Nonlinear Vibration of Buckled Beams

1972 ◽  
Vol 94 (2) ◽  
pp. 637-645 ◽  
Author(s):  
Gwo-Bao Min ◽  
J. G. Eisley
Keyword(s):  
Equations Of Motion ◽  
Single Mode ◽  
Analog Computer ◽  
State Response ◽  
Buckled Beams ◽  
Mode Solution ◽  
Instability Regions ◽  
The Stability ◽  
Parametric Response ◽  
Assumed Mode

The steady state response and stability of free and forced vibration of simply supported, axially restrained, buckled beams is investigated. The equations of motion dealing with the buckled state include the effect of an initial displacement either by initial load or by initial temperature. By an assumed mode solution, the response and stability of two types of vibration are determined—snap-through (symmetrical) and one-sided (unsymmetric) vibration. The theoretical results of the response and stability are verified by analog computer simulation. It is concluded that the stability of the unsymmetric vibration is not a problem and that the different orders of parametric response of the rest modes (the modes originally not excited) in symmetric vibration correspond to the instability regions determined for the approximate single mode response.

Helicopter Ground Resonance Phenomenon With Blade Stiffness Dissimilarities: Experimental and Theoretical Developments

2012 ◽  
Author(s):  
Leonardo Sanches ◽  
Guilhem Michon ◽  
Alain Berlioz ◽  
Daniel Alazard
Keyword(s):  
Equations Of Motion ◽  
Resonance Phenomenon ◽  
Aging Effects ◽  
Ground Resonance ◽  
Different Types ◽  
Stability Chart ◽  
Instability Regions ◽  
The Stability ◽  
Theoretical Results ◽  
Theoretical Developments

Recent works study the ground resonance in helicopters under the aging effects. Indeed, the blades lead-lag stiffness may vary randomly with time and be different from each other (i.e.: anisotropic rotor). The influence of stiffness dissimilarities between blades on the stability of the ground resonance phenomenon is determined through numerical investigations on the periodical equations of motion, treated by using Floquet’s theory. Stability chart highlights the appearance of new instability zones as function of the perturbation introduced on the lead-lag stiffness of one blade. In order to validate the theoretical results, a new experimental setup is designed and developed. The ground resonance instabilities are investigated for different types of rotor configurations (i.e.: isotropic and anisotropic rotors) and the boundaries of stability are determined. A good correlation between both theoretical and experimental results is obtained and the new instability zones, found in asymmetric rotors, are verified experimentally. The temporal responses of the measured signals highlight the exponential divergence at the instability regions.

The Influence of Internal Friction on the Stability of High Speed Rotors With Anisotropic Supports

1969 ◽  
Vol 91 (4) ◽  
pp. 1105-1113 ◽  
Author(s):  
E. J. Gunter ◽  
P. R. Trumpler
Keyword(s):  
Internal Friction ◽  
High Speed ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Analog Computer ◽  
Three Dimensional Model ◽  
Stiffness Properties ◽  
Stiffness Characteristics ◽  
The Stability ◽  
Internal Rotor

This paper evaluates the stability of the single mass rotor with internal friction on damped, anisotropic supports. The paper shows under what conditions the rotor stability may be improved by an undamped support with anisotropic stiffness properties. A three dimensional model is presented to show the influence of rotor and support stiffness characteristics on stability. Curves are also presented on how support damping may also improve or even reduce rotor stability. An analog computer solution of the governing equations of motion is presented showing the shaft transient motion for various speed ranges, and also plots of the rotor steady state motion are given for various speeds up to and including the stability threshold. The analysis is used to explain many of the experimental observations of B. L. Newkirk concerning stability due to internal rotor friction.

Parametric Excitation of a Non-Homogeneous Bernoulli-Euler Beam

1968 ◽  
Vol 10 (3) ◽  
pp. 205-212 ◽  
Author(s):  
P. H. Francis
Keyword(s):  
Mathieu Equation ◽  
Coupled System ◽  
Euler Beam ◽  
Galerkin Procedure ◽  
Acceptable Accuracy ◽  
First Approximation ◽  
Instability Regions ◽  
The Stability ◽  
Thermal Sources ◽  
Parametric Response

In this paper is considered the problem of the stability of the parametric response of a simply supported Bernoulli-Euler beam having an elastic modulus that varies continuously and monotonically throughout its length. The beam is excited by axial harmonic forces applied to the ends. The Galerkin procedure is used, which, in the first approximation, leads to a single Mathieu equation representing the stability regions for an equivalent uniform beam having averaged properties. For the second and higher approximations, the co-ordinate functions used in the Galerkin procedure couple, leading to a coupled system of Mathieu equations. Results from the first and second approximations are compared with, a view toward establishing the degree of non-homogeneity for which the first approximation predicts the instability regions with acceptable accuracy. It is shown that for moderate non-homogeneities, such as might be introduced by thermal sources, the first approximation leads to results of quite tolerable accuracy. In an Appendix are presented some computed data for the free vibrational frequencies of the non-homogeneous beam under static end forces.

A Note on a New Stability Method for the Linear Modes of Nonlinear Two-Degree-of-Freedom Systems

1962 ◽  
Vol 29 (2) ◽  
pp. 258-262 ◽  
Author(s):  
Jack Porter ◽  
C. P. Atkinson
Keyword(s):  
Equations Of Motion ◽  
Analog Computer ◽  
Degree Of Freedom ◽  
Oscillatory Systems ◽  
Coupled Equations ◽  
Theoretical Predictions ◽  
Stability Method ◽  
The Stability ◽  
Two Degree Of Freedom

This paper presents a method for analyzing the stability of the linearly related modes of nonlinear two-degree-of-freedom oscillatory systems. For systems described by the coupled equations x¨1 = f(x1, x2) and x¨2 = g(x1, x2) there exist solutions related by the linear modal restraint x1 = cx2 where c is a constant. Such oscillations are not always stable. The method of this paper allows the prediction of the stability of the modes in terms of the amplitudes of the oscillations and the parameters of the equations of motion. Analog-computer results are presented which confirm the theoretical predictions.

On nonlinear perturbation analysis of a structure carrying a circular cylindrical liquid tank under horizontal excitation

2018 ◽  
Vol 25 (5) ◽  
pp. 1058-1079 ◽  
Author(s):  
N. K. A. Attari ◽  
F. R. Rofooei ◽  
Z. Waezi
Keyword(s):  
Differential Equations ◽  
Perturbation Analysis ◽  
Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Structural Response ◽  
Single Mode ◽  
Excitation Amplitude ◽  
Combined System ◽  
The Stability

The lateral response of a single degree of freedom structural system containing a rigid circular cylindrical liquid tank under harmonic and earthquake excitations at a 1:2 autoparametric resonance case is considered. The governing nonlinear differential equations of motion for the combined system are solved by means of a multiple scales method considering the first three liquid sloshing modes (1,1), (0,1), and (2,1), under horizontal excitation. The fixed points of the gyroscopic type of governing differential equations are determined and their stability is investigated employing the perturbation method. The obtained results reveal an increase in the stability region for a single-mode response with respect to the excitation amplitude. The saturation phenomenon is observed in the decoupled modes of the system; however, the structural mode and the first anti-symmetric mode of liquid are a combination of the saturated mode and another mode whose scale factor is crucial for the structural response. The results of perturbation analysis are in good agreement with results obtained from numerical methods.

scholarly journals Dynamic Stability in a Cracked Turbo Blade-Disk

1999 ◽  
Author(s):  
J. H. Kuang ◽  
B. W. Huang
Keyword(s):  
Multiple Scales ◽  
Rotation Speed ◽  
Equations Of Motion ◽  
Crack Depth ◽  
Disk System ◽  
Rotating Blade ◽  
Blade System ◽  
Instability Regions ◽  
The Stability ◽  
Multiple Scales Perturbation Method

Analysis of the stability in a cracked blade-disk system is proposed. The effect of modal localization on the stability in a rotating blade-disk was studied. A crack near the root of a blade is regarded as a local disorder in this periodically coupled blade system. Hamilton’s principle and Galerkin’s method were used to formulate the equations of motion for the cracked blade-disk. The instability regions of this cracked blade-disk system were specified by employing the multiple scales perturbation method. Numerical results indicate that the rotation speed, shroud stiffness and crack depth in the blades affect the stability regions of this mistuned system significantly.

Efficient Dynamic Computer Simulation of Simple-Closed Spatial Linkages

1990 ◽  
Author(s):  
L. T. Wang
Keyword(s):  
Computer Simulation ◽  
Computational Algorithm ◽  
Equations Of Motion ◽  
Joint Space ◽  
Kinematic Chains ◽  
Spatial Linkages ◽  
Dynamic Computer Simulation ◽  
The Stability ◽  
Coordinate Partitioning ◽  
Generalized Coordinate

Abstract A new method of formulating the generalized equations of motion for simple-closed (single loop) spatial linkages is presented in this paper. This method is based on the generalized principle of D’Alembert and the use of the transformation Jacobian matrices. The number of the differential equations of motion is minimized by performing the method of generalized coordinate partitioning in the joint space. Based on this formulation, a computational algorithm for computer simulation the dynamic motions of the linkage is developed, this algorithm is not only numerically stable but also fully exploits the efficient recursive computational schemes developed earlier for open kinematic chains. Two numerical examples are presented to demonstrate the stability and efficiency of the algorithm.

Sensitivity Analysis for the Steady-State Response of Damped Linear Elastodynamic Systems Subject to Periodic Loads

1990 ◽  
Author(s):  
Daniel A. Tortorelli
Keyword(s):  
Steady State ◽  
Vibration Amplitude ◽  
Equations Of Motion ◽  
Steady State Response ◽  
Analysis Techniques ◽  
Direct Differentiation ◽  
State Response ◽  
Need To Evaluate ◽  
The Time Domain ◽  
General Response

Abstract Adjoint and direct differentiation methods are used to formulate design sensitivities for the steady-state response of damped linear elastodynamic systems that are subject to periodic loads. Variations of a general response functional are expressed in explicit form with respect to design field perturbations. Modal analysis techniques which uncouple the equations of motion are used to perform the analyses. In this way, it is possible to obtain closed form relations for the sensitivity expressions. This eliminates the need to evaluate the adjoint response and psuedo response (these responses are associated with the adjoint and direct differentiation sensitivity problems) over the time domain. The sensitivities need not be numerically integrated over time, thus they are quickly computed. The methodology is valid for problems with proportional as well as non-proportional damping. In an example problem, sensitivities of steady-state vibration amplitude of a crankshaft subject to engine firing loads are evaluated with respect to the stiffness, inertial, and damping parameters which define the shaft. Both the adjoint and direct differentiation methods are used to compute the sensitivities. Finite difference sensitivity approximations are also calculated to validate the explicit sensitivity results.

Influence of Interaction Between Torsion and Bending on Long-Term Nonlinear Rotor Dynamics

1999 ◽  
Author(s):  
Jiazhong Zhang ◽  
Bram de Kraker ◽  
Dick H. van Campen
Keyword(s):  
Critical Speed ◽  
Floquet Theory ◽  
Rotor Dynamics ◽  
Steady State Response ◽  
Bending Vibrations ◽  
Bending Vibration ◽  
State Response ◽  
Stability And Bifurcation ◽  
The Stability

Abstract An elementary system with gears and excited by unbalance mass has been constructed for analyzing the interaction between torsion and bending vibration in rotor dynamics. For this system, only the interaction caused primarily by unbalance mass has been investigated. The stability and bifurcation characteristics of the system have been studied by numerical computation based on Hopf bifurcation and Floquet theory. The results show that the interaction between torsion and bending vibrations can affect the stability and bifurcation of the unbalance response, in particular the onset speed of instability. In addition to the above, the interaction also affects the steady-state response. To investigate the influence of unbalance mass, the periodic solution and its stability have been studied near the first bending critical speed of the decoupled system. All the results show that the coupling of torsion and bending vibrations can have a significant influence on the nonlinear dynamics of the whole system.

scholarly journals A Study of the Stability of a Plate-Like Load Towed beneath a Helicopter

1971 ◽  
Vol 13 (5) ◽  
pp. 330-343 ◽  
Author(s):  
D. F. Sheldon
Keyword(s):  
Equations Of Motion ◽  
Physical Parameters ◽  
Recent Experience ◽  
Wind Tunnel Testing ◽  
Stability Derivatives ◽  
Direct Indication ◽  
Metric Dimensions ◽  
Set Up ◽  
The Stability ◽  
Derivative Information

Recent experience has shown that a plate-like load suspended beneath a helicopter moving in horizontal forward flight has unstable characteristics at both low and high forward speeds. These findings have prompted a theoretical analysis to determine the longitudinal and lateral dynamic stability of a suspended pallet. Only the longitudinal stability is considered here. Although it is strictly a non-linear problem, the usual assumptions have been made to obtain linearized equations of motion. The aerodynamic derivative data required for these equations have been obtained, where possible, for the appropriate ranges of Reynolds and Strouhal number by means of static and dynamic wind tunnel testing. The resulting stability equations (with full aerodynamic derivative information) have been set up and solved, on a digital computer, to give direct indication of a stable or unstable system for a combination of physical parameters. These results have indicated a longitudinal unstable mode for all practical forward speeds. Simultaneously the important stability derivatives were found for this instability and modifications were made subsequently in the suspension system to eliminate the instabilities in the longitudinal sense. Throughout this paper, all metric dimensions are given approximately.

Sign in / Sign up

Export Citation Format

Share Document