《物理双语教学课件》Chapter 9 Oscillations 振动_大学物理振动课件

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Chapter 9 Oscillations

We are surrounded by oscillations─motions that repeat themselves.(1).There are swinging chandeliers, boats bobbing at anchor, and the surging pistons in the engines of cars.(2).There are oscillating guitar strings, drums, bells, diaphragms in telephones and speaker systems, and quartz crystals in wristwatches.(3).Le evident are the oscillations of the air molecules that transmit the sensation of sound, the oscillations of the atoms in a solid that convey the sensation of temperature, and the oscillations of the electrons in the antennas of radio and TV transmitters.Oscillations are not confined to material objects such as violin strings and electrons.Light, radio waves, x-rays, and gamma rays are also oscillatory phenomena.You will study such oscillations in later chapters and will be helped greatly there by analogy with the mechanical oscillations that are about to study here.Oscillations in the real world are usually damped;that is, the motion dies out gradually, transferring mechanical energy to thermal energy by the action of frictional force.Although we cannot totally eliminate such lo of mechanical energy, we can replenish the energy from some source.9.1 Simple Harmonic Motion 1.The figure shows a sequence of “snapshots” of a simple oscillating system, a particle moving repeatedly back and forth about the origin of the x axis.2.Frequency:(1).One important property of oscillatory motion is its frequency, or number of oscillations

that

are completed each second.(2).The symbol for frequency is f, and(3)its SI unit is hertz(abbreviated Hz), where 1 hertz = 1 Hz = 1 oscillation per second = 1 s-1.3.Period: Related to the frequency is the period T of the motion, which is the time for one complete oscillation(or cycle).That is T1f.4.Any motion that repeats itself at regular intervals is called period motion or harmonic motion.We are interested here in motion that repeats itself in a particular way.It turns out that for such motion the displacement x of the particle from the origin is given as a function of time by

x(t)xmcos(t), in which xm,,and

are constant.The motion is called simple harmonic motion(SHM), the term that means that the periodic motion is a sinusoidal of time.5.The quantity

xm, a positive constant whose value depends on how the motion was started, is called the amplitude of the motion;the subscript m stands for maximum displacement of the particle in either direction.6.The time-varying quantity

(t)

is called the phase of the motion, and the constant  is called the phase constant(or phase angle).The value of  depends on the displacement and velocity of the particle at t=0.7.It remains to interpret the constant .The displacement

x(t)

must return to its initial value after one period T of the motion.That is,x(t)

must equal

0x(tT)for all t.To simplify our analysis, we put.So we then have xmcostxmcos[(tT)].The cosine function first repeats itself when its argument(the phase)has increased by that we must have 22fT2 rad, so

(tT)t2orT2.It means.The quantity  is called the angular frequency

of the motion;its SI unit is the radian per second.8.The velocity of SHM:(1).Take derivative of the displacement with time, we can find an expreion for the velocity of the particle moving with simple harmonic motion.That is, v(t)dx(t)xmsin(t)vmcos(t/2).(2).The dtpositive quantity

vmxm in above equation is called the velocity amplitude.9.The acceleration of SHM: Knowing the velocity for simple harmonic motion, we can find an expreion for the acceleration of the oscillation particle by differentiating once more.Thus we have

a(t)dv(t)vmsin(t/2)amcos(t)dtThe positive quantity

amvm2xm is called the acceleration

a(t)2x(t)amplitude.We can also to get , which is the hallmark of simple harmonic motion: the acceleration is proportional to the displacement but opposite in sign, and the two quantities are related by the square of the angular frequency.9.2 The Force Law For SHM 1.Once we know how the acceleration of a particle varies with time, we can use Newton’s second law to learn what forcemust act on the particle to give it that acceleration.For simple harmonic motion, we have

Fma(m2)xkx.This result-a force proportional to the displacement but opposite in sign-is something like Hook’s law for a spring, the spring constant here being km2.2.We can in fact take above equation as an alternative definition of simple harmonic motion.It says: Simple harmonic motion is the motion executed by a particle of ma m subject to a force that is proportional to the displacement of the particle but opposite in sign.3.The block-spring system forms a linear simple harmonic oscillator

(linear oscillator for short), where linear indicates that F is proportional to x rather than to some other power of x.(1).The angular frequency  of the simple harmonic motion of the block is oscillator is

9.3 Energy in Simple Harmonic Motion 1.The potential energy of a linear oscillator depends on how much the spring is stretched or compreed, that is, on

k/m.(2).The period of the linear T2m/k.x(t).We have U(t)1212kxkxmcos2(t).222.The kinetic energy of the system depends on haw fast the block is moving, that is on

K(t)v(t).We have 1212mvm2xmsin2(t)22

1k212m()xmsin2(t)kxmsin2(t)2m23.The mechanical energy is

EUK121212kxmcos2(t)kxmsin2(t)kxm 222The mechanical energy of a linear oscillator is indeed a constant, independent of time.9.4 An Angular simple Harmonic Oscillator 1.The figure shows an angular version of a simple harmonic oscillator;the element of springine or elasticity is aociated with the twisting of a suspension wire rather than the extension and compreion of a spring as we previously had.The device is called a torsion pendulum, with torsion referring to the twisting.2.If we rotate the disk in the figure from its rest position and release it, it will oscillate about that position in angular simple harmonic motion.Rotating the disk through an angle  in either direction introduce a restoring torque given by Here (Greek kappa)is a constant, called the .torsion constant, that depends on the length, diameter, and material of the suspension wire.3.From the parallelism between angular quantities and linear quantities(give a little more explanation), we have

T2I

for the period of the angular simple harmonic oscillator, or torsion pendulum.9.5 Pendulum We turn now to a cla of simple harmonic oscillators in which the springine is aociated with the gravitational force rather than with the elastic properties of a twisted wire or a compreed or stretched spring.1.The Simple Pendulum(1).We consider a simple pendulum, which consists of a particle of ma m(called the bob of the pendulum)suspended from an un-stretchable, male string of length L, as in the figure.The bob is free to swing back and forth in the plane of the page, to the left and right of a vertical line through the point at which the upper end of the string is fixed.(2).The forces acting the particle, shown in figure(b), are its weight and the tension in the string.The restoring force is the tangent component of the weight

mgsin, which is always acts opposite the displacement of the particle so as to bring the particle back toward its central location, the equilibrium(0).We write the restoring force as Fmgsin, where the minus sign indicates that F acts opposite the displacement.(3).If we aume that the angle is small, the

sin is very nearly equal to  in radians, and the displacement s of the particle measured along its arc is equal to FmgmgL.Thus, we have

smg()s.Thus if a simple pendulum swings LLthrough a small angle, it is a linear oscillator like the block-spring oscillator.(4).Now the amplitude of the motion is measure as the angular amplitude m, the maximum angle of swing.The period of

a

simple

pendulum

is T2m/k2m/(mg/L)2L/g.This result hods only if the angular amplitude m is small.2.The Physical Pendulum

(1).The figure shows a generalized physical pendulum, as we shall call realistic pendulum, with its weight mg

acting at the center of ma C.(2).When the pendulum is displaced through an angle  in either direction from its equilibrium position, a restoring torque appears.This torque acts about an axis through the suspension point O in the figure and has the magnitude (mgsin)(h).The minus sign indicates that the torque is a restoring torque, which always acts to reduce the angle  to zero.(3).We once more decide to limit our interest to small amplitude, so that (mgh).sin.Then the torque becomes

T2I/mgh,(4).Thus the period of a physical pendulum is when m is small.Here I is the rotational inertia of the pendulum.(5).Corresponding to any physical pendulum that oscillates about a given suspension point O with period T is a simple pendulum of length L0 with the same period T.The point along the physical pendulum at distance L0 from point O is called the center of oscillation of the physical pendulum for the given suspension point.3.Measuring g: We can use a physical pendulum to measure the free-fall acceleration g through measuring the period of the pendulum.9.6 Simple Harmonic Motion and Uniform circular Motion 1.Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs.2.The figure(a)gives an example.It shows a reference moving circular particle in

P’

uniform with motion angular speed  in a reference circle.The radius xm of the circle is the magnitude of the particle’s position vector.At any time t, the angular position of the particle is t.3.The projection of particle P’ onto the x axis is a point P.The projection of the position vector of particle P’ onto the x axis gives the location x(t)of P.Thus we find

x(t)xmcos(t).Thus if reference particle P’ moves in uniform circular motion, its projection particle P moves in simple harmonic motion.4.The figure(b)shows the velocity of the reference particle.The magnitude of the velocity is xm, and its projection on the x axis is

v(t)xmsin(t).The minus sign appears because the velocity component of P points to the left, in the direction of decreasing x.5.The figure(c)shows the acceleration of the reference particle.The magnitude of the acceleration vector is projection on the x axis is

a(t)2xmcos(t).2xm

and its 6.Thus whether we look at the displacement, the velocity, or the acceleration, the projection of uniform circular motion is indeed simple harmonic motion.9.7 Damped Simple Harmonic Motion A pendulum will swing hardly at all under water, because the water exerts a drag force on the pendulum that quickly eliminates the motion.A pendulum swinging in air does better, but still the motion dies out because the air exerts a drag force on the pendulum, transferring energy from the pendulum’s motion.1.When the motion of an oscillator is reduced by an external force, the oscillator and its motion are said to be damped.An idealized example of a damped oscillator is shown in the figure: a block with ma m oscillates on a spring with spring constant k.From the ma, a rod extends to a vane(both aumed male)that is submerged in a liquid.As the vane moves up and down, the liquid exerts an inhibiting drag force on it and thus on the entire oscillating system.With time, the mechanical energy of the block-spring system decreases, as energy is transferred to thermal energy of the liquid and vane.2.Let us aume that the liquid exerts a damped forceFd that is

proportional in magnitude to the velocity v of the vane and

block.Then

Fdbv,where b is a damped constant that depends on the characteristics of both the vane and the liquid and has the SI unit of kilogram per second.The minus sign indicates that

Fd

opposes the motion.3.The total force acting on the block is Fkxbvkxbdx.dtSo we have equation

d2xdxm2bkx0,dtdtwhose solution is x(t)xmebt/2mcos('t), where ', the angular frequency of the 12 damped oscillator, is given by 'kb2m4m2.4.We can regard the displacement of the damped oscillator as a cosine function whose amplitude, which is decreases with time.5.The mechanical energy of a damped oscillator is not constant but decreases with time.If the damping is small, we can find E(t)

xmebt/2m, gradually by replacing

xm

with

xmebt/2m,the amplitude of the

E(t)12bt/mkxme, which 2damped oscillation.Doing so, we find tells us that the mechanical energy decreases exponentially with time.9.8 Forced Oscillations and Resonance 1.A person swing paively in a swing is an example of free oscillation.If a kind friend pulls or pushes the swing periodically, as in the figure, we have forced, or driven, oscillations.There are now two angular frequencies with which to deal with:(1)the natural angular frequency  of the system, which is the angular frequency at which it would oscillate if it were suddenly disturbed and then left to oscillate freely, and(2)the angular frequency d of the external driving force.2.We can use the right figure to represent an idealized forced simple harmonic oscillator if we allow the structure marked “rigid support” to move up and down at a variable angular frequency d.A forced oscillator oscillates at the angular frequency d of driving force, and its displacement is given by

x(t)xmcos(dt), where xm

is the amplitude of the oscillations.How large the displacement amplitude and .3.The velocity amplitude

vm xm

is depends on a complicated function of d

of the oscillations is easier to describe: it is greatest when d, a condition called resonance.Above equation is also approximately the condition at which the displacement amplitude

xm

of oscillations is greatest.The figure shows how the displacement amplitude of an oscillator depends on the angular frequency

d of the driving force, for three values of the damped coefficient b.4.All mechanical structures have one or more natural frequencies, and if a structure is subjected to a strong external driving force that matches one of these frequencies, the resulting oscillations of structure may rupture it.Thus, for example, aircraft designers must make sure that none of the natural frequencies at which a wing can vibrate matches the angular frequency of the engines at cruising speed.15