Nonlinear Dynamic Coupling in a Beam Vibration

1955 ◽  
Vol 22 (4) ◽  
pp. 573-578
Author(s):  
P. H. McDonald
Keyword(s):  
Nonlinear Dynamic ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Lateral Force ◽  
Dynamic Coupling ◽  
Beam Vibration ◽  
Uniform Beam

Abstract This investigation treats the vibration of a uniform beam with hinged ends which are restrained, and which has arbitrary initial conditions of motion. A representative example is discussed in which the beam is subjected to a concentrated lateral force at the mid-point of its span and released from rest at the deflected position. The equations of motion are found to be inherently nonlinear, even for small vibrations, and there is dynamic coupling of the modes. It is found that the frequencies of the various modes are functionally related to the initial conditions, particularly the amplitudes of all the modes.

The Effects of Harmonic Drive Gears on Robot Dynamics

1991 ◽  
Author(s):  
Tsung-Chieh Lin ◽  
K. Harold Yae
Keyword(s):  
Mathematical Models ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Robot Dynamics ◽  
Harmonic Drive ◽  
Dynamic Coupling ◽  
Manipulator Dynamics ◽  
Dynamic Formulation ◽  
System Equations ◽  
Modelling Assumptions

Abstract Mathematical models of the harmonic drive have been developed, and their effects on manipulator dynamics have been examined. The harmonic drive is modelled as a flexible gear with a high gear reduction ratio. The recursive Newton-Euler dynamic formulation is applied to deriving the system equations of motion that include the effects of the geared actuation. The equations include not only the nonlinear dynamic coupling between rotors and links but the gyroscopic effect due to the spinning rotors. Different modelling assumptions creates four models and their time responses are compared. As an example, a seven degree of freedom robot was chosen to make comparisons in time responses.

Assessing the influence of the road-tire friction coefficient on the yaw and roll stability of articulated vehicles

2018 ◽  
Vol 233 (12) ◽  
pp. 2987-2999 ◽  
Author(s):  
André de Souza Mendes ◽  
Agenor de Toledo Fleury ◽  
Marko Ackermann ◽  
Fabrizio Leonardi ◽  
Roberto Bortolussi
Keyword(s):  
Friction Coefficient ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Lateral Force ◽  
Longitudinal Force ◽  
Convergence Region ◽  
The Road ◽  
Articulated Vehicles ◽  
Tire Friction

This article addresses the yaw stability of articulated vehicles by assessing the influence of the road-tire friction coefficient on the convergence region of a particular equilibrium condition. In addition, the boundaries of this region are compared to the boundaries of the non-jackknife and non-rollover regions to distinguish the instability phenomenon, jackknife or roll-over, responsible for this delimitation. The vehicle configuration considered in this analysis is composed by one tractor unit and one towed unit connected through an articulation point, for instance, a tractor-semitrailer combination. A nonlinear articulated bicycle model with four degrees of freedom is used together with a nonlinear lateral force tire model. To estimate the convergence region, the phase trajectory method is used. The equations of motion of the mathematical model are numerically integrated for different initial conditions in the phase plane, and the state orbits are monitored in order to verify the convergence point and the occurrence of instability events. In all cases, the longitudinal force on each tire, such as traction and braking, is not considered. The results show the existence of convergence regions delimited only by jackknife events, for low values of the friction coefficient, and only by rollover events, for high values of the friction coefficient. Moreover, the transition between these two conditions as the friction coefficient is changed is graphically presented. The main contributions of this article are the identification of the abrupt reduction of the convergence region as the value of the friction coefficient increases and the distinction of the instability events, jackknife or rollover, that define the boundaries of the convergence region.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Nonlinear Dynamic Study on Effects of Rub-Impact Caused by Oil-Rupture in a Multi-Shafts Turbine With a Squeeze Film Damper

2014 ◽  
Author(s):  
T. N. Shiau ◽  
C. R. Wang ◽  
D. S. Liu ◽  
W. C. Hsu ◽  
T. H. Young
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Rotor System ◽  
Squeeze Film ◽  
Speed Ratio ◽  
Frequency Spectra ◽  
Squeeze Film Damper ◽  
Assumed Mode Method ◽  
Nonlinear Dynamic Behavior

An investigation is carried out the analysis of nonlinear dynamic behavior on effects of rub-impact caused by oil-rupture in a multi-shafts turbine system with a squeeze film damper. Main components of a multi-shafts turbine system includes an outer shaft, an inner shaft, an impeller shaft, ball bearings and a squeeze film damper. In the squeeze film damper, oil forces can be derived from the short bearing approximation and cavitated film assumption. The system equations of motion are formulated by the global assumed mode method (GAMM) and Lagrange’s approach. The nonlinear behavior of a multi-shafts turbine system which includes the trajectories in time domain, frequency spectra, Poincaré maps, and bifurcation diagrams are investigated. Numerical results show that large vibration amplitude is observed in steady state at rotating speed ratio adjacent to the first natural frequency when there is no squeeze film damper. The nonlinear dynamic behavior of a multi-shafts turbine system goes in its way into aperiodic motion due to oil-rupture and it is unlike the usual way (1T = >2T = >4T = >8T etc) as compared to one shaft rotor system. The typical routes of bifurcation to aperiodic motion are observed in a multi-shafts turbine rotor system and they suddenly turn into aperiodic motion from the periodic motion without any transition. Consequently, the increasing of geometric or oil parameters such as clearance or lubricant viscosity will improve the performance of SFD bearing.

Influence of Bistable Plunge Stiffness On Nonlinear Airfoil Flutter

2021 ◽  
Author(s):  
Renan F. Corrêa ◽  
Flávio D. Marques
Keyword(s):  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Vibration ◽  
Nonlinear Behavior ◽  
Limit Cycle Oscillation ◽  
Restoring Force ◽  
Aeroelastic System ◽  
Technical Literature ◽  
Practical Applications

Abstract Aeroelastic systems have nonlinearities that provide a wide variety of complex dynamic behaviors. Nonlinear effects can be avoided in practical applications, as in instability suppression or desired, for instance, in the energy harvesting design. In the technical literature, there are surveys on nonlinear aeroelastic systems and the different manners they manifest. More recently, the bistable spring effect has been studied as an acceptable nonlinear behavior applied to mechanical vibration problems. The application of the bistable spring effect to aeroelastic problems is still not explored thoroughly. This paper contributes to analyzing the nonlinear dynamics of a typical airfoil section mounted on bistable spring support at plunging motion. The equations of motion are based on the typical aeroelastic section model with three degrees-of-freedom. Moreover, a hardening nonlinearity in pitch is also considered. A preliminary analysis of the bistable spring geometry’s influence in its restoring force and the elastic potential energy is performed. The response of the system is investigated for a set of geometrical configurations. It is possible to identify post-flutter motion regions, the so-called intrawell, and interwell. Results reveal that the transition between intrawell to interwell regions occurs smoothly, depending on the initial conditions. The bistable effect on the aeroelastic system can be advantageous in energy extraction problems due to the jump in oscillation amplitudes. Furthermore, the hardening effect in pitching motion reduces the limit cycle oscillation amplitudes and also delays the occurrence of the snap-through.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

scholarly journals Elastic shocks in relativistic rigid rods and balls

2019 ◽  
Vol 475 (2225) ◽  
pp. 20180858 ◽  
Author(s):  
João L. Costa ◽  
José Natário
Keyword(s):  
Free Boundary ◽  
Minkowski Space ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Space Time ◽  
Spherically Symmetric ◽  
Soft Phase ◽  
Minkowski Space Time ◽  
Dimensional Minkowski Space

We study the free boundary problem for the ‘hard phase’ material introduced by Christodoulou in (Christodoulou 1995 Arch. Ration. Mech. Anal. 130 , 343–400), both for rods in (1 + 1)-dimensional Minkowski space–time and for spherically symmetric balls in (3 + 1)-dimensional Minkowski space–time. Unlike Christodoulou, we do not consider a ‘soft phase’, and so we regard this material as an elastic medium, capable of both compression and stretching. We prove that shocks must be null hypersurfaces, and derive the conditions to be satisfied at a free boundary. We solve the equations of motion of the rods explicitly, and we prove existence of solutions to the equations of motion of the spherically symmetric balls for an arbitrarily long (but finite) time, given initial conditions sufficiently close to those for the relaxed ball at rest. In both cases we find that the solutions contain shocks if and only if the pressure or its time derivative do not vanish at the free boundary initially. These shocks interact with the free boundary, causing it to lose regularity.

Nonlinear Dynamic of Cantilever Functionally Graded Plates Under the Thermalmechanical Loads

2009 ◽  
Author(s):  
Yu-xin Hao ◽  
Wei Zhang ◽  
Jian-hua Wang
Keyword(s):  
Nonlinear System ◽  
Functionally Graded Materials ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Governing Equations ◽  
Graded Materials ◽  
Two Degree Of Freedom

An analysis on nonlinear dynamic of a cantilevered functionally graded materials (FGM) plate which subjected to the transverse excitation in the uniform thermal environment is presented for the first time. Materials properties of the constituents are graded in the thickness direction according to a power-law distribution and assumed to be temperature dependent. In the framework of the Third-order shear deformation plate theory, the nonlinear governing equations of motion for the functionally graded materials plate are derived by using the Hamilton’s principle. For cantilever rectangular plate, the first two vibration mode shapes that satisfy the boundary conditions is given. The Galerkin’s method is utilized to discretize the governing equations of motion to a two-degree-of-freedom nonlinear system under combined thermal and external excitations. By using the numerical method, the two-degree-of-freedom nonlinear system is analyzed to find the nonlinear responses of the cantilever FGMs plate. The influences of the thermal environments on the nonlinear dynamic response of the cantilevered FGM plate are discussed in detail through a parametric study.

Vibrational Characteristics of Ball Bearings

1977 ◽  
Vol 99 (2) ◽  
pp. 284-287 ◽  
Author(s):  
P. K. Gupta ◽  
L. W. Winn ◽  
D. F. Wilcock
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Vibrational Analysis ◽  
Dominant Frequency ◽  
Hertzian Contact ◽  
Analytical Simulation ◽  
Contact Frequency ◽  
Ball Contact ◽  
Vibrational Characteristics ◽  
General Motion

The classical differential equations of motion of the ball mass center in an angular contact thrust loaded ball bearing are integrated with prescribed initial conditions in order to simulate the natural high frequency vibrational characteristics of the general motion. Two distinct frequencies are identified in the analytical simulation and their existence is also confirmed experimentally. One of the frequencies is found to be associated with the Hertzian contact spring at the ball race contact and it is therefore defined as the “elastic contact frequency”, Ωe. The other dominant frequency corresponding to oscillatory motion of the ball in the raceway groove appears to be kinematic in nature and it is, therefore, termed as the “bearing kinematic frequency”, Ωk. It is shown that for a given bearing Ωe and Ωk, vary as, respectively, 1/6 and 1/2 powers of the ball contact load and, therefore, for a given load these frequencies correspond to the natural frequencies of the bearing as applied in any vibrational analysis or simulation.

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