The Damped Harmonic Oscillator

THE DAMPED HARMONIC OSCILLATOR Reading: Main 3.1, 3.2, 3.3 Taylor 5.4 Giancoli 14.7, 14.8 Free, undamped oscillators – other examples k m L No friction I C k m q 1 x m!x! = !kx q!! = ! q LC ! ! r; r L = θ Common notation for all g !! 2 T ! " # ! !!! + " ! = 0 m L 0 mg k friction m 1 LI! + q + RI = 0 x C 1 Lq!!+ q + Rq! = 0 C m!x! = !kx ! bx! ! r L = cm θ Common notation for all g !! ! 2 T ! " # ! # b'! !!! + 2"!! +# ! = 0 m L 0 mg Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ ! !kx ! k Need a model for this. m Try restoring force proportional to velocity k m x !bx! How do we choose a model? Physically reasonable, mathematically tractable … Validation comes IF it describes the experimental system accurately Natural motion of damped harmonic oscillator Force = mx˙˙ restoring force + resistive force = mx˙˙ !kx ! bx! = m!x! ! Divide by coefficient of d2x/dt2 ! and rearrange: x 2 x 2 x 0 !!+ ! ! + " 0 = inverse time β and ω0 (rate or frequency) are generic to any oscillating system This is the notation of TM; Main uses γ = 2β. Natural motion of damped harmonic oscillator 2 x˙˙ + 2"x˙ +#0 x = 0 Try x(t) = Ce pt C, p are unknown constants ! x˙ (t) = px(t), x˙˙ (t) = p2 x(t) p2 2 p 2 x(t) 0 Substitute: ( + ! + " 0 ) = ! 2 2 Now p is known (and p = !" ± " ! # 0 there are 2 p values) p t p t x(t) = Ce + + C'e " Must be sure to make x real! ! Natural motion of damped HO Can identify 3 cases " < #0 underdamped ! " > #0 overdamped ! " = #0 critically damped time ---> ! underdamped " < #0 # 2 !1 = ! 0 1" 2 ! 0 ! time ---> 2 2 p = !" ± " ! # 0 = !" ± i#1 x(t) = Ce"#t+i$1t +C*e"#t"i$1t Keep x(t) real "#t x(t) = Ae [cos($1t +%)] complex <-> amp/phase System oscillates at "frequency" ω1 (very close to ω0) ! - but in fact there is not only one single frequency associated with the motion as we will see. ! underdamped " < #0 ! Damping time or "1/e" time is τ = 1/β > 1/ω0 % (>> 1/"0 if ! is very small) How many T0 periods elapse in the damping time? This number (times π) is the Quality factor or Q of the system. " # Q = ! = 0 large if β is small compared to ω0 T0 2$ LRC circuit dI q V = L ;V = IR;V = L L dt R C C I C dI q R !L ! IR ! = 0 ! dt C q L (inductance), C (capacitance), Lq!!+ Rq! + = 0 cause oscillation, R (resistance) C causes damping 2 R 1 q˙˙ + 2"q˙ +#0 q = 0 q!!+ q! + q = 0 L LC ! LRC circuit LCR circuit obeys precisely the L same equation as the damped mass/spring. I C Q factor: 1 Natural (resonance) R Q = frequency determined by ! 0 RC the inductor and capacitor 1 Typical numbers: ! 0 = LC L≈500µH; C≈100pF; R≈50Ω ω ≈106s-1 (f ≈700 kHz) Damping determined by 0 0 resistor & inductor τ=1/β≈2µs; Q≈45 (your lab has different parameters) R ! = 2L Does the model fit? Does the model fit? Summary so far: • Free, undamped, linear (harmonic) oscillator • Free, undamped, non-linear oscillator • Free, damped linear oscillator Next • Driven, damped linear oscillator • Laboratory to investigate LRC circuit as example of driven, damped oscillator • Time and frequency representations • Fourier series .

Copy Link

Recommended publications

Forced Mechanical Oscillations
169 Carl von Ossietzky Universität Oldenburg – Faculty V - Institute of Physics Module Introductory laboratory course physics – Part I Forced mechanical oscillations Keywords: HOOKE's law, harmonic oscillation, harmonic oscillator, eigenfrequency, damped harmonic oscillator, resonance, amplitude resonance, energy resonance, resonance curves References: /1/ DEMTRÖDER, W.: „Experimentalphysik 1 – Mechanik und Wärme“, Springer-Verlag, Berlin among others. /2/ TIPLER, P.A.: „Physik“, Spektrum Akademischer Verlag, Heidelberg among others. /3/ ALONSO, M., FINN, E. J.: „Fundamental University Physics, Vol. 1: Mechanics“, Addison-Wesley Publishing Company, Reading (Mass.) among others. 1 Introduction It is the object of this experiment to study the properties of a „harmonic oscillator“ in a simple mechanical model. Such harmonic oscillators will be encountered in different fields of physics again and again, for example in electrodynamics (see experiment on electromagnetic resonant circuit) and atomic physics. Therefore it is very important to understand this experiment, especially the importance of the amplitude resonance and phase curves. 2 Theory 2.1 Undamped harmonic oscillator Let us observe a set-up according to Fig. 1, where a sphere of mass mK is vertically suspended (x-direc- tion) on a spring. Let us neglect the effects of friction for the moment. When the sphere is at rest, there is an equilibrium between the force of gravity, which points downwards, and the dragging resilience which points upwards; the centre of the sphere is then in the position x = 0. A deflection of the sphere from its equilibrium position by x causes a proportional dragging force FR opposite to x: (1) FxR ∝− The proportionality constant (elastic or spring constant or directional quantity) is denoted D, and Eq.
[Show full text]
VIBRATIONAL SPECTROSCOPY • the Vibrational Energy V(R) Can Be Calculated Using the (Classical) Model of the Harmonic Oscillator
VIBRATIONAL SPECTROSCOPY • The vibrational energy V(r) can be calculated using the (classical) model of the harmonic oscillator: • Using this potential energy function in the Schrödinger equation, the vibrational frequency can be calculated: The vibrational frequency is increasing with: • increasing force constant f = increasing bond strength • decreasing atomic mass • Example: f cc > f c=c > f c-c Vibrational spectra (I): Harmonic oscillator model • Infrared radiation in the range from 10,000 – 100 cm –1 is absorbed and converted by an organic molecule into energy of molecular vibration –> this absorption is quantized: A simple harmonic oscillator is a mechanical system consisting of a point mass connected to a massless spring. The mass is under action of a restoring force proportional to the displacement of particle from its equilibrium position and the force constant f (also k in followings) of the spring. MOLECULES I: Vibrational We model the vibrational motion as a harmonic oscillator, two masses attached by a spring. nu and vee! Solving the Schrödinger equation for the 1 v h(v 2 ) harmonic oscillator you find the following quantized energy levels: v 0,1,2,... The energy levels The level are non-degenerate, that is gv=1 for all values of v. The energy levels are equally spaced by hn. The energy of the lowest state is NOT zero. This is called the zero-point energy. 1 R h Re 0 2 Vibrational spectra (III): Rotation-vibration transitions The vibrational spectra appear as bands rather than lines. When vibrational spectra of gaseous diatomic molecules are observed under high-resolution conditions, each band can be found to contain a large number of closely spaced components— band spectra.
[Show full text]
Harmonic Oscillator with Time-Dependent Effective-Mass And
Harmonic oscillator with time-dependent effective-mass and frequency with a possible application to 'chirped tidal' gravitational waves forces affecting interferometric detectors Yacob Ben-Aryeh Physics Department, Technion-Israel Institute of Technology, Haifa,32000,Israel e-mail: [email protected] ; Fax: 972-4-8295755 Abstract The general theory of time-dependent frequency and time-dependent mass ('effective mass') is described. The general theory for time-dependent harmonic-oscillator is applied in the present research for studying certain quantum effects in the interferometers for detecting gravitational waves. When an astronomical binary system approaches its point of coalescence the gravitational wave intensity and frequency are increasing and this can lead to strong deviations from the simple description of harmonic oscillations for the interferometric masses on which the mirrors are placed. It is shown that under such conditions the harmonic oscillations of these masses can be described by mechanical harmonic-oscillators with time- dependent frequency and effective-mass. In the present theoretical model the effective- mass is decreasing with time describing pumping phenomena in which the oscillator amplitude is increasing with time. The quantization of this system is analyzed by the use of the adiabatic approximation. It is found that the increase of the gravitational wave intensity, within the adiabatic approximation, leads to squeezing phenomena where the quantum noise in one quadrature is increased and in the other quadrature it is decreased. PACS numbers: 04.80.Nn, 03.65.Bz, 42.50.Dv. Keywords: Gravitational waves, harmonic-oscillator with time-dependent effective- mass 1 1.Introduction The problem of harmonic-oscillator with time-dependent mass has been related to a quantum damped oscillator [1-7].
[Show full text]
The Q of Oscillators
The Q of oscillators References: L.R. Fortney – Principles of Electronics: Analog and Digital, Harcourt Brace Jovanovich 1987, Chapter 2 (AC Circuits) H. J. Pain – The Physics of Vibrations and Waves, 5th edition, Wiley 1999, Chapter 3 (The forced oscillator). Prerequisite: Currents through inductances, capacitances and resistances – 2nd year lab experiment, Department of Physics, University of Toronto, http://www.physics.utoronto.ca/~phy225h/currents-l-r-c/currents-l-c-r.pdf Introduction In the Prerequisite experiment, you studied LCR circuits with different applied signals. A loop with capacitance and inductance exhibits an oscillatory response to a disturbance, due to the oscillating energy exchange between the electric and magnetic fields of the circuit elements. If a resistor is added, the oscillation will become damped. 1 In this experiment, the LCR circuit will be driven at resonance frequencyωr = , when the LC transfer of energy between the driving source and the circuit will be a maximum. LCR circuits at resonance. The transfer function. Ohm’s Law applied to a LCR loop (Figure 1) can be written in complex notation (see Appendix from the Prerequisite) Figure 1 LCR circuit for resonance studies v( jω) Ohm’s Law: i( jω) = (1) Z - 1 - i(jω) and v(jω) are complex instantaneous values of current and voltage, ω is the angular frequency (ω = 2πf ) , Z is the complex impedance of the loop: 1 Z = R + jωL − (2) ωC j = −1 is the complex number The voltage across the resistor from Figure 1, as a result of current i can be expressed as: vR ( jω) = Ri( jω) (3) Eliminating i(jω) from Equations (1) and (3) results into: R v ( jω) = v( jω) (4) R R + j(ωL −1/ωC) Equation (4) can be put into the general form: vR ( jω) = H ( jω)v( jω) (5) where H(jω) is called a transfer function across the resistor, in the frequency domain.
[Show full text]
Arxiv:1704.05328V1 [Cond-Mat.Mes-Hall] 18 Apr 2017 Tuning the Mechanical Mode Frequencies by More Than Silicon Wafers
Quantitative Determination of the Mechanical Properties of Nanomembrane Resonators by Vibrometry In Continuous Light Fan Yang,∗ Reimar Waitz,y and Elke Scheer Department of Physics, UniversitätKonstanz, 78464 Konstanz, Germany We present an experimental study of the bending waves of freestanding Si3N4 nanomembranes using optical profilometry in varying environments such as pressure and temperature. We introduce a method, named Vibrometry in Continuous Light (VICL) that enables us to disentangle the response of the membrane from the one of the excitation system, thereby giving access to the eigenfrequency and the quality (Q) factor of the membrane by fitting a model of a damped driven harmonic oscillator to the experimental data. The validity of particular assumptions or aspects of the model such as damping mechanisms, can be tested by imposing additional constraints on the fitting procedure. We verify the performance of the method by studying two modes of a 478 nm thick Si3N4 freestanding membrane and find Q factors of 2 × 104 for both modes at room temperature. Finally, we observe a linear increase of the resonance frequency of the ground mode with temperature which amounts to 550 Hz=◦C for a ground mode frequency of 0:447 MHz. This makes the nanomembrane resonators suitable as high-sensitive temperature sensors. I. INTRODUCTION frequencies of bending waves of nanomembranes may range from a few kHz to several 100 MHz [6], those of Nanomechanical membranes are extensively used in a thickness oscillation may even exceed 100 GHz [21], re- variety of applications including among others high fre- quiring a versatile excitation and detection method able quency microwave devices [1], human motion detectors to operate in this wide frequency range and to resolve [2], and gas sensors [3].
[Show full text]
22.51 Course Notes, Chapter 9: Harmonic Oscillator
9. Harmonic Oscillator 9.1 Harmonic Oscillator 9.1.1 Classical harmonic oscillator and h.o. model 9.1.2 Oscillator Hamiltonian: Position and momentum operators 9.1.3 Position representation 9.1.4 Heisenberg picture 9.1.5 Schrödinger picture 9.2 Uncertainty relationships 9.3 Coherent States 9.3.1 Expansion in terms of number states 9.3.2 Non-Orthogonality 9.3.3 Uncertainty relationships 9.3.4 X-representation 9.4 Phonons 9.4.1 Harmonic oscillator model for a crystal 9.4.2 Phonons as normal modes of the lattice vibration 9.4.3 Thermal energy density and Specific Heat 9.1 Harmonic Oscillator We have considered up to this moment only systems with a finite number of energy levels; we are now going to consider a system with an infinite number of energy levels: the quantum harmonic oscillator (h.o.). The quantum h.o. is a model that describes systems with a characteristic energy spectrum, given by a ladder of evenly spaced energy levels. The energy difference between two consecutive levels is ∆E. The number of levels is infinite, but there must exist a minimum energy, since the energy must always be positive. Given this spectrum, we expect the Hamiltonian will have the form 1 n = n + ~ω n , H | i 2 | i where each level in the ladder is identified by a number n. The name of the model is due to the analogy with characteristics of classical h.o., which we will review first. 9.1.1 Classical harmonic oscillator and h.o.
[Show full text]
Exact Solution for the Nonlinear Pendulum (Solu¸CÃoexata Do PêndulonÃolinear)
Revista Brasileira de Ensino de F¶³sica, v. 29, n. 4, p. 645-648, (2007) www.sb¯sica.org.br Notas e Discuss~oes Exact solution for the nonlinear pendulum (Solu»c~aoexata do pêndulon~aolinear) A. Bel¶endez1, C. Pascual, D.I. M¶endez,T. Bel¶endezand C. Neipp Departamento de F¶³sica, Ingenier¶³ade Sistemas y Teor¶³ade la Se~nal,Universidad de Alicante, Alicante, Spain Recebido em 30/7/2007; Aceito em 28/8/2007 This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is di®erent from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75¼ (135±) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ! in terms of the complete elliptic integral of the ¯rst kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude).
[Show full text]
Solving the Harmonic Oscillator Equation
Solving the Harmonic Oscillator Equation Morgan Root NCSU Department of Math Spring-Mass System Consider a mass attached to a wall by means of a spring. Define y=0 to be the equilibrium position of the block. y(t) will be a measure of the displacement from this equilibrium at a given time. Take dy(0) y(0) = y0 and dt = v0. Basic Physical Laws Newton’s Second Law of motion states tells us that the acceleration of an object due to an applied force is in the direction of the force and inversely proportional to the mass being moved. This can be stated in the familiar form: Fnet = ma In the one dimensional case this can be written as: Fnet = m&y& Relevant Forces Hooke’s Law (k is FH = −ky called Hooke’s constant) Friction is a force that FF = −cy& opposes motion. We assume a friction proportional to velocity. Harmonic Oscillator Assuming there are no other forces acting on the system we have what is known as a Harmonic Oscillator or also known as the Spring-Mass- Dashpot. Fnet = FH + FF or m&y&(t) = −ky(t) − cy&(t) Solving the Simple Harmonic System m&y&(t) + cy&(t) + ky(t) = 0 If there is no friction, c=0, then we have an “Undamped System”, or a Simple Harmonic Oscillator. We will solve this first. m&y&(t) + ky(t) = 0 Simple Harmonic Oscillator k Notice that we can take K = m and look at the system: &y&(t) = −Ky(t) We know at least two functions that will solve this equation.
[Show full text]
RF-MEMS Load Sensors with Enhanced Q-Factor and Sensitivity In
Microelectronic Engineering 88 (2011) 247–253 Contents lists available at ScienceDirect Microelectronic Engineering journal homepage: www.elsevier.com/locate/mee RF-MEMS load sensors with enhanced Q-factor and sensitivity in a suspended architecture ⇑ Rohat Melik a, Emre Unal a, Nihan Kosku Perkgoz a, Christian Puttlitz b, Hilmi Volkan Demir a, a Departments of Electrical Engineering and Physics, Nanotechnology Research Center, and Institute of Materials Science and Nanotechnology, Bilkent University, Ankara 06800, Turkey b Department of Mechanical Engineering, Orthopaedic Bioengineering Research Laboratory, Colorado State University, Fort Collins, CO 80523, USA article info abstract Article history: In this paper, we present and demonstrate RF-MEMS load sensors designed and fabricated in a suspended Received 31 August 2009 architecture that increases their quality-factor (Q-factor), accompanied with an increased resonance fre- Received in revised form 7 July 2010 quency shift under load. The suspended architecture is obtained by removing silicon under the sensor. Accepted 29 October 2010 We compare two sensors that consist of 195 lm Â 195 lm resonators, where all of the resonator features Available online 9 November 2010 are of equal dimensions, but one’s substrate is partially removed (suspended architecture) and the other’s is not (planar architecture). The single suspended device has a resonance of 15.18 GHz with 102.06 Q-fac- Keywords: tor whereas the single planar device has the resonance at 15.01 GHz and an associated Q-factor of 93.81. Fabrication For the single planar device, we measured a resonance frequency shift of 430 MHz with 3920 N of applied IC Resonance frequency shift load, while we achieved a 780 MHz frequency shift in the single suspended device.
[Show full text]
Notes on the Critically Damped Harmonic Oscillator Physics 2BL - David Kleinfeld
Notes on the Critically Damped Harmonic Oscillator Physics 2BL - David Kleinfeld We often have to build an electrical or mechanical device. An understand- ing of physics may help in the design and tuning of such a device. Here, we consider a critically damped spring oscillator as a model design for the shock absorber of a car. We consider a mass, denoted m, that is connected to a spring with spring constant k, so that the restoring force is F =-kx, and which moves in a lossy manner so that the frictional force is F =-bv =-bx˙. Prof. Newton tells us that F = mx¨ = −kx − bx˙ (1) Thus k b x¨ + x + x˙ = 0 (2) m m The two reduced constants are the natural frequency k ω = (3) 0 m and the decay constant b α = (4) m so that we need to consider 2 x¨ + ω0x + αx˙ = 0 (5) The above equation describes simple harmonic motion with loss. It is dis- cussed in lots of text books, but I want to consider a formulation of the solution that is most natural for critical damping. We know that when the damping constant is zero, i.e., α = 0, the solution 2 ofx ¨ + ω0x = 0 is given by: − x(t)=Ae+iω0t + Be iω0t (6) where A and B are constants that are found from the initial conditions, i.e., x(0) andx ˙(0). In a nut shell, the system oscillates forever. 1 We know that when the the natural frequency is zero, i.e., ω0 = 0, the solution ofx ¨ + αx˙ = 0 is given by: x˙(t)=Ae−αt (7) and 1 − e−αt x(t)=A + B (8) α where A and B are constants that are found from the initial conditions.
[Show full text]
Teaching the Difference Between Stiffness and Damping
LECTURE NOTES « Teaching the Difference Between Stiffness and Damping Henry C. FU and Kam K. Leang In contrast to the parallel spring-mass-damper, in the necessary step in the design of an effective control system series mass-spring-damper system the stiffer the system, A is to understand its dynamics. It is not uncommon for a the more dissipative its behavior, and the softer the sys- well-trained control engineer to use such an understanding to tem, the more elastic its behavior. Thus teaching systems redesign the system to become easier to control, which requires modeled by series mass-spring-damper systems allows an even deeper understanding of how the components of a students to appreciate the difference between stiffness and system influence its overall dynamics. In this issue Henry C. damping. Fu and Kam K. Leang discuss the diff erence between stiffness To get students to consider the difference between soft and damping when understanding the dynamics of mechanical and damped, ask them to consider the following scenario: systems. suppose a bullfrog is jumping to land on a very slippery floating lily pad as illustrated in Figure 2 and does not want to fall off. The frog intends to hit the flower (but he simple spring, mass, and damper system is ubiq- cannot hold onto it) in the center to come to a stop. If the uitous in dynamic systems and controls courses [1]. TThis column considers a concept students often have trouble with: the difference between a “soft” system, which has a small elastic restoring force, and a “damped” system, x c which dissipates energy quickly.
[Show full text]
The Harmonic Oscillator
Appendix A The Harmonic Oscillator Properties of the harmonic oscillator arise so often throughout this book that it seemed best to treat the mathematics involved in a separate Appendix. A.1 Simple Harmonic Oscillator The harmonic oscillator equation dates to the time of Newton and Hooke. It follows by combining Newton’s Law of motion (F = Ma, where F is the force on a mass M and a is its acceleration) and Hooke’s Law (which states that the restoring force from a compressed or extended spring is proportional to the displacement from equilibrium and in the opposite direction: thus, FSpring =−Kx, where K is the spring constant) (Fig. A.1). Taking x = 0 as the equilibrium position and letting the force from the spring act on the mass: d2x M + Kx = 0. (A.1) dt2 2 = Dividing by the mass and deﬁning ω0 K/M, the equation becomes d2x + ω2x = 0. (A.2) dt2 0 As may be seen by direct substitution, this equation has simple solutions of the form x = x0 sin ω0t or x0 = cos ω0t, (A.3) The original version of this chapter was revised: Pages 329, 330, 335, and 347 were corrected. The correction to this chapter is available at https://doi.org/10.1007/978-3-319-92796-1_8 © Springer Nature Switzerland AG 2018 329 W. R. Bennett, Jr., The Science of Musical Sound, https://doi.org/10.1007/978-3-319-92796-1 330 A The Harmonic Oscillator Fig. A.1 Frictionless harmonic oscillator showing the spring in compressed and extended positions where t is the time and x0 is the maximum amplitude of the oscillation.
[Show full text]