Investigation of Restitution Coefficient and Spring-Damper Models for the Bouncing Ball Problem

2011 ◽  
Author(s):  
Alexandre Carbonelli ◽  
Joe¨l Perret-Liaudet ◽  
Emmanuel Rigaud ◽  
Mohamed-Sai¨d Feki
Keyword(s):  
Contact Stiffness ◽  
Damping Ratio ◽  
Dynamic Responses ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Restitution Coefficient ◽  
Bouncing Ball ◽  
Ball Problem ◽  
Non Linear System ◽  
The Impact

The non linear system under study consists on the bouncing ball problem. Focusing on the n-T periodic solutions and the permanent contact motion, simulations performed underline the effect of spring-damper contact model in comparison with the classical restitution coefficient. Both approaches are implemented in an adimensional way. For the restitution coefficient approach, iterating maps are easy to obtain after some assumptions. On the contrary, the spring-damper model leads to transcendental equations needing the use of numerical continuation methods. The damping ratio is defined as a function of the restitution coefficient. The effect of the contact stiffness is studied. For high values of the contact stiffness, the spring-damper model has the same behavior as the restitution coefficient model as the impact duration gets shorter. Predictions of the two models diverge when the contact stiffness decreases. Results are illustrated by time histories and Poincare´ Maps of dynamic responses. This paper aims to be guideline to quantify the error made by making the assumptions required for a restitution coefficient model.

Quantifying parametric uncertainty in the Rothermel model

2008 ◽  
Vol 17 (5) ◽  
pp. 638 ◽  
Author(s):  
Edwin Jimenez ◽  
M. Yousuff Hussaini ◽  
Scott Goodrick
Keyword(s):  
Variance Reduction ◽  
Input Parameter ◽  
Parametric Uncertainty ◽  
Fire Spread ◽  
The United States ◽  
Linear System Of Equations ◽  
Spread Model ◽  
Non Linear System ◽  
The Impact ◽  
Sensitivity Derivative

The purpose of the present work is to quantify parametric uncertainty in the Rothermel wildland fire spread model (implemented in software such as BehavePlus3 and FARSITE), which is undoubtedly among the most widely used fire spread models in the United States. This model consists of a non-linear system of equations that relates environmental variables (input parameter groups) such as fuel type, fuel moisture, terrain, and wind to describe the fire environment. This model predicts important fire quantities (output parameters) such as the head rate of spread, spread direction, effective wind speed, and fireline intensity. The proposed method, which we call sensitivity derivative enhanced sampling, exploits sensitivity derivative information to accelerate the convergence of the classical Monte Carlo method. Coupled with traditional variance reduction procedures, it offers up to two orders of magnitude acceleration in convergence, which implies that two orders of magnitude fewer samples are required for a given level of accuracy. Thus, it provides an efficient method to quantify the impact of input uncertainties on the output parameters.

Dynamics of the Gear Train Set in Wind Turbine

2011 ◽  
Vol 697-698 ◽  
pp. 701-705
Author(s):  
D.D. Ji ◽  
Y.M. Song ◽  
J. Zhang
Keyword(s):  
Wind Turbine ◽  
Dynamic Model ◽  
Harmonic Balance Method ◽  
Planetary Gear ◽  
Gear Train ◽  
Dynamic Responses ◽  
Multi Stage ◽  
Lumped Parameter ◽  
Cartesian Coordinate ◽  
The Impact

A lumped-parameter dynamic model for gear train set in wind turbine is proposed to investigate the dynamics of the speed-increasing gear box. The proposed model is developed in a universal Cartesian coordinate, which includes transversal and torsional deflections of each component, time-varying mesh stiffness, gear profile errors and external excitations. By solving the dynamic model, a modal analysis is performed. The results indicate that the modal properties of the multi-stage gear train in wind turbine are similar to those of a single-stage planetary gear set. A harmonic balance method (HBM) is used to obtain the dynamic responses of the gearing system. The responses give insight into the impact of excitations on the vibrations.

Impact of Stefan blowing and magnetic dipole on bio-convective flow of Maxwell nanofluid over a stretching sheet

2021 ◽  
pp. 095440892110581
Author(s):  
A. Alhadhrami ◽  
Hassan A. H. Alzahrani ◽  
B. M. Prasanna ◽  
N. Madhukeshwara ◽  
K. C. Rajendraprasad ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Brownian Motion ◽  
Magnetic Dipole ◽  
Stretching Sheet ◽  
High Heat ◽  
Hard Drives ◽  
Linear System Of Equations ◽  
High Heat Transfer ◽  
Non Linear System ◽  
The Impact

The features of ferromagnetic fluids make it supportive for an extensive usage in loudspeakers, magnetic resonance imaging, computer hard drives, directing of magnetic drug and magnetic hyperthermia. Owing to all such potential applications, the current investigation is to understand the relationship between the thermal distribution, magnetic field and resulting fluid flow of Maxwell liquid over a stretching sheet. Investigation of thermal energy and concentration is carried out in the presence of thermal radiation, non-uniform heat sink/source, chemical reaction, Stefan blowing, magnetic dipole, thermophoresis and Brownian motion. Also, microorganisms are considered just to stabilize the suspended nanoparticles. Boundary layer approximation is employed during mathematical derivation. Based on a new constitutive relation, the governing equations are formulated and are reduced into a coupled non-linear system of equations using appropriate transformations. Further, these equations are solved numerically using fourth-order Runge–Kutta method with shooting technique. The impact of involved parameters is discussed and analysed graphically. Outcomes disclose that Newtonian liquid shows high heat transfer when compared to non-Newtonian (Maxwell) liquid for increased values of Brownian motion and thermophoresis parameters. Increased values of Peclet number declines the rate of gyrotactic microorganisms. Finally, an increase in Brownian and thermophoresis motion parameters declines the rate of heat transfer.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

scholarly journals Stationary localized structures and the effect of the delayed feedback in the Brusselator model

2018 ◽  
Vol 376 (2135) ◽  
pp. 20170385 ◽  
Author(s):  
B. Kostet ◽  
M. Tlidi ◽  
F. Tabbert ◽  
T. Frohoff-Hülsmann ◽  
S. V. Gurevich ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Dissipative Structures ◽  
Delayed Feedback ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Delayed Feedback Control ◽  
Brusselator Model ◽  
Localized Structures ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions

The Brusselator reaction–diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

scholarly journals Influence of Structure on the Aseismic Stability and Dynamic Responses of Liquefiable Soil

2021 ◽  
Author(s):  
Pengfei Dou ◽  
Chengshun Xu ◽  
Xiuli Du ◽  
Su Chen
Keyword(s):  
Earthquake Engineering ◽  
Damping Ratio ◽  
Free Field ◽  
Shaking Table ◽  
Dynamic Responses ◽  
Seismic Stability ◽  
Geotechnical Earthquake Engineering ◽  
Liquefiable Soil ◽  
Porewater Pressure ◽  
Lateral Displacements

Abstract In previous major earthquakes, the damage and collapse of structures located in liquefied field which caused by site failure a common occurrence, and the problem of evaluation and disscusion on site liquefaction and the seismic stability is still a key topic in geotechnical earthquake engineering. To study the influence of the presence of structure on the seismic stability of liquefiable sites, a series of shaking table tests on liquefiable free field and non-free field with the same soil sample was carried out. It can be summarized from experimental results as following. The natural frequency of non-free field is larger and the damping ratio is smaller than that of free field. For the weak seismic loading condition, the dynamic response of sites show similar rules and trend. For the strong ground motion condition, soils in both experiments all liquefied obviously and the depth of liquefaction soil in the free field is significantly greater than that in the non-free field, besides, porewater pressure in the non-free field accumulated relately slow and the dissapited quikly from analysis of porewater pressure ratios(PPRs) in both experiments. The amplitudes of lateral displacements and acceleration of soil in the non-free field is obviously smaller than that in the free field caused by the effect of presence of the structure. In a word, the presence of structures will lead to the increase of site stiffness, site more difficult to liquefy, and the seismic stability of the non-free site is higher than that of the free site due to soil-structure interaction.

scholarly journals The Impact Behavior between Coal Gangue Particles and the Tail Beam Based on Rock Failure

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Zhengyuan Xin ◽  
Qingliang Zeng ◽  
Yang Yang
Keyword(s):  
Energy Consumption ◽  
Impact Force ◽  
Rock Failure ◽  
Coal Gangue ◽  
Dynamic Responses ◽  
Particle Model ◽  
Brittle Damage ◽  
Failure Conditions ◽  
Fracture Particle ◽  
The Impact

In top coal caving mining, common impact occurs between coal gangue particles and tail beam. Little attention has been paid to the effects of coal gangue particles failure on impact force and tail beam response theoretically, numerically, and experimentally. This paper aims to reveal the influence of coal gangue particles failure on the impact effect of tail beam. First, this paper incorporates the theory of rock failure and energy consumption to assess the impact process of coal gangue particles on the tail beam. A new model to simulate the actual failure conditions of rock particles was developed: the brittle damage-fracture particle model. By comparing damage phenomena and simulation data, the brittle damage-fracture particle model was proved to be correct. Based on this model, a dynamic simulation of brittle coal gangue particles impacting the tail beam was conducted. Then, the dynamic responses of the particles and tail beam were analyzed. The results show that particle failure significantly affects the impact force and dynamic response of the tail beam. The impact effects of coal and gangue particles on the tail beam and their failure energy consumption also differed significantly. This paper stresses the importance of coal gangue particle failure conditions for research on top coal caving mining. Theoretical support is provided for the research of coal gangue identification technology based on the tail beam vibration signal.

Dynamic Responses of Buried Pipelines Under Impact Loading Caused by Landslide

2012 ◽  
Vol 204-208 ◽  
pp. 3476-3479 ◽  
Author(s):  
Xiu Xing Zhu ◽  
Shi Feng Xue ◽  
Xing Hua Tong ◽  
Chuan Qi Liu
Keyword(s):  
Impact Loading ◽  
Impact Force ◽  
Large Diameter ◽  
Influence Factors ◽  
Dynamic Responses ◽  
Thin Wall ◽  
Buried Pipelines ◽  
Steel Pipes ◽  
Mountainous Regions ◽  
The Impact

Cases of pipeline damage caused by landslide are common in coastal or mountainous regions, where the design of buried pipelines should be improved in order to reduce the risk of damage or failure. Dynamic responses of large diameter thin wall steel pipes under impact loading were analyzed using a nonlinear contact model of pipe-soil coupling in this paper. Several influence factors were studied, such as the impact velocity of rockfall, buried depth of pipeline, ratio of diameter to thickness and style of soil. The results show that an ellipsoid induces much more impact force than a sphere which has the same volume, and the larger one in volume have greater impact force for two spheres. Dangerous compressive areas of pipeline occupy 1/6 of the whole area, so the pipelines subject to landslide occur local failure. Based on results, some useful suggestions for the design of pipelines in landslide region are given

scholarly journals Inversion for Damping Ratio of Flat Blade Based on BP Neural Network

Proceedings ◽  
2018 ◽  
Vol 2 (8) ◽  
pp. 547
Author(s):  
Xiamei Zhang ◽  
Shudan Xia
Keyword(s):  
Neural Network ◽  
Decay Time ◽  
Bp Neural Network ◽  
Damping Ratio ◽  
Drop Test ◽  
Analysis Model ◽  
Bp Network ◽  
The Real ◽  
Neural Network Inversion ◽  
The Impact

Aero engine is impacted by foreign objects frequently during daily usage, including runway gravel, birds, fuselage components and so on, so the fan and compressor may damage, resulting in serious air crash. Thus, simulating the impact of blades and establishing the numerical analysis model of dynamic response demand immediate attention. In the analysis model, damping coefficient is one of the most important physical parameters of the blade structure and cannot be directly measured. Rayleigh damping is widely applied and can be converted to direct modal damping in ABAQUS. BP neural network is a multi-layer feedforward neural network using back propagation algorithm to adjust the network weights. It can be proved that there exists a three-layer BP network to realize the mapping of arbitrary continuous functions with arbitrary precision. In this study, a novel method for obtaining the damping ratio of the flat blade which applies BP neural network inversion is proposed. In order to demonstrate this method, a simplified experiment was conducted. Firstly, fix a section of aluminum plate and then conduct two set of drop tests on different positions with different impact velocities by a steel ball. At the same time, vibration response was recorded by displacement sensor. Secondly, establish a finite element model using ABAQUS to simulate the drop test. Adopt twenty groups of models with different damping ratio and then obtain their amplitudes and decay time, respectively. Thirdly, train a BP neural network using MATLAB program and then establish the mapping relationship between amplitude, decay time and damping ratio. Fourth, a set of experimental amplitude and decay time is substituted into the previously obtained BP neural network mapping model, and then the real damping ratio is obtained by inference. Finally, the real damping ratio is applied to the flat blade impact simulation of the other set of drop test for validation. The numerical results are consistent with the experimental data, which indicates that the damping ratio obtained by BP neural network inversion is reasonable and reliable.

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