Fundamentals of Physics School of Physical Science and Technology Mechanics (Bilingual Teaching) 张昆实 School of Physical Science and Technology Yangtze University

Chapter 14 Oscillations 16-1 Oscillations 16-2 Simple Harmonic Motion 16-3 The Force Law for Simple Harmonic Motion 16-4 Energy in Simple Harmonic Motion 16-5 An Angular Simple Harmonic Oscillator 16-6 Pendulums 16-7 Simple Harmonic Motion and Uniform Circular Motion 16-8 Damped Simple Harmonic Motion 16-9 Forced Oscillations and Resonance

We are surounded by oscillations motions that repeat themselves. There are swinging chandeliers, boats bobbing at anchor, and the surging pistons in the engines of cars.

16-1 Oscillations There are oscillating guitar strings, drums, bells, diaphragms in telephones and speaker systems, and quartz crystals in wristwatches.

16-1 Oscillations Less 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,

16-1 Oscillations the oscillations of the electrons in the antennas of radio and TV transmitters that convey information.

16-2 Simple Harmonic Motion In a oscillating system a particle moves repeatidly back and forth about the origin of an axis. Properties of oscillation frequency (symbol ): the number of oscillations completed in one second. SI unit: 1 hertz = 1 Hz = 1 oscillation per second = 1 s-1 period (symbol ): the time for one complete oscillation (sycle) (16-2) harmonic motion: Any motion that repeats itself at regular intervals is called periodic motiom or harmonic motion.

16-2 Simple Harmonic Motion Simple Harmonic Motio (SHM) The displacement of the particle from the origin is given as a function of time by (16-3) ( displacement )

16-2 Simple Harmonic Motion The amplitude of the motion : is a positive constant whose value depends on how the motion was started. The subscript stands for maximum because the amplitude is the magnitude of the maximum displacement of the particle in either direction. The cosine function in Eq.16-3 varies between the limits , so the displacement x(t) varies between the limits .

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

16-2 Simple Harmonic Motion The constant is called the angular frequency of the motion. Note, the desplacement must return to its initial value after one period of the motion; that is must equal for all . For simplicity, let then (16-4)

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

16-2 Simple Harmonic Motion Compare for two simple harmonic motions (a) Different amplitude (b) Different period (c) Different phase constant

16-2 Simple Harmonic Motion The Velocity of SHM (16-3) ( displacement ) By differentiating (16-6) (velocy) positive quantity is called the velocity amplitude : The curve of is shifted to the left from the curve of by one-quarter period.

16-2 Simple Harmonic Motion The Acceleration of SHM (velocy) (16-6) By differentiating (16-7) (acceleration) positive quantity is the acceleration amplitude : Varies between the limits

16-2 Simple Harmonic Motion The Acceleration of SHM (16-3) (16-6) (16-7) Combine Eqs. 161-3 and 16-7 (16-8) In SHM, the acceleration is pro- portional to the displacement but opposite in sign, and the two quantities are related by the square of angular frequen- cy. 漆安慎 《力学》P290 例题1

16-3 The Force Law for Simple Harmonic Motion Combine Newton’s second law and Eq.16-8, the force acting on a body in SHM can be found (16-9) is a restoring force that is proportional to the displacement but opposite in sign. (16-10) It is Hooke’s law For a spring, the spring constant here being (16-11) 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.

16-3 The Force Law for Simple Harmonic Motion The block-spring system forms a linear simple harmonic oscillator. (linear means ) (16-12) By Eq.16-11, the angular frequency of the SHM is ( angular frequency) Combining Eq.16-5 (16-13) ( period ) 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 ).

16-4 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 ). (16-18) The kinetic energy of a linear oscillator is associa- ted entirely with the block. Its value depends on how fast the block is moving (no ). (16-19)

16-4 Energy in Simple Harmonic Motion Consider the mechanical energy of the oscillator Substitute for (16-20) The mechanical energy of the oscillator is (16-21) The mechanical energy of a linear oscillator is indeed constant and independent of time.

16-4 Energy in Simple Harmonic Motion Potential energy , kinetic energy and mechanical energy as a function of time for a linear harmonic oscillator. All energies are positive. Potential energy , kinetic energy and mechanical energy as a function of position for a linear harmonic oscillator with amplitude .

16-5 An Angular Simple Harmonic Oscillator The device in Fig.16-7 is called torsion pendulum. The element of elasticity is associated with the twisting of a suspension wire. If we rotate the 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 either direction introduces a restoring torque given by (16-22)

16-5 An Angular Simple Harmonic Oscillator restoring torque (16-22) Constant (kappa) is called the torsion constant, that depends on the length, diameter, and material of the suspension wire. (16-22) comparing (16-10) Hooke’s law Hooke’s law (angular form) linear SHM angular SHM The rotational inertial of the oscillating desk period (16-13) (16-23) ( torsion pendulum ) 漆安慎 《力学》P286

16-6 Pendulums Discussing a class of simple harmonic oscillators in which the springiness is associated with the gravitational force. the Simple Pendulum an unstretchable, massless string of length L . a bob of the pendulum (mass ) can swing back and forth freely in the virtical plane. Two forces act on the bob : from the sthing gravitational force tangential radial

16-6 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 . (16-24) (16-25) take For small then (16-26) (16-8)

16-6 Pendulums but they are opposite in sign Hallmark of SHM Hallmark of SHM but they are opposite in sign Thus as the bob moves to the right, its acceleration to the left increaces 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 forth in SHM. Period of simple pendulum (16-27)

16-6 Pendulums The rotational inertia of the pendulum is (16-27) (16-28) (Period of simple pendulum, small amplitude) 漆安慎《力学》P285

16-5 An Angular Simple Harmonic Oscillator The Physical Pendulum: Is the real pendulum with complicated distribution of mass (Fig. 16-10). the gravitational force acts at its center of mass C at a distance h from the pivot point O. simple pendulum tangetial component moment arm period for small in SHM in SHM physical pendulum

16-5 An Angular Simple Harmonic Oscillator If a physical pendulum and a simple pendulum has the same period T, center of oscillation the length of the simple pendulum is L0, 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.

16-5 An Angular Simple Harmonic Oscillator Measuring Take the pendulum to be a uniform rod of length L, suspended from one end. The period of the physical pendulum is (16-29) (16-30) Sample Problem 16-5 P357 (16-32) (16-31)

16-7 Simple Harmonic Motion and Uniform Circular Motion Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs. (16-3) Vector rotates counterclockwise with uniform angular speed , at time it makes an angle of with axis. The projection of ’ end( ) on the axis is point P and its displacement is ( SHM ) (16-6) The speed of is ; its projection on the axis is ( SHM ) The radial acceleration of is ; its projection on the axis is (16-7) ( SHM )

16-8 Damped Simple Harmonic Motion When the motion of an oscillator is reduced by an external force, the oscillator and its motion are said to be damped. Fig.16-15 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: (16-37) damping force (for small v) b is a damping constant. SI unit: kg/s Newton’s second law: restoring force The gravitational force is negligible compared to and . (16-38)

16-8 Damped Simple Harmonic Motion Newton’s second law: (16-38) Substituting for and for , get the differential equation (16-39) (16-40) The solution of this equation is If (16-3) damped oscillator angular frequency If undamped oscillator angular frequency mechanical energy mechanical energy

16-9 Forced Oscillations and Resonance Two angular frequencies are associated with a system undergoing forced (driven) oscillations . 1. the natural angular frequency of the system 2. the angular frequency of the external driving force causing the driven oscillations. Such a forced oscillator oscillates at the angular frequency of the driving force, and its displacement is (16-43) The amplitude depends on a com- plicated function of and . The velocity amplitude of the oscilla-tions is greatest when (16-44) (resonance)

16-9 Forced Oscillations and Resonance (16-44) (resonance) this condition is called resonance During resonance the amplitude of the oscilla-tions is also approximately greatest. Fig.16-17 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 when that is, when the resonance condition is satisfied. The curves show that less damping gives a taller and narrower rasonance peak.

9.4 同方向不同频率简谐振动的合成 漆安慎《力学》P299

9.4同方向不同频率简谐振动的合成 拍的形成 漆安慎《力学》P302

9.4 振动合成原理的模拟 方波的傅立叶合成与分解 9.4 振动合成原理的模拟 方波的傅立叶合成与分解 漆安慎《力学》P308

9.4相互垂直的简谐振动的合成 利萨如图形1 漆安慎《力学》P307 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 漆安慎《力学》P307 1:2 2:3 4:5