同济大学：《大学物理》课程教学资源（电子教案，英文版）第八章 振动

Oscillations

Oscillations

Agenda Today 1 Simple harmonic motion 2 Pendulum 3 Damped harmonic motion

Agenda Today 1 Simple harmonic motion 2 Pendulum 3 Damped harmonic motion

Oscillation(振动)： The movement that repeat itself Simple harmonic motion:(简谐振动) Any oscillation that could be expressed as a sinusoidal function

Oscillation(振动）: The movement that repeat itself. Simple harmonic motion:(简谐振动） Any oscillation that could be expressed as a sinusoidal function

x=Acos(@t+p A:amplitude(振幅) p:phase constant(振动的“初位相”) (otp):phase(振动的“位相”) o:angular frequency(圆频率)

x  Acos(t  ) A ：amplitude(振幅）  ：phase constant(振动的“初位相 ”) （ t+ ) ：phase(振动的“位相 ”)  ：angular frequency(圆频率）

Frequency (频率)v:the number of oscillations that finished in one second. 2π Period(周期)T:the time needed to finish one oscillation 2π

Period （周期）T：the time needed to finish one oscillation    2  Frequency （频率）：the number of oscillations that finished in one second.    1 2 T  

Velocity and acceleration of SHM s+p)=eota+p+》 dx V= dt a 血=-02Acos(o1+p)=0.c0so1+9士π） 1、Vm=@4 velocity amplitude“速度幅”。 2、am=o2A acceleration amplitude m “加速度幅

) 2 sin( ) cos(     A  t   v  t   dt dx v m cos( ) cos( ) 2    A  t   a  t   dt dv a m 1、 vm  A velocity amplitude“速度幅 ”。 2、 acceleration amplitude “加速度幅 ”。 am A 2  Velocity and acceleration of SHM

Spring oscillator(弹簧振子) △△△个个M X By Hook's law F=- By Newton's second law of motion: d'x F=m dt" =kx

Spring oscillator（弹簧振子） F By Hook’s law F kx     o x x By Newton’s second law of motion: k x dt d x F  m   2 2

that is 0x=0 dt2 The solution: x=Acos(@t+o) Where A,and o are decided by the initial conditions

0 2 2 2  x  dt d x  that is x  Acos(t  ) x t m k   Where A, and  are decided by the initial conditions The solution:

Does the oscillator has same period when it free end hangs from the ceiling? Yo Equilibrium position mg

0 y free end Equilibrium position y o y mg F Does the oscillator has same period when it hangs from the ceiling?

The energy of SHM X X Kinetic energy: 女-mmsi) Potential E-3k-3k cos() energy: mo2 k

The energy of SHM sin ( ) 2 1 2 1 2 2 2 2 Ek  m v  m A  t  cos ( ) 2 1 2 1 2 2 2 Ep  k x  k A  t  m  k 2  Kinetic energy: Potential energy: o x x v