Equation 1 is the very famous damped, forced oscillator equation. The output of a simple harmonic oscillator is a pure sinusoid. Forced harmonic oscillator institute for nuclear theory. Thus, the most general solution to the driven damped harmonic oscillator equation, consists of two parts. Sep 18, 2015 video for my teams oral presentation of the physics 362 intermediate laboratory independent laboratory project. This type of motion is characteristic of many physical phenomena. Oo a simple harmonic oscillator subject to linear damping may oscillate with exponential decay, or it may decay biexponentially without oscillating, or it may decay most rapidly when it is critically damped. The arbitrary constants, and, are determined by the initial conditions. Classic examples are pendulum driven clocks which need winding, or a child on a swing who needs pushing. The initial sections deal with determining a model for the tting function. Solving the damped harmonic oscillator using green functions.
Complex fourier harmonic oscillator mathematics stack exchange. The nal section gives a description of the chisquare that is minimized in the t. Damped simple harmonic oscillator if the system is subject to a linear damping force, f. Browse other questions tagged homeworkandexercises harmonicoscillator or ask your own question. We will see how the damping term, b, affects the behavior of the system. Quantum dynamics of the damped harmonic oscillator. Driven harmonic oscillators are damped oscillators further affected by an externally applied force. Fourier analysis university of miami physics department. Several new concepts such as the fourier integral representation. The fractional fourier transform and harmonic oscillation. This is why fourier transformation is so useful in spectroscopy. Where if the signal contains that harmonic then the fourier coefficient for this harmonic is different than 0 and 0 otherwise i simplify.
Next, we fourier transform both sides of the greens function equation 2, and make. The transform for which you seek is the laplace transform. Fourier transform, and then by using the results of the above section. Here xt is the displacement of the oscillator from equilibrium. It consists of a mass m, which experiences a single force f, which pulls the mass in the direction of the point x 0 and depends only on the position x of the mass and a constant k. In this chapter we treat the quantum damped harmonic oscillator, and study mathematical structure of the model, and construct general solution with any initial condition, and give a quantum counterpart in the case of taking coherent state as an initial condition. The transient solution is the solution to the homogeneous differential equation of motion which has been combined with the particular solution and forced to fit the physical boundary conditions of the problem at hand.
The quantum theory of the damped harmonic oscillator has been a subject of continual investigation since the 1930s. Lcr circuits driven damped harmonic oscillation we saw earlier, in section 3. This is a much fancier sounding name than the springmass dashpot. The timedependent wave function the evolution of the ground state of the harmonic oscillator in the presence of a timedependent driving force has an exact solution. Laplace is a very close cousin to fourier, but takes into account initial conditions and allows you to inspect transients and the final state of the system. Equation 1 is a nonhomogeneous, 2nd order differential equation. Damped driven harmonic oscillator and linear response theory physics 258259 last revised december 4, 2005 by ed eyler purpose. To measure and analyze the response of a mechanical damped harmonic oscillator. It is also shown that the damped harmonic oscillator is susceptible to the analysis. The strength of controls how quickly energy dissipates. The oscillator we have in mind is a springmassdashpot system. Exponential decays and oscillations are typical patterns of experimental timedomain data which are fourier transformed to produce spectra. Response of a damped system under harmonic force the equation of motion is written in the form.
When we add damping we call the system in 1 a damped harmonic oscillator. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. The fractional fourier transform and harmonic oscillation 171 9. The objective of this experiment is to characterize the behavior of a damped harmonic oscillator driven by a harmonic force. Thanks to damping, it is often desirable to purposely drive harmonic oscillation by inputting energy. Resonance examples and discussion music structural and mechanical engineering waves sample problems. From the line position we see at what frequencies and hence energies an interaction occurs, and from the. Driven harmonic motion lets again consider the di erential equation for the damped harmonic oscil. Solving di erential equations with fourier transforms. Fourier series allow you to expand a function on a finite interval as an infinite series of. Damped harmonic oscillator university of connecticut. When we get to fourier analysis, we will see why this is actually a very general type of force to consider. Describe a driven harmonic oscillator as a type of damped oscillator. Harmonic oscillator and fourier series mathematics stack.
Driven harmonic oscillator equation a driven harmonic oscillator satis es the following di ential equation. Both the impulse response and the response to a sinusoidal driving force are to be measured. Understand the behaviour of this paradigm exactly solvable physics model that appears in numerous applications. A simple harmonic oscillator is an oscillator that is neither driven nor damped. I have found the complex fourier series for my desired force. Video for my teams oral presentation of the physics 362 intermediate laboratory independent laboratory project. Phase shifting and truncation artefacts fourier wiggles are features common in spectra derived from digitised experimental data. The physics of the damped harmonic oscillator matlab.
Resonance lineshapes of a driven damped harmonic oscillator. However, if there is some from of friction, then the amplitude will decrease as a function of time g t a0 a0 x if the damping is sliding friction, fsf constant, then the work done by the. The complex differential equation that is used to analyze the damped driven massspring system is. Harmonic analysis and transform techniques in damped. The damped, forced harmonic oscillator differential equation is. The plots show solid lines the frequency dependence of the amplitude, the phase, the inphase component, and the quadrature component of a driven damped harmonic oscillator. In fact, because of the linearity of integration, it is a.
Deriving the particular solution for a damped driven. The fourier integral and its applications mcgrawhill. Understand the connection between the response to a sinusoidal driving force and intrinsic oscillator properties. Generally in references where explanation are not that rigorous, there is often an attempt at explaining the principle behind fourier series, as making the signal resonate with an harmonic oscillator.
I now need to find the steadystate forced vibration of my oscillator as a fourier series. The equation of motion for a driven damped oscillator is. We show that the hermite functions, the eigenfunctions of the harmonic oscillator, are an orthonormal basis for l2, the space of squareintegrable functions. As an introduction to the greens function technique, we study the driven harmonic oscillator, which is a damped harmonic oscillator subjected to an arbitrary driving force.
Solving differential equations with fourier transforms. In fact, because the preceding solution contains two arbitrary constants, we can be sure that it is the most general solution. When driven sinusoidally, it resonates at a frequency near the nat. The driven steady state solution and initial transient behavior. If a harmonic oscillator, instead of vibrating freely, is driven by a periodic force, it will vibrate harmonically with the period of the force. It emphasizes an important fact about using differential equa. The fractional fourier transform and harmonic oscillation 159 the kernel of the fractional fourier transform corresponds to the greens function associated with the quantummechanical harmonic oscillator differential equation schrodingers equation. However, the amplitude of the oscillation grows monotonically as, and so takes a time of order to attain its final value, which is, of course, larger that the driving amplitude by the resonant amplification factor or quality factor. Notes on the harmonic oscillator and the fourier transform arthur ja. Module 3 damped and driven harmonic oscillations per wiki.
The second order linear harmonic oscillator damped or undamped with sinusoidal forcing can be solved by using the method of undetermined coe. The obstacle to quantization created by the dissipation of energy is usually dealt with by including a discrete set of additional harmonic oscillators as a reservoir. These systems appear over and over again in many different fields of physics. Oscillations in this lab you will look in detail at two of the most important physical systems in nature, the damped harmonic oscillator and the coupled oscillator. The offset from the original line position is equal to the frequency of the oscillation while the decay determines the line width, the frequency determines the line position. First, the solution, which oscillates at the driving frequency with a constant. Damped harmonic oscillator article about damped harmonic. Solving differential equations with fourier transforms physics.
Lets again consider the differential equation for the damped harmonic oscil. Fourier transformation 3 transforming in practice maths. Consider a damped simple harmonic oscillator with damping. Complex fourier harmonic oscillator mathematics stack. This demonstration analyzes in which way the highlimit lorentzian lineshapes of a driven damped harmonic oscillator differ from the exact resonance lineshapes. Solving di erential equations with fourier transforms consider a damped simple harmonic oscillator with damping and natural frequency. It is known that harmonic oscillations correspond to circular or elliptic motions in the. The equation of motion of a damped harmonic oscillator with mass, eigenfrequency, and damping constant driven by a periodic force is. This is a simple and good model of quantum mechanics with dissipation which is important to understand real world, and readers will. In the undamped case, beats occur when the forcing frequency is close to but not equal to the natural frequency of the oscillator.
The damped, driven oscillator is governed by a linear differential equation section 5. We can think of the fourier transform as an operator that acts on one func. The mass is at equilibrium at position x 1 when it is at rest. Lecture notes for mathematical methods in materials science mit. Notes on the periodically forced harmonic oscillator. This example explores the physics of the damped harmonic oscillator by solving the equations of motion in the case of no driving forces, investigating the cases of under, over, and criticaldamping. The harmonic motion of the drive can be thought of as the real part of circular motion in the complex plane. Shm using phasors uniform circular motion ph i l d l lphysical pendulum example damped harmonic oscillations forced oscillations and resonance. Fourier transform solution to the dampedforced linear harmonic oscillator. The final section gives a description of the chisquare that is minimized in the fit. Oscillations occur about x 1 at the driving frequency. Transient solution, driven oscillator the solution to the driven harmonic oscillator has a transient and a steadystate part. Physics 106 lecture 12 oscillations ii sj 7th ed chap 15.
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