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開始行:
*振動:Oscillation [#j3339bb4]
Oscillation is the repetitive variation, typically in tim...
**単振動子:Simple harmonic oscillator [#m9484ffb]
The simplest mechanical oscillating system is a mass atta...
The specific dynamics of this spring-mass system are desc...
An undamped spring–mass system is an oscillatory system.
#ref(http://upload.wikimedia.org/wikipedia/commons/9/9d/S...
*フックの法則 [#qbda2aa0]
In physics, simple harmonic motion (SHM) is the motion of...
A body in simple harmonic motion experiences a single for...
Mathematically, Hooke's law states that
F=-K・x
where
x is the displacement of the end of the spring from its e...
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...
-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...
The motion is periodic: the body oscillates about an equi...
--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...
*運動:Dynamics of simple harmonic motion [#s80b20e7]
For oscillation in a single dimension, combining Newton's...
F = m d2x/dt2 = −kx
where m is the mass of the body, x is its displacement fr...
The solutions to this differential equation are sinusoida...
#ref(http://upload.wikimedia.org/math/1/3/e/13ec1e5adcd1e...
where A, ω, and φ are constants, and the equilibrium posi...
*加速度と周期 [#q543ffde]
Using the techniques of differential calculus, the veloci...
#ref(http://upload.wikimedia.org/math/6/e/6/6e6331e9fb740...
#ref(http://upload.wikimedia.org/math/9/f/0/9f04d3219c799...
Position, velocity and acceleration of a SHM as phasorsAc...
Acceleration can also be expressed as a function of displ...
a・x = - ω^2・x
Now since ma = −mω^2x = −kx,
ω^2 = k/m.
Then since ω = 2πf,
#ref(http://upload.wikimedia.org/math/0/0/9/009260845b0c2...
and since T = 1/f where T is the time period,
#ref(http://upload.wikimedia.org/math/f/e/e/fee6809cefc85...
These equations demonstrate that period and frequency are...
*位置、速度、位相 [#x711ee9d]
Simple harmonic motion shown both in real space and phase...
#ref(http://upload.wikimedia.org/wikipedia/commons/thumb/...
*例題1.固有振動数を求めよ [#l110a658]
-バネ定数k=9.807kN/m 、重量m=5kgの時、1秒...
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...
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...
平衡位置ー自然長=バネの伸びである。kX=mgより自然長x...
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...
*線形微分方程式:linear differential equation [#t42bee07]
In mathematics, a linear differential equation is of the ...
Dy(t)= f(t)
where the differential operator D is a linear operator, 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 res...
(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 "...
#ref(http://upload.wikimedia.org/math/6/7/2/672a3eb17da69...
and
#ref(http://upload.wikimedia.org/math/c/9/e/c9e0ce7ee8bfc...
These last two trigonometric solutions are linearly indep...
yH = C0cos(kx) + C1sin(kx).
終了行:
*振動:Oscillation [#j3339bb4]
Oscillation is the repetitive variation, typically in tim...
**単振動子:Simple harmonic oscillator [#m9484ffb]
The simplest mechanical oscillating system is a mass atta...
The specific dynamics of this spring-mass system are desc...
An undamped spring–mass system is an oscillatory system.
#ref(http://upload.wikimedia.org/wikipedia/commons/9/9d/S...
*フックの法則 [#qbda2aa0]
In physics, simple harmonic motion (SHM) is the motion of...
A body in simple harmonic motion experiences a single for...
Mathematically, Hooke's law states that
F=-K・x
where
x is the displacement of the end of the spring from its e...
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...
-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...
The motion is periodic: the body oscillates about an equi...
--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...
*運動:Dynamics of simple harmonic motion [#s80b20e7]
For oscillation in a single dimension, combining Newton's...
F = m d2x/dt2 = −kx
where m is the mass of the body, x is its displacement fr...
The solutions to this differential equation are sinusoida...
#ref(http://upload.wikimedia.org/math/1/3/e/13ec1e5adcd1e...
where A, ω, and φ are constants, and the equilibrium posi...
*加速度と周期 [#q543ffde]
Using the techniques of differential calculus, the veloci...
#ref(http://upload.wikimedia.org/math/6/e/6/6e6331e9fb740...
#ref(http://upload.wikimedia.org/math/9/f/0/9f04d3219c799...
Position, velocity and acceleration of a SHM as phasorsAc...
Acceleration can also be expressed as a function of displ...
a・x = - ω^2・x
Now since ma = −mω^2x = −kx,
ω^2 = k/m.
Then since ω = 2πf,
#ref(http://upload.wikimedia.org/math/0/0/9/009260845b0c2...
and since T = 1/f where T is the time period,
#ref(http://upload.wikimedia.org/math/f/e/e/fee6809cefc85...
These equations demonstrate that period and frequency are...
*位置、速度、位相 [#x711ee9d]
Simple harmonic motion shown both in real space and phase...
#ref(http://upload.wikimedia.org/wikipedia/commons/thumb/...
*例題1.固有振動数を求めよ [#l110a658]
-バネ定数k=9.807kN/m 、重量m=5kgの時、1秒...
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...
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...
平衡位置ー自然長=バネの伸びである。kX=mgより自然長x...
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...
*線形微分方程式:linear differential equation [#t42bee07]
In mathematics, a linear differential equation is of the ...
Dy(t)= f(t)
where the differential operator D is a linear operator, 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 res...
(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 "...
#ref(http://upload.wikimedia.org/math/6/7/2/672a3eb17da69...
and
#ref(http://upload.wikimedia.org/math/c/9/e/c9e0ce7ee8bfc...
These last two trigonometric solutions are linearly indep...
yH = C0cos(kx) + C1sin(kx).
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