DOCSLIB.ORG

Oscillations of Mechanical Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Oscillations of Mechanical Systems Oscillations of Mechanical Systems Math 240 Free oscillation No damping Damping Forced Oscillations of Mechanical Systems oscillation No damping Damping Math 240 | Calculus III Summer 2015, Session II Thursday, July 30, 2015 Oscillations of Mechanical Agenda Systems Math 240 Free oscillation No damping Damping Forced oscillation 1. Free oscillation No damping Damping No damping Damping 2. Forced oscillation No damping Damping Oscillations of Mechanical Introduction Systems Math 240 Free oscillation No damping Damping Forced oscillation We have now learned how to solve constant-coefficient linear No damping Damping differential equations of the form P (D)y = F for a sizeable class of functions F . We are going to use this knowledge to study the motion of mechanical systems consisting of a mass attached to a spring. Let's begin by modeling our system using a differential equation. Oscillations of Mechanical The setup Systems Math 240 Free oscillation No damping Damping Forced oscillation A mass of m kg is attached to No damping Damping the end of a spring with spring constant k N/m whose natural length is l0 m. At equilibrium, the mass hangs without moving at a displacement of L0 m, so that mg = kL0. Spring-mass system in static equilibrium. Oscillations of Mechanical The setup Systems Math 240 Free To analyze the system in motion, we let y(t) denote the oscillation No damping position of the mass at time t and take y = 0 to coincide with Damping the equilibrium position. The forces that act on the mass are Forced oscillation No damping Damping 1. The force of gravity, Fg = mg. 2. The spring force, Fs = −k(y(t) + L0). 3. A damping force proportional to the velocity of the mass, dy Fd = −c dt . 4. Any external driving A damped spring-mass system. force, F (t). Oscillations of Mechanical Our equation Systems Math 240 Free oscillation Newton says, the equation governing motion of the mass is No damping Damping d2y Forced m = Fg + Fs + Fd + F (t) oscillation dt2 No damping dy Damping = mg − k(L + y) − c + F (t): 0 dt Rearranging gives us the linear differential equation c k 1 y00 + y0 + y = F (t): m m m We may also have initial conditions 0 y(0) = y0 and y (0) = v0: These indicate that at t = 0 the mass is displaced a distance of y0 m and released with a downward velocity of v0 m/s. Oscillations of Mechanical Free oscillation Systems Math 240 Free oscillation No damping Damping Forced First consider the case where there are no external forces acting oscillation No damping on the system, that is, set F (t) = 0. Our differential equation Damping reduces to c k y00 + y0 + y = 0: m m We will study the two subcases I no damping: c = 0 and I damping: c > 0. Oscillations of Mechanical No damping Systems Math 240 Setting c = 0 in our equation, we have Free 00 2 oscillation y + !0y = 0; No damping p Damping where !0 = k=m. This equation has the general solution Forced oscillation y(t) = c1 cos !0t + c2 sin !0t: No damping Damping From the constants c1 and c2 we can derive p 2 2 A I the amplitude, A0 = c + c , 0 1 2 c 2 I the phase, φ = arctan(c2=c1). c 1 Using these new constants, the equation of our motion is y(t) = A0 cos(!0t − φ): This is simple harmonic motion. The constant !0 is called the circular frequency. Oscillations of Mechanical Simple harmonic motion Systems Math 240 Free oscillation No damping Damping Forced oscillation No damping Damping 2π p m This function is periodic with a period of T = ! = 2π k : q 0 1 !0 1 k Its frequency is f = T = 2π = 2π m : Note that these quantities are independent of the initial conditions. They are properties of the system itself. Oscillations of Mechanical Damped motion Systems Math 240 Free The motion of the system is damped when c > 0. Our oscillation equation is then No damping Damping c k y00 + y0 + : Forced oscillation m m No damping The auxiliary polynomial has roots Damping p −c ± c2 − 4km r = : 2m The behavior of the system will depend on whether there are distinct real roots, a repeated real root, or complex conjugate roots. This can be determined using the (dimensionless) quantity c2=(4km). We say that the system is 2 I underdamped if c =(4km) < 1 (complex conjugate roots), 2 I critically damped if c =(4km) = 1 (repeated real root), 2 I overdamped if c =(4km) > 1 (distinct real roots). Oscillations of Mechanical Underdamping Systems Math 240 The two complex roots of the auxiliary polynomial give rise to Free oscillation the general solution No damping −ct=(2m) Damping y(t) = e (c1 cos µt + c2 sin µt); p Forced 2 oscillation where µ = 4km − c =(2m). Using amplitude and phase, it's No damping −ct=(2m) Damping y(t) = A0e cos(µt − φ): Although the amplitude decays exponentially, this motion has a constant quasiperiod T = 2π = p 4πm : µ 4km−c2 Oscillations of Mechanical Critical damping Systems Math 240 Free 2 oscillation Critical damping happens when c =(4km) = 1. Then the No damping Damping equation 2 Forced 00 c 0 c oscillation y + y + 2 y = 0 No damping m 4m Damping has general solution −ct=(2m) y(t) = e (c1 + c2t): The motion is not oscillatory|it will pass through y = 0 at most once. Oscillations of Mechanical Overdamping Systems Math 240 When c2=(4km) > 1 we have two real roots of the auxiliary Free oscillation polynomial: No damping p p Damping −c + c2 − 4km −c − c2 − 4km Forced r1 = and r2 = : oscillation 2m 2m No damping Thus, our general solution is Damping −ct=(2m) µt −µt y(t) = e (c1e + c2e ); p where µ = c2 − 4km=(2m). Overdamped motion is qualitatively similar to critically damped|it is not oscillatory and passes through the equilibrium position at most once. Oscillations of Mechanical Sanity check Systems Math 240 Free oscillation No damping Damping Forced oscillation No damping Damping Notice that in all three cases of damped motion, the amplitude diminishes to zero as t ! 1. This is certainly what we expect in such a system. Oscillations of Mechanical Forced oscillation Systems Math 240 Free oscillation No damping Damping Forced Let's investigate the nonhomogeneous situation when an oscillation No damping external force acts on the spring-mass system. We will focus on Damping periodic applied force, of the form F (t) = F0 cos !t; for constants F0 and !. Our general equation is now c k F y00 + y0 + y = 0 cos !t: m m m Oscillations of Mechanical No damping Systems Math 240 Free oscillation No damping Damping Forced oscillation Setting c = 0, we want to solve No damping F Damping y00 + !2y = 0 cos !t; 0 m p where again !0 = k=m is the circular frequency. We have seen that the complementary function is yc(t) = A0 cos(!0t − φ): Our trial solution will depend on whether ! = !0. Oscillations of Mechanical Forced harmonic oscillation Systems Math 240 Free oscillation No damping Damping When ! 6= !0, we can use the trial solution Forced oscillation yp(t) = A cos !t + B sin !t: No damping Damping Then we can find the particular solution F0 yp(t) = 2 2 cos !t; m(!0 − ! ) so the general solution is F0 y(t) = A0 cos(!0t − φ) + 2 2 cos !t: m(!0 − ! ) From this we see that the motion will look like a superposition of two simple harmonic oscillations. Oscillations of Mechanical Forced harmonic oscillation Systems Math 240 Free oscillation If !=!0 is a rational number, say, p=q, then the resulting No damping Damping motion will have a period of Forced 2πq 2πp oscillation T = = : No damping !0 ! Damping Otherwise, it will be oscillatory but not periodic. Oscillations of Mechanical Forced harmonic oscillation Systems Math 240 Free oscillation No damping Damping Interesting things happen when ! is very close to ! : Forced 0 oscillation No damping Damping Oscillations of Mechanical Resonance Systems Math 240 If instead we have ! = !0, then we must use the trial solution Free oscillation yp(t) = t(A cos !0t + B sin !0t): No damping Damping This leads to the particular solution Forced F oscillation y (t) = 0 t sin ! t No damping p 0 Damping 2m!0 and general solution F0 y(t) = A0 cos(!0t − φ) + t sin !0t: 2m!0 Notice that the amplitude increases without bound as t ! 1. Oscillations of Mechanical Damping Systems Math 240 Free Finally, we will consider a damped nonhomogeneous equation oscillation No damping 00 c 0 k F0 Damping y + y + y = cos !t; Forced m m m oscillation No damping with c > 0. The trial solution yp(t) = A cos !t + B sin !t Damping yields the particular solution F y (t) = 0 (k − m!2) cos !t + c! sin !t : p (k − m!2)2 + c2!2 Letting q 2 2 2 2 2 2 c! H = m (!0 − ! ) + c ! and η = arctan 2 2 m(!0 − ! ) turns it into F y (t) = 0 cos(!t − η): p H Oscillations of Mechanical Damping Systems Math 240 As before, the system can be underdamped, critically damped, Free or overdamped. Which one will determine the complementary oscillation No damping function. Damping Forced In each case of damped harmonic motion, the amplitude dies oscillation No damping out as t gets large. But the driving force has a constant Damping amplitude and thus it will dominate. We therefore refer to the complementary function as the transient part of the solution and call yp the steady-state solution..
Load more

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]
  • 14 Mass on a Spring and Other Systems Described by Linear ODE
    14 Mass on a spring and other systems described by linear ODE 14.1 Mass on a spring Consider a mass hanging on a spring (see the figure). The position of the mass in uniquely defined by one coordinate x(t) along the x-axis, whose direction is chosen to be along the direction of the force of gravity. Note that if this mass is in an equilibrium, then the string is stretched, say by amount s units. The origin of x-axis is taken that point of the mass at rest. Figure 1: Mechanical system “mass on a spring” We know that the movement of the mass is determined by the second Newton’s law, that can be stated (for our particular one-dimensional case) as ma = Fi, X where m is the mass of the object, a is the acceleration, which, as we know from Calculus, is the second derivative of the displacement x(t), a =x ¨, and Fi is the net force applied. What we know about the net force? This has to include the gravity, of course: F1 = mg, where g is the acceleration 2 P due to gravity (g 9.8 m/s in metric units). The restoring force of the spring is governed by Hooke’s ≈ law, which says that the restoring force in the opposite to the movement direction is proportional to the distance stretched: F2 = k(s + x), here minus signifies that the force is acting in the opposite − direction, and the distance stretched also includes s units. Note that at the equilibrium (i.e., when x = 0) and absence of other forces we have F1 + F2 = 0 and hence mg ks = 0 (this last expression − often allows to find the spring constant k given the amount of s the spring is stretched by the mass m).
    [Show full text]
  • Vibration and Sound Damping in Polymers
    GENERAL ARTICLE Vibration and Sound Damping in Polymers V G Geethamma, R Asaletha, Nandakumar Kalarikkal and Sabu Thomas Excessive vibrations or loud sounds cause deafness or reduced efficiency of people, wastage of energy and fatigue failure of machines/structures. Hence, unwanted vibrations need to be dampened. This article describes the transmis- sion of vibrations/sound through different materials such as metals and polymers. Viscoelasticity and glass transition are two important factors which influence the vibration damping of polymers. Among polymers, rubbers exhibit greater damping capability compared to plastics. Rubbers 1 V G Geethamma is at the reduce vibration and sound whereas metals radiate sound. Mahatma Gandhi Univer- sity, Kerala. Her research The damping property of rubbers is utilized in products like interests include plastics vibration damper, shock absorber, bridge bearing, seismic and rubber composites, and soft lithograpy. absorber, etc. 2R Asaletha is at the Cochin University of Sound is created by the pressure fluctuations in the medium due Science and Technology. to vibration (oscillation) of an object. Vibration is a desirable Her research topic is NR/PS blend- phenomenon as it originates sound. But, continuous vibration is compatibilization studies. harmful to machines and structures since it results in wastage of 3Nandakumar Kalarikkal is energy and causes fatigue failure. Sometimes vibration is a at the Mahatma Gandhi nuisance as it creates noise, and exposure to loud sound causes University, Kerala. His research interests include deafness or reduced efficiency of people. So it is essential to nonlinear optics, synthesis, reduce excessive vibrations and sound. characterization and applications of nano- multiferroics, nano- We can imagine how noisy and irritating it would be to pull a steel semiconductors, nano- chair along the floor.
    [Show full text]
  • Oscillator Circuit Evaluation Method (2) Steps for Evaluating Oscillator Circuits (Oscillation Allowance and Drive Level)
    Technical Notes Oscillator Circuit Evaluation Method (2) Steps for evaluating oscillator circuits (oscillation allowance and drive level) Preface In general, a crystal unit needs to be matched with an oscillator circuit in order to obtain a stable oscillation. A poor match between crystal unit and oscillator circuit can produce a number of problems, including, insufficient device frequency stability, devices stop oscillating, and oscillation instability. When using a crystal unit in combination with a microcontroller, you have to evaluate the oscillator circuit. In order to check the match between the crystal unit and the oscillator circuit, you must, at least, evaluate (1) oscillation frequency (frequency matching), (2) oscillation allowance (negative resistance), and (3) drive level. The previous Technical Notes explained frequency matching. These Technical Notes describe the evaluation methods for oscillation allowance (negative resistance) and drive level. 1. Oscillation allowance (negative resistance) evaluations One process used as a means to easily evaluate the negative resistance characteristics and oscillation allowance of an oscillator circuits is the method of adding a resistor to the hot terminal of the crystal unit and observing whether it can oscillate (examining the negative resistance RN). The oscillator circuit capacity can be examined by changing the value of the added resistance (size of loss). The circuit diagram for measuring the negative resistance is shown in Fig. 1. The absolute value of the negative resistance is the value determined by summing up the added resistance r and the equivalent resistance (Re) when the crystal unit is under load. Formula (1) Rf Rd r | RN | Connect _ r＋R e ..
    [Show full text]
  • Hydraulics Manual Glossary G - 3
    Glossary G - 1 GLOSSARY OF HIGHWAY-RELATED DRAINAGE TERMS (Reprinted from the 1999 edition of the American Association of State Highway and Transportation Officials Model Drainage Manual) G.1 Introduction This Glossary is divided into three parts: · Introduction, · Glossary, and · References. It is not intended that all the terms in this Glossary be rigorously accurate or complete. Realistically, this is impossible. Depending on the circumstance, a particular term may have several meanings; this can never change. The primary purpose of this Glossary is to define the terms found in the Highway Drainage Guidelines and Model Drainage Manual in a manner that makes them easier to interpret and understand. A lesser purpose is to provide a compendium of terms that will be useful for both the novice as well as the more experienced hydraulics engineer. This Glossary may also help those who are unfamiliar with highway drainage design to become more understanding and appreciative of this complex science as well as facilitate communication between the highway hydraulics engineer and others. Where readily available, the source of a definition has been referenced. For clarity or format purposes, cited definitions may have some additional verbiage contained in double brackets [ ]. Conversely, three “dots” (...) are used to indicate where some parts of a cited definition were eliminated. Also, as might be expected, different sources were found to use different hyphenation and terminology practices for the same words. Insignificant changes in this regard were made to some cited references and elsewhere to gain uniformity for the terms contained in this Glossary: as an example, “groundwater” vice “ground-water” or “ground water,” and “cross section area” vice “cross-sectional area.” Cited definitions were taken primarily from two sources: W.B.
    [Show full text]
  • Oscillations
    CHAPTER FOURTEEN OSCILLATIONS 14.1 INTRODUCTION In our daily life we come across various kinds of motions. You have already learnt about some of them, e.g., rectilinear 14.1 Introduction motion and motion of a projectile. Both these motions are 14.2 Periodic and oscillatory non-repetitive. We have also learnt about uniform circular motions motion and orbital motion of planets in the solar system. In 14.3 Simple harmonic motion these cases, the motion is repeated after a certain interval of 14.4 Simple harmonic motion time, that is, it is periodic. In your childhood, you must have and uniform circular enjoyed rocking in a cradle or swinging on a swing. Both motion these motions are repetitive in nature but different from the 14.5 Velocity and acceleration periodic motion of a planet. Here, the object moves to and fro in simple harmonic motion about a mean position. The pendulum of a wall clock executes 14.6 Force law for simple a similar motion. Examples of such periodic to and fro harmonic motion motion abound: a boat tossing up and down in a river, the 14.7 Energy in simple harmonic piston in a steam engine going back and forth, etc. Such a motion motion is termed as oscillatory motion. In this chapter we 14.8 Some systems executing study this motion. simple harmonic motion The study of oscillatory motion is basic to physics; its 14.9 Damped simple harmonic motion concepts are required for the understanding of many physical 14.10 Forced oscillations and phenomena. In musical instruments, like the sitar, the guitar resonance or the violin, we come across vibrating strings that produce pleasing sounds.
    [Show full text]
  • Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
    Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time.
    [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]
  • Low Frequency Damping Control for Power Electronics-Based AC Grid Using Inverters with Built-In PSS
    energies Article Low Frequency Damping Control for Power Electronics-Based AC Grid Using Inverters with Built-In PSS Ming Yang 1, Wu Cao 1,*, Tingjun Lin 1, Jianfeng Zhao 1 and Wei Li 2 1 School of Electrical Engineering, Southeast University, Nanjing 210096, China; [email protected] (M.Y.); [email protected] (T.L.); [email protected] (J.Z.) 2 NARI Group Corporation (State Grid Electric Power Research Institute), Nanjing 211106, China; [email protected] * Correspondence: [email protected] Abstract: Low frequency oscillations are the most easily occurring dynamic stability problem in the power system. With the increasing capacity of power electronic equipment, the coupling coordination of a synchronous generator and inverter in a low frequency range is worth to be studied further. This paper analyzes the mechanism of the interaction between a normal active/reactive power control grid-connected inverters and power regulation of a synchronous generator. Based on the mechanism, the power system stabilizer built in the inverter is used to increase damping in low frequency range. The small-signal model for electromagnetic torque interaction between the grid-connected inverters and the generator is analyzed first. The small-signal model is the basis for the inverters to provide damping with specific amplitude and phase. The additional damping torque control of the inverters is realized through a built-in power system stabilizer. The fundamentals and the structure of a built-in power system stabilizer are illustrated. The built-in power system stabilizer can be realized through the active or reactive power control loop.
    [Show full text]
  • Tuned Mass Damper Systems
    ConCh04v2.fm Page 217 Thursday, July 11, 2002 4:33 PM C H A PT E R 4 Tuned Mass Damper Systems 4.1 INTRODUCTION A tuned mass damper (TMD) is a device consisting of a mass, a spring, and a damper that is attached to a structure in order to reduce the dynamic response of the structure. The frequency of the damper is tuned to a particular structural frequency so that when that frequency is excited, the damper will resonate out of phase with the structural motion. Energy is dissipated by the damper inertia force acting on the structure. The TMD concept was first applied by Frahm in 1909 (Frahm, 1909) to reduce the rolling motion of ships as well as ship hull vibrations. A theory for the TMD was presented later in the paper by Ormondroyd and Den Hartog (1928), followed by a detailed discussion of optimal tuning and damping parameters in Den Hartog’s book on mechanical vibrations (1940). The initial theory was applicable for an undamped SDOF system subjected to a sinusoidal force excitation. Extension of the theory to damped SDOF systems has been investigated by numerous researchers. Significant contributions were made by Randall et al. (1981), Warburton (1981, 1982), Warburton and Ayorinde (1980), and Tsai and Lin (1993). This chapter starts with an introductory example of a TMD design and a brief description of some of the implementations of tuned mass dampers in building structures. A rigorous theory of tuned mass dampers for SDOF systems subjected to harmonic force excitation and harmonic ground motion is discussed next.
    [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]