kc

�

�

A

0

0±A

T

fT = 1

fu

A2A

60km/h

Tf

h

f = 12π

√km

Δt

δΠH = 0

δ (t1, t2)

ΠH =ˆ t2

t1

(Ep − Ec)dt +ˆ t2

t1

Eddt

Ep Ec Ed

mu + cu + ku = p(t)

m

c

k

u

u

u

mu

cu

ku

p(t)

F (t)I + F (t)D + F (t)K = p(t)

m c k

{M}u + {C}u + {K}u = p(t)

u u u

t

p(x, t)

m∂2u

∂t2+

∂2

∂x2(EI

∂2u

∂x2) = p(x, t)

u

∂2

∂x2(EI

∂2v

∂x2+ c I

∂3v

∂t∂x2) + m

∂2v

∂t2+ c

∂v

∂t= p(x, t)

u(y, t)ψi(y) βi(t)

u(y, t) =∞∑i=1

ψi(y)βi(t)

ψi(y)

βi(t)

F (t)I + F (t)D + F (t)K = {p(t)}

{u}{a}i Yi(t)

{u} = [a] {Y }i

[a]{u}

M {u} + C {u} + K {u} = {p(t)}

Yi + 2DiωiYi + ω2i Yi =

Pi(t)Mi

Pi(t) = {a}Ti {p(t)}

Mi {a}Ti M {a}i

ωi

Di

{u} = {a}1 Y1 + {a}2 Y2 + ... + {a}n Yn

Δt

Δt Δt

Δt

T

Δt

ΔtT ≤ 1

10Δt

x(t)p(t)

H(Ω)P (Ω) H(Ω)P (Ω)

X(Ω)x1(t) p1(t)

x2(t) p2(t)

X1(Ω)P1(Ω)

=X2(Ω)P2(Ω)

= .........Xn(Ω)n(Ω)

= H(Ω)

X1(Ω) P1(Ω)

H(Ω)p(t)

g(t) Tp

g(t) = g(t + nTp)

Tp

g(t) = a0 +∞∑

n=1

ancos(2πnf1t) +∞∑

n=1

bnsin2π(nf1t)

f1 = 1Tp

an = 2Tp

´ Tp2

−Tp2

g(t)cos(nΩ1t)dt

bn = 2Tp

´ Tp2

−Tp2

g(t)sin(nΩ1t)dt

a0 = 1Tp

´ Tp2

−Tp2

g(t)dt

Ω1 = 2πf1

g(t) = a0 +∞∑

n=1

ancos(nΩ1t) +∞∑

n=1

bnsin(nΩ1t)

g(t)

g(t) (−Tp

2 ,Tp

2 )

g(t)g(α+0) g(α−0) α

t = 0

p0

u(t) =p0

k

(p0sinΩ1t)

u(t) =p0

k

11 − β2

sin(Ω1t)

(p0cosΩ1t)

u(t) =p0

k

11 − β2

cos(Ω1t)

β = Ω1ω

ω :Ω1 :

(p0sinΩ1t)

u(t) =p0

k

1(1 − β2)2 + (2ξβ)2

{(1 − β2)sin(Ω1t) − 2ξβcos(Ω1t)

}(p0cosΩ1t)

u(t) =p0

k

1(1 − β2)2 + (2ξβ)2

{2ξβsin(Ω1t) + (1 − β2)cos(Ω1t)

}

g(t) = a0 +∞∑

n=1

ancos(nΩ1t) +∞∑

n=1

bnsin(nΩ1t)

u(t) =a0

k+

∞∑n=1

an

k

11 − β2

n

cos(nΩ1t) +∞∑

n=1

bn

k

11 − β2

n

sin(nΩ1t)

βn = nΩ1ω

u(t) =a0

k+

∞∑n=1

an

k

1(1 − β2

n)2 + (2ξβn)2{2ξβnsin(nΩ1t) + (1 − β2

n)cos(nΩ1t)}

+∞∑

n=1

bn

k

1(1 − β2

n)2 + (2ξβn)2{(1 − β2

n)sin(nΩ1t) − 2ξβncos(nΩ1t}

=a0

k+

∞∑n=1

1k

1(1 − β2

n)2 + (2ξβn)2[{an2ξβn + bn(1 − β2

n)}

sin(nΩ1t)+

{an(1 − βn)2 − bn2ξβn

}cos(nΩ1t)]

ξξ = c

cc

sin(nΩ1t) =einΩ1t − e−inΩ1t

2i

cos(nΩ1t) =einΩ1t + e−inΩ1t

2

g(t) =∞∑

n=−∞cneinΩ1t

p = eiΩ1t

mu + cu + ku = eiΩ1t

u = e−ξωt(Acos(ωddt) + Bsin(ωdt))

ωd = ω√

1 − ξ2 ω

u = HeiΩ1t

u u u

(−mΩ21 + icΩ1 + k)HeiΩ1t = eiΩ1t

u = e−ξωt(Acos(ωdt) + Bsin(ωdt)) +1

−mΩ21 + icΩ1 + k

eiΩ1t

eiΩ1t

H(Ω1)eiΩ1t

H(Ω1) =1

−mΩ21 + icΩ1 + k

=1

k(−β2 + 2iξβ + 1)

β = Ω1ωH(Ω1)

g(t) =∞∑

n=−∞cneinΩ1t

u(t) =∞∑−∞

cnH(nΩ1)einΩ1t

cn =1Tp

ˆ Tp2

−Tp2

g(t)e−inΩ1tdt

g(t) =∞∑−∞

cneinΩ1t

cn =1Tp

ˆ Tp2

−Tp2

g(t)e−inΩ1tdt

Tp

Ω1 = 2πTp

= ΔΩ nΩ1 = Ωn = nΔΩ

cn

g(t) =∞∑−∞

cneinΩ1t =1Tp

∞∑−∞

(cnTp)eiΩnt =12π

∞∑−∞

(cnTp)eiΩntΔΩ

cnTp =ˆ Tp

2

−Tp2

g(t)e−inΩ1tdt =ˆ Tp

2

−Tp2

g(t)e−iΩntdt

Tp ΔΩ nΩ1 = Ωn = nΔΩ Ω.

cnTp = G(Ω) =ˆ ∞

−∞g(t)e−iΩtdt

g(t) =12π

ˆ ∞

−∞G(Ω)eiΩtdΩ

G(Ω) g(t)G(Ω)

∞∑−∞

cneiΩnt

u(t) =1Tp

∞∑−∞

(cnTp)H(Ωn)eiΩnt =12π

∞∑−∞

(cnTp)H(Ωn)eiΩnt�Ω

Tp

u(t) =12π

ˆ ∞

−∞G(Ω)H(Ω)eiΩtdΩ

δ(t)

δ(t) = 0 t �= 0

δ(t) t = 0

´∞−∞ δ(t)dt = 1

h(t) =12π

ˆ ∞

−∞G(Ω)H(Ω)eiΩtdΩ

´∞−∞ f(z)δ(z − t0)dt = f(t0)

G(Ω) =ˆ ∞

−∞δ(t)e−iΩtdt = 1

G(Ω) h(t)

h(t) ⇔ H(Ω)

U(Ω) u(t)

u(t) =12π

ˆ ∞

−∞U(Ω)eiΩtdΩ

U(Ω) = G(Ω)H(Ω)

g(t) h(t) g(t) ∗ h(t)

u(t) = g(t) ∗ h(t) =ˆ ∞

−∞g(τ)h(t − τ)dτ

h(τ)

g(τ)

g(τ)

U(Ω) =ˆ ∞

−∞[ˆ ∞

−∞g(τ)h(t − τ)dτ ]e−iΩtdt =

ˆ ∞

−∞[ˆ ∞

−∞h(t − τ)e−iΩtdt]g(τ)dτ

t − τ = y

U(Ω) =ˆ ∞

−∞[ˆ ∞

−∞h(y)e−iΩtdy]e−iΩτg(τ)dτ =

ˆ ∞

−∞H(Ω)e−iΩτg(τ)dτ = H(Ω)G(Ω)

U(Ω) u(t)

u(t) =12π

ˆ ∞

−∞G(Ω)H(Ω)eiΩtdΩ

Tp

G(Ω) =ˆ ∞

−∞g(t)e−iΩtdt

Ω1 = ΔΩ =2π

Tp

Tp Δt,tm = mΔt.

g(t) =ΔΩ2π

∞∑−∞

G(Ω)eiΩnt

Ωn = nΩ1

eiΩnt = einΔΩmΔt = ein 2π

TpmΔt = ein 2π

NΔtmΔt = e2πin m

N

g(tm) =ΔΩ2π

N−1∑n=0

G(Ω)e2πin mN

(N − 1)ΔΩg(t), G(Ω)

G(Ω) = Tpcn =ˆ Tp

2

−Tp2

g(t)e−iΩntdt

G(nΔΩ) = Δt

N−1∑m=0

g(mΔt)e−2πin mN

ΔΩ

Tp = NΔt

u(nΔt) =N−1∑n=0

g(nΔt)h {(m − n)Δt}Δt

g(t) h(t) Δt

g(mΔt)h(mΔt)

G(nΔΩ) =N−1∑m=0

g(mΔt)e−2πim nN

h(t)h(t)

”T0”

G(nΔΩ) H(nΔΩ)

U(nΔΩ) = G(nΔΩ)H(nΔΩ)

u(mΔt) =12π

N−1∑m=0

U(nΔΩ)e2πim nN ΔΩ

h(t)H(Ω)

H(Ω)

Tp = 1,6

Tp

mΔt m Δt = 0,1

G(nΔΩ) =N−1∑m=0

g(mΔt)e−imΔtnΔΩΔt =N−1∑m=0

g(mΔt)e−2πim nN Δt

nΔΩnΔf Δf

Δf =ΔΩ2π

=1Tp

Tp = 2πΔΩ = 1

Δf = 1, 6

ΔΩ = 1,25π Δf = 0,625

n = N2

n = N2 .

kΔΩ

G(kΔΩ) =N−1∑m=0

g(mΔt)e−2πim kN Δt

kΔΩ N ΔΩ2 lΔΩ l = N

2 + (N2 − k) = N − k

G {(N − k)ΔΩ} =N−1∑m=0

g(mΔt)e−2πim(N−k)

N Δt =

=N−1∑m=0

g(mΔt)e2πim kN Δt

n = −k

G {(−k)ΔΩ} =N−1∑m=0

g(mΔt)e2πim kN Δt

N ΔΩ2

N ΔΩ2

N ΔΩ2 = π

Δt Δt

G(Ω) =ˆ t1

0e−iΩtdt

G(Ω) =ˆ t1

0cos(Ωt)dt − i

ˆ t1

0sin(Ωt)dt =

sin(Ωt1)Ω

+ icos(Ωt1) − 1

Ω

NΔΩN ΔΩ

2

N ΔΩ2

Δt

πΔt

Δt

ΔtTp

H(Ω)

H(Ω)

h(t)

G(Ω) =ˆ t1

0e−iΩtdt =

sin(Ωt1)Ω

+ icos(Ωt1) − 1

Ω

ΔΩ

NΔΩN ΔΩ

2

N2

N2

g(mΔt) =12π

N−1∑n=0

G(nΔΩ)e2πim nN ΔΩ

N = 16 ΔΩ = 1,25π.

N ΔΩ2

N

H(Ω) =1

m(ω2 − Ω2)

ωh(t)

H(Ω)

ΔΩ = 0,15625Tp = 64 × 0,15625 = 10

H(nΔΩ)

H(nΔΩ) =1

m(ω2 − Ω2)

H(Ω).

g(mΔt) =12π

N−1∑n=0

G(nΔΩ)e2πim nN ΔΩ

h(mΔt) =12π

N−1∑n=0

H(nΔΩ)e2πim nN ΔΩ

H(Ω)h(t)

h(t) =1

mΩsin(Ωt) t > 00 t < 0

h(t)

h

H(Ω) h(t)

h(t)”ξ”

h(t)

ΔΩ

Tp = 2πΔΩ

h(t) 2πrΔΩ r

−∞ ∞

h(t) =1

mωde−ξω(t− 2π

ΔΩm)sinωd(t− 2π

ΔΩm)

m

h(t)

h(t)

h(t) =1

mωd

0∑r=−∞

e−ξω(t−rTp)sin(ωd(t − rTp))

sinθ = 12(eiθ − e−iθ)

h(t) =1

mωde−ξωt

0∑r=−∞

eξωrTpeiωd(t−rTp) − e−iωd(t−rTp)

2i=

=1

mωd

e−ξωt

2i

{eiωdt

0∑r=−∞

e−rTp(−ξω+iωd) − e−iωdt0∑

r=−∞e−rTp(−ξω−iωd)

}

−r r

h(t) =1

mωd

e−ξωt

2i

{eiωdt

∞∑r=0

erTp(−ξω+iωd) − e−iωdt∞∑

r=0

erTp(−ξω−iωd)

}

h(t) =1

mωd

e−ξωt

2i

{eiωdt

1 − eTp(−ξω+iωd)− e−iωdt

1 − eTp(−ξω−iωd)

}

h(t) =e−ξωt

mωd

{sin(ωdt) − e−ξωTpsin(ωd(t − Tp)1 − 2e−ξωTpcos(ωdTp) + e−2ξωTp

}

h(t) h(t)Tp Tp h(t)

h(t)

Tp

ξ = 0

h(t) =1

2mω(sin(ωt) +

cos(ωt)sin(ωTp)1 − cos(ωTp)

h(t)

h(t) h(t) Tp

h(t)

g(t) Tp

h

Th,

h

g h

Tp = pΔt Th = qΔt T0 T0 = NΔt = (p + q + 1)Δt

Th

Th

Th

h

Th

Tp +Th

g tp

g

tpTh g

g

′t′p

th

g Tp

Tp

T0 = Tp + T2

T0.

h(t),0 < t < T0

t = 0,−T0,−2T0... h(t)

h(t) =e−ξωt

mωd

{sin(ωdt) − e−ξωT0sin(ωd(t − T0))1 − 2e−ξωT0cos(ωdT0) + e−2ξωT0

}

u(kΔt)g(t)

u(mΔt) =N−1∑j=0

g(jΔt)h {(m − j)Δt}Δt

u(mΔt) =12π

N−1∑n=0

G(nΔΩ)H(nΔΩ)e2πim nN ΔΩ

G(nΔΩ) g(mΔt) H(nΔΩ)h(mΔt)

u(mΔt) u(mΔt)

t = 0 u(mΔt) u(mΔt)0 < t < T0

T0

u(mΔt)

u(mΔt)u(mΔt)

t = 0 u(0) u(0)

u(mΔt) u(0)t = 0 u(0)

u(0) u(0)

t = 0 u(0)

u(mΔt) =12π

N−1∑n=0

G(nΔΩ)H(nΔΩ)e2πim nN ΔΩ

H(nΔΩ) h(t)

h(t) =1

mωd

e−ξωt

2i

{eiωdt

1 − eT0(−ξω+iωd)− e−iωdt

1 − eT0(−ξω−iωd)

}

U(Ω).

U(Ω).

U(nΔΩ) = G(nΔΩ)H(nΔΩ)

U(nΔΩ)−N

2N2 ,

u(mΔt) =1T0

N2∑

n=−N2

U(nΔΩ)e2πin tT0

t = mΔt

T0 = NΔt

ΔΩ = 2πT0

t = 0 t = 0

u(0) =2πi

T 20

N2∑

n=−N2

nU(nΔΩ) =2πi

T 20

N2∑

n=−N2

nRe {U(nΔΩ)} + inIm {U(nΔΩ)}

Re {U(nΔΩ)} ⇒ U(nΔΩ)Im {U(nΔΩ)} ⇒ U(nΔΩ)

Re {U(nΔΩ)} nRe {U(nΔΩ)}

Im {U(nΔΩ)} nIm {U(nΔΩ)}

u(0) =−4π

T 20

N2∑

n=0

nIm {U(nΔΩ)}

Δu(0) = u(0) − u(0)

Δu(0) = u(0) − u(0)

u(kΔt)Δu(0) Δu(0)

t = 0

r(t) = e−ξωt(cos(ωdt) +ξω

ωdsin(ωdt))

t = 0

s(t) =e−ξωt

ωdsin(ωdt)

η1(t) = Δu(0)r(t)

η2(t) = Δu(0)s(t)

u(mΔt) = u(mΔt) + η1(mΔt) + η2(mΔt)

δ(t) N − 2u1 δ(t) N − 1

u2

δ(t) u1 h (N−2)Δtu2 h (N − 1)Δt

R1 R2

u(mΔt) = u(mΔt) + R1u1(mΔt) + R2u2(mΔt)

R1u1(0) + R2u2(0) = u(0) − u(0)R1 ˙u1(0) + R2 ˙u2(0) = u(0) − ˙u(0)

u(0) u(0)

u(0)

u(mΔt) =12π

N−1∑n=0

G(nΔΩ)H(nΔΩ)e2πim nN ΔΩ

˙u(0)

˙u(0) = − 4π

(T0)2

N2∑

n=0

nIm {U(nΔΩ)}

˙u1(0) ˙u2(0) h(t)

˙h(t) =1

mωdΔe−ξωt

{ωdcosωdt − ωde

−ξωT0cosωd(t − T0)}− ξωh(t)

Δ = 1 − 2e−ξωT0cosωdT0 + e−2ξωT0

˙u1(t) ˙h(t) (N − 2)Δt

˙u2(t) ˙h(t) (N−1)Δt

u1(0) = h(2Δt) ˙u1(0) = ˙h(2Δt)u2(0) = h(Δt) ˙u2(0) = ˙h(Δt)

R1 R2

u(nΔt) =N−1∑n=0

g(nΔt)h {(m − n)Δt}Δt

G(nΔΩ) =N−1∑m=0

g(mΔt)e−2πin mN Δt

G(nΔΩ) g(t).

H(nΔΩ) =N−1∑k=0

h(kΔt)e−2Πin kN Δt

H(nΔΩ)h(t).

U(nΔΩ) = G(nΔΩ)H(nΔΩ)

U(nΔΩ)

u(mΔt) =12π

N−1∑n=0

U(nΔΩ)e2πin mN

u(mΔt)

N2

N2

N2

N = 2γ γ

X(n) =N−1∑m=0

x(m)e−2πin mN

Δt.γ = 3.

W = e−2π iN

n m

n = 4n2 + 2n1 + n0

m = 4m2 + 2m1 + m0

n0 n1 n2 m0 m1 m2

X(n2, n1, n0) =1∑

m0=0

1∑m1=0

1∑m2=0

x(m2, m1, m0)Wnm

Wnm

Wnm = W (4n2+2n1+n0)(4m2+2m1+m0) = W (4n2+2n1+n0)4m2W (4n2+2n1+n0)2m1W (4n2+2n1+n0)m0

WmN = e−2πim NN = cos(2mπ) − isin(2mπ) = 1

W 16n2m2 = W 8n1m2 = W 8n2m1 = 1

Wnm = W 4n0m2W (2n1+n0)2m1W (4n2+2n1+n0)m0

X(n2, n1, n0) =1∑

m0=0

1∑m1=0

{1∑

m2=0

x(m2, m1, m0)W 4n0m2

}W (2n1+n0)2m1W (4n2+2n1+n0)m0

x1(n0, m1, m0) =1∑

m2=0

x(m2, m1, m0)W 4n0m2

m1 m0 n0

x1(n0, m1, m0) = x(0, m1, m0)W 0 + x(1, m1, m0)W 4n0

x1 xn0 = 1 m1 = 0 m0 = 1

x1(1, 0, 1) = x1(5) =

x(0, 0, 1)W 4×1×0 + x(1, 0, 1)W 4×1×1 = x(1) + x(5)

x x1

x(1) x(5) W 4

x(5) x1(5)4n0m2 m2 = 0 4n0 m2 = 1

W 4n0 , n0 = 0,W 0

x x1

(N − 1)x1

x1(0) x1(4)

x1(0) = x(0) + x(4)W 0

x1(4) = x(0) + x(4)W 4

x1

m1 m0 n0 = m2 = 0n0 = m2 = 1. N

2 = 4W s W s+(N

2)

W s+(N2

) = e(2π iN

)(N2

+s) = eiπe2iπ sN = −W s

x1(j) = x(j) + x(N2 + j)W s

x1(N2 + j) = x(j) − x(N

2 + j)W s

x x1N2

x2(n0, n1, m0) =1∑

m1=0

x1(n0, m1, m0)W (2n1+n0)2m1

n0 n1 m0 x2(n0, n1, m0)x2 x1

W (2n1+n0)2m1

x1 x2N2 N − 1

x3(n0, n1, n2) =1∑

m0=0

x2(n0, n1, m0)W (4n2+2n1+n0)m0

X x3

X(n2, n1, n0) = x3(n0, n1, n2)

X

X x3

x1 x2 x3

γ = log2NN2 (N

2 )log2N.

m = 0,25

t1 = 0,6

ξ = 0,06.

π2

ω = 2π

p = sin( 21,2 t)

ωd = ω√

1 − ξ2

mu + cu + ku = p(t)

mu + cu + ku = 0

u = e−ωξt[C1sinωdt + C2cosωdt] +p0

k

(1 − (Ωω )2)sinΩt − 2ξ Ω

ω cosΩt

[1 − (Ωω )2]2 + 4ξ2(Ω

ω )2

C1 C2

−0,2423 0,097

u = e−0,377t − 0,2423sin(6,2718t) + e−0,377t0,097cos(6,2718t) + 0,299sin(5,235t) − 0,097cos(5,235t).

u = e−0,377t(−0,040sin(6,2718t) + 0,1475cos(6,2718t).

t

2,9s0,1s

t u t u

2,9s

¯h(t) g(t)H(Ω)

G(Ω)U(Ω) u(t)

t u u t u u

2,9h(t)

h(t)

5s

h(t) 10s

30s

15s

h(t)

u(mΔt) = u(mΔt) + η1(mΔt) + η2(mΔt)

η1 μ2

2,9

t u u u t u u u

{u}φn(t)

{u} =M∑

n=1

φnyn

[φ]M n M

n

yn + 2ξnωnyn + ω2nyn = pn

y0n = φTnMu0

y0n = φTnMv0

n

M ¨{u} + C{u} + K{u} = fg(t)

f

g(t)

H(Ω)eiΩt

{u} = HeiΩt

(−Ω2M + iΩC + K)H = f

Ω HHR + iHI 2N

N

Ω H 2N

Hj j

(u)G × Hj, G

g(t)

uj(mΔt) =12π

L−1∑l=0

Hj(lΔΩ)G(lΔΩ)e2πim lL ΔΩ

LΔtΔt

uj G(lΔΩ)uj

uj uj

Hj Hj

Hj Hj

Hj(Ω) =∞∑

s=−∞Hj(Ω + s

2π

Δt)

s 2πΔt Hj(Ω)

H(Ω)

πΔt

Hj(Ω) Ω � πΔt Hj(Ω) � Hj

Hj(Ω)

uj(mΔt) =12π

L−1∑l=0

Hj(lΔΩ)G(lΔΩ)e2πim lL ΔΩ

Rp tpT0

Rp

up(tp) = Rphj(tp)

hj Hj

hj =12π

L−1∑l=0

Hj(lΔΩ)ΔΩ

uj(0) = uj(0) +2N∑p=1

Rphj(tp)

2N∑p=1

Rphj(tp) = uj(0) − uj(0)

2N∑p=1

Rp˙hj(tp) = uj(0) − ˙uj(0)

uj(mΔt) = uj(mΔt) +2N∑p=1

Rphj(tp + mΔt)

hj(t)˙hj(t)

hj Hj(lΔΩ)˙hj(t)

p

˙hj(tp) =hj(tp + Δt) − hj(tp − Δt)

2Δt

h(t)

(p1

p2

)=

(1,000,75

)p

m2 = 2 m1 = 3

k1 = 100 k2 = 150

t1 = 1,00

ξ = 0,00.

p = 20

Δt = 0,05

t

M {u} + K {u} = {p(t)}

i

yi + ω2i yi = pi

K =(

250 −100−100 100

)M =

(3 00 2

)

|K − ω2i M | = 0

ω1 = 4,75rad/s ω2 = 10,52rad/s

(K − ωiM) {ai} = 0

a1 =(

1,001,82

)a2 =

(1,00−0,82

)

y1 + ω21y1 = p1

y2 + ω22y2 = p2

ω1 = 4,75 ω2 = 10,52

hi(t) =1

2Miωi(sinωit +

cosωitsinωiT0

1 − cosωiT0)

Mi i Mi = {ai}T M {ai}

t y y y t y y y

T0 = 3s.

(−Ω2M + K)H = f

f

f =(

10,75

)M K f

H

[250 − 3Ω2 −100

−100 100 − 2Ω2

]H =

(1

0,75

)

H(Ω)H(Ω) Ω

Δt = 0,05s.

ΔΩ = 2πT0

= 2,09N2 ΔΩ = 60

2 2,09 = 62,70rad/s

H(Ω)

H(Ω) H1(Ω) H2(Ω)

H1 (Ω) � H1(Ω) H2 (Ω) � H2(Ω)

H1(Ω) H2(Ω)g(kΔt)

G(kΔΩ)

{U(kΔΩ)} = {G(kΔΩ)}(

H1(kΔΩ)H2(kΔΩ)

)

R1h1(0,40) + R2h1(0,30) + R3h1(0,40) + R4h1(0,40) = u1(0)

R1h2(0,40) + R2h2(0,30) + R3h2(0,40) + R4h2(0,40) = u2(0)

R1˙h1(0.40) + R2

˙h1(0.30) + R3˙h1(0.20) + R4

˙h1(0.40) = ˙u1(0)

R1˙h2(0.40) + R2

˙h2(0.30) + R3˙h2(0.20) + R4

˙h2(0.40) = ˙u2(0)

u1(0) = −0,049 u2(0) = 0,084 ˙u1(0) = 2,8228 ˙u2(0) = 0,463.

R1 R2 R3 R4

R1 = 458,952 R2 = −972,150 R3 = 994,566 R4 = −444,381

uj(mΔt) = uj(mΔt) +2N∑p=1

Rphj(tp + mΔt)

fn1 =π

2

√EaI

mL4

Ea

I

m

L

Rx(τ)

Rx(τ)

DEP (Ω) =12π

ˆ +∞

−∞Rx(τ)e−iΩτdτ

Rx(τ) =ˆ +∞

−∞DEP (Ω)eiΩτdΩ

m

αβ α β

DEP =1

2πΔfA(Ω)A∗(Ω)

ΔfA(Ω) a(t)

A∗(Ω) A(Ω)

Ea = 210000 Nmm2

Ixx = 450,10−8m4

m = 20,4kgm

±30g.

N = 26322400Hz

2400

1200Hz.

DEP =1

2πΔfA(Ω)A∗(Ω)

357,4Hz

fn1 =π

2

√EaI

mL4

fn1 =π

2

√2,1E11 × 450E−8

20,4 × 0,904= 417,39Hz

R = ku R u

ξ =δ√

4π2 + δ2

δ

(ti, ai),

yi = L(ai).

τi = tif f

(τi, yi)

y = aτ + b

δ = −a

δ.

δ = 0,070ξ = 0,070√

4π2+0,0702= 0,070

6,283 = 0,011 ξ = 1,1 %