制作张昆实 Yangtze University

BilingualMechanics Chapter 8 Oscillations 制作张昆实 Yangtze University

Chapter 8 Oscillations 8-1 What Is Physics? 8-2 Simple Harmonic Motion 8-3 The Force Law for Simple Harmonic Motion 8-4 Energy in Simple Harmonic Motion 8-5 An Angular Simple Harmonic Oscillator

Chapter 8 Oscillations 8-6 Pendulums 8-7 Simple Harmonic Motion and Uniform Circular Motion 8-8 Damped Simple Harmonic Motion 8-9 Forced Osillations and Resonance *8-10 The combination of Simple Harmonic Motions

There are swinging power lines, boats bobbing at anchor, and the surging pistons in the engines of cars. 8-1 What Is Physics? We are surounded by oscillations in which objects move back and forth repeatedly

There are oscillating guitar strings, drums, bells, diaphragms in telephones and 8-1 What Is Physics? speaker systems, and quartz crystals in wristwatches.

Less evident are the oscillations of the air molecules that transmit the sensation of sound, 8-1 What Is Physics? the oscillations of the atomsin a solid that convey the sensation of temperature,

the oscillations of the electrons in the antennas of radio and TV transmitters that convey information. 8-1 What Is Physics?

8-1 What Is Physics? ★The study and control of oscillations are two of the primary goals of physics and engineering. ★In this chapter we learnwhat is physics through discussing a basic type of oscillation calledsimple harmonic motion.

Properties of oscillation (8-2) 8-2Simple Harmonic Motion In a oscillating system a particle moves repeatidly back and forth about the origin of an axis. frequency ( ): the number of oscillations completed in one second. SI unit: 1 hertz = 1 Hz = 1 oscillation per second = 1 s-1 period ( ):the time for one complete oscillation (sycle) Any motion that repeats itself at regular intervals is called periodic motiom or harmonic motion.

(8-3) ( displacement ) 8-2Simple Harmonic Motion Simple Harmonic Motio (SHM) The displacement of the particle from the origin is given as a function of time by

The cosine function in Eq.8-3 varies between the limits , so the displacement x(t)varies between the limits . 8-2 Simple Harmonic Motion The amplitude of the motion :is a positive constant whose value depends on how the motionwas started. The subscript stands for maximum because the amplitude is the magnitude of the maximum displacement of the particle in either direction.

8-2Simple Harmonic Motion The value of depends onthe displacement and velocity of the particle at time i . For the plots of Fig.16-3a, the phase constant is zero. The time-varying quantity ( ) in Eq.16-3 is called the phase of the motion, and the constant is called the phase constant (or phaseangle).

8-2 Simple Harmonic Motion The constant is called the angular frequency of the motion. Note, the desplacementmust return to its initial value after one period of the motion; that is must equal for all . For simplicity, let then (8-4)

8-2Simple Harmonic Motion The cosine functionfirst repeats itself when its phase has increased by , so Eq. 16-4 gives us For simplicity, let then From (8-5) The SI unit of angular frequency is the radian per second . (8-2) (8-4)

Compare for two simple harmonic motions (a) Different amplitude (b) Different period (c) Different phase constant 8-2 Simple Harmonic Motion

(8-3) ( displacement ) (velocy) (8-6) 8-2 Simple Harmonic Motion The Velocity of SHM By differentiating positive quantity is called the velocity amplitude : The curve of is shiftedto the left from thecurve of by one-quarter period.

8-2 Simple Harmonic Motion The Acceleration of SHM (velocy) (8-6) By differentiating (8-7) (acceleration) positive quantity is the acceleration amplitude : Varies between the limits

(8-3) (8-6) (8-7) Combine Eqs. 8-3 and 8-7 (8-8) In SHM, the acceleration is proportional to the displacement but opposite in sign, and the two quantities are related by the square of angular frequency. 8-2Simple Harmonic Motion The Acceleration of SHM

is a restoring force that is proportional to the displacement but opposite in sign. (8-9) It is Hooke’s law (8-10) For a spring, the spring constant here being Simple harmonic motion is the motion executed by a particle of mass m subject to a force that is proportional to the displacement of the particle but opposite in sign. (8-11) 8-3The Force Law for Simple Harmonic Motion CombineNewton’s secondlaw and Eq.8-8, the force acting on a body in SHMcan be found

The block-spring system forms a linear simple harmonic oscillator. (linear means ) By Eq.8-11, the angular frequency of the SHM is ( angularfrequency) (8-12) Combining Eq.8-5 ( period ) (8-13) Equations 16-12, 16-5 and 16-13 tell us that a large angular frequency (and thus a small period ) goes with a stiff spring (large ) and a light block (small ). 8-3The Force Law for Simple Harmonic Motion

8-4 Energy in Simple Harmonic Motion Consider the mechanical energy of the oscillator The potential energy of a linear oscillator is associated entirely with the spring. Its value depends on how much the spring is stret- ched or compressed (no ). (8-18) The kinetic energy of a linear oscillator is associated entirely with the block. Its value depends on how fast the block is moving (no ). (8-19)

8-4 Energy in Simple Harmonic Motion Consider the mechanical energy of the oscillator Substitute for (8-19) The mechanical energy of the oscillatoris (8-20) The mechanical energy of a linear oscillator is indeed constant and independent of time.

Potential energy , kinetic energy and mechanical energy as a function of time for a linear harmonic oscillator. Potential energy , kinetic energy and mechanical energy as a function of position for a linear harmonic oscillator with amplitude . 8-4Energy in Simple Harmonic Motion All energies are positive.

If we rotatethe desk by some angular displacement from its rest position(where the reference line is at ) and release it, it will oscillate about that position in angular simple harmonic motion. Rotating the desk through an angle in eitherdirection introduces a restoring torque given by (8-21) 8-5An Angular Simple Harmonic Oscillator The device in Fig.8-7 is called torsion pendulum. The element of elasticity is associated with the twisting of a suspensionwire.

restoring torque Constant (kappa) is called the torsion constant, that depends on the length, diameter, and material of the suspension wire. (8-21) comparing (8-21) (8-10) Hooke’s law Hooke’s law (angular form) The rotational inertial of the oscillating desk period ( torsion pendulum ) (8-22) (8-13) 8-5An Angular Simple Harmonic Oscillator linear SHM angular SHM

Discussing a class of simple harmonic oscillators in which the springiness is associated with the gravitational force. a bob of the pendulum (mass ) can swing back and forth freely in the virtical plane. from the sthing gravitational force 8-6 Pendulums the Simple Pendulum an unstretchable, massless string of length L . Two forces act on the bob : tangential radial

8-6 Pendulums The tangential component produces a restoring torque about the pendulum’s pivot point, because it always acts opposite the displacement of the bob so as to bring the bob back toward its equilibrium position . (8-23) (8-24) taking For small then (8-25) (8-8)

8-6 Pendulums Hallmark of SHM Hallmark of SHM but they are opposite in sign Thus as the bob moves to the right, its acceleration to the leftincreaces until it stopes and bigins moving to the left. The same thing happens when it is on the left, and so on, as it swing back and forthin SHM. Period of simple pendulum (8-26)

The rotational inertia of the pendulum about the pivot is (8-27) (Period of simple pendulum, small amplitude) 8-6 Pendulums (8-26)

simple pendulum physical pendulum tangetial component moment arm period for small in SHM in SHM 8-6 Pendulums The Physical Pendulum: Is the real pendulum with complicated distribution of mass (Fig. 8-10). the gravitational force acts at its center of mass C at a distance h from the pivot point O.

8-6 Pendulums If a physical pendulum and a simple pendulum has he same period T, center of oscillation the length of the simple pendulumisL0, the point along the physical pendulum at distance L0 from the pivot point O is called the center of oscillation of the physical pendulum for the given suspension point.

8-6 Pendulums Measuring Take the pendulum to be a uniformrod of length L, suspended from one end. The period of the physicalpendulum is (8-28) (8-29) (8-30)

Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the circular motion occurs. Vector rotates counterclockwise with uniform angular speed , at time it makes an angle of with axis. The projection of ’ end( ) on theaxis is point P and its displacement is The speed of is ; its projection on the axis is The radial acceleration of is ; its projection on the axis is 8-7Simple Harmonic Motion and Uniform Circular Motion ( SHM ) ( SHM ) ( SHM )

Whenthe motion of an oscillator is reduced by an external force, the oscillator and its motion are said to be damped. Fig.8-14 shows a damped oscillator, where a block (m) oscillates vertically on a spring (k). A rod and a vane (massless) is fixed to the block. The vane is submerged in a liquid. As the vane moves up and down, the liquid exerts a drag force on the osillating system. The forces acting on the system: damping force (for small v) (8-37) Newton’s second law: restoring force The gravitational force is negligible compared to and . (8-38) 8-8Damped Simple Harmonic Motion b is a damping constant. SI unit: kg/s

Substituting for and for , (8-39) The solution of this equation is (8-40) If dampedoscillator angular frequency If undamped oscillatorangular frequency mechanical energy mechanical energy 8-8Damped Simple Harmonic Motion Newton’s second law: (8-38) get the differential equation

1. the natural angular frequencyof the system 2. the angular frequency of the external driving force causing the driven oscillations. Sucha forced oscillatoroscillates at the angular frequency of the driving force, and its displacement is (8-43) The amplitude depends ona complicated functionof and . The velocity amplitude of the oscilla-tions is greatestwhen (resonance) (8-44) 8-9Forced Oscillations and Resonance Two angular frequencies are associated with a system undergoing forced (driven)oscillations .

During resonance the amplitude of the oscilla-tions is also approximately greatest. Fig.8-16 shows hwo the displacement amplitude of an oscillator depends on the angular frequency of the driving force, for three values of the damping coefficient . Note that for all three the amplitude is approximately greatest whenthat is, when the resonance condition is satisfied. (resonance) (8-44) 8-9Forced Oscillations and Resonance this condition is called resonance The curves show that less damping gives a taller and narrowerrasonance peak.

*8-10 Superposition of two SHMs in the same direction The resultant motion is still a SHM with the same angular frequency

9.4 同方向不同频率简谐振动的合成

9.4同方向不同频率简谐振动的合成 拍的形成

9.4 振动合成原理的模拟 方波的傅立叶合成与分解

*8-11 Superposition of two SHMs in perpendicular directions (a SHM in the x direction) (a SHM in the y direction) Only when The resultant motion is a stable figure called the “Lissajou’s figure”.

9.4相互垂直的简谐振动的合成利萨如图形1 1:1 1:2 1:3

9.4相互垂直的简谐振动的合成利萨如图形2 1:2 2:3 4:5

制作张昆实 Yangtze University