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International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
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International Journal of Non-LinearMechanics
journal homepage: www.e lsev ier .com/ locate /n lm
Modified shooting approach to the non-linear periodic forced response ofisotropic/composite curved beams
S.M. Ibrahim, B.P. Patel, Y. Nath∗Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi 110016, India
A R T I C L E I N F O A B S T R A C T
Article history:Received 25 December 2008Received in revised form 15 August 2009Accepted 21 August 2009
Keywords:ShootingNon-linearCurved beam
Large amplitude periodic forced vibration of curved beams under periodic excitation is investigatedusing a three-noded beam element. The element is based on the higher-order shear deformation theorysatisfying interlayer continuity of displacements and transverse shear stress, and top-bottom conditionson the latter. The periodic responses are obtained using shooting technique coupled with Newmarktime marching and arc length continuation algorithm developed. The second order governing differentialequations of motion are solved without transforming to the first order differential equations therebyresulting in a computationally more efficient algorithm. The effects of excitation amplitude, supportconditions and beam curvature on the frequency versus response amplitude relation are highlighted.The typical frequency response curves for isotropic and cross-ply laminated curved beams are presented.Phenomenon of strong modal interactions is observed.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction
The necessity of analysing non-linear dynamics of curved beamsarises not only from the point of view of their application in civil,aerospace and other structural systems but also due to the interestthey stimulate as classical problems in vibration and stability. Fur-ther, the periodic excitation near to the resonance condition maycause large amplitude response resulting in stresses leading to thefailure. The steady state response is of great importance for dynamicdesign as well as a noise and vibration control.
In spite of the importance of non-linear dynamic analysis ofcurved beams, non-linear forced response studies are relatively few.The mode-coupled dynamics of curved beams and arches has beenstudied by Bolotin [1] theoretically based on two mode approxima-tion and experimental simulation. Sheinmann [2] has studied thedynamic buckling behaviour of shallow and deep circular arches.The solution methodology used could not capture the convergedsolutions adequately for the deep arches. Thomsen [3] has studiedthe auto-parametric resonance of the non-shallow arches using twomodes approximation. The chaotic motion with 2:1 internal reso-nance between the lowest symmetric and antisymmetric modes wasobserved. Steady state solutions in the presence of external excita-tion leading to 1:2 and 1:1 internal resonances were obtained by
∗ Corresponding author. Tel.: +911126581220; fax: +911126581119.E-mail address: [email protected] (Y. Nath).
0020-7462/$ - see front matter © 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijnonlinmec.2009.08.004
Tein and co-workers [4,5]. Malhotra and Namachchivaya [6,7] stud-ied the chaotic behaviour of shallow arches. The preceding analyses[4–7] were carried out employing the modified Melnikov method[8] using 2-DOF model and the studies were limited to the shallowarches. The non-linear response curves of shallow curved beamsresting on elastic foundation have been obtained by Oz et al. [9].In the study, a two mode solution has been assumed and thenon-linear equations have been solved by the method of multiplescales. Softening non-linearity due to the curvature was predictedfor the first mode, whereas the curvature effect was of hardeningtype for second mode. Bi and Dai [10] have studied the non-lineardynamics and bifurcation of the shallow arches subjected to pe-riodic excitation. Internal resonance, period-3 solutions and theroute to chaos by period doubling bifurcation were observed. Thedynamic stability of buckled beam considering the snap throughmotion under sinusoidal loading was investigated by Poon et al.[11] and an effort was made to obtain the non-linear frequency-response amplitude curves. Non-linear steady state response ofbeams, frames and shallow arches has been analysed in the fre-quency domain using the h-version straight beam finite elementby Chen et al. [12]. The geometrically non-linear thermoelasticvibration analysis of straight and curved beams has been carriedout using p-version hierarchical finite elements by Ribeiro andManoach [13]. The influence of temperature variation and curva-ture on the non-linear dynamics of curved beam has been studied.Periodic as well as aperiodic motions were observed. The vibrationof a curved beam in antisymmetric mode, due to auto-parametricexcitation, has been studied by Lee et al. [14] experimentally and
1074 S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
numerically. It has been found that the auto-parametric resonancephenomenon makes the primary region of dynamic instability sig-nificantly broader, and thus the instabilities occur under smallerforces and over wider range of excitation frequencies. Chen and Yang[15] have studied, both theoretically and experimentally, the non-linear vibration of shallow arch under the harmonic excitation at oneend. A geometrically exact approach to study the non-linear dynam-ics of rods undergoing finite bending, shearing and extension hasbeen formulated by Simo and Vu-Quoc [16] and numerical resultshave been presented for transient dynamics of straight beam, framesand bent rod. Based on the above formulation [16], Mata et al. [17]have developed fully geometric and constitutive non-linear modelfor the dynamic behaviour of beam structures considering an inter-mediate curved reference configuration. The model has been used tostudy the transient response of straight beams, frames and curvedbeams.
It is observed from the literature review that most of the stud-ies on forced response have been carried out by assuming solutiona priori—either a single mode solution or a two mode solution andemploying the harmonic balance or direct time integration to solvethe governing equations. Choosing a single or 2-mode solution hasthe advantage of being small and computationally efficient, but maynot be accurate enough [13]. The harmonic balance and incrementalharmonic balance methods lead to erroneous results if the appro-priate number of harmonics is not included. Limited time domainstudies are insufficient to draw conclusion about the unstable peri-odic response of curved beams. It is further observed that no studyis available which deals with the non-linear dynamic behaviour oflaminated curved beams.
It is further inferred from the literature that FEM has beensuccessfully employed in the non-linear analysis of curved beams[12,13]. Although the FE discretization results in a large system ofequations, but this weakness is offset by the resulting smaller band-width of set of equations for large dynamical systems. It may benoted that the solution of banded system of equations is computa-tionally more efficient. It can also be noted from the literature thatthe FE based system of equations for beams/plates/shells are solvedby the traditional shooting method [18–20] in which the secondorder governing equations of motion are transformed to first orderequations. This results in doubling the number of equations andthe banded nature of the system of equations is destroyed. In thispaper, these two issues are taken care of by directly applying theshooting method to the second order governing equations.
It is apt to make a mention here that to obtain steady statesolution using direct time integration, time marching for a largenumber of excitation cycles is generally required and the steadystate condition is not guaranteed. Thus significant computationaltime is required to get steady state solution for a particular forc-ing frequency. Further, direct time integration alone cannot pre-dict the steady state solution corresponding to unstable and evensome portion of stable regions of forcing frequency versus ampli-tude curves. But, the shooting method based approach predicts thesteady state response in about 8–10 shooting iterations, thus govern-ing equations are required to be integrated only for 8–10 cycles ofexcitation force. The total computational time is significantly lesserthan the direct time integration without shooting. Further, completefrequency–amplitude curve can be traced using arc-length continu-ation and shooting method.
To ascertain the steady state response employing the shootingmethod for second order systems, a FEM based methodology hasbeen developed for the first time in this paper. Its efficacy has beendemonstrated for the non-linear dynamics of isotropic and compos-ite curved beams. Stability of the solution and bifurcation pointsare also highlighted based on the eigenvalues of monodromy ma-trix (obtained as a by product of shooting method). For obtaining
the complete response, the analysis is carried out in two phases: (i)first starting from far enough from resonance, forcing frequency isincremented and the periodic solutions are obtained and (ii) whenthe bifurcation points are encountered, the solutions are continuedwith the arc length continuation method. For the second phase, forc-ing frequency is treated as an unknown and incremental arc lengthis specified [21].
2. Formulation
A curved beam (Fig. 1), having radius of curvature R, is consideredwith the co-ordinates x along the meridional direction and z alongthe thickness direction, respectively. The displacements uk and wk ata distance s from left end are expressed as functions of mid surfacedisplacement uo and w and independent rotation � of the normal inxz plane, as
uk(s, z, t) = uo(s, t)(1 + z/R) − zw,s(s, t)
+ [f (z) + gk(z)](w,s(s, t) + �(s, t)) (1a)
wk(s, z, t) = w(s, t) (1b)
The functions f (z) and gk (z) are defined as
f (z) = h/� Sin(�z/h) − h/�b44 Cos(�z/h) (2a)
gk(z) = akz + bk (2b)
where h is the thickness of the beam and t denotes time.The constants ak, bk and b44 (Eq. (2)) are determined by satisfying
interlayer continuity of displacements and transverse shear stressand top-bottom conditions on the latter [22]. The expressions of theconstants b44, ak and bk are given in Appendix A.
The strain displacement relations are based on kinematic approx-imations: (i) large deflection but small strains, (ii) moderately largerotation, and (iii) thin beam (z/R>1) such that 1+z/R ≈ 1. Transverseshear deformation is considered due to higher E/G ratio of compositebeam. Strain fields based on the above assumptions can be written as
{�}k ={
�xx
�xz
}k
= {�L}k + {�NL}k (3)
where
{�L}k ={
�pk
0
}+{
�bk
�sk
}and {�NL}k =
{ 12w
2,s
0
}
L
z, wk
x, uk
+
F=F0 Cos �Ft
R
s
Fig. 1. Geometry and coordinate system of the curved beam.
S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084 1075
The membrane �pk , bending �bk and shear �sk strains in Eq. (3) arewritten as
�pk = uo,s + w/R (4a)
�bk = −zw,ss + [f (z) + gk(z)](w,ss + �,s) + zuo,s/R (4b)
�sk = (f,z + gk,z)(w,s + �) (4c)
where the subscript comma denotes the partial derivative with re-spect to spatial co-ordinate succeeding it.
The stress–strain relation is written as
{r}k ={
�xx
�xz
}k
= [Q ]k{�}k (5)
where Qk14 = Qk
41 = 0; Qk11 = E1 and Qk
44 = G13 for 0◦ fiber angle;Qk11 = E2 and Qk
44 = G23 for 90◦ fiber angle; E and G denote Young'sand shear moduli, respectively; subscripts 1 and 2 are longitudinaland transverse fiber directions for a composite laminated beam.
The potential energy functional U for a curved beam of lengthL is written as
U(d) = (1/2)∫ L
0
N∑k=1
∫ hk+1
hk{�}kT [Q ]k{�}kbdzds
−∫ L
0{u0 w �}{f x fw m}T ds (6)
where d is the vector of degrees of freedom associated with thedisplacement field in a finite element discretization and b is thewidth of the beam; fx and fw are the forces per unit length in x andz directions, respectively; m is the moment per unit length of thebeam; N is the number of layers and hk, hk+1 denote z coordinatesof the bottom and top surfaces of the kth layer.
Following the procedure of Rajasekaran and Murray [23], the po-tential energy functional can be rewritten as
U(d) = {d}T [(1/2)[K] + (1/6)[K1(d)] + (1/12)[K2(d)]]{d} − {d}T {F} (7)
where [K] is the linear stiffness matrix; [K1] and [K2] are non-linearstiffness matrices linearly and quadratically dependent on the fieldvariables, respectively; {F} is the load vector.
The kinetic energy of the beam is given by
T(d) = (1/2)∫ L
0
N∑k=1
∫ hk+1
hk�(uk
2 + w2)bdsdz (8a)
where dot over the variable denotes the partial derivative with re-spect to time and � is the mass density. Using Eq. (1), Eq. (8a) canbe rewritten as
T(d) = (1/2)∫ L
0
N∑k=1
∫ hk+1
hk�{d}T[Zm]
T[Zm]{d}bdsdz (8b)
where
[Zm] =[1 + z / R 0 f + gk − z f + gk
0 1 0 0
]and
{d} = {u0 w w,s �}T
Substituting Eqs. (7) and (8b) in Lagrange's equation of motion, thegoverning equations for dissipative system become
[M]{d} + [C]{d} + [[K] + (1/2)[K1(d)] + (1/3)[K2(d)]]{d} = {F} (9)
where {d}, {d} and {d} are acceleration, velocity and displacementvectors, respectively. In the present study, proportional damping
[C] = [K] is assumed; = 2m/�m, m and �m being the modaldamping factor and natural frequency.
3. Solution methodology
Eq. (9) is solved using the methodology based on shooting tech-nique, Newmark time integration scheme [24] and Newton–Raphsoniteration method. In this approach, state vector at time t ( = 0 say){d(0)
d(0)
}={g} and solution
{d(t,g;�F)
d(t,g;�F)
}with a minimal period T are
sought [25] such that{d(T,g;�F)
d(T,g;�F)
}2N×1
= {g}2N×1 (10)
The value of time period (T) is taken to be known for systemswith harmonic excitation (T = 2�/�F) for response with fundamen-tal and higher harmonics. For obtaining the branches of the periodicresponse curve with subharmonic participation, response time pe-riod T is taken as integer multiple of fundamental period. For au-tonomous systems, time period (T) is treated as an unknown and anadditional equation such as phase condition/amplitude is required.
Since the state vector {g} is not known a priori, the solution isstarted from an initial guess {g0} and correction {Dg} is applied tothis such that{d(T,g0 +Dg;�F)
d(T,g0 +Dg;�F)
}− {g0 +Dg} ≈ {0} (11)
Using Taylor series and keeping only linear terms, Eq. (11) can berewritten as⎡⎢⎢⎢⎣
�d�g
(T,g0;�F)
�d�g
(T,g0;�F)
− I
⎤⎥⎥⎥⎦ {Dg} = {g0} −
{d(T,g0;�F)
d(T,g0;�F)
}(12)
where �d/�g and �d/�g are N×2N matrices and I is 2N×2N identitymatrix.
For the evaluation of �d/�g and �d/�g, differentiation of Eq. (9)with respect to � gives
[M]
[�d�g
]t+�t
+ [C]
[�d�g
]t+�t
+ [K]t+�t
[�d�g
]t+�t
= [0] (13)
where [K]t+�t = [[K] + [K1(d)] + [K2(d)]]t+�t is the tangent stiffnessmatrix.
The initial conditions associated with Eq. (13) are:⎡⎢⎢⎢⎣
�d�g
�d�g
⎤⎥⎥⎥⎦t=0
= [I] (14)
[�d�g
]t=0
= −[M]−1
[[C]
[�d�g
]t=0
+ [K]
[�d�g
]t=0
](15)
With marching in time, the solution of Eq. (9) starting from initialstate vector {g0} and that of Eq. (13) with initial condition given byEq. (14) is obtained at any time t using Newmark's time integrationapproach.
For the solution of Eq. (13) using Newmark's time integrationapproach,[
�d�g
]t+�t
and
[�d�g
]t+�t
1076 S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
and are written as:[�d�g
]t+�t
= a0
[[�d�g
]t+�t
−[
�d�g
]t
]− a2
[�d�g
]t
− a3
[�d�g
]t
(16)
[�d�g
]t+�t
= a1
[[�d�g
]t+�t
−[
�d�g
]t
]− a4
[�d�g
]t
− a5
[�d�g
]t
(17)
where a0 = 1 �t2 ; a1 = �
�t ; a2 = 1 �t ; a3 = 1
2 − 1; a4 = � − 1;
a5 = �t2
[� − 2
]; �t is time step size; � and are integration con-
stants.Using Eqs. (16) and (17), Eq. (13) can be rewritten as
[a0[M] + a1[C] + [K]]
[�d�g
]t+�t
= [a0[M] + a1[C]]
[�d�g
]t
+ [a2[M] + a4[C]]
[�d�g
]t
+ [a3[M] + a5[C]]
[�d�g
]t
(18)
Eqs. (16)–(18) are used to evaluate coefficient matrix in Eq. (12).Then Eq. (12) is solved for correction of initial state vector and thisprocedure is repeated until the state vector converges within thetolerance limit of 0.001%. The simultaneous equations are solvedusing Choleskey factorization in the implementation of the presentapproach. It can be noted that Choleskey factored left hand side ofEq. (18) is already available while solving Eq. (9). The eigenvalues
of the monodromy matrix [Y] =[
�d�g
�d�g
]Tare obtained using QR
algorithm to study the stability of periodic responses.
4. Element description
The element used is a 3-noded beam element (Fig. 2). Each endnode of the element has four degrees of freedom (u0, w, w,s, �) and
1
2
3
θ2
u02
θ1
w,s1
w1
u01
θ3
w,s3
w3
u03
�
Fig. 2. Element description showing the degrees of freedom at the three nodes.
Table 1Linear free vibration frequencies of the excited modes of the curved beams.
Boundary conditions Frequency (rad/s)
L/R = 0.25 L/R = 0.125 L/R = 0.0625
Immovable clamped–clamped Isotropic 1247.80 988.34 579.28Two-layered (0◦/90◦) 1174.63 1095.73 –Three-layered (0◦/90◦/0◦) 2263.99 1630.13 –
Immovable simply supported Isotropic 973.75 910.92 567.79Two-layered (0◦/90◦) 925.36 939.05 –Three-layered (0◦/90◦/0◦) 1799.10 1585.84 –
central node has two degrees of freedom (u0,�). The interpolationfunctions for the element are used directly to interpolate all fieldvariables in deriving the shear andmembrane strains. The Lagrangianinterpolation functions used for field variables u and � are
N1(�) = −1/2�(1 − �); N2(�) = (1 − �2); N3(�) = 1/2�(1 + �)
Hermitian functions used to interpolate w are
H1(�) = 1/4(2 − 3� + �3); H2(�) = 1/4(1 − � − �2 + �3)Le/2
H3(�) = 1/4(2 + 3� − �3); H4(�) = 1/4(−1 − � + �2 + �3)Le/2
where Le is the length of an element.The element is free from shear locking as w,s and � terms are
interpolated consistently.
5. Numerical results
The effect of curvature, loading amplitude and support conditionson the non-linear vibration response of isotropic and cross-ply lam-inated composite curved beams subjected to the central harmonicforce is investigated.
The material properties considered are
Isotropic: E1 = E2 = 71.72GPa,G13 = G23 = 26.96GPa, � = 2800kg/m3
Composite: E1 = 147.0GPa, E2 = 10.7GPa, G13 = G23 = 7.0GPa,� = 1600kg/m3
Boundary conditions considered are
Immovable simply supported: u0 = w = 0 at s = 0, LImmovable clamped: u0 = w = w,s = � = 0 at s = 0, L
The geometrical parameters of the beam are: length (L) = 0.5m;width (b) = 0.02m; thickness (h) = 0.002m; and L/R is taken as 0.25,0.125, 0.0625. Linear free vibration frequencies of the excited modes(�) for the beams considered are given in Table 1 .
Concentrated force acting at the centre of the curved beam istaken as: F = F0 cos�Ft, where F0 is the force amplitude and �F isthe forcing frequency. Modal damping factor m = 0.005 is assumed.
Based on the convergence study, 10 and 20 elements discretiza-tion is found to be sufficient for modelling isotropic and compositecurved beams, respectively.
Validation of the present formulation and solution methodologyis carried out considering forced response of straight isotropic beam[19] and results are shown in Fig. 3. It can be noted that the resultsare in good agreement.
In all the cases considered, first symmetric mode is excited andthe curves relating the non-dimensional transverse displacement(w/h) at the loading point and the non-dimensional frequency(�F/�m) are obtained. In the response versus forcing frequencycurves, the stable regions are shown by solid curves and the unstable
S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084 1077
Present
Ref. [19]
Forcing Frequency (�F/�m)
Am
plitu
de R
atio
(w/h
)
0
0.3
0.6
0.9
1.2
1.5
1.8
0.8 1 1.2 1.4 1.6
Fig. 3. Comparison of steady state amplitude versus frequency curve for straightisotropic beam (E = 71.72GPa, � = 0.33, � = 2800kg/m3, L = 0.406m, b = 0.02,h = 0.002m, = 0, �m = 396.54 rad/s).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
(L/R = 0.25)
(L/R = 0.125)
(L/R = 0.0625)Am
plitu
de R
atio
(w/h
)
Period doublingCyclic fold Symmetry breaking
Forcing Frequency (�F/�m)0.8 0.9 1 1.1 1.2
Fig. 4. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable clamped–clamped isotropic curved beams (F0 = 0.5N).
ones are shown by dashed curves. The unstable regions are identifiedwhenever at least one eigenvalue of the monodromy matrix crossesunit circle. The bifurcation points marked in the response curvesare identified either as period doubling if at least one eigenvalue ofthe monodromy matrix crosses unit circle along negative real axisor as symmetry breaking/cyclic fold if it leaves the unit circle alongpositive real axis or as secondary Hopf bifurcation if two complexconjugate eigenvalues cross the unit circle. The results are presentedand discussed for different cases in the following sub-sections.
5.1. Immovable clamped–clamped isotropic curved beams
The effect of length to radius ratio (L/R) on the forced responseof isotropic curved beams is shown in Figs. 4 and 5 for load
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1
(L/R = 0.25)
(L/R = 0.125)
(L/R = 0.0625)
Am
plitu
de R
atio
(w/h
)
25
46
Period doubling
Cyclic fold
Secondary Hopf
3
Forcing Frequency (�F/�m)0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Fig. 5. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable clamped–clamped isotropic curved beams (F0 = 1.0N).
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-2
-1.5
-1
-0.5
0
0.5
1
-2.5
-1.5
-0.5
0.5
1.5
-2
-1.5
-1
-0.5
0
0.5
1
-3
-2
-1
0
1
2
-2.5
-1.5
-0.5
0.5
1.5
w/h t
w/h
t
w/h
w/h
w/h
w/h
tt
tt
0 0.25 0.5 0.75 10 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
Fig. 6. Steady state central response amplitude versus time curves for differentforcing frequency ratios corresponding to points marked in Fig. 5 for immovableclamped–clamped isotropic curved beams (F0 = 1.0N, L/R = 0.0625).
1078 S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
(L/R = 0.0625)
Am
plitu
de R
atio
(w/h
)A
mpl
itude
Rat
io (w
/h)
Period doubling
Cyclic fold Secondary Hopf
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(L/R = 0.125)
(L/R = 0.25)
Period doubling
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2
Forcing Frequency (�F/�m)
Forcing Frequency (�F/�m)
Fig. 7. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable simply supported isotropic curved beams (F0 = 0.5N).
amplitude F0 = 0.5 and 1.0N, respectively. It is seen that as the ini-tial curvature decreases, the beam becomes stiffer and consequently,the vibration amplitude decreases. The non-linearity is hardeningtype for L/R = 0.25 and softening one for L/R = 0.125, 0.0625. It isinteresting to observe that for F0 = 0.5 and 1N, the response curvesare qualitatively similar for L/R = 0.25 and 0.125, whereas they aresignificantly different for L/R = 0.0625 with multiple peaks for 1Nload amplitude. These multiple local peaks may be attributed tothe exchange of energy among modes. It is further noticed that thenumber of unstable regions is increased for L/R = 0.0625 subjectedto the load with 1N amplitude. Central transverse displacement ver-sus time curves are plotted in Fig. 6 for L/R = 0.0625 and F0 = 1.0Nfor one cycle of the steady state response. It can be seen from thisfigure that the inward displacement is significantly greater than theoutward displacement. In particular, around the peak region in therange of �F/�m = 0.6–0.66, the inward displacement magnitude is3 times of the outward displacement with the presence of higherharmonics.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Am
plitu
de R
atio
(w/h
)
i
ii
iii
iv
b
c
d(L/R = 0.25)
(L/R = 0.125)
a
Period doublingCyclic fold
Forcing Frequency (�F/�m)0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Fig. 8. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable clamped–clamped two-layered cross-ply (0◦/90◦) composite curvedbeams (F0 = 1.0N).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Am
plitu
de R
atio
(w/h
)
(L/R = 0.25)
(L/R = 0.125)
Period doublingCyclic fold
0.6 0.7 0.8 0.9 1 1.1 1.2Forcing Frequency (�F/�m)
Fig. 9. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable clamped–clamped three-layered cross-ply (0◦/90◦/0◦) compositecurved beams (F0 = 2.0N).
5.2. Immovable simply supported isotropic curved beam
The effect of curvature (L/R) on the steady state response is de-picted in Fig. 7 for loadmagnitude of F0 = 0.5N. Softening is observedeven for L/R = 0.25. For L/R = 0.0625, a complicated behaviour is ob-served with the occurrence of multiple peaks. The response ampli-tude, frequency of occurrence of unstable regions and non-linearityfor L/R = 0.25 and 0.125 are comparablewith those for correspondingclamped cases. However, they differ significantly for L/R = 0.0625.
5.3. Immovable clamped–clamped composite curved beams
The responses of cross-ply laminated beams (L/R = 0.25, 0.125)are shown in Figs. 8 and 9 for two-layer (0◦/90◦) and three-layer(0◦/90◦/0◦), respectively. The softening behaviour is observed except
S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084 1079
-600
-300
0
300
600
w/h
w/h
-1200
-600
0
600
1200
w/h
w/h.
w/h
t
0 0.25 0.5 0.75 1
w/h
t
0 0.25 0.5 0.75 1
s/L
w/h
0 0.25 0.5 0.75 1
s/Lw/h
0 0.25 0.5 0.75 1
s/Lw/
h
0 0.25 0.5 0.75 1
-0.4
0
0.4
-0.5
0
0.5
-0.3
0
0.3
-0.2
0
0.2
s/L
w/h
0 0.25 0.5 0.75 1
w/h t
0 0.25 0.5 0.75 1
-0.6
-0.4
-0.2
0
0.2
0.4
-0.8
-0.4
0
0.4
-0.4
-0.2
0
0.2
0.4
-0.2
-0.1
0
0.1
0.2
w/h t
0 0.25 0.5 0.75 1
-600
-300
0
300
600
w/h
w/h.
-0.5 0 0.5
-0.5 0 0.5
-250
-150
-50
50
150
250
w/h
w/h.
-0.2 0 0.2
-1 0
Fig. 10. Time history , phase-plane and deformation shape plots corresponding to points marked in Fig. 8 for immovable clamped–clamped two-layered cross-ply (0◦/90◦)composite curved beam (F0 = 1.0N, L/R = 0.25).
for L/R = 0.25 case of three-layered (0◦/90◦/0◦) beams. For this case,a slight hardening behaviour is observed. It is seen that the responsefor two-layered (0◦/90◦) beam with L/R = 0.25 becomes unstable at�F/�m = 0.911 with peak amplitude of 0.34 and regains the stabilityat �F/�m = 0.919 with the occurrence of another peak at the fre-quency ratio of �F/�m = 1.04. It is interesting to note that unstableregion is observed for a wide frequency range �F/�m = 0.9–1.2 fortwo-layered beam having L/R = 0.125.
Time history, phase-plane and deformation shape plots corre-sponding to the maximum outward displacement, at typical pointsmarked in Fig. 8, are shown in Fig. 10. It can be seen from this fig-ure that the beam vibrates with different inward and outward dis-placements. In the peak response amplitude region, the difference
between inward and outward amplitudes at a particular location (s)of the beam is significantly greater. The loops observed in the phaseplane plots indicate the presence of higher harmonics.
Similar results for L/R = 0.125, for different frequency ratios, areshown in Fig. 11. Starting with a lower frequency ratio �F/�m =0.905, a harmonic motion of the symmetrically excited mode isshown. At larger vibration amplitude (w/h ≈ 0.63, �F/�m = 0.92),second harmonic appears as depicted in the phase–plane plot.For w/h = 0.52 and �F/�m = 0.96, the coupling between modesbecomes stronger. With further increase in the frequency ratio(�F/�m = 1.081), the beam vibrates with the dominant participa-tion of a single mode. This qualitative change in the deformationshape is not observed for L/R = 0.25 (Fig. 10).
1080 S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
-400
-200
0
200
400
-800
-400
0
400
800
-800
-400
0
400
800
-0.6
-0.2
0.2
0.6
-1.2
-0.8
-0.4
0
0.4
0.8
-1.2
-0.8
-0.4
0
0.4
0.8
-0.4
-0.2
0
0.2
0.4
-300
-150
0
150
300
-0.2
0
0.2
0.4
0
0.2
0.4
0.6
0.8
-0.2
0
0.2
0.4
0.6
-0.1
0
0.1
0.2
0.3
w/h
w/h
w/h
w/h
s/L
s/L
s/L
s/L
t
t
t
t
w/h
w/h
w/h
w/h
w/h.
w/h.
w/h.
w/h.
w/h
w/h
w/h
w/h
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
-0.5 0 0.5
-1.5 -0.5 0.5
-1 -0.5 0 0.5 1
-0.3 -0.1 0.1 0.3
Fig. 11. Time history, phase-plane and deformation shape plots corresponding to points marked in Fig. 8 for immovable clamped–clamped two-layered cross-ply (0◦/90◦)composite curved beam (F0 = 1.0N, L/R = 0.125).
Am
plitu
de R
atio
(w/h
)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
i
iiiii
iv
a
b
d
c
(L/R = 0.25)
(L/R = 0.125)
Period doubling
Cyclic fold Secondary Hopf
Forcing Frequency (�F/�m)0.6 0.7 0.8 0.9 1 1.1 1.2
Fig. 12. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable simply supported two-layered cross-ply (0◦/90◦) composite curvedbeams (F0 = 1.0N).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Am
plitu
de R
atio
(w/h
)
(L/R = 0.25)
(L/R = 0.125)
Period doublingSecondary Hopf Cyclic fold
0.6 0.7 0.8 0.9 1 1.1 1.2Forcing Frequency (�F/�m)
Fig. 13. Non-linear steady state response amplitude versus forcing frequency curvesfor immovable simply supported three-layered cross-ply (0◦/90◦/0◦) compositecurved beams (F0 = 2.0N).
S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084 1081
-300
-150
0
150
300
-800
-400
0
400
800
-1000
-500
0
500
1000
-0.4
-0.2
0
0.2
0.4
-1.2
-0.8
-0.4
0
0.4
0.8
-1.5
-1
-0.5
0
0.5
1
-0.3
-0.15
0
0.15
0.3
5 5
-300
-150
0
150
300
-0.3
-0.1
0.1
0.3
-1
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0
0.5
1
-0.8
-0.4
0
0.4
0.8
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
w/h
w/h
w/h
w/h
s /L
s/L
s/L
s/L
t
t
t
t
w/h
w/h
w/h
w/h
w/h.
w/h.
w/h.
w/h.
w/h
w/h
w/h
w/h
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.5 10.2 0.7
-0.5 0 0.5
-1.5 -0.5 0.5
-2 -1 0 1
-0.3 0 0.3
Fig. 14. Time history, phase-plane and deformation shape plots corresponding to points marked in Fig. 12 for immovable simply supported two-layered cross-ply (0◦/90◦)composite curved beam (F0 = 1.0N, L/R = 0.25).
5.4. Immovable simply supported composite curved beams
The response curves for immovable simply supported two-layered (0◦/90◦) and three-layered (0◦/90◦/0◦) composite curvedbeams are shown in Figs. 12 and 13, respectively. Softeningnon-linear behaviour is predicted for both two-layered beamwith L/R = 0.25 and three-layered beam with L/R = 0.125. Forthree-layered beam with L/R = 0.25 and two-layered one withL/R = 0.125, the response curves with two distinct peaks are pre-dicted. For the three-layered case, softening behaviour is observedfor �F/�m � 1.04 and hardening one for �F/�m > 1.04. The re-sponse is unstable for a wide forcing frequency range for two-and three-layered beams with L/R = 0.125 similar to immovableclamped–clamped curved beams.
Time history, phase-plane and deformation shape plots (cor-responding to the points marked in Fig. 12) are shown in
Figs. 14 and 15 for L/R = 0.25 and 0.125, respectively. The influenceof higher modes is more pronounced compared to the correspond-ing clamped–clamped beam cases. The relative participation ofdifferent modes during one cycle of steady state response is given inTable 2, and at the same time instants, the variations of transversedisplacement along length are shown in Fig. 16 for two-layeredimmovable simply supported beam (L/R = 0.125, �F/�m = 1.023).It is interesting to observe that the deformation shape changessignificantly within the cycle. The modal participation factors fordifferent forcing frequencies are given in Table 3 for two-layeredimmovable simply supported beam (L/R = 0.125, t = T). It is revealedfrom this table that at least five symmetric modes participate in thenon-linear response of the curved beams, and the changes in defor-mation shapes at the time instants, corresponding to the maximumoutward displacement, are attributed to the changes in the relativeparticipation of different modes.
1082 S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
-400
-200
0
200
400
-0.2
0
0.2
0.4
-1000
-500
0
500
1000
0
0.2
0.4
0.6
0.8
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-400
0
400
800
-0.2
0
0.2
0.4
0.6
-0.4
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0
0.2
0.4
-1.5
-1
-0.5
0
0.5
1
-0.9
-0.6
-0.3
0
0.3
0.6
-0.6
-0.4
-0.2
0
0.2
0.4
-600
-300
0
300
600
-0.4
-0.2
0
0.2
0.4
w/h
w/h
w/h
w/h
s/L
s/L
s/L
s/L
t
t
t
t
w/h
w/h
w/h
w/h
w/h.
w/h.
w/h.
w/h.
w/h
w/h
w/h
w/h
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
0 0.25 0.5 0.75 1
-0.5 0 0.5
-1.5 -1 -0.5 0 0.5 1
-1 0
-0.5 0 0.5
Fig. 15. Time history, phase-plane and deformation shape plots corresponding to points marked in Fig. 12 for immovable simply supported two-layered cross-ply (0◦/90◦)composite curved beam (F0 = 1.0N, L/R = 0.125).
6. Conclusions
The modified shooting method based on Newmark time march-ing scheme and arc length continuation algorithm is successfullydeveloped and its efficacy is demonstrated for the study of non-linear vibration behaviour of curved beams. The study is carriedout using FEM based solution approach wherein unlike analyticalapproaches, modes are not assumed a priori. The higher order sheardeformation theory (HSDT) satisfying the interlayer continuity oftransverse shear stress and top-bottom boundary conditions isused for the finite element formulation. Proposed solution strategyis computationally more efficient compared to existing methodswhich have usually been used for systems with fewer degreesof freedom. The modal interaction involving the participation of
multiple modes can be studied using FEM based solution requir-ing, in some cases, discretization with a quite large number ofdegrees of freedom. The participation of multiple modes in theproblems studied is a clear indication that this important non-linearphenomenon has fundamental importance in the study of curvedbeams. Some of the conclusions drawn from the detailed studyare:
(i) With the change in support conditions, there is a change innon-linearity, especially for beams of small curvature.
(ii) Deeper beams vibrate with distinct higher harmonics, especiallynear the peak regions of response amplitudes.
(iii) Occurrence of unstable regions increases with the increase inthe excitation force magnitude.
S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084 1083
Table 2Relative modal participation factors for immovable simply supported two-layered cross-ply (0◦/90◦) composite curved beam (�F/�m = 1.023, L/R = 0.125).
Time (T) Modal participation factors (symmetric modes)
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
0.1 0.2490 0.2096 −0.0371 −0.0067 0.00340.2 0.1067 −0.0075 0.0461 −0.0022 0.00130.3 −0.0953 −0.1419 0.0035 0.0069 −0.00340.4 −0.3412 −0.0902 −0.0239 0.0135 −0.00530.5 −0.6736 0.2359 0.0191 −0.0041 −0.00080.6 −0.7636 0.3629 0.0151 −0.0031 0.00150.7 −0.4699 0.0341 −0.0324 0.0054 −0.00340.8 −0.2081 −0.1517 −0.0029 0.0170 −0.00420.9 −0.0104 −0.0596 0.0513 −0.0030 0.00001.0 0.1829 0.1438 −0.0147 0.0004 0.0030
*T denotes response time period.
-0.1
0
0.1
0.2
0.3
-1.5
-0.5
0.5
-0.35
-0.2
-0.05
-1
-0.2
-0.8
-0.4
0
-0.7
-0.3
-1.2
-0.6
0
-0.25
-0.1
0
0.2
0.4
0
0.3
0.6
w/h
s/L
s/Ls/L
s/L
s/Ls/L
s/Ls/L
s/L s/L
w/h
w/h
w/h
w/h
w/h
w/h
w/h
w/h
w/h
0 0.5 1
0 0.5 1
0 0.5 10 0.5 1
0 0.5 1 0 0.5 1
0 0.5 10 0.5 1
0 0.5 10 0.5 1
Fig. 16. Deformation shape plots in one time period T (corresponding to Table 2) for immovable simply supported two-layered cross-ply (0◦/90◦) composite curved beam(�F/�m = 1.023, L/R = 0.125).
Table 3Modal participation factors for immovable simply supported two-layered cross-ply (0◦/90◦) composite curved beam (L/R = 0.125).
Frequency ratio (�F/�m) Modal participation factors (symmetric modes)
Mode 1 Mode 2 Mode 3 Mode 4 Mode 5
0.714 0.1260 −0.0437 −0.0068 0.0012 0.00050.85 0.1304 −0.0008 −0.0060 −0.0002 0.00111.166 −0.1794 0.0997 −0.0086 0.0004 0.00081.029 0.1168 0.1111 0.0204 0.0007 0.00271.023 0.1829 0.1438 −0.0147 0.0004 0.00301.2 −0.1298 0.0776 −0.0077 0.0005 0.0008
1084 S.M. Ibrahim et al. / International Journal of Non-Linear Mechanics 44 (2009) 1073 -- 1084
(iv) At least five symmteric modes participate in the non-linear be-haviour of curved beams considered in the study, and the rela-tive participation of modes changes with time.
Appendix A.
The details of constants used in Eq. (2):
b44 =∑N−1
k=1 (Qk+144 − Qk
44)Cos�zk
h∑N−1k=1 (Q
k44 − Qk+1
44 )Sin�zk
h+ Q1
44 + QN44
(A.1)
a1 = b44; aN = −b44 (A.2)
ak+1 = Qk44
Qk+144
(Cos
�zk
h+ b44 Sin
�zk
h+ ak
)
−(Cos
�zk
h+ b44 Sin
�zk
h
)(A.3)
bk+1 = bk + (ak − ak+1)zk (A.4)
bk0 = h�b44 (A.5)
where zk is the z coordinate of the kth interface, N is the number oflayers and k0 is the layer number containing point z = 0.
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