Large Amplitude Oscillations of Thin Circular Rings

1987 ◽  
Vol 54 (2) ◽  
pp. 315-322 ◽  
Author(s):  
S. P. Maganty ◽  
W. B. Bickford
Keyword(s):  
Large Amplitude ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Circular Ring ◽  
Bending Mode ◽  
Out Of Plane ◽  
Circular Rings ◽  
Linear Problems ◽  
Plane Bending ◽  
Nonlinear Equations Of Motion

Using an intrinsic formulation, an accurate set of geometrically nonlinear equations of motion is derived for the large amplitude oscillations of a thin circular ring. Non-dimensionalization of the equations of motion and the compatibility conditions indicates clearly that certain terms involving the extensional deformation, the shear deformation, and the rotatory inertia are relatively small and can be discarded. The resulting equations of motion are analyzed by the method of multiple scales with a single bending mode approximation to the linear problems indicating a softening type of nonlinearity for both the in-plane and the out-of-plane problems with the out-of-plane flexural motion experiencing a greater degree of softening when compared to that of the in-plane flexural motion. The results for the nonresonant case indicate that the frequency of an out-of-plane bending mode is significantly reduced by the presence of a nonzero in-plane bending amplitude, whereas the results for the resonant case indicate the presence of unsteady oscillations with an exchange of energy between the in-plane and the out-of-plane modes.

Load Transfer Control for a Gantry Crane with Arbitrary Delay Constraints

2002 ◽  
Vol 8 (2) ◽  
pp. 135-158 ◽  
Author(s):  
Paolo Dadone ◽  
Hugh F. Vanlandingham
Keyword(s):  
Load Transfer ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Gantry Crane ◽  
Phase Plane Analysis ◽  
Plane Analysis ◽  
Nonlinear Approach ◽  
Linearized Model ◽  
Nonlinear Equations Of Motion ◽  
Quadratic And Cubic Nonlinearities

This paper describes a method to move the load of a gantry crane to a desired position in the presence of known, but arbitrary, motion-inversion delays as well as cart acceleration constraints. The method idea is based on a phase-plane analysis of the linearized model. In order to limit residual pendulation at the goal position, the method is extended to account for quadratic and cubic nonlinearities. The method of multiple scales is used to determine an approximate solution to the nonlinear equations of motion, thus providing a more accurate measure of the frequency of the oscillations. The nonlinear approach is very successful in limiting residual oscillations to very small values (less than 1 degree of amplitude), offering a reduction, with respect to the linear case, of as much as two orders of magnitude. Finally, this method offers a rationale for the future development of a controller for suppression of load oscillations in ship-mounted cranes in the presence of arbitrary delays.

Asymptotic analysis of the forced vibrations of a one-disc rotor on a non-linear flexible base

2010 ◽  
Vol 224 (8) ◽  
pp. 1593-1604 ◽  
Author(s):  
K V Avramov
Keyword(s):  
Asymptotic Analysis ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Forced Vibrations ◽  
Linear Dynamics ◽  
Non Linear ◽  
Out Of Plane ◽  
Symmetric Rotor ◽  
Flexible Base ◽  
Steady Motions

Equations of motion for a four-degree-of-freedom dynamical system describing the vibrations of a one-disc elastic rotor taking into account gyroscopic moments on a non-linear flexural base are derived. A new version of the multiple scales method is developed and applied to analyse the non-linear dynamics of such a system for different resonances. The steady motions of the rotor are analysed. From the asymptotic analysis, it is shown that out-of-plane motions of the disc exist in the symmetric rotor.

Multimode Interactions in Suspended Cables

2002 ◽  
Vol 8 (3) ◽  
pp. 337-387 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat ◽  
Char-Ming Chin ◽  
Walter Lacarbonara
Keyword(s):  
Internal Resonance ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Internal Resonances ◽  
Out Of Plane ◽  
Space Formulation ◽  
Suspended Cables ◽  
One To One ◽  
Plane Mode

We investigate the nonlinear nonplanar responses of suspended cables to external excitations. The equations of motion governing such systems contain quadratic and cubic nonlinearities, which may result in two-to-one and one-to-one internal resonances. The sag-to-span ratio of the cable considered is such that the natural frequency of the first symmetric in-plane mode is at first crossover. Hence, the first symmetric in-plane mode is involved in a one-to-one internal resonance with the first antisymmetric in-plane and out-of-plane modes and, simultaneously, in a two-to-one internal resonance with the first symmetric out-of-plane mode. Under these resonance conditions, we analyze the response when the first symmetric in-plane mode is harmonically excited at primary resonance. First, we express the two governing equations of motion as four first-order (i.e., state-space formulation) partial-differential equations. Then, we directly apply the methods of multiple scales and reconstitution to determine a second-order uniform asymptotic expansion of the solution, including the modulation equations governing the dynamics of the phases and amplitudes of the interacting modes. Then, we investigate the behavior of the equilibrium and dynamic solutions as the forcing amplitude and resonance detunings are slowly varied and determine the bifurcations they may undergo.

The Influence of Nonlinear Boundary Conditions on the Nonplanar Autoparametric Responses of an Inextensible Beam

2001 ◽  
Author(s):  
Haider N. Arafat ◽  
Ali H. Nayfeh
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Nonlinear Boundary Conditions ◽  
Nonlinear Springs ◽  
Out Of Plane

Abstract The nonplanar responses of a beam clamped at one end and restrained by nonlinear springs at the other end is investigated under a primary resonance base excitation. The beam’s geometry and the springs’ linear stiffnesses are such that the system possesses a one-to-one autoparametric resonance between the nth in-plane and out-of-plane modes. The beam is modeled using Euler-Bernoulli theory and includes cubic geometric and inertia nonlinearities. The objective is to assess the influence of the nonlinear boundary conditions on the beam’s oscillations. To this end, the method of multiple scales is directly applied to the integral-partial-differential equations of motion and associated boundary conditions. The result is a set of four nonlinear ordinary-differential equations that govern the slow dynamics of the system. Solutions of these modulation equations are then used to characterize the system’s nonlinear behavior.

The effect of intermolecular forces upon the vibrations of molecules in the crystalline state I. The out-of-plane bending of the carbonate ion in aragonite minerals

1963 ◽  
Vol 275 (1362) ◽  
pp. 295-309 ◽  
Keyword(s):  
Intermolecular Forces ◽  
Absolute Intensities ◽  
Carbonate Ion ◽  
Nearest Neighbours ◽  
Bending Mode ◽  
Infra Red ◽  
Out Of Plane ◽  
Plane Bending ◽  
Oxygen Bond ◽  
Dipole Energy

The infra-red absorption of polycrystalline BaCO 3 , SrCO 3 , and CaCO 3 , the latter in both the aragonite and calcite modifications, has been measured in the region 600 to 2000 cm -1 . Absolute intensities were determined for each of the three fundamental bands of the carbonate ion, by the method of extrapolation to infinite dilution of the carbonate in the alkali halide matrix. The band due to the out-of-plane bending mode was examined under high resolution, the samples employed having been enriched to a 50/50 ratio of 12 C/ 13 C. Intermolecular coupling in this band, which is revealed by the isotopic solid solution, has been interpreted as arising primarily from interaction of the dipoles produced during the vibration. The dipole derivatives calculated from this coupling agree fairly well with those estimated from the absolute intensities. Various sources of error are discussed, and in particular, an estimate is made of the dipole-dipole energy summed over the entire lattice, instead of merely for nearest neighbours. The carbon-oxygen bond moment during the vibration has an effective value of from 1.3 to 1.7 debyes.

Nonlinear vibration and stability analysis of an electrically actuated piezoelectric nanobeam considering surface effects and intermolecular interactions

2015 ◽  
Vol 23 (12) ◽  
pp. 1873-1889 ◽  
Author(s):  
S Mehrdad Pourkiaee ◽  
Siamak E Khadem ◽  
Majid Shahgholi
Keyword(s):  
Stability Analysis ◽  
Nonlinear Vibration ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Van Der Waals ◽  
Surface Effects ◽  
Van Der Waals Forces ◽  
System Response ◽  
Nonlinear Equations Of Motion ◽  
Piezoelectric Nanobeam

This paper investigates the nonlinear vibration and stability analysis of a doubly clamped piezoelectric nanobeam, as a nano resonator actuated by a combined alternating current and direct current loadings, including surface effects and intermolecular van der Waals forces. The governing equation of motion is obtained using the extended Hamilton principle. The multiple scales method is used to solve nonlinear equations of motion. The influence of van der Waals forces, piezoelectric voltages and surface effects are investigated on the static equilibria, pull-in voltages and dynamic primary resonances of the nano resonator. It is shown that for accurate and exact investigation of the system response, it is necessary to consider the surface effects. To validate the analytical results, numerical simulation is performed. It is seen that the perturbation results are in accordance with numerical results.

Nonlinear Dynamics of Closed Spherical Shells

2005 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Haider N. Arafat
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Primary Resonance ◽  
Resonance Excitation ◽  
Internal Resonances ◽  
Spherical Shells ◽  
Linear Eigenvalue Problem ◽  
Nonlinear Responses ◽  
Nonlinear Equations Of Motion ◽  
Primary Resonance Excitation

We investigate the axisymmetric dynamics of forced closed spherical shells. The nonlinear equations of motion are formulated using a variational approach and surface analysis. First, we revisit the linear eigenvalue problem. Then, using the method of multiple scales, we assess the possibility of the activation of two-to-one internal resonances between the different types of modes. Lastly, we examine the shell’s nonlinear responses to an axisymmetric primary-resonance excitation and analyze their bifurcations.

INTERNAL RESONANCES AND BIFURCATIONS OF AN ARRAY BELOW THE FIRST PULL-IN INSTABILITY

2010 ◽  
Vol 20 (03) ◽  
pp. 605-618 ◽  
Author(s):  
STEFANIE GUTSCHMIDT ◽  
ODED GOTTLIEB
Keyword(s):  
Nearest Neighbor ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Internal Resonances ◽  
Large Size ◽  
Element Array ◽  
Nonlinear Equations Of Motion ◽  
Chaotic Transients ◽  
Nonlinear Continuum ◽  
Continuation Technique

A nonlinear continuum model is used to investigate the dynamic behavior of an array of N nonlinearly coupled microbeams. Investigations concentrate on the region below the array's first pull-in instability, which is shown to be governed by several internal three-to-one and combination resonances. The nonlinear equations of motion for a two-element system are solved using the asymptotic multiple-scales method for the weak nonlinear system. The analytically obtained periodic response of two coupled microbeams is numerically evaluated by a continuation technique and complemented by a numerical analysis of a three-element array which exhibits quasi-periodic responses and lengthy chaotic transients. This study of small-size microbeam arrays serves for design purposes and the understanding of nonlinear nearest-neighbor interactions of medium- and large-size arrays.

A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
U. H. Hegazy ◽  
M. H. Eissa ◽  
Y. A. Amer
Keyword(s):  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Combined Resonance

This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

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