- 1 -

Md 2x

dt2= −Kx

K/M = ω2

d 2x

dt2+ω 2x = 0

A x = Acos(ωt +φ)

B x = Bp cosωt +Bq sinωt

D x = Re De jωt⎡⎣ ⎤⎦

A, φ, Bp, Bq, D

A [m]

φ [rad]

ω [1/s]

ν =ω2π

[Hz]

KM

xM�������K���� ���x������

- 2 -

T =1

2M

dx

dt

⎛

⎝⎜

⎞

⎠⎟2

[J]

V =1

2Kx2 [J]

W = T +V =1

2M

dx

dt

⎛

⎝⎜

⎞

⎠⎟2

+1

2Kx2 [J]

A

W =1

2M −Aω sin(ωt +φ){ }

2+1

2K Acos(ωt +φ){ }

2

=1

2KA2 sin2(ωt +φ)+

1

2KA2 cos2(ωt +φ)

=1

2KA2 =

1

2Mω 2A2

2

Md 2x

dt2= −Kx − R

dx

dt

K/M = ω2 R/M = k

d 2x

dt2+ k dx

dt+ω 2x = 0

KM

x

R

M�������K���� ���R������ �����x������

- 3 -

A x = Ae−12ktcos(ω f t +φ)

B x = e−12ktBp cosω f t +Bq sinω f t( )

D x = e−12ktRe De jω f t⎡⎣

⎤⎦

ω f = ω 2 −1

4k 2 =ω 1−

k2ω

⎛

⎝⎜

⎞

⎠⎟2

ω 2 −1

4k 2 > 0 k < 2ω

ωf [1/s] ω

k [1/s]

τ = 2/k [s] 1/e

W = T +V =1

2M

dx

dt

⎛

⎝⎜

⎞

⎠⎟2

+1

2Kx2

A

W =1

2M −Ae

−12kt k2cos(ω f t +φ)+ω f sin(ω f t +φ)

⎧⎨⎩

⎫⎬⎭

⎡

⎣⎢

⎤

⎦⎥

2

+1

2K Ae

−12ktcos(ω f t +φ)

⎧⎨⎩

⎫⎬⎭

2

=1

2MA2e− kt

k 24cos(2ω f t + 2φ)+

k2ω f sin(2ω f t + 2φ)+ω

2⎧⎨⎩

⎫⎬⎭

W t

1 (Tf = 2π/ωf)

W =1

TfW (t)dt

Tf

∫ =1

2MA2ω 2e− kt =

1

2KA2e− kt

- 4 -

Q Q = ω/k = ωτ/2

Q

k<2ω

ω 2 −1

4k 2 < 0 k > 2ω

x =C1e−k2+ωh

⎛

⎝⎜

⎞

⎠⎟t

+C2e−k2−ωh

⎛

⎝⎜

⎞

⎠⎟t

ωh =1

4k 2 −ω 2 =ω

k2ω

⎛

⎝⎜

⎞

⎠⎟2

−1

C1, C2

ω 2 −1

4k 2 = 0 k = 2ω

x = C1t +C2( )e−ωt

- 5 -

Md 2x

dt2= −Kx − R

dx

dt+F cos pt

K/M = ω2 R/M = k

d 2x

dt2+ k dx

dt+ω 2x =

F

Mcos pt

A

x = Acos(pt +φ) =F

M

1

ω 2 − p2( )2+ k 2p2

cos(pt +φ)

cosφ =ω 2 − p2

ω 2 − p2( )2+ k 2p2

, sinφ =−kp

ω 2 − p2( )2+ k 2p2

D

x = Re De jpt⎡⎣ ⎤⎦= ReF

M⋅

1

ω 2 − p2 + jkpe jpt

⎡

⎣⎢

⎤

⎦⎥

KM Fcospt

x

R

M�������K���� ���R������ �����x������

F�������� �p��������������

- 6 -

A =F

M

1

ω 2 − p2( )2+ k 2p2

=F

RpH (p)

H (p) =kp

ω 2 − p2( )2+ k 2p2

= −sinφ( )

pA =F

RH (p)

p2A =Fp

RH (p)

ω p

pA

F/R

0

ω !"#

$%&'(

ω p

A

F/K

0

ω)*+,-./01 !"#

$%&'(

QF/K

ω p

p2A

F/M

0

ω)*+,2./01 !"#

F/M%&'(

QF/M

����

34��

�34��

- 7 -

Q A p2A

ω p0

−π/2

−π

φ

-π%&'(

ω p

A

F/K

0

5678.9:;6<=.>

56<=.9:;678.>

ω p0

−π/2

−π

φ

5678.9:;6<=.>

56<=.9:;678.>

- 8 -

Pw = F cos pt ×dx

dt

= F cos pt ×−F

RH (p)sin pt +φ( )

= −F 2

RH (p)

sin 2pt +φ( )− sin −φ( )2

= −F 2

2RH (p) sin 2pt +φ( )+ sin φ( ){ }

[J/s] [W]

Pw t

1 (T = 2π/p)

Pw =1

TP(t)dt

T

∫ = −F 2

2RH (p)sinφ =

F 2

2RH 2 (p)

Pw k

k

k

pω0

F2/4R

F2/2R

Pw

- 9 -

=

Z p( ) = F

jpD

=F

jpFM

⋅1

ω 2 − p2 + jkp

=M k + 1jp

ω 2 − p2( )⎧⎨⎩

⎫⎬⎭

= R+ j pM −K

p

⎛

⎝⎜

⎞

⎠⎟

ω p

|Z(p)|

R

0

0

Z(p)�?@�ABC �DE

pFGF∞

pFGF0

p = ω

p ≒ ω + k/2

p ≒ ω - k/2

RHI

J�

π/4

π/4k