DYNAMIC STABILITY OF TWO HARMONICALLY EXCITED THIN-WALLED STRUCTURES CARRYING A TOP MASS R.H.B. Fey(1)*, N.J. Mallon(2), C.S. Kraaij(3), H. Nijmeijer(1) (1)
Eindhoven University of Technology, Department of Mechanical Engineering, PO Box 513, 5600 MB Eindhoven, the Netherlands (2) TNO-CMC, PO Box 49, 2600 AA Delft, the Netherlands (3) IHC Lagersmit BV, PO Box 5, 2960 AA Kinderdijk, the Netherlands *Presenting/corresponding author:
[email protected], tel. +31 40 2475406, fax +31 40 2461418
ABSTRACT In this paper the dynamic stability of two imperfect thin-walled structures is investigated both theoretically and experimentally. The first structure is a vertical, slender beam carrying a top mass and the second structure is a cylindrical shell also carrying a top mass. In both cases, the top mass can only move in vertical direction; the top mass weight does not cause static buckling. However, both the beam and the cylindrical shell are also harmonically excited at their base in vertical (i.e. axial) direction. In both structures, for certain dynamic excitation set-points, this will induce dynamic buckling. The numerical results are obtained using a semi-analytical approach, enabling accurate as well as fast parameter studies. Shaker and amplifier dynamics are included in both models. New insights in the steady-state behavior of both structures are obtained. Good quantitative (beam) and qualititative (cylinder) correspondence between numerical and experimental results is found. 1. INTRODUCTION In general, thin-walled structures possess very high inplane stiffness, whereas their out-of-plane stiffness is relatively low. Aerospace structures are often designed such that the loading mainly assesses the in-plane stiffness. In this way, load-carrying structures with very high stiffness-to-mass ratios can be achieved. Although there are still some open issues, the analysis of the (nonlinear) response and buckling of thin-walled structured subjected to static loading is well established in engineering science. However, in practice thin-walled structures are often subjected not only to a static load, but also to a dynamic load. The resistance of structures liable to buckling, to withstand time dependent loading, is addressed as the dynamic stability of these structures. In this paper the dynamic stability of two thin-walled structures (a slender beam and a cylindrical shell) with top mass will be investigated. Both structures are not buckled in the static situation. An additional harmonic
base excitation, however, may cause a dynamic instability, which may be induced by resonances. Harmonic excitation is also frequently used in lab tests of prototypes (often in the form of frequency sweeps). In order to describe post-buckling behavior, it is essential to take geometrically nonlinear behavior into account in dynamic stability investigations of thinwalled structures. In our case of in-plane dynamic loading, parametric excitation is also an important issue. Earlier work, related to the current paper, involves a study of the nonlinear normal modes of a parametrically excited cantilever beam [1]. The nonlinear response of a post-buckled beam subjected to a harmonic axial excitation is investigated in [2]. In [3] linear and nonlinear dynamics of a circular cylindrical shell connected to a free rigid disk are studied. Reference [4] studies the dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads. In this paper, for both nonlinear structures the steadystate behavior resulting from the harmonic excitation will be analyzed using modern numerical nonlinear dynamics tools [5, 6]. This includes calculation of periodic solutions and their stability by solving twopoint boundary value problems and using Floquet theory. Path following is applied to determine branches of periodic solutions for a varying system parameter (e.g. excitation frequency or amplitude) and to detect bifurcation points. The objectives of this paper, which is based on [7, 8], are as follows. First, as will be shown, the used semianalytic modeling approach offers easy coupling between structural dynamics and amplifier and shaker dynamics. Secondly, and more important, in combination with the modern nonlinear dynamic analysis tools mentioned above, the semi-analytical modeling approach enables very efficient investigation of the steady-state behavior of thin-walled structures. Especially when parameter studies are needed, this approach may realize a much shorter time-to-market
than a Finite Element approach, because in general analysis of semi-analytic models is computationally faster, and yet gives accurate results. Thirdly, new insights in the dynamic stability of two specific thinwalled structures are presented. The outline of the paper is as follows. In section 2, a semi-analytic model for a slender beam with top mass will be derived and coupled to an amplifier and shaker model, the experimental set-up will be described, and the steady-state behavior will be investigated. In the low frequency range, super-, sub-, and harmonic resonances are calculated and verified experimentally. In section 3, first, a semi-analytic model of a cylindrical shell with top mass system will be derived. Subsequently, the analysis of the steady-state behavior of the system will be concentrated on the frequency range, where the first harmonic resonance peak occurs. When increasing the excitation amplitude, while keeping the excitation frequency constant, a robust dynamic instability will be found, both numerically and experimentally. In section 4, conclusions and recommendations will be given. 2. BEAM WITH TOP MASS 2.1 Semi-analytic model Fig. 1 shows the steel beam with a top mass of mt = 0.51 kg, which can only move in y -direction. The beam properties are respectively length L = 180 mm, width b = 15 mm, thickness h = 0.5 mm, mass density ρ = 7850 kg/m3, and Young’s modulus E = 2.0 ⋅ 1011 Pa. The initial shape of the imperfect beam is indicated by v0 ( y ) . The transversal displacement relative to v0 ( y ) is given by v(t , y ) . The base excitation is represented by U b (t ) . The axial field relative to U b (t ) is indicated by u (t , y ) , and U t (t ) = U b (t ) + u (t , L ) . Gravity acceleration is equal to g = 9.81 m/s2.
Since the beam is slender ( h
