振動:Oscillation

Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples include a swinging pendulum and AC power.

単振動子:Simple harmonic oscillator

The simplest mechanical oscillating system is a mass attached to a linear spring subject to no other forces. Such a system may be approximated on an air table or ice surface. The system is in an equilibrium state when the spring is static. If the system is displaced from the equilibrium, there is a net restoring force on the mass, tending to bring it back to equilibrium.

The specific dynamics of this spring-mass system are described mathematically by the simple harmonic oscillator and the regular periodic motion is known as simple harmonic motion. In the spring-mass system, oscillations occur because, at the static equilibrium displacement, the mass has kinetic energy which is converted into potential energy stored in the spring at the extremes of its path.

An undamped spring–mass system is an oscillatory system.

フックの法則

In physics, simple harmonic motion (SHM) is the motion of a simple harmonic oscillator, a periodic motion that is neither driven nor damped.

A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in the opposite direction.

Mathematically, Hooke's law states that

F=-K・x

where

x is the displacement of the end of the spring from its equilibrium position; F is the restoring force exerted by the material; and k is the force constant (or spring constant).

  • xの単位はm、Fの単位はニュートンN = kg·m·s−2、k はばね定数と呼ばれる定数。個々のばね固有の値であり、ばねの強さを表している。[ニュートン毎メートル]
  • この法則が適用できるとき、その挙動は線型と呼ばれ、グラフに表すと正比例の直線グラフとなる。 The motion is periodic: the body oscillates about an equilibrium position in a sinusoidal pattern.
    • フックの法則は17世紀のイギリスの物理学者、ロバート・フックが提唱したものであり、彼の名を取ってフックの法則と名づけられた。

運動:Dynamics of simple harmonic motion

For oscillation in a single dimension, combining Newton's second law (F = m d2x/dt2) and Hooke's law (F = −kx, as above) gives the second-order linear differential equation

F = m d2x/dt2 = −kx

where m is the mass of the body, x is its displacement from the mean position, and k is a constant. The solutions to this differential equation are sinusoidal; one solution is

where A, ω, and φ are constants, and the equilibrium position is chosen to be the origin.[1] Each of these constants represents an important physical property of the motion: A is the amplitude, ω = 2πf is the angular frequency, and φ is the phase.

加速度と周期

Using the techniques of differential calculus, the velocity and acceleration as a function of time can be found:

Position, velocity and acceleration of a SHM as phasorsAcceleration can also be expressed as a function of displacement.

Acceleration can also be expressed as a function of displacement:

a・x = - ω^2・x

Now since ma = −mω^2x = −kx,

ω^2 = k/m.

Then since ω = 2πf,

and since T = 1/f where T is the time period,

These equations demonstrate that period and frequency are independent of the amplitude and the initial phase of the motion.

位置、速度、位相

Simple harmonic motion shown both in real space and phase space. The orbit is periodic.

例題1.固有振動数を求めよ

  • バネ定数k=9.807kN/m 、重量m=5kgの時、1秒間に何回振動するか?
    ω=√k/m=√9807/5 
    単位はニュートンがN=kg・m/s^2 なのでωはラジアン/sの単位である。
    有振動数は f=ω/(2π)より f=7.05(Hz)
    1秒間に7.05回振動します
  • バネの自然長(荷重ゼロ)の位置から物体を離すと静的平衡位置を通過するとき最大速度となるが、その速度は?
    平衡位置ー自然長=バネの伸びである。kX=mgより自然長x=mg/k=5(mm)
    変位x=Acosωt+Bsinωt
    速度は、微分して dx/dt=ーAωsinωt+BωCOSωt
    t=0でx0=-5(mm)より A=-5である。
    t=0でdx/dt=0より B=0
    変位はx=-5cosωt、dx/dt=5ωsinωtなので、最大速度は5ω=221.5(mm/s)
     

線形微分方程式:linear differential equation

In mathematics, a linear differential equation is of the form

Dy(t)= f(t) 

where the differential operator D is a linear operator, y is the unknown function (such as a function of time y(t)), and the right hand side ƒ is a given function of the same nature as y .

The second order differential equation

D^2y = − k^2y, 
  • 簡単のため、m=1としている。 which represents a simple harmonic oscillator, can be restated as
(D^2 + k^2)y = 0.

The expression in parenthesis can be factored out, yielding

(D + ik)(D − ik)y = 0, 

which has a pair of linearly independent solutions, one for

(D − ik)y = 0 

and another for

(D + ik)y = 0.

The solutions are, respectively,

y0 = A0・e^ik・x 

and

y1 = A1・e ^(− ik)・x. 

These solutions provide a basis for the two-dimensional "solution space" of the second order differential equation: meaning that linear combinations of these solutions will also be solutions. In particular, the following solutions can be constructed

and

These last two trigonometric solutions are linearly independent, so they can serve as another basis for the solution space, yielding the following general solution:

yH = C0cos(kx) + C1sin(kx).

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Last-modified: 2012-04-26 (木) 11:41:00 (2223d)