Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

Random Forces Resulting From Internal Two-Phase Flow on Bends and Tees

2006 ◽  
Author(s):  
E. de Langre ◽  
J. L. Riverin ◽  
M. J. Pettigrew
Keyword(s):  
Two Phase Flow ◽  
Experimental Results ◽  
Time Dependent ◽  
Phase Flow ◽  
Water Mixture ◽  
Dimensionless Form ◽  
Two Phase ◽  
Practical Applications ◽  
Random Forces

The time dependent forces resulting from a two-phase air-water mixture flowing in an elbow and a tee are measured. Their magnitudes as well as their spectral contents are analyzed. Comparison is made with previous experimental results on similar systems. For practical applications a dimensionless form is proposed to relate the characteristics of these forces to the parameters defining the flow and the geometry of the piping.

Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

scholarly journals Laminar Burning Velocities of Hydrogen-Blended Methane–Air and Natural Gas–Air Mixtures, Calculated from the Early Stage of p(t) Records in a Spherical Vessel

Energies ◽  
2021 ◽  
Vol 14 (22) ◽  
pp. 7556
Author(s):  
Maria Mitu ◽  
Domnina Razus ◽  
Volkmar Schroeder
Keyword(s):  
Natural Gas ◽  
Initial Conditions ◽  
Numerical Study ◽  
Early Stage ◽  
Linear Trend ◽  
Burning Velocity ◽  
Pressure Rise ◽  
Spherical Vessel ◽  
Practical Applications ◽  
Hydrogen Addition

The flammable hydrogen-blended methane–air and natural gas–air mixtures raise specific safety and environmental issues in the industry and transportation; therefore, their explosion characteristics such as the explosion limits, explosion pressures, and rates of pressure rise have significant importance from a safety point of view. At the same time, the laminar burning velocities are the most useful parameters for practical applications and in basic studies for the validation of reaction mechanisms and modeling turbulent combustion. In the present study, an experimental and numerical study of the effect of hydrogen addition on the laminar burning velocity (LBV) of methane–air and natural gas–air mixtures was conducted, using mixtures with equivalence ratios within 0.90 and 1.30 and various hydrogen fractions rH within 0.0 and 0.5. The experiments were performed in a 14 L spherical vessel with central ignition at ambient initial conditions. The LBVs were calculated from p(t) data, determined in accordance with EN 15967, by using only the early stage of flame propagation. The results show that hydrogen addition determines an increase in LBV for all examined binary flammable mixtures. The LBV variation versus the fraction of added hydrogen, rH, follows a linear trend only at moderate hydrogen fractions. The further increase in rH results in a stronger variation in LBV, as shown by both experimental and computed LBVs. Hydrogen addition significantly changes the thermal diffusivity of flammable CH4–air or NG–air mixtures, the rate of heat release, and the concentration of active radical species in the flame front and contribute, thus, to LBV variation.

scholarly journals Combination of a Green and a Traditional Method for Estimating Relative and Absolute Ink Age: A Case Study of Ballpoint Pen Ink Dating in Vietnam

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Anh Duc Hoang ◽  
Minh Binh Tu ◽  
Thi Thao Ta ◽  
Manh Hung Hoang
Keyword(s):  
Crystal Violet ◽  
High Performance ◽  
Raman Spectrometry ◽  
Acceptable Error ◽  
Practical Applications ◽  
Relative Dating ◽  
Examination Methods ◽  
Absolute Age ◽  
Green Technique ◽  
Layer Chromatography

The dating of ink in questioned documents remains a significant challenge in forensic investigations in Vietnam and other countries. Many forensic examination methods have been usually applied to ensure the highest accuracy of the assessment results while maintaining high environment awareness. In this study, paper characteristics were physically tested to confirm source similarity, and the relative ink dating was established by high-performance thin-layer chromatography (HPTLC). Absolute ink dating by solvent and dye identification was performed by Raman spectrometry—a green technique, using a time-dependent degradation model for crystal violet and the comparison between 2-phenoxyethanol peak intensities. We found that the relative dating of the questioned document was 14 ± 3 months lesser than that of the reference samples, i.e., the absolute age of the questioned samples was estimated to be 24 ± 3 months. The combination of the conventional HPTLC method with the dynamic crystal violet degradation Raman model provides promising results for relative and absolute ink dating of ballpoint pens, which can be applied for documents written 1–15 years prior to the time of examination. The combination of the abovementioned methods demonstrated an acceptable error margin, affording highly practical applications in modern forensic science.

scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

scholarly journals Determine of the wave equation in the task of electrical oscillations

2018 ◽  
Vol 184 ◽  
pp. 01023
Author(s):  
Gordana V. Jelić ◽  
Vladica Stanojević ◽  
Dragana Radosavljević
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Mathematical Physics ◽  
Equation Of Motion ◽  
Independent Variables ◽  
Equations Of Mathematical Physics ◽  
Basic Equations ◽  
Derivatives Of ◽  
Two Independent Variables ◽  
Transverse Oscillations

One of the basic equations of mathematical physics (for instance function of two independent variables) is the differential equation with partial derivatives of the second order (3). This equation is called the wave equation, and is provided when considering the process of transverse oscillations of wire, longitudinal oscillations of rod, electrical oscillations in a conductor, torsional vibration at waves, etc… The paper shows how to form the equation (3) which is the equation of motion of each point of wire with abscissa x in time t during its oscillation. It is also shown how to determine the equation (3) in the task of electrical oscillations in a conductor. Then equation (3) is determined, and this solution satisfies the boundary and initial conditions.

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.

Dynamic Analysis of a Viscoelastic Nanobeam

2019 ◽  
Vol 799 ◽  
pp. 223-229 ◽  
Author(s):  
Mustafa Arda ◽  
Metin Aydogdu
Keyword(s):  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Elasticity Theory ◽  
Nonlocal Elasticity ◽  
Initial Conditions ◽  
Nonlocal Elasticity Theory ◽  
Equation Of Motion ◽  
Critical Buckling Load ◽  
Damping Effect ◽  
Material Equation

Vibration of an axially loaded viscoelastic nanobeam is analyzed in this study. Viscoelasticity of the nanobeam is modeled as a Kelvin-Voigt material. Equation of motion and boundary conditions for viscoelastic nanobeam are provided with help of Eringen’s Nonlocal Elasticity Theory. Initial conditions are used in solution of governing equation of motion. Damping effect of the viscoelastic nanobeam structure is investigated. Nonlocal effect on natural frequency and damping of nanobeam and critical buckling load is obtained.

scholarly journals Evaluating Approximations and Heuristic Measures of Integrated Information

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 525 ◽  
Author(s):  
André Sevenius Nilsen ◽  
Bjørn Erik Juel ◽  
William Marshall
Keyword(s):  
Binary Systems ◽  
Initial Conditions ◽  
Transition Probability ◽  
Ground Truth ◽  
Transition Probability Matrix ◽  
Efficient Estimation ◽  
Model Systems ◽  
Practical Applications ◽  
Linear Threshold ◽  
Integrated Information

Integrated information theory (IIT) proposes a measure of integrated information, termed Phi (Φ), to capture the level of consciousness of a physical system in a given state. Unfortunately, calculating Φ itself is currently possible only for very small model systems and far from computable for the kinds of system typically associated with consciousness (brains). Here, we considered several proposed heuristic measures and computational approximations, some of which can be applied to larger systems, and tested if they correlate well with Φ. While these measures and approximations capture intuitions underlying IIT and some have had success in practical applications, it has not been shown that they actually quantify the type of integrated information specified by the latest version of IIT and, thus, whether they can be used to test the theory. In this study, we evaluated these approximations and heuristic measures considering how well they estimated the Φ values of model systems and not on the basis of practical or clinical considerations. To do this, we simulated networks consisting of 3–6 binary linear threshold nodes randomly connected with excitatory and inhibitory connections. For each system, we then constructed the system’s state transition probability matrix (TPM) and generated observed data over time from all possible initial conditions. We then calculated Φ, approximations to Φ, and measures based on state differentiation, coalition entropy, state uniqueness, and integrated information. Our findings suggest that Φ can be approximated closely in small binary systems by using one or more of the readily available approximations (r > 0.95) but without major reductions in computational demands. Furthermore, the maximum value of Φ across states (a state-independent quantity) correlated strongly with measures of signal complexity (LZ, rs = 0.722), decoder-based integrated information (Φ*, rs = 0.816), and state differentiation (D1, rs = 0.827). These measures could allow for the efficient estimation of a system’s capacity for high Φ or function as accurate predictors of low- (but not high-)Φ systems. While it is uncertain whether the results extend to larger systems or systems with other dynamics, we stress the importance that measures aimed at being practical alternatives to Φ be, at a minimum, rigorously tested in an environment where the ground truth can be established.

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