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Duffing Oscillator
The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by \ddot + \delta \dot + \alpha x + \beta x^3 = \gamma \cos (\omega t), where the (unknown) function x = x(t) is the displacement at time , \dot is the first derivative of x with respect to time, i.e. velocity, and \ddot is the second time-derivative of x, i.e. acceleration. The numbers \delta, \alpha, \beta, \gamma and \omega are given constants. The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion (which corresponds to the case \beta=\delta=0); in physical terms, it models, for example, an elastic pendulum whose spring's stiffness does not exactly obey Hooke's law. The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover, the Duffing system presents in t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duffing Oscillator Strange Attractor With Color
Georg Wilhelm Christian Caspar Duffing (April 11, 1861 in Waldshut – April 5, 1944 in Schwedt/Oder) was a German engineer and inventor. In 1918, he described vibrations and their resonances mathematically, as the Duffing equation. Georg Duffing's equation for vibration theory is a standard model for nonlinear vibration. Since the 1970s, it has been popular in chaos theory.Ivana Kovacic, Michael J. Brennan: The Duffing Equation: Nonlinear Oscillators and Their Behaviour, John Wiley & Sons, 2011, p. 1 Family and career Georg Duffing was born in 1861 in the Baden town of Waldshut as the eldest son of merchant Christian Duffing and his wife Julie Spies. In 1862, the family moved to Mannheim, where the father-in-law owned a carpentry business. In high school, Duffing showed a particular talent for mathematics and music. Due to a heart defect, he abandoned his initial intention of pursuing a military career and enrolled from 1878 to 1883 at the Karlsruhe Institute of Technology, su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Damping Ratio
In physical systems, damping is the loss of energy of an oscillating system by dissipation. Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. Examples of damping include viscous damping in a fluid (see Viscosity, viscous Drag (physics), drag), Friction, surface friction, radiation, Electrical resistance and conductance, resistance in electronic oscillators, and absorption and scattering of light in optical oscillators. Damping not based on energy loss can be important in other oscillating systems such as those that occur in ecology, biological systems and Bicycle_and_motorcycle_dynamics#Lateral_motion_theory, bikes (ex. Suspension (mechanics)). Damping is not to be confused with friction, which is a type of dissipative force acting on a system. Friction can cause or be a factor of damping. Many systems exhibit oscillatory behavior when they are disturbed from their position of static equilibrium. A mass su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of Explicit and implicit methods, implicit and explicit iterative methods, List of Runge–Kutta methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding y value is y_0. The function f and the initial conditions t_0, y_0 are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Method
In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book '' Institutionum calculi integralis'' (published 1768–1770). The Euler method is a first-order method, which means that the local error (error per step) is proportional to the square of the step size, and the global error (error at a given time) is proportional to the step size. The Euler method often serves as the basis to construct more complex methods, e.g., predictor–corrector method. Geometrical description Purpose and why it works Consider the problem of calculating the shape of an unknown curve which starts at a given point and s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frobenius Method
In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form z^2 u'' + p(z)z u'+ q(z) u = 0 with u' \equiv \frac and u'' \equiv \frac. in the vicinity of the regular singular point z=0. One can divide by z^2 to obtain a differential equation of the form u'' + \fracu' + \fracu = 0 which will not be solvable with regular power series methods if either or is not analytic at . The Frobenius method enables one to create a power series solution to such a differential equation, provided that ''p''(''z'') and ''q''(''z'') are themselves analytic at 0 or, being analytic elsewhere, both their limits at 0 exist (and are finite). History Frobenius' contribution was not so much in all the possible ''forms'' of the series solutions involved (see below). These forms had all been established earlier, by Lazarus Fuchs. The ''indicial polynomial'' (see bel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perturbation Theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbative" parts. In regular perturbation theory, the solution is expressed as a power series in a small parameter The first term is the known solution to the solvable problem. Successive terms in the series at higher powers of \varepsilon usually become smaller. An approximate 'perturbation solution' is obtained by truncating the series, often keeping only the first two terms, the solution to the known problem and the 'first order' perturbation correction. Perturbation theory is used in a wide range of fields and reaches its most sophisticated and advanced forms in quantum field theory. Perturbation theory (quantum mechanics) describes the use of this method in quantum mechanics. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Conditions
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For a system of order ''k'' (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension ''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional coordinate vector), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Buckingham π Theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number ''n'' physical variables, then the original equation can be rewritten in terms of a set of ''p'' = ''n'' − ''k'' dimensionless parameters 1, 2, ..., ''p'' constructed from the original variables, where ''k'' is the number of physical dimensions involved; it is obtained as the rank of a particular matrix. The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown. The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
