Oscillations

Oscillations

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  Waves: Oscillations Oscillations Introduction: Mechanicalvibration Simple Harmonic Motion Some oscillating systems Damped Oscillations Driven oscillations and resonance Traveling waves Wave motion. The wave equation Periodic Waves: on a string, sound and electromagnetic waves Waves in Three dimensions. Intensity Waves encountering barriers. Reflection, refraction, diffraction The Doppler effect Superposition and standing waves Superposition and interference Standing waves
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  Oscillations • Simple HarmonicMotion. Energy • Some oscillating systems Vertical String The simple pendulum The physical pendulum • Damped Oscillations • Driven (Forced) oscillations and resonance
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  INTRODUCTION. MECHANICAL VIBRATIONS Amechanical vibration is the motion of a particle or a body which oscilates about a position of equilibrium. A mechanical vibration generally results when a system is displaced from a position of stable equilibrium. The system tends to return to this position under the action of restoring forces (either elastic forces as the case of springs or gravitational forces, as the case of pendulum) Period of vibration. The time interval required by the system to complete a full cycle of motion. Frequency: The number of cycles per unit of time Amplitude: The maximum displacement of the system from its position of equilibrium Most vibrations are undesirable, wasting energy and creating unwanted sound – noise. For example, the vibrational motions of engines, electric motors, or any mechanical device in operation are typically unwanted. Such vibrations can be caused by imbalances in the rotating parts, uneven friction, the meshing of gear teeth, etc. Careful designs usually minimize unwanted vibrations. The study of sound and vibration are closely related. Sound, or "pressure waves", are generated by vibrating structures (e.g. vocal cords); these pressure waves can also induce the vibration of structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to reduce vibration. Drum vibration VIBRATION Free (Driven) Forced Undamped Damped
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  SIMPLE HARMONIC MOTION Considerthe forces exerted on the block that is placed above a table without friction. The net (resultant) force on the block is that exerted by the spring. This force is proportional to the displacement x, measured from the equilibrium position. Applying the Newton´s Second Law, we have xkF  x m k dt xd xk dt xd mF   2 2 2 2 This equation is a second-order linear constant coefficient ordinary differential equation describing the harmonic oscillator A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines Verify that each of the functions satisfies the differential equation           t m kCx t m kCx sin cos 22 11 Constant of the spring Visualizing the simple harmonic motion through the motion of a block on a spring
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  Simple Harmonic Motion x,position;A, amplitude, (ωt+δ) phase of the motion v, velocity acceleration Case study: harmonic motion of an object on a spring f , frequency, T period, ω, angular frequency (natural circular frecuency), δ, phase angle or constant phase m k
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  Simple Harmonic Motionand Circular Motion Position, [m]; Amplitude [m]; phase (ωt+δ) [rad] Velocity, [m/s] f , frequency, [cycles/s], T period,[s] ω, [rad/s]angular frequency (natural circular frecuency), δ, phase angle [rad] Simple harmonic motion can be visualized as the motion of the projection onto the x axis from a point which moves in a circular motion at constant speed 1.-A 0.8-kg object is attached to a spring of force constant k = 400 N/m. The block is held a distance 5 cm from the equilibrium position and is released at t =0. Find the angular frequenccy and the period T. (b) Write the position x and velocity of the object as a function of time.(c) Calculate the maximum speed the block reaches. (d) The energy of the oscillating system 2.- An object oscillates with angular frequency 8.0 rad/s. At t = 0, the object is at x = 4 cm with an initial velocity v = -25 cm/s. (a) Find the amplitude and the phase constant for the motion; (b) Write the position x and velocity of the object as a function of time.(c) Calculate the maximum speed the object reaches (e) The energy of the oscillating system m k
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  Simple Harmonic Motion.Energy 2 0 2 1 )( xkdxxkU x x   Potential Energy Kinetic energy  22 )sin( 2 1 2 1   tAmvmK Total mechanical energy in Simple Harmonic Motion 222 2 1 2 1 AmAkKUEtotal  The total mechanical energy in simple harmonic motion is proportional to the square of the amplitude
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  Some oscillating systems SpringThe simple pendulum The physical pendulum k mT m k   2 ;   Free-body diagram The motion of a pendulum approximates simple harmonic motion for small angular displacements Free-body diagram      L g L g dt d dt d mLmg Lmmg amF TT     sin sin sin 2 2 2 2 g LT L g   2        I MgD I MgD dt d dt d IMgD I     sin sin 2 2 2 2
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  Show that forthe situations depicted the object oscillates, in the case (a) as if it were a spring with a force constant of k1+k2, and, in the case (b) 1/k = 1/k1 +1/k2 The figure shows the pendulum of a clock. The rod of length L=2.0 m has a mass m = 0.8 kg. The attached disk kas a mass M= 1.2 kg, and radius 0.15 m. The period of the clock is 3.50 s. What should be the distance d so that the period of this pendulum is 2.5 s Find the resonance frequency for each of the three systems
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  Damped Oscillations Spring force Viscous force       t m b o t m b o o o t m b o t m b o eEemAmAE m b andeAA teAx xk dt dx b dt xd m amvbxkF               22 2 122 2 1 2 2 2 2 2 2 1´ )´cos( 0     x Equilibrium position
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  Driven (Forced) Oscillationsand resonance tFF oext cos External driving force (harmonic) In addition to restoring forces and dumping forces are acting external (periodic) forces