method_dynamics

Ɂɚɞɚɱɚ 4. ȼɵɱɢɫɥɢɬɶ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ

ɨɫɟɣ ɤɨɨɪɞɢɧɚɬ

x, y, z ɬɨɧɤɨɣ ɨɞɧɨɪɨɞɧɨɣ ɤɪɭɝɨɜɨɣ ɩɥɚɫɬɢɧɵ ɪɚɞɢɭɫɚ r ,

ɜɧɭɬɪɢ ɤɨɬɨɪɨɣ

ɇɚɣɬɢ ɨɫɟɜɵɟ ɦɨɦɟɧɬɵ ɢɧɟɪɰɢɢ

J x , J y

ɞɥɹ

ɨɞɧɨɪɨɞɧɨɝɨ ɬɨɧɤɨɝɨ ɤɪɭɝɥɨɝɨ ɞɢɫɤɚ

ɪɚɞɢɭɫɚ

R ɢ

ɉɪɢ t 0: x x0

P

, x x0 0

,

c

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, Fx c(d x) .

(1)

mx

P Fx ,

(2)

ɢɫɩɨɥɶɡɭɹ (1), ɩɨɥɭɱɢɦ ɢɡ (2):

P

x

P cd cx .

(3)

g

Ɂɚɩɢɲɟɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ (3) ɜ ɜɢɞɟ

x k2 x

0 ,

(4)

ɝɞɟ

k

cg

– ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ ɤɨɥɟɛɚɧɢɣ (ɭɝɥɨɜɚɹ ɱɚɫɬɨɬɚ).

P

Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4)

Ο2 k2

0.

Ʉɨɪɧɢ ɭɪɚɜɧɟɧɢɹ

Ο1,2 ρki .

Ɋɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɡɚɩɢɲɟɦ ɜ ɜɢɞɟ

x

c1 cos(kt) c2 sin(kt).

(5)

Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɨɫɬɨɹɧɧɵɯ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ, ɜɵɱɢɫɥɢɦ ɫɤɨɪɨɫɬɶ

x c1k sin(kt) c2k cos(kt) .

P

(6)

ɉɨɞɫɬɚɜɢɦ(5) ɢ(6) ɜɧɚɱɚɥɶɧɵɟɭɫɥɨɜɢɹ t

0

x x0

,

x

x0

0 .

P

c

ɇɚɯɨɞɢɦ ɫ1

x0

,

c2

0.

c

ɍɪɚɜɧɟɧɢɹɞɜɢɠɟɧɢɹɝɪɭɡɚɩɪɢɦɟɬɜɢɞ x

P cos(kt)

P cos(

cgt).

c

c

P

ɉɨɞɫɬɚɜɥɹɟɦ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ k

cg

12 ɫ 1,

P

6,8 ɫɦ.

P

c

Ⱥɦɩɥɢɬɭɞɚ ɤɨɥɟɛɚɧɢɣ

a = 6,8 cɦ;

ɧɚɱɚɥɶɧɚɹ ɮɚɡɚ ɤɨɥɟɛɚɧɢɣ

Į = –ʌ/2;

ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ

k = 12 ɫ-1.

ɇɚɩɪɚɜɢɦ ɨɫɶ ɯ ɜɟɪɬɢɤɚɥɶɧɨ ɜɧɢɡ;

ɬɨɱɤɚ Ɉ – ɧɚɱɚɥɨ ɨɬɫɱɟɬɚ ɜ ɩɨɥɨɠɟɧɢɢ

ɪɚɜɧɨɜɟɫɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ ɫɭɞɧɚ. ɉɪɢ

ɷɬɨɦ ɜɵɫɨɬɚ ɩɨɞɜɨɞɧɨɣ ɱɚɫɬɢ ɫɭɞɧɚ ɪɚɜɧɚ

d.

Ʉ ɫɭɞɧɭ ɩɪɢɥɨɠɟɧɵ: Ɋ – ɜɟɫ ɜ ɰɟɧɬɪɟ

ɬɹɠɟɫɬɢ ɋ ɫɭɞɧɚ, Rɫɬ – ɧɨɪɦɚɥɶɧɚɹ

ɫɬɚɬɢɱɟɫɤɚɹ ɪɟɚɤɰɢɹ ɜɨɞɵ ɜ ɰɟɧɬɪɟ

ɬɹɠɟɫɬɢ Ʉ ɨɛɴɟɦɚ ɜɨɞɵ, ɜɵɬɟɫɧɟɧɧɨɣ

ɫɭɞɧɨɦ.

Ɇɨɞɭɥɶ Rɫɬ ɪɚɜɟɧ ɜɟɫɭ ɨɛɴɟɦɚ V

ɜɨɞɵ,

ɜɵɬɟɫɧɟɧɧɨɣ

ɫɭɞɧɨɦ,

ɬ.ɟ. Rɫɬ

ϑ V ϑ S d ,

ɫɥɟɞɨɜɚɬɟɥɶɧɨ,

ɭɫɥɨɜɢɟ

ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ ɢɦɟɟɬ

ɜɢɞ

P ϑ S d 0 .

(7)

ɇɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ

ɩɪɢ t

0:

x

0,

x v0 .

(8)

ɂɡ-ɡɚ ɧɚɥɢɱɢɹ ɧɚɱɚɥɶɧɨɣ ɫɤɨɪɨɫɬɢ v0 ɫɭɞɧɨ ɧɚɱɢɧɚɟɬ ɞɜɢɝɚɬɶɫɹ

ɜɟɪɬɢɤɚɥɶɧɨ

ɜɧɢɡ.

Ɉɛɴɟɦ

ɜɨɞɵ, ɜɵɬɟɫɧɟɧɧɨɣ

ɫɭɞɧɨɦ,

ɪɚɜɟɧ

S(d x).Ɂɧɚɱɢɬ, ɩɪɨɟɤɰɢɹ ɧɚ ɨɫɶ ɯ ɧɨɪɦɚɥɶɧɨɣ ɪɟɚɤɰɢɢ R ɪɚɜɧɚ

Rx

ϑ S (d x) .

(9)

ɋɨɫɬɚɜɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɰɟɧɬɪɚ ɬɹɠɟɫɬɢ ɋ ɜ

ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ

mx

Px Rx .

Ɍɚɤ ɤɚɤ Ɋ=Ɋɯ, ɢɫɩɨɥɶɡɭɹ ɭɪɚɜɧɟɧɢɹ (1) ɢ (3), ɩɨɥɭɱɢɦ

P

x

ϑ Sx .

g

Ɂɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ ɫɜɨɛɨɞɧɵɯ ɤɨɥɟɛɚɧɢɣ ɜ ɤɚɧɨɧɢɱɟɫɤɨɦ ɜɢɞɟ:

x k 2 x

0 ,

(10)

ɝɞɟ

gϑ S

k –

ɤɪɭɝɨɜɚɹ ɱɚɫɬɨɬɚ.

P

Ɂɚɩɢɲɟɦ ɢɫɤɨɦɨɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (4) ɜ ɜɢɞɟ

x

asin(kt

) ,

(11)

2

kx0

ɝɞɟ a

x02 x02 – ɚɦɩɥɢɬɭɞɚɤɨɥɟɛɚɧɢɣ,

arctg

– ɧɚɱɚɥɶɧɚɹɮɚɡɚ.

x0

k

ɂɫɩɨɥɶɡɭɹɧɚɱɚɥɶɧɵɟɭɫɥɨɜɢɹ(2), ɧɚɣɞɟɦɚɦɩɥɢɬɭɞɭɢɧɚɱɚɥɶɧɭɸɮɚɡɭ:

v0

v

P

a

gSϑ

,

0.

(12)

k

0

ɂɫɩɨɥɶɡɭɹ (6),(7) ɢ (5), ɨɤɨɧɱɚɬɟɥɶɧɨ ɩɨɥɭɱɢɦ

P

§

gSϑ

·

x v0

¨

t

¸

(13)

gSϑ

sin¨

P

¸.

©

¹

ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ

T

2Σ

2Σ

P .

k

gsϑ

t 0, x

x0

4 ɫɦ,

x

x0

4 ɫɦ/ c.

Fx c(d x) .

(1)

P

x P cd cx ΕΞx .

(2)

g

P cd

0.

(3)

ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ

x 2nx k2 x 0,

(4)

ɝɞɟ Ξx

x, k

cg , n

Εg

, ɩɨɥɭɱɚɟɦ k

=

10

c-1, n =

8

ɫ-1, ɬɚɤɢɦ

ɨɛɪɚɡɨɦ, n < k.

P

2P

Ɂɚɩɢɲɟɦ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (4):

Ο2 2nΟ k2

0,

Ο

n ρ n2

k2

n ρi k2 n2 .

1,2

ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ

x

e nt (c1 cos(

k2 n2 t) c2 sin(

k2 n2 t)) .

(5)

ɂɫɩɨɥɶɡɭɹ ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ, ɩɨɥɭɱɚɟɦ ɩɨɫɬɨɹɧɧɵɟ ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ

c1

x0 ,

c

x0

nx0

.

2

2 n 2

k

ɉɪɟɨɛɪɚɡɭɟɦ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ, ɨɩɪɟɞɟɥɹɹ

x0

Asin

,

x0

nx0

Acos .

(6)

Ɍɟɩɟɪɶ ɭɪɚɜɧɟɧɢɟ ɩɪɢɦɟɬ ɜɢɞ

k 2 n2

x

Ae nt sin(

k 2 n2 t

).

(7)

Ⱦɜɢɠɟɧɢɟ ɝɪɭɡɚ ɹɜɥɹɟɬɫɹ ɡɚɬɭɯɚɸɳɢɦ (ɬ.ɤ. ɩɪɢ t x

0) ɫ ɤɪɭɝɨɜɨɣ

ɱɚɫɬɨɬɨɣ

kc

k 2 n2

. ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ

ɡɧɚɱɟɧɢɹ

ɜ

ɮɨɪɦɭɥɵ,

ɧɚɯɨɞɢɦ Ⱥ=7,2 ɫɦ, Į=0,59 ɪɚɞ, kc=6 ɫ-1.

ɂɬɚɤ, ɝɪɭɡ ɫɨɜɟɪɲɚɟɬ ɡɚɬɭɯɚɸɳɢɟ ɤɨɥɟɛɚɧɢɹ ɩɨ ɡɚɤɨɧɭ

x

7,2e 8t sin(6t 0,59) ɫɦ.

(8)

ɉɟɪɢɨɞ ɤɨɥɟɛɚɧɢɣ ɪɚɜɟɧ Tc

2Σ

1,05 ɫ.

kc

ɧɚɱɚɥɶɧɵɟ

ɭɫɥɨɜɢɹ

ɞɜɢɠɟɧɢɹ

ɝɪɭɡɚ

ɢɦɟɸɬ

ɜɢɞ

ɩɪɢ t

↑

x

x0

4 ɫɦ

.

0 →

x0

240 ɫɦ/

↓ x

ɫ

ɋɥɟɞɭɹ ɪɟɲɟɧɢɸ ɩɪɟɞɵɞɭɳɟɣ ɡɚɞɚɱɢ, ɩɨɥɭɱɢɦ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ

ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ

x 2nx kx

0, ɝɞɟ Ξx

x,

k

cg ,

n Εg .

P

2P

ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɚɟɦ k=10 c-1, n=26 ɫ-1, ɬɚɤɢɦ

ɨɛɪɚɡɨɦ, n>k (ɫɥɭɱɚɣ ɛɨɥɶɲɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ).

Ɂɚɩɢɲɟɦ

ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ

ɭɪɚɜɧɟɧɢɟ

Ο2 2nΟ k2

0, ɟɝɨ

ɤɨɪɧɢ

ɪɚɜɧɵ Ο1

n

n2

k 2 , Ο2 n

k 2

n2 .

Ɍɚɤ ɤɚɤ n > k, ɬɨ ɤɨɪɧɢ Ȝ1 ɢ Ȝ2 ɹɜɥɹɸɬɫɹ ɜɟɳɟɫɬɜɟɧɧɵɦɢ ɢ

ɨɬɪɢɰɚɬɟɥɶɧɵɦɢ. ɍɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɝɪɭɡɚ ɢɦɟɟɬ ɜɢɞ

x

ɫ1eΟ1t c2eΟ2t .

(9)

c

Ο2 x0 x0

, c

2

Ο1x0 x0

.

Ο Ο

1

Ο

Ο

2

2

1

1

ɉɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (1) ɫ ɭɱɟɬɨɦ ɧɚɣɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ:

x

1

>(Ο x

0

x

0

)eΟ2t

(Ο

2

x

0

x

0

)e

Ο1t

.

(10)

Ο1 Ο2

1

ȼɨɫɩɨɥɶɡɨɜɚɜɲɢɫɶ ɡɧɚɱɟɧɢɹɦɢ Ȝ1 ɢ

Ȝ2

ɢ

ɝɢɩɟɪɛɨɥɢɱɟɫɤɢɦɢ

ɮɭɧɤɰɢɹɦɢ, ɡɚɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ (2) ɜ ɜɢɞɟ

x

e nt

>(x0

nx0 )sh

n2 k 2

x0

n2

k 2 ch

n2 k 2 t .

(11)

n2 k 2

ɩɨɥɭɱɢɦ x

1

e

26t

(29e

24t

5e

24t

)

6

1

ɢɥɢ x

e

26t

(12ch24t 17sh24t)

3

Ɂɚɞɚɱɚ ʋ 1. ɇɚ

ɪɢɫɭɧɤɟ

ɢɡɨɛɪɚɠɟɧɚ ɫɯɟɦɚ ɩɪɢɛɨɪɚ ɞɥɹ

ɢɡɦɟɪɟɧɢɹ ɞɚɜɥɟɧɢɹ. Ʉ ɩɨɥɡɭɧɭ Ⱥ

ɜɟɫɨɦ Ɋ=196 Ƚ ɩɪɢɤɪɟɩɥɟɧɚ

ɫɬɪɟɥɤɚ

ȼ,

ɨɬɦɟɱɚɸɳɚɹ

ɩɨɤɚɡɚɧɢɹ

ɧɚ

ɧɟɩɨɞɜɢɠɧɨɣ

ɲɤɚɥɟ

ɋ.

ɉɨɥɡɭɧ

Ⱥ,

ɩɪɢɤɪɟɩɥɟɧɧɵɣ ɤ ɤɨɧɰɭ ɩɪɭɠɢɧɵ

D,

ɩɟɪɟɦɟɳɚɟɬɫɹ

ɩɨ

Ɉɩɪɟɞɟɥɢɬɶ:

ɝɨɪɢɡɨɧɬɚɥɶɧɨɣ

ɢɞɟɚɥɶɧɨ

1)

ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ

ɝɥɚɞɤɨɣ

ɩɥɨɫɤɨɫɬɢ. Ʉ

ɩɨɥɡɭɧɭ

ɜ ɫɥɭɱɚɟ

ɨɬɫɭɬɫɬɜɢɹ

ɫɢɥɵ

ɩɪɢɥɨɠɟɧɚ ɝɨɪɢɡɨɧɬɚɥɶɧɚɹ ɫɢɥɚ

ɫɨɩɪɨɬɢɜɥɟɧɢɹ;

S = H·sin(pt),

ɝɞɟ ɇ = 1,6 ɤȽ,

2)

ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɹ ɫɬɪɟɥɤɢ ȼ

ɪ = 60

ɫ-1.

Ʉɨɷɮɮɢɰɢɟɧɬ

ɭɩɪɭɝɨɫɬɢ ɪɚɜɟɧ ɫ = 2

ɤȽ/ɫɦ. ȼ

ɩɪɢ

ɧɚɥɢɱɢɢ

ɫɢɥɵ

ɫɨɩɪɨɬɢɜɥɟɧɢɹ,

ɧɚɱɚɥɶɧɵɣ

ɦɨɦɟɧɬ

ɩɨɥɡɭɧ

ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ

ɩɟɪɜɨɣ

ɧɚɯɨɞɢɥɫɹ ɜ ɩɨɤɨɟ, ɜ ɩɨɥɨɠɟɧɢɢ

ɫɬɟɩɟɧɢ

ɫɤɨɪɨɫɬɢ ɩɨɥɡɭɧɚ

ɫɬɚɬɢɱɟɫɤɨɝɨ ɪɚɜɧɨɜɟɫɢɹ.

R=ȕȣ, ɝɞɟ ȕ=25,6 Ƚ ɫ/ɫɦ.

ɯ:

Hg

cg

mx

Sx Fx

ɢɥɢ

x

sin pt

x , ɨɬɤɭɞɚ

P

P

x k 2 x

k sin pt,

(1)

ɝɞɟ k

cg ,

h

hg

. ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k=100 ɫ-1, h=8000 ɫɦ/ɫ-2.

P

P

x1

c1 cos(kt) c2 sin(kt).

(2)

ɑɚɫɬɧɨɟ

ɪɟɲɟɧɢɟ

ɭɪɚɜɧɟɧɢɹ

(1)

ɩɪɢɦɟɦ

ɜ

ɜɢɞɟ

x2 Asin(pt) Bcos(pt), ɬɨɝɞɚ

h

x2

sin( pt) .

(3)

k

2

p

2

Ɂɚɩɢɲɟɦ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɭɪɚɜɧɟɧɢɹ (1), ɫɥɨɠɢɜ (2) ɢ (3):

x

c1 cos(kt) c2 sin(kt)

h

sin( pt) .

(4)

k

2

p

2

ɂɫɩɨɥɶɡɭɹ

ɧɚɱɚɥɶɧɵɟ

ɭɫɥɨɜɢɹ,

ɨɩɪɟɞɟɥɢɦ

ɩɨɫɬɨɹɧɧɵɟ

ɢɧɬɟɝɪɢɪɨɜɚɧɢɹ:

p

h

c1 0, c2

.

k

k

2

p

2

ɂɬɚɤ, ɭɪɚɜɧɟɧɢɟ ɞɜɢɠɟɧɢɟ ɫɬɪɟɥɤɢ

x

p

h

sin(kt)

h

sin( pt) .

(5)

k k

2

2

k

2

p

2

p

ɉɨɞɫɬɚɜɢɜ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ, ɩɨɥɭɱɢɦ

x ( 0,75sin(100t) 1,25sin(60t)) cɦ .

(6)

mx

Sx Fx

Rx ɢɥɢ x

Hg

sin pt

cg

x

Εg

x

, ɨɬɤɭɞɚ

P

P

P

x 2nx k 2 x

k sin pt,

(7)

ɝɞɟ k

cg ,

n Εg , h

hg . ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ k = 100 ɫ-1, h = 80 ɫɦ ɫ-2,

P

2P

P

k2 n2 76,8

a

h

0,8 Η

arctg

2np

0,87 .

(k2 p2 )2 4n2 p2

k

2

2

p

Ⱥ ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɵɛɢɪɚɹ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɜ

ɜɢɞɟ

x2

asin(pt Η),

ɩɨɥɭɱɢɦ

e 64t (c1 cos76,8t c2 sin 76,8t) 0,8sin(60t 0,87).

x

(8)

x e 64t (0,62cos76,8t 0,12sin 76,8t) 0,8sin(60t 0,87).

(9)

x >0,63e 64t sin(76,8t 1,74) 0,8sin(60t 0,87) ɫɦ.

(10)

§6. Ɋɟɡɨɧɚɧɫ

Ɂɚɞɚɱɚ ʋ

1.

Ɉɩɪɟɞɟɥɢɬɶ

ȼ ɧɚɱɚɥɶɧɵɣ

ɦɨɦɟɧɬ

ɬɨɱɤɚ

ɭɪɚɜɧɟɧɢɟ

ɞɜɢɠɟɧɢɹ

ɧɚɯɨɞɢɥɚɫɶ ɜ ɩɨɤɨɟ ɜ ɧɚɱɚɥɟ

ɦɚɬɟɪɢɚɥɶɧɨɣ

ɬɨɱɤɢ Ɇ

ɜɟɫɨɦ

ɨɬɫɱɟɬɚ

ɨɫɢ

ɯ.

ɋɢɥɨɣ

Ɋ=196 Ƚ, ɞɜɢɠɭɳɟɣɫɹ ɜɞɨɥɶ ɨɫɢ ɯ

ɫɨɩɪɨɬɢɜɥɟɧɢɹ

ɞɜɢɠɟɧɢɸ

ɩɨɞ ɞɟɣɫɬɜɢɟɦ ɫɢɥɵ ɭɩɪɭɝɨɫɬɢ F

ɩɪɟɧɟɛɪɟɱɶ.

ɢ ɜɨɡɦɭɳɚɸɳɟɣ ɫɢɥɵ S. ɉɪɨɟɤɰɢɢ

ɷɬɢɯ ɫɢɥ ɧɚ ɨɫɶ ɯ ɪɚɜɧɵ: Fx = –cx,

Sx = H·sin(pt),

ɝɞɟ

ɫ = 2

ɤȽ/ɫɦ,

ɩɪɨɟɤɰɢɢ ɧɚ ɨɫɶ ɯ mx

Fx

Sx

ɢɥɢ mx

cx H sin pt

x k 2 x

hsin pt ,

(1)

ɝɞɟ k2

c

10000 ɫ 1,

p

101ɫ 1, h

H

8000 ɫɦ/ ɫ2..

m

m

(p>k).

Ɂɧɚɱɢɬ, ɪɟɲɟɧɢɟ ɩɪɢɧɢɦɚɟɬ ɜɢɞ

x

p

h

sin kt

h

sin pt .

(2)

k k

2

p

2

k

2

p

2

p

| 1, p k | 2k | 2p.

(3)

k

ɂɫɩɨɥɶɡɭɹ

ɩɟɪɜɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (3), ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ

(2),

x |

h

(sin pt sin kt) .

(k p)(k p)

ɂɫɩɨɥɶɡɭɹ ɜɬɨɪɨɟ ɫɨɨɬɧɨɲɟɧɢɟ (3), ɩɟɪɟɩɢɲɟɦ ɭɪɚɜɧɟɧɢɟ:

h

x

(sin pt sin kt) .

(4)

2k(k p)

ɉɪɟɨɛɪɚɡɨɜɚɜ ɜɵɪɚɠɟɧɢɟ, ɩɨɥɭɱɢɦ

x a(t)cos pt ,

(5)