Nonlinear Bending-Torsion Flutter of a Cantilever Wing Subjected to Parametric Excitation

2005 ◽  
Author(s):  
R. A. Ibrahim ◽  
S. C. Castravete
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Air Flow ◽  
Parametric Instability ◽  
Flow Speed ◽  
Nonlinear Bending ◽  
Nonlinear Flutter ◽  
Nonlinear Equations Of Motion

This study deals with the nonlinear flutter of a cantilever wing in the absence and presence of parametric excitation that acts in the plane of highest rigidity. The nonlinear equations of motion in the presence of an incompressible fluid flow are derived using Hamilton’s principle. The regions of parametric instability are obtained for different values of flow speed. In the neighborhood of combination parametric resonance, the nonlinear response is determined using the multiple scales method for different values of flow speed. In the absence of parametric excitations, numerical simulation is performed for flow speeds at the critical flutter speed. It is found that the nonlinear flutter of the two modes depends on initial conditions, and exhibits symmetric periodic oscillations. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the presence of air flow, the two modes possess the same frequency response. Depending on the flow speed the response could be periodic, quasi-periodic, or chaotic.

On the Parametric Excitation of Pendulum-Type Vibration Absorber

1961 ◽  
Vol 28 (3) ◽  
pp. 330-334 ◽  
Author(s):  
Eugene Sevin
Keyword(s):  
Energy Transfer ◽  
Numerical Solution ◽  
Parametric Excitation ◽  
Equations Of Motion ◽  
Vibration Absorber ◽  
Single Cycle ◽  
Linearized System ◽  
Nonlinear Equations Of Motion ◽  
Beam Oscillation ◽  
Energy Interchange

The free motion of an undamped pendulum-type vibration absorber is studied on the basis of approximate nonlinear equations of motion. It is shown that this type of mechanical system exhibits the phenomenon of auto parametric excitation; a type of “instability” which cannot be accounted for on the basis of the linearized system. Complete energy transfer between modes is shown to occur when the beam frequency is twice the simple pendulum frequency. On the basis of a numerical solution, approximately 150 cycles of the beam oscillation take place during a single cycle of energy interchange.

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.

Dynamic Response of Spinning Blades Subjected to Gyroscopic Motion

1994 ◽  
Vol 116 (1) ◽  
pp. 6-15 ◽  
Author(s):  
T. H. Young ◽  
G. T. Liou
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Spin Axis ◽  
System Response ◽  
First Order ◽  
System Equations ◽  
Element Technique ◽  
Gyroscopic Motion

This paper presents an investigation into the vibration and stability of a blade spinning with respect to a nonfixed axis. Due to the motion of the spin axis, parametric instability of the blade may occur in certain situations. In this work, the discretized equations of motion are first formulated by the finite element technique. Then the system equations are transformed, by a special modal analysis procedure, into independent sets of first-order simultaneous differential equations. Each set of differential equations is solved analytically by the method of multiple scales if the precessional speed of the spin axis is assumed to be small compared to the spin rate of the blade, yielding the system response and the expressions for the boundaries of the unstable regions. Finally, the effects of system parameters on the changes in these boundaries are studied numerically.

scholarly journals MATHEMATICAL MODELING OF PROCESSES OF SEPARATION OF COMPONENTS OF GRAIN MATERIAL IN THE COMBINED VIBRATION-AIR SEPARATOR

2020 ◽  
pp. 51-59
Author(s):  
Sergey Stepanenko ◽  
Borys Kotov
Keyword(s):  
Mathematical Modeling ◽  
Equations Of Motion ◽  
Air Flow ◽  
Flow Speed ◽  
Material Point ◽  
Experiment Planning ◽  
Theoretical Studies ◽  
Fluidized Layer ◽  
Aerodynamic Properties ◽  
Air Separator

Development of a mathematical model and calculated analytical dependencies for determining the trajectories and parameters of grain movement in a vibro-fluidized layer of grain material components under the action of a pulsating air flow. They are based on the methods of deterministic mathematical modeling and theoretical mechanics based on the equations of motion of a material point at a variable air flow speed and the action of a pulsating air flow. Theoretical studies were carried out using the methods of mathematical analysis and modeling. The research results were processed using elements of the theory of probability and mathematical statistics using software packages; to determine the rational parameters of the process, the method of statistical experiment planning was used. A mathematical description of the motion of the grain material particles in a combined vibration-air separator under the action of a pulsating air flow of variable speed is given. The trajectories of motion of particles with different sizes are obtained. The obtained equation of motion of a particle under the influence of a pulsating air flow makes it possible to determine the dependence of the speed of movement of the material in a vibro-fluidized layer of grain material on a number of factors: the geometric parameters of the sieve-free sieve, the feed angle of the material, the initial kinematic mode of the material, the index of the kinematic mode of the sieve-free sieve, as well as the coefficient of windage of the grain. On the basis of theoretical studies, the possibility of separating particles of grain material into fractions according to aerodynamic properties with vibropneumatic loading of grain into the channel has been determined. The use of a pulsating air flow as a separating carrier, and taking into account the deflecting forces, made it possible to significantly increase the splitting of the trajectories and the criterion for dividing the grain into fractions.

Landing Posture Control of a Generalized Twin-Body System Via Methods of Input-Output Linearization and Computed Torque

2007 ◽  
Author(s):  
Yi-Ling Yang ◽  
Paul C.-P. Chao ◽  
Cheng-Kuo Sung
Keyword(s):  
Structural Damage ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Posture Control ◽  
Body System ◽  
Input Output ◽  
Bipedal Robot ◽  
Nonlinear Equations Of Motion ◽  
Computed Torque ◽  
Input Output Linearization

This study is dedicated to achieve landing posture control of a generalized twin-body system using the methods of input-output linearization and computed torque. The twin-body system is a simplified model of bipedal robot, and the success in landing posture control would prevent structural damage. To the end, the dynamic equations are built based on Newton-Euler formulation. The technique of input-output linearization is next applied to the original nonlinear equations of motion, which is followed by adopting the method of computed torque to achieve desired landing postures. While designing the controller, system singularities are circumvented by choosing controllable set of initial conditions and stable landing postures. There are two uncontrollable postures that are immovable under input torques or/and the coupling centripetal and Coriolis forces. Finally, simulation results show that the designed controller is capable of performing desired landing posture control.

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.

Dynamic Stability of Axially Moving Plates With Periodically Varying Speeds Under Partially Distributed Edge Tensions

2012 ◽  
Author(s):  
T. H. Young ◽  
S. J. Huang ◽  
A. C. Liu
Keyword(s):  
Dynamic Stability ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Parametric Instability ◽  
Plane Elasticity ◽  
Moving Plate ◽  
System Equations ◽  
Axially Moving ◽  
Axially Moving Web ◽  
Moving Speed

This paper investigates the dynamic stability of an axially moving web which translates with periodically varying speeds and is subjected to partially distributed tensions on two opposite edges. The web is modeled as a rectangular plate simply supported at two opposite edges where the tension is applied, and free at the other two edges. The plate is assumed to possess internal damping, which obeys the Kelvin-Voigt model. The moving speed of the plate is expressed as the sum of a constant speed and a periodical perturbation with a zero mean. Due to the periodically varying speed of the moving plate, terms with time-dependent coefficients appear in the equations of motion, which may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the partially distributed edge tensions is determined exactly by the theory of plane elasticity. Then, the dependence on the spatial coordinates in the equations of motion is eliminated by the Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve this set of system equations analytically if the periodical perturbation of the moving speed is much smaller as compared with the average speed of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the sum-type appear between modes having the same symmetry class in the transverse direction. Unstable regions of main resonances are generally larger than those of sum-type resonances.

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