dynamic buckling of a base-excited thin cylindrical …dynamic buckling of a base-excited thin...
TRANSCRIPT
Dynamic buckling of a base-excited thin cylindrical shell carrying a
top mass ⋆
N.J. Mallon, R.H.B. Fey, H. NijmeijerDepartment of Mechanical Engineering, Eindhoven University of Technology
PO Box 513, 5600 MB Eindhoven, the Netherlands
[email protected], [email protected], [email protected]
Abstract
This paper considers dynamic buckling of a harmonically base-excited vertical cylindrical shell carrying a top mass.Based on Donnell’s nonlinear shell theory, a semi-analytical model is derived which exactly satisfies the (in-plane)boundary conditions. This model is numerically validated through a comparison with quasi-static and modal analysisresults obtained using finite element modelling. The steady-state nonlinear dynamics of the base-excited cylindricalshell with top mass are examined using both numerical continuation of periodic solutions and standard numericaltime integration. In these dynamic analyses the cylindrical shell is preloaded by the weight of the top mass. Thispreloading results in a single unbuckled stable static equilibrium state. A critical value for the amplitude of theharmonic base-excitation is determined. Above this critical value, the shell exhibits an instationary beating type ofresponse with time intervals showing severe out-of-plane deformations (it buckles dynamically). Similar as for thestatic buckling case, the critical value highly depends on the initial imperfections present in the shell.
Key words: Dynamic Stability, Cylindrical Shells, Semi-Analytical approach.
1. Introduction
Due to their favourable stiffness-to-mass ratio, thin cylindrical shells are encountered in a wide variety ofapplications. It is well known that under compressive loading, such structures may loose their stability, thatis they buckle. The classical static buckling analyses of axially compressed perfect cylindrical shells (basedon linearized small deflection theory) [1, 2] predict many closely spaced buckling loads with buckling modesbeing sinusoidal both in axial direction and in circumferential direction. Experiments on the static bucklingof axially compressed cylindrical shells reveal a large scatter in the obtained buckling loads and notoriousdiscrepancies with the results obtained from the classical static buckling analyses [1, 3]. Both the inevitablesmall deviations from the nominal cylindrical shape (geometric imperfections) and the boundary conditionshave been widely accepted as contributing to the poor correlation between the experimental results andthe classical static buckling results. In practice, thin-walled structures are often not only subjected to apure static loading but to a combination of static loading and dynamic loading. For example, aerospace
⋆ This research is supported by the Dutch Technology Foundation STW, applied science division of NWO and the technologyprogramme of the Ministry of Economic Affairs (STW project EWO.5792).
Artic Summer Conference on Dynamics, Vibrations and Control August 6-10, 2007, Ivalo, Finland
structures are often dimensioned based on the loading conditions during launch [4]. During launch, thepropulsion forces will accelerate the structure resulting in inertia forces consisting of a summation of staticloads and vibrational type of loads. The resistance of structures liable to buckling, to withstand such dynamicloading is often addressed as the dynamic stability of these structures. In the past, many studies alreadyhave been performed concerning the dynamic stability of thin-walled structures. Design strategies, takingrigourously the dynamics of such structures under dynamic loading into account are, however, still lacking.The research described in this paper is intended as a (first) step in deriving such design strategies and dealswith the dynamic stability problem of a base-excited thin cylindrical shell with top mass, see Fig. 1.
Due to the harmonic base-excitation and the weight of the top mass, the cylindrical shell under considera-tion is loaded in axial direction by a combination of a static load and a time dependent load. Indeed, alreadyin many papers (see for example [5–10]) the dynamic stability of harmonic axially loaded thin cylindricalshells is considered. However, these studies consider parametric instabilities of cylindrical shells without atop mass, i.e. axi-asymmetric vibration modes are excited through a Mathieu type of instability aroundexcitation frequencies equal to two times the eigen-frequency of an axi-asymmetric vibration mode. Dueto the top mass, a relatively low frequent axi-symmetric resonance is introduced (far below the parametricinstability regions). In this paper, the dynamic stability of the shell around this resonance is studied usinga semi-analytical approach.
The semi-analytical model of the shell is based on Donnell ’s nonlinear shell theory [11]. The effect ofin-plane inertia is neglected, resulting in two static equilibrium equations for the in-plane fields and onedynamic equilibrium equation for the out-of-plane field. For the case of in-plane boundary conditions interms of membrane forces (as considered for example in [5–10]), the equilibrium equations are usually writtenin terms of an in-plane stress function and the out-of-plane displacement field, see [5–9] for more details.By solving the stress-function analytically for an assumed expression of the out-of-plane displacement field,the number of independent displacement fields is reduced from three to one, resulting in a model with aminimum of degrees of freedom. The stress-function approach is, however, less straightforward for the casewhen the boundary conditions involve the in-plane displacements (as considered here).
Since the in-plane boundary conditions have a significant influence on the thin shell behaviour [2, 12–14],an alternative static reduction approach is followed here which solves the in-plane displacement fields interms of the out-of-plane displacement field. The obtained expressions for the in-plane displacement fieldssatisfy exactly the in-plane boundary conditions for the cylindrical shell with rigid end-disks. Note that acomparable approach, however only for one specific three mode expansion of the out-of-plane displacementfield, is followed in [15]. The resulting nonlinear model is numerically validated through a comparison withstatic and modal analysis results obtained using finite element modelling (FEM). The steady-state nonlineardynamics of the base-excited cylindrical shell with top mass is examined using numerical continuation ofperiodic solutions for a varying excitation frequency and by using standard numerical integration.
During the analysis, the influence of geometrical imperfections is taken into account. For the case ofparametrically induced instabilities it is known that geometrical imperfections tend to increase the dynamiccritical loads [8, 10, 16]. Base-excited cylindrical shells carrying a top mass are previously studied for thecase of shock loading in [15] and for harmonic loading in [17]. However, in [17] different boundary conditionsfor the top mass are considered (completely free instead of radially and rotationally restrained) and onlybeam-type modes are considered (no axi-asymmetrical modes). Results for a base-excited thin beam withtop mass can be found in [16]. An extensive review of the research performed on the nonlinear vibrations ofcylindrical shells until 2003 can be found in [18].
The outline for the paper is as follows. The next section will deal with the derivation of the equationsof motion. In section 3, buckling of the cylindrical shell under a static loading will be discussed and amodal analysis will be performed. The influence of initial imperfections will be illustrated and results willbe compared with FEM results. Dynamic buckling of the base-excited cylindrical shell with top mass willbe discussed in section 4. Finally, in section 5 conclusions will be presented.
2
Ub(t) = rdg sin (2πft)
top mass
g
thin cyl. shell
Fig. 1. Base excited cylindrical shell with top mass.
x
u
h
x = 0
x = L
R θ
w, w0
v
Fig. 2. Cylindrical shell geometry.
2. Semi-analytical model
In this section, a semi-analytical model is derived for the thin cylindrical shell which carries a rigidmass mtop on top and is loaded in axial direction by a prescribed base excitation Ub(t) and by gravityg, see figure 1. The dimensions of the cylindrical shell are defined by a radius R, thickness h and lengthL. Considering the cylindrical coordinate system [r = R, θ, x] (see Fig. 2), the axial in-plane displacementfield is denoted by u(t, x, θ), the circumferential in-plane displacement field by v(t, x, θ), the radial out-of-plane displacement field by w(t, x, θ) and the radial imperfection shape by w0(x, θ). For readability, thenotations for the displacement fields and radial imperfection shape will be abbreviated to u, v, w and w0,respectively. The axial coordinate x and axial displacement field u are measured relative with respect to theprescribed base-motion Ub(t). Donnell’s shallow shell theory is adopted [2, 11] as kinematic model for thethin cylindrical shell. According to Donnell ’s assumptions, the nonlinear strain-displacement relations read
εx = u,x + 1
2w,2x +w,x w0,x , κx = −w,xx ,
εθ = 1
R (v,θ +w) + 1
2R2 w,2θ + 1
R2 w,θ w0,θ , κθ = − 1
R2 w,θθ ,
γxθ = 1
Ru,θ +v,x + 1
R (w,x w,θ +w,x w0,θ +w0,x w,θ ) , κxθ = − 1
Rw,xθ ,
(1)
where ,x means ∂∂x and ,θ means ∂
∂θ . Note that in Eq. (1), the radial displacement field w and the radialimperfection shape w0 are measured positively inwards. Considering linear isotropic homogeneous materialproperties, the stress resultants and stress couples per unit length are defined by
Nx = J (εx + νεθ) , Mx = D (κx + νκθ) ,
Nθ = J (εθ + νεx) , Mθ = D (κθ + νκx) ,
Nxθ = 1
2J(1 − ν)γxθ, Mxθ = D(1 − ν)κxθ,
(2)
where J = Eh/(1 − ν2) and D = Eh3/(12(1 − ν2)). The following boundary conditions for the cylindricalshell with rigid end-disks are considered (’-’ means not prescribed)
u u,θ v w Mx
x = 0 0 0 0 0 0
x = L − 0 0 0 0
. (3)
3
Note that the prescribed base-motion does not appear in the boundary conditions, since u and x are measuredrelatively with respect to Ub(t). Furthermore, the boundary condition in terms of the membrane force Nx
at x = L due to the inertia forces of the top mass are not included. This force will be included via the kineticenergy. Indeed, for a thin cylindrical shell mounted between two rigid end-disks, the clamping condition (inpractice) is probably closer to the case w,x = 0 (instead of Mx = 0). Nevertheless, the clamping condition isassumed to be Mx = 0 since this allows for a simple expansion of w. This assumption is supported by thefact that previous studies show that the rotational restraint (Mx = 0 vs. w,x = 0) has only a mild influenceon the static buckling [2], eigenfrequencies [12, 13] and nonlinear vibrations [13, 14].
The strain energy of the shell, corresponding to Donnell ’s assumptions, reads as
Us =1
2J
∫
2π
0
∫ L
0
(
ε2
x + ε2
θ + 2νεxεθ +1 − ν
2γ2
xθ
)
dxRdθ
+1
2D
∫
2π
0
∫ L
0
(
κ2
x + κ2
θ + 2νκxκθ +1 − ν
2κ2
xθ
)
dxRdθ.
(4)
The case mtop ≫ mshell, where mtop is the top mass and mshell the mass of the shell, is considered.Consequently, the effect of the mass of the shell is neglected in the potential energy of the structure due togravity
Ug = mtopg (Ub(t) + u(t, L, θ)) . (5)
Furthermore, also the influence of in-plane inertia of the shell is neglected in the kinetic energy
T = 1
2ρh
∫
2π
0
∫ L
0
w2 dxRdθ + 1
2mtopu
2
t , (6)
where ut = Ub(t)+ u(t, L, θ) (note that u,θ = 0 for x = L). Previous studies concerning nonlinear vibrationsof cylindrical shells show that neglecting the in-plane inertia results in a moderately overestimated softeningbehaviour [19] and a moderately overestimated dynamic critical load (considering a parametric instability)[9, 10]. These studies, however, consider cylindrical shells without a top mass.
Using Hamilton’s variation principle on the basis of the sum of Eqs. (4), (5) and (6), the correspondingnonlinear equilibrium equations follow to be
RNx,x +Nxθ = 0, (7)
RNxθ + Nθ,θ = 0, (8)
D∇4w +1
RNθ − Nx (w,xx +w0,xx ) +
2
RNxθ (w,xθ +w0,xθ ) +
1
R2Nθ (w,θθ +w0,θθ ) = ρhw, (9)
where the biharmonic operator ∇4 is defined by ∇4w = w,xxxx + 2
R2 w,xxθθ + 1
R4 w,θθθθ. Since the effects ofin-plane inertia are neglected, Eqs. (7)-(9) constitute a set of two static (in-plane) equilibrium equations(Eqs. (7)-(8)) and one dynamic (out-of-plane) equilibrium equation (Eq. (9)).
The out-of-plane displacement field is expanded as
w(t, x, θ) =
N∑
i=1
M∑
j=0
[
Qsij(t) sin (jnθ) + Qc
ij(t) cos (jnθ)]
sin (λix) , (10)
where λi = iπ/L, i is the number of axial half-waves, n is the number of circumferential waves and Qs,cij (t)
are N(2M + 1) generalized degrees of freedom (dof). Note that Eq. (10) satisfies exactly the boundaryconditions for w, see Eq. (3). The N dof Qc
i0(t) correspond to axi-symmetric radial displacements and the2NM dof Q
s,cij (t) (j 6= 0) to axi-asymmetrical displacement fields. The presence of pairs of modes (with dof
Qsij and Qc
ij) with the same shape but with a different angular orientation is due to axi-symmetry of the(perfect) shell. Considering, for example, a radial imperfection with shape sin(jnθ) sin(λix), the mode Qs
ij
will be excited directly by the axial loading [8]. However, depending on the amplitude and frequency of theaxial excitation, also the companion mode Qc
ij can appear in the response with a certain difference in phasewith respect to the driven mode Qs
ij. The latter phenomenon may result in a travelling-wave vibration incircumferential direction [7, 18, 20].
4
The following (axi-asymmetrical) expansion of w0 is considered
w0(x, θ) = h
Ne∑
i=1
ei1 sin (nθ) sin
(
iπx
L
)
, (11)
where ei1 are dimensionless imperfection amplitudes. Note that axi-symmetric imperfection shapes appearedto have less influence on the obtained results and are, therefore, not considered.
Using the assumed expressions for w and w0, the in-plane equilibrium equations (Eqs. (7)-(8)) now consistof a set of linear inhomogeneous static partial differential equations in terms of only u and v. In order toperform a reduction of the independent displacement fields, this set of PDE’s is solved a priori. The solutionprocedure for this purpose results in expressions for u and v satisfying exactly the in-plane equilibriumequations (Eqs. (7) - (8))) and the in-plane boundary conditions (Eq. (3)). During this step, an extra dofUt(t) is introduced which corresponds to the unknown axial displacement of the top mass. Details of thesolution procedure are omitted here for the sake of brevity.
2.1. Equations of motion
Now the displacement fields are known, the equations of motion in terms of the dof
Q(t) =[
Qcij(t),Q
sik(t),Ut(t)
]T, i = 1..N, j = 0..M, k = 1..M, (12)
are determined using the Rayleigh-Ritz procedure. First the energy expressions (Eqs. (4)-(6)) are evaluatedsymbolically. Subsequently, the equations of motion are determined using Lagrange’s equation
d
dtT,Q −T,Q +V,Q = 0, (13)
where V = Us + Ug. Damping in the structure is modelled by adding to each equation of motion a linear
viscous damping term in the form −cijQs,cij or −ctUt, respectively.
In summary, the equations of motion are derived by the following steps(i) Discretize the out-of-plane displacement field w as in Eq. (10) and the radial imperfection shape w0
as in Eq. (11).(ii) Solve the corresponding in-plane fields u and v. During this step, an extra dof Ut(t) is introduced
which corresponds to the unknown axial displacement of the top mass.(iii) Substitute the resulting expressions for w, w0, u and v in the energy and work expressions Eqs. (4)-(6),
and evaluate the integrals symbolically.(iv) Derive the equations of motion using Lagrange’s equation Eq. (13) and add to each equation of motion
a linear viscous damping term in the form −cijQs,cij or −ctUt, respectively.
This procedure is implemented in Maple routines [21], allowing to derive the equations of motions in anautomatic manner after the expressions for w and w0 have been supplied. The equations of motions areexported from Maple [21] to both Fortran code and Matlab code [22] for further analysis.
To illustrate the key features of the model, the equations of motion of the perfect shell (w0 = 0 [m]) andan expansion of w with N = M = 1 (see Eq. (10)) is given
m1Qc10 + c10Q
c10 +
[
k1 + k2 (Qs11)2 + k2 (Qc
11)2 + k3Ut
]
Qc10+
k4 (Qc10)
3− k5Q
c10
2 − k6 (Qc11)
2− k6 (Qs
11)2− k7Ut = 0
m2Qs11 + c11Q
s11 +
[
k8 + k9 (Qc11)
2+ k10Ut − k11Q
c10 + k12 (Qc
10)2]
Qs11 + k13 (Qs
11)3
= 0
m2Qc11 + c11Q
c11 +
[
k8 + k9 (Qs11)
2+ k10Ut − k11Q
s10 + k12 (Qc
10)2]
Qc11 + k13 (Qc
11)3
= 0
mtopUt + ctUt + k14Ut − k15Qc10 + k16 (Qc
11)2
+ k17 (Qc10)
2+ k18 (Qs
11)2
= −mtop
(
Ub + g)
,
5
where ki are positive constants. As can be noted, the dof Ut, which has both a linear coupling and anonlinear coupling with the axi-symmetrical mode Qc
10 (the linear coupling is due to the Poisson effect), isdirectly excited by the prescribed base-acceleration Ub. The coupling of the axi-asymmetrical modes Q
s,c11
is only attained via the non-linear stiffness terms (since w0 = 0 [m]).
3. Static and modal analysis
In this section, the static response of the cylindrical shell with top mass is examined. Subsequently, amodal analysis is performed on the linearized model of the cylindrical shell with top mass. The goal of thesetwo studies is to analyze the static buckling behaviour and the linear eigen-frequencies of the cylindricalshell including the influence of geometric imperfections and to test the convergence of the results for variousexpansions of the out-of-plane displacement field w. The results are numerically validated via a comparisonwith results obtained using the Finite Element (FE) package MSC.Marc. The (nonlinear) kinematic rela-tions used in the FE model are based on Kirchhoff theory (element 139, see [23]) and are valid for largedisplacements and moderate rotations. As a test case, a cylindrical shell which static buckling behaviourdue to compressive loading is experimentally studied in [2], is considered. The cylinder is made of polyesterfilm with E = 5.56 [GPa], ρ = 1370 [kg/m3], ν = 0.3, R = 100 [mm], L = 160.9 [mm] and h = 0.247 [mm](R/h = 405). During the experimental axially compression test of this shell [2], the shell buckles from theaxi-symmetric pre-buckling state to an axi-asymmetric post-buckled state which is dominated by one fullsine-wave (i.e. i = 2, see Eq. (10)) in axial direction and a circumferential wave number of n = 11. Therefore,the considered wave number during the analysis is n = 11.
3.1. Static analysis
For the static analysis, we set all time-derivatives in the equations of motion to zero and consider the caseUb(t) = 0. In the static case, the only force exerted on the cylindrical shell is due to the weight of the topmass
∫ 2π
0
Nx(x = L)Rdθ = −mtopg = −P · Pc [N], (14)
where P is a newly introduced dimensionless load-parameter and
Pc = 2πEh2/√
3(1 − ν2) [N], (15)
is the classical static buckling load of the axially compressed cylindrical shell [1] (Pc = 1289 [N]). By solvingthe resulting algebraic equations for a quasi-static varying load P , using the continuation package AUTO[24], the static (post) buckling behaviour of the cylindrical shell is examined.
For the static analysis, geometrical imperfections are taken into account. The considered expansion of theimperfection w0 (see Eq. (11)) will trigger the modes with dofs Qs
i1. Consequently, for a discretization of w(see Eq. (10)) with M = 2, only the modes with Qc
i0, Qsi1 and due to internal couplings the modes with Qc
i2
will appear in the static response. The modes with Qci1 and Qs
i2 do not contribute to the static responseand are, therefore, removed from the model.
In Fig. 3, the static response of the axially loaded imperfect shell (e11 = e21 = 0.284, other ei1 = 0), basedon a 31-dof model (N = 10, M = 2 with Qc
i1 = Qsi2 = 0 [m], the accuracy of this model is discussed below)
is depicted. Note that the load-path is presented in terms of the dimensionless load P (Eq. (14)) and thedimensionless axial displacement
uL = u(t, L, θ)/L. (16)
Starting at the initial unloaded state (P = 0), buckling of the shell occurs via the limit-point at approximatelyP = Pb = 0.77, see enlargement A in Fig. 3. The computed buckling load is lower than the classical bucklingload due to the included small imperfection (max |w0|/h = 0.5). The post-critical behaviour shows manycoexisting stable and unstable post-buckled states. The deformed shapes for six (stable) states (indicatedin Fig. 3 with a-f) are shown in Fig. 4. As can be noted, along the complex load-path depicted in Fig. 3,
6
0 0.5 1 1.5 2 2.5 3
x 10−3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
z
PbP
[-]
a
b
c
d
e
f
−uL [-]
stableunstable
Fig. 3. Static load-path imperfect cylindrical shell (e11 = e21 = 0.284, n = 11).
0.5 0.5 0.5
0.5 0.5 0.5
θ [rad]θ [rad]θ [rad]
θ [rad]θ [rad]θ [rad]
x/L
x/L
x/L
x/L
x/L
x/L
a, wm = 0.2 b, wm = 7.7 c, wm = 8.0
d, wm = 10.0 e, wm = 9.3 f, wm = 8.9
00
00
00
00
00
00
111
111
-0.2-0.2-0.2
-0.2-0.2-0.2
0.20.20.2
0.20.20.2
Fig. 4. Contour plots of w (only 1/nth segment shown) for labelled states in Fig. 3 (max w =white, minw =black,wm = max |w|/h).
successive destabilization and restabilization events take place, resulting in a so-called ’cellular’ bucklingsequence [25].
The post-critical load-path exhibits a minimum at point f (P = 0.230). This minimum is of practicalimportance since above this load, it is possible that due to some external disturbance, the shell may jumpfrom the initial non-buckled state to a stable buckled state. At the local minimum at point c in Fig. 3(P = 0.245, uL = −1.17 · 10−3), the buckled shape is dominated by one full sine-wave in axial direction, seeFig. 4-c. In the experiments performed with the same shell but with slightly different boundary conditions(w,x = 0 instead of Mx = 0) [2], a comparable (local) minimum post-buckling load with corresponding axialshortening and buckled shape is obtained (P ≈ 0.24 and uL ≈ −1.3 · 10−3, see Fig. 3.52 in [2]). Obviously,also the initial imperfection considered in the semi-analytical model and the imperfection in the actual shellin [2] will differ. Small imperfections have, however, only a mild influence on such (far) post-critical responses[2, 8, 10].
The post-buckled state dominated by three axial half sine-waves (see Fig. 4-f) is not found in the ex-
7
f1 f2 f3
(axi-sym. and axial-sym.) (axi-asym. and axial-sym.) (axi-asym. and axial-asym.)
Fig. 5. First three vibration modes cylindrical shell (n = 11, top mass not shown).
periments [2]. Instead, a buckling event to a state with another circumferential distribution (n = 10) isfound for further increasing axial shortening [2]. This buckling event can, obviously, not be captured withthe current semi-analytical model, since during the computations the circumferential wave number remainsfixed to n = 11.
Primary static buckling loads (Pb) for other imperfection shapes, all with only one nonzero ei1 andmax |w0|/h = 0.5, are compared in Table 1. Note that the buckling loads are obtained in a similar fashionas in Fig. 3, i.e. by determining the limit-point in the initial load-path. Next to results obtained using the31-dof model and a FE model, also results for other discretizations of w (Eq. (10)) are included. As can benoted, the imperfection shapes with the higher number of axial half wave numbers give the largest decreasein buckling load. For comparison of the results obtained using the semi-analytical approach (for a givendiscretization of w) and results obtained using FEM, the mean relative differences ∆, defined by
∆ = mean
(
P ib − P i,FEM
b
P i,FEM
b
)
, (17)
where P ib and P i,FEM
b denote the buckling load for ei1 6= 0, computed using the semi-analytical model andthe FE model, respectively, are included in Table 1. From the models considered, the 31-dof model shows thebest overall correspondence with the FE results. The necessity to include the modes with a double harmonicin circumferential direction (Qc
i2) is illustrated in Table 1 for N = 10, i.e. using M = 1 (21-dof) instead ofM = 2 (31-dof) in Eq. (10) results in a large increase in buckling loads. Due to computational limitations,extension of the 31-dof model with more modes was not possible.
Table 1Primary static buckling loads of the imperfect cylindrical shell (∆ defined by Eq. (17))).
Model Pb [-]
N M dof e11 = 0.5 e21 = 0.5 e31 = 0.5 e41 = 0.5 e51 = 0.5 e61 = 0.5 ∆
6 2 19 0.9366 0.6591 0.6440 0.6492 0.6177 0.6144 27.5%
8 2 25 0.8684 0.6546 0.5314 0.5460 0.5573 0.5263 13.0 %
10 1 21 0.9303 0.7179 0.5961 0.5971 0.6562 0.6206 27.2 %
10 2 31 0.8607 0.6686 0.5407 0.4952 0.5195 0.4628 7.8 %
FEM 0.8001 0.6422 0.5117 0.4818 0.4559 0.4115 -
3.2. Modal Analysis
Next, the undamped eigen-frequencies of the perfect cylindrical shell with top mass are determined.Hereto, first the static equilibrium state for a given pre-load is determined. Subsequently, the equations ofmotion are linearized around this equilibrium state and a modal analysis is performed. The top mass is fixed
8
to mt = 10 [kg] and two levels of pre-load are considered, i.e. g = 0 [m/s2] (P = 0) and g = 9.81 [m/s2](P = 0.076). The results are shown in Table 2, using the same models as used for the static analysis.
The lowest vibration mode is an axi-symmetrical suspension type of mode and the higher vibration modescorrespond to axi-asymmetrical modes, see also Fig. 5. The pre-load (g = 9.81 [m/s2]) results in a smalldecrease (2-3%) of the eigen-frequencies corresponding to axi-asymmetrical vibrational eigen-modes. Theresults are in good agreement with the FEM results, i.e. the maximum difference between the semi-analyticalresults and the FEM results is less than 0.6%. All models predict approximately the same values for the firstthree eigen-frequencies. Apparently, for accurate calculation of the first three vibration modes, less modesare required than for an accurate static buckling analysis (see Table 1).
Including the same geometrical imperfections as considered in Table 1, the eigen-frequencies f1, f2 and f3
of the pre-loaded shell decrease maximally 1.4%, 2.9% and 4.5% (compared to the pre-loaded perfect case),respectively. Similar as for the buckling loads (see Table 1), the largest decreases occur for the imperfectionwith the largest number of axial half waves (e61). In conclusion, the imperfections have more influence onthe primary static buckling loads than on the linearized eigen-frequencies.
Table 2First three eigen-frequencies perfect cylindrical shell with top mass (n = 11, mtop = 10 [kg]).
g = 0 [m/s2] g = 9.81 [m/s2]
N M dof f1 [Hz] f2 [Hz] f3 [Hz] f1 [Hz] f2 [Hz] f3 [Hz]
6 2 19 116.9 357.6 567.9 116.9 350.6 551.5
8 2 25 116.8 357.5 567.9 116.8 350.5 551.2
10 1 21 116.8 357.5 567.9 116.7 350.5 551.2
10 2 31 116.8 357.5 567.9 116.7 350.5 551.2
FEM 116.6 355.7 568.9 116.6 348.5 552.0
4. Dynamic analysis
In this section, the nonlinear dynamic steady-state response of the cylindrical shell subjected to the com-bination of a static load (due to the weight of the top mass) and a harmonic time varying load (due to theprescribed base-acceleration) will be studied. For small amplitudes of the base-excitation (rd) the responseis expected to be harmonic with small (out-of-plane) displacements. However, for increasing values of rd theharmonic response may become unstable and severe large (out-of-plane) amplitude vibrations may appearinstead. The goal of the dynamic analysis is to determine where such instabilities will occur (i.e. for whichcombinations of excitation frequency and amplitude of base-acceleration) and how these results depend onpossible geometric imperfections in the cylindrical shell.
Numerical continuation of periodic solutions [24] with the excitation frequency as continuation parameteris adopted to study the steady-state behaviour of the cylindrical shell. The local stability of the periodicsolutions is determined using Floquet theory [24]. Furthermore, also numerical integration of the equationsof motion is performed using a Runge-Kutta integration scheme with adaptive step-sizing (NAG routineD02PDF [26]). In all simulations, a little amount of damping is taken into account by setting the linearviscous damping parameters cij and ct such, that all the linear vibration modes have the same relativedamping ratio (ξ = 0.01). The top mass is fixed to mt = 10 [kg] and only the preloaded case g = 9.81 [m/s2]is considered. Note that for this level of pre-load (P = 0.076, see Eq. (14)), the unbuckled configuration isstill a unique stable equilibrium state of the cylindrical shell (see Fig. 3).
Among the semi-analytical models considered in the static and modal analyses (see section 3), the 31-dofmodel (expanding w with N = 10, M = 2 see Eq. (10), excluding companion modes) showed to be themost accurate (especially for determining the primary static buckling load, see Table 1). This model will,therefore, also be considered for the dynamic analysis. However, prior to the analysis based on the 31-dof
9
50 100 150 200 250 30010
−5
10−4
10−3
10−2 rd = 0.27
f [Hz]
log
Um
Fig. 6. Frequency-amplitude plot perfect shell for rd = 0.27.
115.9 116.1 116.3 116.5 116.7 116.9 117.1 117.3 117.54.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
6.4x 10
−3
stableunstable
rd = 0.30
rd = 0.29
rd = 0.28
rd = 0.27
f [Hz]
Um
Fig. 7. Frequency-amplitude plot perfect shell for four increas-ing values of rd and f close to f1.
semi-analytical model, first an analysis will be performed using two models with less dof. These models areobtained by setting all dof in the expansion of w with N = 10 and M = 2, which correspond to modesbeing asymmetric with respect to x/L = 1/2 to zero (i.e. Qs
i1 and Qci2 with i = 2, 4, ..), resulting in a model
with 26-dof if companion modes are included and in a model with 16-dof if companion modes are excluded.This simplification is supported by the fact that the primary static buckling loads (see Table 1) obtained forthe imperfection shapes e11, e31, e51 (resulting in initially buckled shapes being symmetric with respect tox/L = 1/2) are not drastically different from the results for the axial asymmetrical imperfection shapes e21,e41, e61 (resulting in initially buckled shapes being asymmetric with respect to x/L = 1/2). Furthermore,the lowest two vibrational eigen-modes are also symmetric with respect to x/L = 1/2 (i.e. axial-symmetric,see Fig. 5).
First, the 16-dof model is used (i.e. no companion modes and no axially asymmetric modes are includedin the model). The steady-state response of the perfect case (w0 = 0 [m]) for a varying excitation frequencyf (computed using the 16-dof model) is depicted in Fig. 6 for rd = 0.27. Note that the steady-state responseis plotted in terms of the following measure
Um = maxT
uL − minT
uL, (18)
where uL denotes the dimensionless axial displacement (see Eq. (16)) and T = 1/f . The response shows aharmonic resonance around the first linear eigen-frequency (f1 = 116.7, see Table 2) and does not exhibitany regions of instability. However, for rd > 0.279, a small region of instability appears in the top of theharmonic resonance, see Fig. 7. The response in this region of instability is further examined using numericalintegration of the equations of motion. As initial condition, a very small perturbation is given to the dofsQ
s,cij /h = 1 · 10−3 to initiate (possible) instabilities. Transients effects are excluded by only considering the
response after t/T = 2000 (T = 1/f). Results of this approach for three values of rd and f = f1 are depictedin Fig. 8 and Fig. 9. Note that the responses are depicted in terms of two measures, i.e. the dimensionlessaxial displacement uL (see Eq. (16)) and the dimensionless out-of-plane displacement
wL/2 = w [t, L/2, π/(2n)]/h. (19)
For rd = 0.27, i.e. just below the loss of stability of the harmonic response at f = f1 (see Fig. 7), theout-of-plane response is in phase with the axial displacements and only small out-of-plane displacementsare initiated due to the Poisson effect, see Fig. 8. For rd = 0.3 the harmonic solution is no longer stable atf = f1 and instead a 1/2 subharmonic response is found (period 2T ), see Fig. 8. By increasing the amplitudeof the base-excitation further to rd = 0.33, a very severe beating type of response is found, see Fig. 9. Thistype of response exhibits short time intervals in which energy is transferred back and forward from the
10
200020022004200620082010−0.4
−0.2
0
0.2
2000 2002 2004 2006 2008 2010−0.4
−0.2
0
0.2
20002002 2004 2006 2008 2010−2
−1
0
1
2x 10
−3
2000 2002 2004 2006 2008 2010−2
−1
0
1
2x 10
−3
rd = 0.27 [-] rd = 0.30 [-]
t/Tt/T
uL
wL
/2
Fig. 8. Harmonic (left) and 1/2 subharmonic response (right)(w0 = 0 [m], f = f1).
2000 2100 2200 2300 2400 2500−5
0
5
10
15
2000 2100 2200 2300 2400 2500−2
−1
0
1
2x 10
−3
A
rd = 0.33 [-]
t/T
uL
wL
/2
Fig. 9. Beating type of response (w0 = 0 [m], f = f1).
2143 2146 2149 2152−4
−2
0
2
4
6
8
10
12t1 t2 t3 t4 t5 t6
t/T
wL
/2
Fig. 10. Enlargement of wL/2 duringinterval of time A in Fig. 8.
0.5 0.5 0.5
0.5 0.5 0.5
θ [rad]θ [rad]θ [rad]
θ [rad]θ [rad]θ [rad]
x/L
x/L
x/L
x/L
x/L
x/L
t1, wm = 0.27 t2, wm = 0.28 t3, wm = 10.4
t4, wm = 8.65 t5, wm = 0.50 t6, wm = 1.42
00
00
00
00
00
00
111
111
-0.2-0.2-0.2
-0.2-0.2-0.2
0.20.20.2
0.20.20.2
Fig. 11. Contour plots of w (only 1/nth segment shown) at indicated time instances inFig. 10 (max w =white, minw =black, wm = max |w|/h).
suspension mode at f = f1 (see Fig. 5), to severe axi-asymmetrical out-of-plane vibrations. For illustration,an enlargement of wL/2 during the interval of time A in Fig. 9 is shown in Fig. 10. The deformed shapesat six time instances during this interval of time are shown in Fig. 11. Although the exact nature of thebeating type of response is not determined, it seems to be nonstationair (i.e. quasi-periodic or chaotic).
As shown for the perfect cylindrical shell under consideration, there exist a threshold value rcd for the
base-excitation rd such that for rd < rcd the harmonic solution is always stable and for rd > rc
d there ex-ists a small region of instability in the top of the harmonic resonance. In this region the response is firstsubharmonic and for further increasing rd, a beating type of response with short time intervals of verysevere out-of-plane vibrations is obtained. The transition for increasing values of rd around f = f1, from theharmonic response with small out-of-plane displacements to the beating type of response with very largeout-of-plane displacements can be considered as dynamic buckling and must be avoided in practise.
In practise, small geometric imperfection are inevitable and the influence of such imperfections on thefound dynamic buckling results must be examined. The steady-state response of the imperfect cylindrical
11
115.9 116.1 116.3 116.5 116.7 116.9 117.1 117.3 117.52.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3x 10
−3
stableunstable
rd = 0.26
rd = 0.255
rd = 0.25
f [Hz]
Um
Fig. 12. Frequency-amplitude plot imperfect shell (e11 = 0.5)for increasing values of rd and f close to f1.
2000 2005 2010−1
0
1
2
2000 2005 2010−2
−1
0
1
2x 10
−3
rd = 0.25 [-]
t/T
uL
wL
/2
Fig. 13. Harmonic response (e11 = 0.5 and f = f1).
shell (e11 = 0.5) with top mass for a varying excitation frequency f is depicted in Fig. 12. Note that forei1 6= 0 (see Eq. (11)), a direct coupling is present between the axi-symmetric modes with dof Qs
i0 and theaxi-asymmetric modes with dof Qs
i1. Similar as for the perfect case, the stability of the harmonic responseis firstly lost at a certain point in the top of the harmonic resonance around f = f1 for increasing valuesof rd. Compared to the perfect case (ei1 = 0, see Fig. 7), the value of rc
d is (slightly) lower for e11 = 0.5(rc
d = 0.252). The response for f = f1 and rd = 0.25 (i.e. just below rcd), obtained using the numerical
integration approach, is depicted in Fig. 13 and in a similar fashion for rd = 0.255 (just above rcd) in Fig.
14. In contrast to the perfect case, for the imperfect case (e11 = 0.5) the response for values of rd just abovercd is directly the beating type of response (instead of first a subharmonic response, see Fig. 8). The sudden
transition between the harmonic response and the beating type of response is more clearly visualized inFig. 15. This bifurcation diagram is obtained by plotting for each value of rd, 1000 times the T -sampledsteady-state value of wL/2.
Similar transitions (i.e. from harmonic response directly to the beating type of response) are found forother imperfection shapes (all with only one nonzero ei1 and max |w0|/h = 0.5). Similar as for the primarystatic buckling loads (see Table 1), the value for rc
d also depends highly on the shape of the imperfection.For example for e51 = 0.5, the harmonic response becomes already unstable at f = f1 for rc
d = 0.135 (i.e 45% lower than for e11 = 0.5).
Next, the effect of extending the semi-analytical model with the companion modes (with dof Qci1 and
Qsi2), on the obtained dynamic buckling results for the imperfect cylindrical shell is examined (i.e. the 26-
dof is used). Similar as found in [10] for the onset to parametric instabilities of an imperfect cylindricalshell without a top mass, it appeared that inclusion of the companion modes do not affect the value of rc
d.Furthermore, for values of rd just above rc
d, the companion modes do also not contribute to the beatingresponse. Consequently, for determining the dynamic critical loads of the imperfect cylindrical shell withtop mass, the companion modes do not have to be included in the semi-analytical model.
Finally, the effect of including the axially asymmetric modes is examined, i.e. the full 31-dof semi-analyticalmodel (based on the expansion of w with N = 10, M = 2 see Eq. (10), excluding companion modes withdof Qc
i1 and Qsi2) is used to study the dynamic response of the base-excited cylindrical shell with top mass.
Considering the response of the perfect shell obtained with this model, stability of the harmonic solutionfor f = f1 is lost at rc
d = 0.274 which is slightly lower than than obtained for the model without theaxially asymmetric modes (16-dof : rc
d = 0.279, see also Fig. 7). The transition from the harmonic solutionto the beating type of solution via an intermediate subharmonic solution is not changed by including the
12
2000 2100 2200 2300 2400 2500−5
0
5
10
15
2000 2100 2200 2300 2400 2500−2
−1
0
1
2x 10
−3
rd = 0.255 [-]
t/T
uL
wL
/2
Fig. 14. Beating type of response (e11 = 0.5 and f = f1).
0.24 0.245 0.25 0.255 0.26 0.265 0.27 0.275 0.28−5
0
5
10
rcd
rd [-]
wL
/2
Fig. 15. Bifurcation diagram (e11 = 0.5, f = f1).
axial-asymmetrical modes. However, also axially asymmetric modes participate in the response obtainedfor rd > rc
d. The values of rcd obtained using the 31-dof model for imperfection shapes, all with only one
nonzero ei1 and max |w0|/h = 0.5 (similar as considered in Table 1), are shown in Table 3. Similar as forthe primary static buckling loads (Pb, see Table 3), the imperfection shapes with the higher number of axialhalf wave numbers give the largest decrease in rc
d. Consequently, for the load-case and cylindrical shell underconsideration, the dynamic critical load exhibits a similar severe imperfection sensitivity as found for theprimary static buckling load.
Table 3Dynamic critical loads rc
dof the imperfect cylindrical shell.
Model rcd
[-]
N M dof e11 = 0.5 e21 = 0.5 e31 = 0.5 e41 = 0.5 e51 = 0.5 e61 = 0.5
10 2 31 0.2518 0.16157 0.1633 0.12285 0.1318 0.11468
5. Conclusions
The objective of this paper is to determine the dynamic stability limits of a base-excited thin cylindricalshell with top mass and how these results are affected by possible geometrical imperfections. First a semi-analytical model is derived which satisfies exactly the in-plane boundary conditions. The resulting modelis numerically validated through a comparison with static and modal results obtained using FE modelling.Although there are still some discrepancies, generally a good correspondence is obtained between the semi-analytical results and the FE results. The dynamic stability of the base-excited cylindrical shell is studiedusing numerical continuation of periodic solutions with the excitation frequency as continuation parameter.Due to the top mass, a relatively low frequent resonance is introduced, corresponding to an axi-symmetricalvibration mode. It is shown that the harmonic response may become unstable in the peak of this resonanceand a beating type of response with time intervals of severe out-of-plane deformations may appear instead.The transition between the mild harmonic response and the severe beating type of response with very largeout-of-plane displacements can be considered as dynamic buckling and must be avoided. Similar as for thestatic buckling case, the critical value for the amplitude of the prescribed harmonic base-acceleration forwhich the harmonic response changes to the severe beating type of response, depends highly on the initialimperfections present in the shell. Future research will include the effect of considering other circumferentialwave numbers and damping ratios and experimental validation of the obtained semi-analytical results.
13
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