7-1 Simple Harmonic Os-cillation

Remember!!!

삼각함수는

공학에서

가장 중요한 함수 !

Simple Harmonic Oscilla-tion (1)

Hooke’s Law for the Restoring Force:

F = - k x Newton’s Second Law of Motion:

F = ma = m(dv/dt) = m(d2x/dt2)

Coupling of the Above Equations:

kxdt

xdm

2

2

Simple Harmonic Oscilla-tion (2)

Let’s define the angular frequency:

Then we have

What is the solution of this equation?

022

2

xdt

xd

m

k

Differentiation ofTrigonometric Func-

tions

)sin()cos( baxabaxdx

d

)cos()sin( baxabaxdx

d

Exercise

Show that

x(t) = A cos(ω t + φ )

Satisfies the following equation:

A, ω, and φ are constants.

022

2

xdt

xd

Notice!!!

신호 (Signal) 는

삼각함수이다 !

Determination of the Constants

x(t) = A cos(ωt+φ)

v(t) = (dx/dt) = −Aω sin(ωt+φ)

x(0) = A cos(φ )

v(0) = −Aω sin(φ)

Sinusoidal Motion

t

AT = 2π /ω

y(t)=Asin(ωt+φ)

Variables

Amplitude A

Angular Frequency ω

Frequency ν = (ω / 2π)

Period T = 1/ν = (2π/ω)

Initial Phase φ

A Quiz

• Sampling Frequency = 16,000 / sec• Data 사이의 시간 간격은 ?• 1,024 Data 사이의 시간 간격은 ?

Simple Harmonic Oscillation

)cos()cos()( tRRtx

)(tx

R 스프링에 매달린 추 물질 내부의 분자들 소리를 내는 공기분자들 안테나 속의 전자들

모두 삼각함수에 의해 기술된다 !

Transduction of Sound & Image through Microphone &

CCD

Modulation (Coding)

Oscillation Circuit

Transmission into Space

Receiving by Resonating Circuit

Demodulation (Decoding)

Transduction of Electrical Signal into Sound and Image

Telecommunication (1898-)