Vol. 3 Main Contents ： 1.Vibration 2.Wave---Propagation of vibration or field. 3.Wave optics---Propagation of light Vibration and Wave Modern Physics 4. Special theory of relativity---The theory for high speed motion. 5. Quantum physics---The theory for the motion of microscopic bodies.

Chapter 14 Vibration A body repeats its motion over and over

§14-5 Superposition of SHM 简谐振动的合成 §14-1 Simple Harmonic Motion (SHM) 简谐振动 §14-4 Damped Vibration& Forced Vibration Resonance 阻尼振动 受迫振动 共振

Harmonic oscillator §14-1 Simple Harmonic Motion Force constant— Mass— I. SHM model— Frictionless surface F

II.Equations of SHM ： 1. Force ： —Restoring force 2. Motion ： ---Dynamic Eq. Here As Then —Angular frequency ， --- 

The solution of Eq.  or Curve of SHM x t o A T ---  Kinematic Eq.

III.The velocity and acceleration of SHM They are periodic functions of t

IV.Characteristic quantities of SHM 1.Amplitude A ： 2. Period T, Frequency For HO ：

3. Phase angular PA : Decide the phase of SHM on time t Initial PA As initial conditions Solve A and : We get

V.Energy in SHM Potential E ： Kinetic E ：

Total mechanical E ： Conservation

o EpEp x E A EkEk x EpEp PE. curve ： The average of E ：

VI. Vector representation of SHM 0 M0M0 M x (t=0) (t)(t) —Rotating vector

[Example] Spring k, dish M are at rest when t=0. Prove that the motion of m+M is SHM after m falls down to the dish and find the kinematic equation. Prove h m Choose the origin point 0 is at the point that the resultant force acting on M+ m is zero. Gravity, elastic force 0 x When M+ m is at any position x, Analyse the forces acting on M+ m then

 the motion of m+M is SHM A,  are determined by initial conditions

So we can get As x 0 0, we can get that  <  < 3  /2 kinematic equation ：

A ： axis I ： moment inertia through A c : center of M. 0 m A c h VII.Angular SHM –Simple pendulum and physical … mg

Angular acceleration PP SP Restoring torque Dynamics Eq. Ang. Fr. Kine. Eq.

Torsion pendulum 扭摆  Torsion torque: Newton’s second law for rotation: Motion of TP can be known

§14-2 Damped vibration Forced … I. Damped Vibr.  Restoring force ：  Drag … ： -- drag coefficient Dynamics Eq. ： Let

Characteristic equation: Cha. values: 1. Underdamped ：  0 The solution of Dyn. Eq. Here Dyn.Eq. ：

Conclusions  Amplitude Ae － β t decreases with time t.  The “period” of damped Vibr ：  Energy  Amplitude 2 E=E 0 e － 2 β t

2.Overdamped ：  0  Damped Vibr. is not periodic motion.

3.Critically damped ：   We get The object returns to equilibrium position with no oscillation in the shortest time possible.

Critically damped Overdamped Underdamped

1.Eq. of forced vibration Driven force ： F 0 : the amplitude of driven force : the angular frequency of driven force Let --dynamic Eq. II. Forced (Driven) vibration

Underdamped (  0 ) ， 齐次方程通解 非齐次方程特解 + SHM  Forced vibration = Damped vibration Discussion  After, the motion tends to SHM.

The kinematic Eq. at stable state ： --- SHM Transitional state Stable state

2. The amplitude and initial phase at stable state substitute to dynamic Eq. We get

When t = 0, We can get When,

Discussion   As  >>  0 or  <<  0, A is smaller  As  0 ， A is larger   getting larger as  getting  smaller smaller larger

3. Resonance 共振 --A( ,  ) has the largest value =A r. Let We get The angular frequency of resonance  r = the angular frequency of driving force when the system is at resonance.

Note Whenβis smaller,ω r tends toω 0 ， A r is larger. Energy loss is larger. smaller larger

Tacoma bridge was destroyed by wind Tacoma bridge was destroyed by wind

§14-3 Superposition of SHM I. Superposition Principle of vibration If an object participates two or more vibrations simultaneously, the resultant position of the object at any time is equal to the vector sum of the positions produced by each of the vibrations separately.

II. Superposition of two SHM with same direction and same frequency 1. Algebraic solution Separate vibrations ： Resultant vibration ： is still SHM with same

The amplitude of resultant vibration ： The initial phase of resultant vibration ：

2. Rotating Vector solution o x

3. Discussion  when ， （ construction ）  when ， （ destruction ）

III. Superposition of two SHM with same direction and different frequency Separate vibrations ： Resultant vibration ： If, we can get

--Beat ( 拍 ) It varies periodically. When, The amplitude of resultant vibration is

IV. Superposition of two perpendicular SHM with same frequency Separate vibrations ： Resultant vibration ： Ellipse equation —— Trajectory equation ：

Discussion --Straight line  o y x  o y x

 -- Ellipse o y x o y x 

V. Superposition of two perpendicular SHM with different frequency If the frequency ratio of the two perpendicular SHM has simple integral relation, the path of the resultant vibration is called as Lissajous curves( 李萨如曲线 ) 李萨如