well with unstable region of simple resonance with the first natural frequency appears at 0 N 201,. The first-, those of Takahashi [7]. In Figs 2-13 the regions of instability have been second- and third-order unstable regions of combination resonance of sum-type occur near plotted for uniaxial (a = 0) and follower (a = OS,1.O, 2.0) forces for representative values of fl, @= (WI + w,), tw, + a,)/29 (WI + %)/3, (% f W), and &. The resonances which are obtained for (co3+ w, ). Combination resonances of difference-type do not exist in the present case. The first-order 5 e 5.0 are omitted. The kth order simple parametric resonances in the vicinities of 2tq/k (i= 1,2,3, regions of simple resonance at 6 = 2wz,2w, are wider k = 1,2,3) and the sum- and difference-type combithan the other regions for all values of p,. Figure 3 nation resonances in the neighbourhoods of depicts the instability regions occurring due to a (wj + o,)/k(i>j,i = 1,2,3,j = 1,2, k = 1,2,3) are subtangential force with a = 0.5.Combination resoobtained simultaneously, where wi is the ith natural nances of sum- and difference-type appear in the frequency of the unloaded beam with associated vicinities of (w2 + 03) and (wz -w,), (q - 0,)/2, boundary conditions. CD,), respectively. Regions of simple resonance (w3 Figures 2-5 show the stability diagrams for a = 0, of the first-order, which were observed in Fig. 2, 0.5, 1 and 2, respectively, with b, = 0 and f10= 0.1. continue to exist at 0 * 20,, 20,, 20,, while the For a = 0 (Fig. 2), the unstable regions of simple second-order regions in the neighbourhoods of o2 parametric resonance with the second natural fre- and w3 are found to have been suppressed. The quency are obtained in the vicinities of 2w, and W, first-order simple resonance regions of second and and those with the third natural frequency in the third modes are more important than the other
6 IT3
Simple resonance
hz9 ~bina~ion
resortonce
Fig. 2. Regions of instability for /I, = 0.0, Be= 0.1, a = 0.0.
472
R. C. KAR and T. SUJATA
115
8 Fig. 4. Regions of instability for 1, = 0.0, fis = 0.1, a = 1.O. Key as for Fig. 2. simple and combination resonance regions. A subtangential force with a = 0.5 has caused a reduction in the width of all the instability regions as compared to those in Fig. 2. The stability diagram for tangential force (i.e. a = 1.0) is shown in Fig. 4. Simple resonance regions occur at 0 2: 2w,, 2w,, w2, 2w, and w,, while combination resonances of sum- and differencetype are obtained in the vicinities of (u+ + q), and (0~~- q), (q - 0,)/2, (wj - q), respectively. It is interesting to note that the (w2 - w,), 2w, and 20, regions are wider than the other instability regions. An increase in the value of c( from 0.5 to 1.0 has increased the width of the (w2 - w, ) region and reduced the width of all other first-order regions. A value of a = 2.0 (Fig. 5) is found to have given rise resonances at to second-order combination 0 = (0, + 0,)/2, (w2 + w,)/2 and (q + w,)/2 in addition to the first-order combination resonances in the vicinities of (w, + q), (q + We) and (We+ w,). Also, simple resonances are observed in the neighbourhoods of 21~0,)2w,, wz, 2w,, 03, 2w,/3. No differencetype combination resonance regions appear at this value of IX.While the unstable regions near 0 = 20,~
0
5
10
20
30
Fig. 6. Regions of instability for /3,= 1.0, & = 0.0, a = 0.0. Key as for Fig. 2.
w2, w, and (q + wl) have become wider, those at 0 N 2w, and 2w, have reduced in width as a result of increase in the value of M:from 1.0 to 2.0. The parametric resonance regions in the vicinities of 2w,, (w, + q) and (wj + w,) are much wider and hence more important than those at 6 N 2w,, 2w, and (% + 0,). The regions of parametric instability for four different values of tl with /?, = 1.0 and PO= 0 are shown in Figs 69. From Fig. 6 it is seen that, for u = 0, first-order regions of combination resonance exist near 0 = (w, + w,), (q +o,) and (w, + w,), while second-order combination resonance of the sum-type is observed in the vicinity of (q + wj)/2. Simple resonances occur at 6 N 20,) 20, and oj . The 20, region is the widest of all the instability regions. At do= 0.5 (Fig. 7), first-order sum- and differencetype combination resonances appear in the neighbourhoods of (q + q) and (q - w,), respectively, and no second-order combination resonances are seen. While the region at 0 E (q + wj)/2 is supregions near 0 = 2w,, pressed, the instability (w2 + wj) and 2w, have reduced in width due to an increase in the value of a from 0 to 0.5. For a value of a = 1.0, only sum-type combination resonances
40
60
75
115
Fig. 5. Regions of instability for j, = 0.0, /$ = 0.1, a = 2.0. Key as for Fig. 2.
Ins~bility of an elastically restrained cantilever beam
60
65
5
15
30
60
35
&I
8
Fig. 7. Regions of instability for j?, = 1.0, & = 0.0, e = 0.5. Key as for Fig. 2.
Fig. 9. Regions of instability for p, = 1.0, /3s= 0.0, a = 2.0. Key as for Fig. 2.
occur near 19= (02 + o,) and (a+ + w,). Regions of simple resonance appear in the vicinities of 2w,, q, Zw, and q. Analogous to the case of a = 0.5, no second-order combination resonances exist in Fig. 8. The first-order simple resonance regions are much wider than those of combination resonance. All the regions have become narrower with an increase in a from 0.5 to 1.0. At a = 2.0 (Fig. 9), the combination resonance region near f3 = (CL+ + w,) is more predominant than all other instability regions. An increase in the value of a from 1.0 to 2.0 has not only widened the (CD* + Ok) region, but also reduced the width of the 204 and 20, regions, besides giving rise to secondorder combination resonances at 6 CC(w2 + 03)/2 and (03 f 01 j/2. Figures 10-13 show the stabihty diagrams for a = 0, OS, 1 and 2, respectively, for p, = 1.O and & = 0.1. At a = 0, sum-type combination resonances
An increase in parametric resonance regions. the value of 01from 0 to 0.5 (Fig. 11) has replaced the (wj + q) region by the (q - w,) region, while the
appear in the vicinities of (w, + co*), (w2 + w,), (or + 0+)/2 and (oj + w, ), but difference-type combi-
nation resonances are not observed. Regions of simple resonance exist near 8 = 204, 2~4 and wj. The first-order region at 8 N 2~0, is the widest of all the
combination sum-type @N (w2 + wX) continues
region at resonance to appear. Here, the 204
2
s
1
0
10
15
30
55
8
Fig. 10. Regions of instability for B, = 1.0, j& = 0.1, a = 0.0. Key as for Fig. 2. 2
4
1
0
Fig. 8. Regions of instability for /I, = 1.0, & = 0.0, do= 1.0. Key as for Fig. 2.
Fig. 1I. Regions of instability for & = 1.0, ,& = 0.1, a = 0.5. Key as for Fig. 2.
R. C. KAR and T.
474 2
SUJATA
w) regions have become more prominent with a = 2.0. A comparison of Figs 2-5 with Figs 10-13, respectively, reveals that all the parametric instability regions have shifted towards lower excitation frequencies due to an increase in the value of /I, from PI 0 to 1.0. In addition, for t( = 0 and 0.5, whereas the 1 first-order combination resonance regions have reduced in width, the simple resonance regions near 0 = 2~4 and 204 have become more important. At a = 1.O, an increase in the value of /I, to 1.0 has caused the replacement of the difference-type combination resonance region at 0 N (We- wr) by a sum0 type one at 0 N (w2 + Ok). The simple resonance 5 10 30 55 regions in the vicinities of 2~~ and 20, have become 8 wider, thus signifying greater instability. On the other Fig. 12. Regions of instability for /I, = 1.0, j?”= 0.1, a = 1.0. hand, for CI= 2.0, as seen in Figs 5 and 13, with Key as for Fig. 2. increase in /I, the regions at 0 E 204, (w, + w2) and (oj + w,) have reduced in width while that near 0 = (w2 + wj) has become wider. region is observed to be the widest. Moreover, there Again, by a comparison of Figs 69 with is a reduction in the width of the regions near Figs 10-13, it is observed that an increase in the value 0 = 2~4, 2~4, w, and (w2 + Ok) and suppression of of f10from 0 to 0.1 has made the system more sensitive the second-order combination resonance region at to periodic forces by shifting all the instability regions 0 N (w2 + w,)/2. At a = 1.0 (Fig. 12), only sum-type towards lower excitation frequencies. At t( = 0, 0.5 combination resonance regions appear in the neighand 1.0, most of the first-order instability regions in bourhoods of (co2+ wj) and (oj + o,), while simple Figs lo-12 are seen to have become wider than the resonances occur near 0 = 204,204 and wj . The 2w, corresponding regions in Figs 68, respectively. From region is again observed to be the widest of all Figs 9 and 13, for a = 2.0, it may be noted that an the instability regions. Due to an increase in the increase in & has increased the width of the combivalue of a from 0.5 to 1.0, whereas the second-order nation resonance regions at e N (co2 + oj), (co, + w,) region of simple resonance at 0 = w, has become and the simple resonance region near 0 = 2w,. Morewider, those of the first-order have reduced in width. over, the second-order simple resonance region at With a further increase in the value of c( to 2.0 0 N wj which did not exist in Fig. 9 has appeared at (Fig. 13), combination resonance regions appear near resonance region 0 = (w, + wq) and (wr + 04)/2 in addition to the /I0 = 0.1, while the combination instability regions existing in Fig. 12. The (w2 + We) near 6’ = (wj + a,)/2 has been suppressed. and (wj + w,) regions are wider than the other regions shown in Fig. 13. As compared to the case of 5. CONCLUSIONS c( = 1.0 (Fig. 12) the 204 and 204 regions have The regions of parametric instability have been reduced in width, while the (w2 + w,), (w, + w, ) and obtained for a beam elastically restrained at one end and free at the other under the influence of pulsating uniaxial and follower forces. The results reveal that A the unstable regions for uniaxial force (a = 0) are wider than those for follower force in most of the cases considered. Combination resonances of sum(w,+w,l type occur for uniaxial as well as supertangential 2 force, but those of difference-type do not occur. In contrast, combination resonances of difference-type appear predominantly for subtangential and tangential force. Whereas most of the regions of simple resonance are wider than those of combination resonance for uniaxial and subtangential force, the combination resonances are observed to have increasing importance under tangential and supertangential force. In the latter cases, some of the combination 05 10 30 55 60 resonance regions are found to be wider than the 8 regions of simple resonance. An increase in the value of either /I, or /I0 causes Fig. 13. Regions of instability for p, = 1.0, /?@= 0.1, a = 2.0. Key as for Fig. 2. shifting of all the instability regions towards lower
-r
Instability of an elastically restrained cantilever beam
415
excitation frequencies, thus making the system more sensitive to periodic forces. Moreover, in most of the cases, the combination resonance regions experience reduction in width while the regions of simple resonance become wider with an increase in either j3, or
against rotation at one end and with concentrated mass ai the other. J. Sound Vibr. 99, 144148 (1985). 3. M. J. Maurizi. D. V. Bambill de Rossit and P. A. A. Laura, Free and forced vibrations of beams elastically restrained against translation and rotation at the ends.
B0’
4. W. H. Liu, Approximate formula for determining the fundamental frequency of a restrained cantilever.
J. Sound Vibr. 120, 626-630
J. Sound Vibr. 124, 204-205
(1988).
(1988).
5. M. Giirgiize, Parametric vibrations of a restrained beam with an end mass under displacement excitation. REFERENCES
I. B. A. H. Abbas, Vibrations of Timoshenko beams with elastically restrained ends. J. Sound Vibr. 97, 541-548 (1984). 2. V. H. Cortinez and P. A. A. Laura. Vibrations and buckling of a non-uniform beam elastically restrained
J. Sound Vibr. 108, 73-84 (1986).
6. R. C. Kar and T. Sujata, Parametric instability of a non-uniform beam with thermal gradient and elastic end support. J. Sound Vibr. 122, 209-215 (1988). 7. K. Takahashi, An approach to investigate the instability of multiple-degree-of-freedom parametric dynamic systems. J. Sound Vibr. 78, 519-529 (1981).
004E7949/90 s3.00 + 0.00 1990 Pqmma pm. plc
PARAMETRIC INSTABILITY OF AN ELASTICALLY RESTRAINED CANTILEVER BEAM R. C. KAR and T. SUJATA Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721 302, India (Received 10 January 1989)
Abstract-The dynamic stability of a beam elastically restrained at one end and free at the other subjected to pulsating uniaxial and follower forces has been studied. The effects of the tangency coefficient of the applied force, and the rotational and translational end-flexibilities of the beam on the regions of parametric instability of simple and combination resonances have been investigated. The results reveal that these parameters have significant influence on the dynamic stability of the system.
1. INTRODUCTION Vibration and stability of elastically restrained beams have been studied by many researchers. Abbas [l] investigated the effects of translational and rotational end-flexibilities on the natural frequencies of free vibration of Timoshenko beams using a finite element method. Cortinez and Laura [2] dealt with the vibration and buckling of non-uniform beams with a rotational restraint at one end and a concentrated mass at the other. Maurizi et al. [3] studied the free and forced vibrations of elastically restrained beams. They observed the effects of both translational and rotational flexibilities on the displacement, bending moments and shear force amplitudes. In a recent work, Liu [4] presented approximate expressions for the fundamental natural frequency of a restrained, slender cantilever beam of uniform cross-section for various combinations of rotational and translational end-flexibilities. GiirgGze [5] considered a simplysupported vertical beam carrying a concentrated mass, restrained at one end and subjected to a periodic axial displacement excitation at the other. He investigated the effects of various parameters such as mass-ratio, spring-stiffness and damping on the boundaries of the main parametric instability zones as well as on the dynamic response. In an earlier paper, the present authors [6] reported the regions of instability for simple parametric resonance of a nonuniform beam with an elastic end support and thermal gradient under a pulsating axial load. The purpose of the present work is to investigate the dynamic stability of a uniform beam elastically restrained at one end and free at the other, subjected to a directionally controlled pulsating longitudinal force. The first-, second- and third-order regions of parametric instability of simple and combination resonance are determined simultaneously by an eigensolution approach. The effects of end-flexibilities and tangency coefficient of the applied force on the 469
instability regions have been studied through the use of graphs.
2. FORMULATIONOF THE PROBLEM Figure 1 shows a uniform beam of length L elastically restrained against translation and rotation at the end X, = 0 by springs of stiffness K, and K. respectively. A pulsating force p(t) = PO+ P, cos wt with a tangency coefficient a acts on the beam at the end X, = L. Euler-Bernoulli theory, with the aid of a conservation law, yields the equation of motion as
0 < X, < L, and the boundary
conditions
t > 0,
(1)
as
K, WO, t) + EZW,,,,(O, r) + Z’(r)W’,,(O, t) = 0 & w,,(t), 1) - EZW,,, (0, t) = 0 EZW,,,(L,f)=O (a - l)P(t)W,,(L,
t) - EZW,,,,(L,
1) = 0,
(2)
where W(X,, t) is the transverse deflection, E is the Young’s modulus, p is the density, A is the area of cross-section, Z is the moment of inertia, D, is the external damping coefficient, and W,, = 8 W/at, w,, = aw/ax,, etc. Introducing the dimensionless parameters ?l = W/L, { = X,/L, T = ct (where c2 = EZ/pAL4), 8 =w/c,p,=P,L2/EZ,po = P,, L2/EZ, d, = D, L4c/EZ, 8, = EZ/K,L’ and PO= EZ/K,L, eqns (1) and (2) re-
duce to tf”‘+p(r)q”+d,i+tj’=O,
O<<
r>O
(3)
R. C. KAR and T. SUJATA
470
P(t)
*
L
x,
w
Fig. 1. Configuration and
of the system.
where rl(O,7)lBI+rl”‘(O,7)+p(7)?‘(0,7)=0 VW
7)//30- r1”(0,7) s”(l,
(a - I)p(z)rl’(l, respectively,
where
etc. A series solution
(
= 0
7) = 0
7) - ~“‘(1,7)
. ) = a(
= 0,
)/at, (
S’0
Vi(thj(5)
d5
(4)
)’ = c?( )/at,
of eqn (3) is sought
rl(r, 7) = i
D,=d,
in the form
f,(T)rl,(t;)>
Ku=
‘111’(5hj’(t)dt I0
+~:‘(O)~~(O)-~~(O)~j(O)
(9
i,j=l,2
r=l
where the
f,(r)s
the ~(5)s
are the co-ordinate
to
satisfy
conditions
are unknown
as many
functions
in eqn (4) as possible.
chosen
of
n.
(9)
of time and
functions
or as much
,...,
the
so as
boundary
It is further
assumed
3. DETERMINATION OF THE INSTABILITY REGIONS The solution
of eqn (8) is sought
in the form
that q,(t) can be written as %(5)=sin1,5
+A,sinh1,5
(f(7))
+B,cos&<
= eW/2{bo)
+ $,
(&Jsin
+ {b,}cos
+ C, cash A,<, (6) where
where A,, A,, B, and C, are determined boundary
using
the
conditions
tl,(O)/S,+rt:“(O)=O,
k@7
{b,},
{bk}
(10)
. .) are vectors in(10) in eqn (8) and application of harmonic balance method yields the quadratic eigenvalue problem in 1
dependent
{a,},
ker)],
(k = 1,2,.
of time. Substitution
of eqn
rl;(Wso-~:(o)=o (cupful+
q:(l)=o,
r]:“(l)=0
W,l
+ PfoWJ
= v%
(11)
(7)
where which, in turn, are obtained (4) by deleting
from those given in eqns
the terms containing
the load parame-
{X} = [{bo}, {b, 1, . . . , {a, 1, . . Jr.
ter p(7). Substituting the extended
eqns (5) and (6) in eqn (3) and using Galerkin
-PI
method,
Wlcos
one obtains
@7){f(7)1
= P>?
(8)
By introducing the new variable {Y} =1(X}, eqn (11) can be reduced to a standard eigenvalue problem of the double-size matrix
Instability of an elastically restrained cantilever beam In eqn (1 l), if all the eigenvalues have negative real parts, the corresponding basic solution is bounded as z -+ 00 and the solution is stable. On the other hand, if any of the eigenvalues has a positive real part, the solution is unstable. Thus the problem of determination of the stability of the non-trivial solutions becomes that of finding the signs of the eigenvaiues of a real non-symmet~c matrix. 4. RESULTS
AND
4
DISCUSSION
The numerical calculations have been carried out on a CYBER 180~8~A computer system for a beam with static load parameter pa = 0.5 and external damping coefficient d,= 0.01.The natural frequen6 cies obtained have been compared with those given Fig. 3. Regions of instability for /3,= 0.0, be = 0.1, u = 0.5. by Abbas [l] and Liu [4] and the static buckling loads Key as for Fig. 2. with those obtained analytically. In both cases, the results were found to be in good agreement. Moreover, the instability regions for the special case of a neighbourhoods of 2w,, wj and 20~ J3. The only cantilever beam (i.e. 8, = 0, & = 0) agree well with unstable region of simple resonance with the first natural frequency appears at 0 N 201,. The first-, those of Takahashi [7]. In Figs 2-13 the regions of instability have been second- and third-order unstable regions of combination resonance of sum-type occur near plotted for uniaxial (a = 0) and follower (a = OS,1.O, 2.0) forces for representative values of fl, @= (WI + w,), tw, + a,)/29 (WI + %)/3, (% f W), and &. The resonances which are obtained for (co3+ w, ). Combination resonances of difference-type do not exist in the present case. The first-order 5 e 5.0 are omitted. The kth order simple parametric resonances in the vicinities of 2tq/k (i= 1,2,3, regions of simple resonance at 6 = 2wz,2w, are wider k = 1,2,3) and the sum- and difference-type combithan the other regions for all values of p,. Figure 3 nation resonances in the neighbourhoods of depicts the instability regions occurring due to a (wj + o,)/k(i>j,i = 1,2,3,j = 1,2, k = 1,2,3) are subtangential force with a = 0.5.Combination resoobtained simultaneously, where wi is the ith natural nances of sum- and difference-type appear in the frequency of the unloaded beam with associated vicinities of (w2 + 03) and (wz -w,), (q - 0,)/2, boundary conditions. CD,), respectively. Regions of simple resonance (w3 Figures 2-5 show the stability diagrams for a = 0, of the first-order, which were observed in Fig. 2, 0.5, 1 and 2, respectively, with b, = 0 and f10= 0.1. continue to exist at 0 * 20,, 20,, 20,, while the For a = 0 (Fig. 2), the unstable regions of simple second-order regions in the neighbourhoods of o2 parametric resonance with the second natural fre- and w3 are found to have been suppressed. The quency are obtained in the vicinities of 2w, and W, first-order simple resonance regions of second and and those with the third natural frequency in the third modes are more important than the other
6 IT3
Simple resonance
hz9 ~bina~ion
resortonce
Fig. 2. Regions of instability for /I, = 0.0, Be= 0.1, a = 0.0.
472
R. C. KAR and T. SUJATA
115
8 Fig. 4. Regions of instability for 1, = 0.0, fis = 0.1, a = 1.O. Key as for Fig. 2. simple and combination resonance regions. A subtangential force with a = 0.5 has caused a reduction in the width of all the instability regions as compared to those in Fig. 2. The stability diagram for tangential force (i.e. a = 1.0) is shown in Fig. 4. Simple resonance regions occur at 0 2: 2w,, 2w,, w2, 2w, and w,, while combination resonances of sum- and differencetype are obtained in the vicinities of (u+ + q), and (0~~- q), (q - 0,)/2, (wj - q), respectively. It is interesting to note that the (w2 - w,), 2w, and 20, regions are wider than the other instability regions. An increase in the value of c( from 0.5 to 1.0 has increased the width of the (w2 - w, ) region and reduced the width of all other first-order regions. A value of a = 2.0 (Fig. 5) is found to have given rise resonances at to second-order combination 0 = (0, + 0,)/2, (w2 + w,)/2 and (q + w,)/2 in addition to the first-order combination resonances in the vicinities of (w, + q), (q + We) and (We+ w,). Also, simple resonances are observed in the neighbourhoods of 21~0,)2w,, wz, 2w,, 03, 2w,/3. No differencetype combination resonance regions appear at this value of IX.While the unstable regions near 0 = 20,~
0
5
10
20
30
Fig. 6. Regions of instability for /3,= 1.0, & = 0.0, a = 0.0. Key as for Fig. 2.
w2, w, and (q + wl) have become wider, those at 0 N 2w, and 2w, have reduced in width as a result of increase in the value of M:from 1.0 to 2.0. The parametric resonance regions in the vicinities of 2w,, (w, + q) and (wj + w,) are much wider and hence more important than those at 6 N 2w,, 2w, and (% + 0,). The regions of parametric instability for four different values of tl with /?, = 1.0 and PO= 0 are shown in Figs 69. From Fig. 6 it is seen that, for u = 0, first-order regions of combination resonance exist near 0 = (w, + w,), (q +o,) and (w, + w,), while second-order combination resonance of the sum-type is observed in the vicinity of (q + wj)/2. Simple resonances occur at 6 N 20,) 20, and oj . The 20, region is the widest of all the instability regions. At do= 0.5 (Fig. 7), first-order sum- and differencetype combination resonances appear in the neighbourhoods of (q + q) and (q - w,), respectively, and no second-order combination resonances are seen. While the region at 0 E (q + wj)/2 is supregions near 0 = 2w,, pressed, the instability (w2 + wj) and 2w, have reduced in width due to an increase in the value of a from 0 to 0.5. For a value of a = 1.0, only sum-type combination resonances
40
60
75
115
Fig. 5. Regions of instability for j, = 0.0, /$ = 0.1, a = 2.0. Key as for Fig. 2.
Ins~bility of an elastically restrained cantilever beam
60
65
5
15
30
60
35
&I
8
Fig. 7. Regions of instability for j?, = 1.0, & = 0.0, e = 0.5. Key as for Fig. 2.
Fig. 9. Regions of instability for p, = 1.0, /3s= 0.0, a = 2.0. Key as for Fig. 2.
occur near 19= (02 + o,) and (a+ + w,). Regions of simple resonance appear in the vicinities of 2w,, q, Zw, and q. Analogous to the case of a = 0.5, no second-order combination resonances exist in Fig. 8. The first-order simple resonance regions are much wider than those of combination resonance. All the regions have become narrower with an increase in a from 0.5 to 1.0. At a = 2.0 (Fig. 9), the combination resonance region near f3 = (CL+ + w,) is more predominant than all other instability regions. An increase in the value of a from 1.0 to 2.0 has not only widened the (CD* + Ok) region, but also reduced the width of the 204 and 20, regions, besides giving rise to secondorder combination resonances at 6 CC(w2 + 03)/2 and (03 f 01 j/2. Figures 10-13 show the stabihty diagrams for a = 0, OS, 1 and 2, respectively, for p, = 1.O and & = 0.1. At a = 0, sum-type combination resonances
An increase in parametric resonance regions. the value of 01from 0 to 0.5 (Fig. 11) has replaced the (wj + q) region by the (q - w,) region, while the
appear in the vicinities of (w, + co*), (w2 + w,), (or + 0+)/2 and (oj + w, ), but difference-type combi-
nation resonances are not observed. Regions of simple resonance exist near 8 = 204, 2~4 and wj. The first-order region at 8 N 2~0, is the widest of all the
combination sum-type @N (w2 + wX) continues
region at resonance to appear. Here, the 204
2
s
1
0
10
15
30
55
8
Fig. 10. Regions of instability for B, = 1.0, j& = 0.1, a = 0.0. Key as for Fig. 2. 2
4
1
0
Fig. 8. Regions of instability for /I, = 1.0, & = 0.0, do= 1.0. Key as for Fig. 2.
Fig. 1I. Regions of instability for & = 1.0, ,& = 0.1, a = 0.5. Key as for Fig. 2.
R. C. KAR and T.
474 2
SUJATA
w) regions have become more prominent with a = 2.0. A comparison of Figs 2-5 with Figs 10-13, respectively, reveals that all the parametric instability regions have shifted towards lower excitation frequencies due to an increase in the value of /I, from PI 0 to 1.0. In addition, for t( = 0 and 0.5, whereas the 1 first-order combination resonance regions have reduced in width, the simple resonance regions near 0 = 2~4 and 204 have become more important. At a = 1.O, an increase in the value of /I, to 1.0 has caused the replacement of the difference-type combination resonance region at 0 N (We- wr) by a sum0 type one at 0 N (w2 + Ok). The simple resonance 5 10 30 55 regions in the vicinities of 2~~ and 20, have become 8 wider, thus signifying greater instability. On the other Fig. 12. Regions of instability for /I, = 1.0, j?”= 0.1, a = 1.0. hand, for CI= 2.0, as seen in Figs 5 and 13, with Key as for Fig. 2. increase in /I, the regions at 0 E 204, (w, + w2) and (oj + w,) have reduced in width while that near 0 = (w2 + wj) has become wider. region is observed to be the widest. Moreover, there Again, by a comparison of Figs 69 with is a reduction in the width of the regions near Figs 10-13, it is observed that an increase in the value 0 = 2~4, 2~4, w, and (w2 + Ok) and suppression of of f10from 0 to 0.1 has made the system more sensitive the second-order combination resonance region at to periodic forces by shifting all the instability regions 0 N (w2 + w,)/2. At a = 1.0 (Fig. 12), only sum-type towards lower excitation frequencies. At t( = 0, 0.5 combination resonance regions appear in the neighand 1.0, most of the first-order instability regions in bourhoods of (co2+ wj) and (oj + o,), while simple Figs lo-12 are seen to have become wider than the resonances occur near 0 = 204,204 and wj . The 2w, corresponding regions in Figs 68, respectively. From region is again observed to be the widest of all Figs 9 and 13, for a = 2.0, it may be noted that an the instability regions. Due to an increase in the increase in & has increased the width of the combivalue of a from 0.5 to 1.0, whereas the second-order nation resonance regions at e N (co2 + oj), (co, + w,) region of simple resonance at 0 = w, has become and the simple resonance region near 0 = 2w,. Morewider, those of the first-order have reduced in width. over, the second-order simple resonance region at With a further increase in the value of c( to 2.0 0 N wj which did not exist in Fig. 9 has appeared at (Fig. 13), combination resonance regions appear near resonance region 0 = (w, + wq) and (wr + 04)/2 in addition to the /I0 = 0.1, while the combination instability regions existing in Fig. 12. The (w2 + We) near 6’ = (wj + a,)/2 has been suppressed. and (wj + w,) regions are wider than the other regions shown in Fig. 13. As compared to the case of 5. CONCLUSIONS c( = 1.0 (Fig. 12) the 204 and 204 regions have The regions of parametric instability have been reduced in width, while the (w2 + w,), (w, + w, ) and obtained for a beam elastically restrained at one end and free at the other under the influence of pulsating uniaxial and follower forces. The results reveal that A the unstable regions for uniaxial force (a = 0) are wider than those for follower force in most of the cases considered. Combination resonances of sum(w,+w,l type occur for uniaxial as well as supertangential 2 force, but those of difference-type do not occur. In contrast, combination resonances of difference-type appear predominantly for subtangential and tangential force. Whereas most of the regions of simple resonance are wider than those of combination resonance for uniaxial and subtangential force, the combination resonances are observed to have increasing importance under tangential and supertangential force. In the latter cases, some of the combination 05 10 30 55 60 resonance regions are found to be wider than the 8 regions of simple resonance. An increase in the value of either /I, or /I0 causes Fig. 13. Regions of instability for p, = 1.0, /?@= 0.1, a = 2.0. Key as for Fig. 2. shifting of all the instability regions towards lower
-r
Instability of an elastically restrained cantilever beam
415
excitation frequencies, thus making the system more sensitive to periodic forces. Moreover, in most of the cases, the combination resonance regions experience reduction in width while the regions of simple resonance become wider with an increase in either j3, or
against rotation at one end and with concentrated mass ai the other. J. Sound Vibr. 99, 144148 (1985). 3. M. J. Maurizi. D. V. Bambill de Rossit and P. A. A. Laura, Free and forced vibrations of beams elastically restrained against translation and rotation at the ends.
B0’
4. W. H. Liu, Approximate formula for determining the fundamental frequency of a restrained cantilever.
J. Sound Vibr. 120, 626-630
J. Sound Vibr. 124, 204-205
(1988).
(1988).
5. M. Giirgiize, Parametric vibrations of a restrained beam with an end mass under displacement excitation. REFERENCES
I. B. A. H. Abbas, Vibrations of Timoshenko beams with elastically restrained ends. J. Sound Vibr. 97, 541-548 (1984). 2. V. H. Cortinez and P. A. A. Laura. Vibrations and buckling of a non-uniform beam elastically restrained
J. Sound Vibr. 108, 73-84 (1986).
6. R. C. Kar and T. Sujata, Parametric instability of a non-uniform beam with thermal gradient and elastic end support. J. Sound Vibr. 122, 209-215 (1988). 7. K. Takahashi, An approach to investigate the instability of multiple-degree-of-freedom parametric dynamic systems. J. Sound Vibr. 78, 519-529 (1981).