Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part I: Theory

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
Ryan J. Monroe ◽  
Steven W. Shaw
Keyword(s):  
Transient Response ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Torque Converter ◽  
Companion Paper ◽  
Transient Dynamics ◽  
Circular Path ◽  
Vibration Absorbers ◽  
Automotive Engines ◽  
Minimal Mass

Transient dynamics of centrifugal pendulum vibration absorbers, which are used for reducing torsional vibrations in rotating machines, are investigated using analysis and simulations of a dynamical model. These absorbers are being implemented in automotive engines to smooth vibrations and aid with fuel saving technologies, such as cylinder deactivation and torque converter lockup. In order for the absorbers to operate effectively with minimal mass, they must be designed to accommodate large amplitude, nonlinear responses, and in automotive engines they will experience a variety of transient environments. Here we consider the most severe transient environment, that of sudden activation near resonance, which leads to beating behavior of a nonlinear oscillator coupled to a driven rotor. An approximate method for predicting the percent overshoot of the beating transient response is derived, based on perturbation analysis of the system equations of motion. The main result is expressed in terms of the system and excitation parameters, and is found to accurately predict results from direct simulations of the model equations of motion. It is shown that absorbers with near-tautochronic paths behave much like linear absorbers, and when lightly damped and start from small initial conditions, they have an overshoot close to 100%. For absorbers with softening paths, such as the commonly used circular path absorbers, the overshoot can reach up to 173%, depending on system and input parameters, far exceeding predictions from linear analysis. These results provide a useful tool for design of absorbers to meet transient response specifications. In the following companion paper an experimental investigation is used to verify the analytical predictions.

Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers

2011 ◽  
Author(s):  
Ryan J. Monroe ◽  
Steven W. Shaw
Keyword(s):  
Equations Of Motion ◽  
Torque Converter ◽  
Transient Dynamics ◽  
Circular Path ◽  
Vibration Absorbers ◽  
Approximate Analytical Expression ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Nonlinear Transient ◽  
System Equations ◽  
Minimal Mass

This paper describes an analytical and experimental investigation of the transient dynamics of centrifugal pendulum vibration absorbers, which are used for reducing torsional vibrations in rotating machines. Recently these absorbers have been proposed for use in automotive engines, to aid with fuel saving technologies such as cylinder deactivation and torque converter lockup. In order for them to operate effectively with minimal mass, they must be designed to allow for large amplitude, nonlinear responses. In this paper we consider the transient dynamics of these absorbers, focusing on the response during startup. During these transient events the absorbers experience a beating type motion, resulting in overshoot of the absorber response before reaching steady state conditions. Using a perturbation analysis of the system equations of motion, an approximate analytical expression for nonlinear overshoot is derived, relating the overshoot to the system and excitation parameters. These predictive results are derived for a general class of absorbers, and are verified by simulations of the full equations of motion and by experiments using a fully instrumented spin rig. It is found that the overshoot for absorbers with softening nonlinearity, such as circular path absorbers, can be well over the 100% upper limit for a linear absorber, and can be as high as 173%. For absorbers with tautochronic paths, the overshoot remains quite close to that of the linearized system, even for large amplitudes. These results provide a useful tool for the design of absorbers to meet transient response specifications.

Nonlinear Transient Dynamics of Pendulum Torsional Vibration Absorbers—Part II: Experimental Results

2013 ◽  
Vol 135 (1) ◽  
Author(s):  
Ryan J. Monroe ◽  
Steven W. Shaw
Keyword(s):  
Torsional Vibration ◽  
Companion Paper ◽  
Experimental Results ◽  
Circular Path ◽  
Vibration Absorbers ◽  
Automotive Engine ◽  
Engine Order ◽  
Applied Torque ◽  
Centrifugal Pendulum Vibration Absorbers ◽  
Nonlinear Transient

This paper presents results from an experimental investigation of the transient response of centrifugal pendulum vibration absorbers, including a comparison with the analytical results derived in the companion paper, Part I. The focus of the study is the overshoot experienced by pendulum-type torsional vibration absorbers when a rotor running at a constant speed is suddenly subjected to an applied fluctuating torque. The experiments are carried out using a fully instrumented spin rig controlled by a servo motor that can provide user-specified engine order disturbances, including those that simulate automotive engine environments. The absorber overshoot depends on the absorber tuning relative to the excitation order, the absorber damping, the amplitude of the applied torque, and on the system nonlinearity, which is set by the absorber path and/or kinematic coupling between the rotor and the absorber. Two types of absorbers are used in the study, a simple circular path pendulum, for which the path nonlinearity is dominant, and a nearly tautochronic path pendulum with a bifilar support, for which the path and coupling nonlinearities are both small. It is found that the experimental results agree very well with the analytical predictions from the companion paper. In addition, it is confirmed that the general path pseudoenergy prediction (which depends on a single parameter) provides a useful, conservative upper bound for most practical absorber designs, provided the absorber damping is small.

Optimal Design of Vibration Absorber Systems Supported by Elastic Base

1991 ◽  
Author(s):  
Hashem Ashrafiuon
Keyword(s):  
Dynamic Models ◽  
Damping Ratio ◽  
Equations Of Motion ◽  
Elastic Base ◽  
Design Parameters ◽  
Primary System ◽  
Vibration Absorbers ◽  
Equivalent Damping ◽  
System Equations ◽  
Different Levels

Abstract This paper presents the effect of foundation flexibility on the optimum design of vibration absorbers. Flexibility of the base is incorporated into the absorber system equations of motion through an equivalent damping ratio and stiffness value in the direction of motion at the connection point. The optimum values of the uncoupled natural frequency and damping ratio of the absorber are determined over a range of excitation frequencies and the primary system damping ratio. The design parameters are computed and compared for the rigid, static, and dynamic models of the base as well as different levels of base flexibility.

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.

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.

scholarly journals Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

1978 ◽  
Vol 88 (4) ◽  
pp. 685-709 ◽  
Author(s):  
U. Schumann ◽  
G. S. Patterson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Numerical Study ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Companion Paper ◽  
Real Space ◽  
Pressure Gradients ◽  
Velocity Statistics ◽  
Decaying Turbulence

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

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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