Dept. of Mechanical and Civil Engineering

The analytical conditions for the applicability of a Non-linear Superposition Technique (NST) are established by using the Normal Form Method (NFM). The superposition technique represents the extension of modal analysis to weakly non-linear vibration systems. In this paper it is shown that the NST leads to a non-uniform perturbation expansion for the solution of dynamical systems when some special non-linear terms are present in the dynamical system. An analytical proof for general dynamical systems and a numerical application to a simple mechanical system is included in the paper.

The nonlinear dynamics of a simply supported beam resting on a nonlinear spring bed with cubic stiffness is analyzed. The continuous differential operator describing the mathematical model of the system is discretized through the classical Galerkin procedure and its nonlinear dynamic behavior is investigated using the method of Normal Forms. This model can be regarded as a simple system describing the oscillations of flexural structures vibrating on nonlinear supports and then it can be considered as a simple investigation for the analysis of more complex systems of the same type. Indeed, the possibility of the model to exhibit actually interesting nonlinear phenomena (primary, superharmonic, subharmonic and internal resonances) has been shown in a range of feasibility of the physical parameters. The singular perturbation approach is used to study both the free and the forced oscillations; specifically two parameter families of stationary solutions are obtained for the forced oscillations.

The study presented is an investigation of the non-linear dynamics and stability of simply supported, circular cylindrical shells containing inviscid incompressible #uid #ow. Non-linearities due to large-amplitude shell motion are considered by using the non-linear Donnell's shallow shell theory, with account taken of the e!ect of viscous structural damping. Linear potential #ow theory is applied to describe the #uid}structure interaction. The system is discretiszd by Galerkin's method, and is investigated by using a model involving seven degrees of freedom, allowing for travelling wave response of the shell and shell axisymmetric contraction. Two di!erent boundary conditions are applied to the #uid #ow beyond the shell, corresponding to: (i) in"nite ba%es (rigid extensions of the shell), and (ii) connection with a #exible wall of in"nite extent in the longitudinal direction, permitting solution by separation of variables; they give two di!erent kinds of dynamical behaviour of the system, as a consequence of the fact that axisymmetric contraction, responsible for the softening non-linear dynamical behaviour of shells, is not allowed if the #uid #ow beyond the shell is constrained by rigid ba%es. Results show that the system loses stability by divergence.

THE RECENT LETTER by Dowell (1998) touched on several aspects of the literature and the physics of nonlinear vibrations of circular cylindrical shells, clarifying some of the issues while attempting to reconcile some hitherto contradictory findings, and at the same time outlining some of the unresolved issues and potentially profitable directions for further research.

The nonlinear free and forced vibrations of a simply supported, circular cylindrical shell in contact with an incompressible and inviscid, quiescent and dense fluid are investigated. Donnell's shallow-shell theory is used, so that moderately large vibrations are analysed. The boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied, while axial constraint is satisfied on the average. The problem is reduced to a system of ordinary differential equations by means of the Galerkin method. The mode shape is expanded by using three degrees of freedom; in particular, two asymmetric modes (driven and companion modes), plus an axisymmetric mode are employed. The time dependence of each term of the expansion is general and the axisymmetric mode is obtained from a series involving all axisymmetric linear modes. Different tangential constraints can be imposed at the shell ends. Effects of both internal and external dense fluid are studied. Internally, the shell is considered completely filled, while externally, an unbounded fluid domain is considered around the shell in the radial direction. The solution is obtained both numerically and by the Method of Normal Forms. Numerical results are obtained for both free and forced vibrations of empty and water-filled shells.

The non-linear response of empty and #uid-"lled circular cylindrical shells to harmonic excitations is investigated. Both modal and point excitations have been considered. The model is suitable to study simply supported shells with and without axial constraints. Donnell's non-linear shallow-shell theory is used. The boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satis"ed. The radial de#ection of the shell is expanded by using a basis of seven linear modes. The e!ect of internal quiescent, incompressible and inviscid #uid is investigated. The equations of motion, obtained in Part I of this study, are studied by using a code based on the collocation method. The validation of the present model is obtained by comparison with other authoritative results. The e!ect of the number of axisymmetric modes used in the expansion on the response of the shell is investigated, clarifying questions open for a long time. The results show the occurrence of travelling wave response in the proximity of the resonance frequency, the fundamental role of the "rst and third axisymmetric modes in the expansion of the radial de#ection with one longitudinal half-wave, and limit cycle responses. Modes with two longitudinal half-waves are also investigated.

The response of simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of one of the lowest natural frequencies is investigated by using improved mode expansions with respect to those assumed in Parts I and II of the present study. Two cases are studied: (1) shells in vacuo; and (2) shells "lled with stagnant water. The improved expansions allow checking the accuracy of the solutions previously obtained and giving de"nitive results within the limits of Donnell's non-linear shallow-shell theory. The improved mode expansions include: (1) harmonics of the circumferential mode number n under consideration, and (2) only the principal n, but with harmonics of the longitudinal mode included. The e!ect of additional longitudinal modes is absolutely insigni"cant in both the driven and companion mode responses. The e!ect of modes with 2n circumferential waves is very limited on the trend of non-linearity, but is signi"cant in the response with companion mode participation in the case of lightly damped shells (empty shells). In particular, the travelling wave response appears for much lower vibration amplitudes and presents a frequency range without stable responses, corresponding to a beating phenomenon. A liquid (water) contained in the shell generates a much stronger softening behaviour of the system. Experiments with a water-"lled circular cylindrical shell made of steel are in very good agreement with the present theory.

The response of a shell conveying #uid to harmonic excitation, in the spectral neighbourhood of one of the lowest natural frequencies, is investigated for di!erent #ow velocities. The theoretical model has already been presented in Part I of the present study. Non-linearities due to moderately large-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory. Linear potential #ow theory is applied to describe the #uid-structure interaction by using the model proposed by PamK doussis and Denise. For di!erent amplitudes and frequencies of the excitation and for di!erent #ow velocities, the following are investigated numerically: (1) periodic response of the system; (2) unsteady and stochastic motion; (3) loss of stability by jumps to bifurcated branches. The e!ect of the #ow velocity on the non-linear periodic response of the system has also been investigated. PoincareH maps and bifurcation diagrams are used to study the unsteady and stochastic dynamics of the system. Amplitude modulated motions, multi-periodic solutions, chaotic responses, cascades of bifurcations as the route to chaos and the so-called &&blue sky catastrophe'' phenomenon have all been observed for di!erent values of the system parameters; the latter two have been predicted here probably for the "rst time for the dynamics of circular cylindrical shells.

This letter is related to the paper of and is written for three reasons: (i) to explain better the approximation on tangential boundary conditions used in this paper, (ii) to complete the literature review therein with additional papers, (iii) to correct a few misprints.

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.

The response-frequency relationship in the vicinity of a resonant frequency, the occurrence of travelling wave response and the presence of internal resonances are investigated for simply supported, circular cylindrical shells. Donnell's nonlinear shallow-shell theory is used. The boundary conditions on radial displacement and the continuity of circumferential displacement are exactly satisfied. The problem is reduced to a system of four ordinary differential equations by means of the Galerkin method. The radial deflection of the shell is expanded by using a basis of four linear modes. The effect of internal fluid is also investigated. The equations of motion are studied by using a code based on the Collocation Method. The present model is validated by comparison of some results with others available. A water-filled shell presenting the phenomenon of 1:1:1:2 internal resonances is investigated for the first time; it shows intricate and interesting dynamics.

The nonlinear ordinary differential equations describing the dynamics of a fluid filled circular cylindrical shell, obtained in Part 1 of the present study, is studied by using a second order perturbation approach and direct simulations. Strong modal interactions are found when the structure is excited with small resonant loads. Modal interactions arise in the whole range of vibration amplitude, showing that the internal resonance condition makes the system non-linearizable even for extremely small amplitudes of oscillation. Stationary and nonstationary oscillations are observed and the complex nature of modal interactions is accurately analyzed. No chaotic motion is observed in the case of 1:1:1:2 internal resonance studied. ͓S0739-3717͑00͒01304-0͔

The aeroelastic stability of simply supported, circular cylindrical shells in supersonic ow is investigated. Nonlinearities caused by large-amplitude shell motion are considered by using the Donnell nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied to the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear piston theory is applied to describe the uid-structure interaction by using two different formulations, taking into account or neglecting the curvature correction term. The system is discretized by Galerkin projections and is investigated by using a model involving seven degrees of freedom, allowing for traveling-wave utter of the shell and shell axisymmetric contraction. Results show that the system loses stability by standing-wave utter through supercritical bifurcation; however, traveling-wave utter appears with a very small increment of the freestream static pressure that is used as the bifurcation parameter. A very good agreement between theoretical and existing experimental data has been found for utter amplitudes. The in uence of internal static pressure has also been studied.

In this paper, the nonlinear normal modes (NNMs) of a thin beam resting on a nonlinear spring bed subjected to an axial tension is studied. An energy-based method is used to obtain NNMs. In conjunction with a matched asymptotic expansion, we analyze, through simple formulas, the local effects that a small bending stiffness has on the dynamics, along with the secular effects caused by a symmetric nonlinearity. Nonlinear mode shapes are computed and compared with those of the unperturbed linear system. A double asymptotic expansion is employed to compute the boundary layers in the nonlinear mode shape due to the small bending stiffness. Satisfactory agreement between the theoretical and numerical backbone curves of the system in the frequency domain is observed.

The stability of circular cylindrical shells with supported ends in compressible, inviscid axial flow is investigated. Nonlinearities due to finite-amplitude shell motion are considered by using Donnell's nonlinear shallow-shell theory; the effect of viscous structural damping is taken into account. Two different in-plane constraints are applied at the shell edges: zero axial force and zero axial displacement; the other boundary conditions are those for simply supported shells. Linear potential flow theory is applied to describe the fluid-structure interaction. Both annular and unbounded external flow are considered by using two different sets of boundary conditions for the flow beyond the shell length: (i) a flexible wall of infinite extent in the longitudinal direction, and (ii) rigid extensions of the shell (baffles). The system is discretized by the Galerkin method and is investigated by using a model involving seven degrees-of-freedom, allowing for traveling-wave response of the shell and shell axisymmetric contraction. Results for both annular and unbounded external flow show that the system loses stability by divergence through strongly subcritical bifurcations. Jumps to bifurcated states can occur well before the onset of instability predicted by linear theory, showing that a linear study of shell stability is not sufficient for engineering applications.

The aeroelastic stability of simply supported, circular cylindrical shells in supersonic flow is investigated by using both linear aerodynamics (first-order piston theory) and nonlinear aerodynamics (third-order piston theory). Geometric nonlinearities, due to finite amplitude shell deformations, are considered by using the Donnell's nonlinear shallow-shell theory, and the effect of viscous structural damping is taken into account. The system is discretized by Galerkin method and is investigated by using a model involving up to 22 degrees-of-freedom, allowing for travelling-wave flutter around the shell and axisymmetric contraction of the shell. Asymmetric and axisymmetric geometric imperfections of circular cylindrical shells are taken into account. Numerical calculations are carried out for a very thin circular shell at fixed Mach number 3 tested at the NASA Ames Research Center. Results show that the system loses stability by travelling-wave flutter around the shell through supercritical bifurcation. Nonsimple harmonic motion is observed for sufficiently high post-critical dynamic pressure. A very good agreement between theoretical and existing experimental data has been found for the onset of flutter, flutter amplitude, and frequency. Results show that onset of flutter is very sensible to small initial imperfections of the shells. The influence of pressure differential across the shell skin has also been deeply investigated. The present study gives, for the first time, results in agreement with experimental data obtained at the NASA Ames Research Center more than three decades ago.

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to ÿnite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the e ect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean de ection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is conÿrmed and the role of higher order asymmetric modes is clariÿed in order to obtain the correct character of the circular cylindrical shell non-linearity. The e ect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries. ?

In this paper, the dynamic response of a simply supported travelling beam subjected to a transverse load is investigated in the super-critical speed range. The well-known axially moving beam theory is considered and a simple viscous damping mechanism has been introduced. The displacement field is expanded in a series of the buckling modes, a sine series, and different techniques have been used in analyzing the response of the dynamical system. Periodic oscillations are studied by means of continuation techniques, while nonstationary dynamics are investigated through direct simulations. A comparison with the literature and a convergence test on the series expansion are performed. A sample case of a physical beam is developed and numerical results are presented concerning bifurcation analysis and stability, and direct simulations of global postcritical dynamics. A complex scenario of alternate regular and chaotic motions is found in a large range of the main parameters.