振動工学に基礎

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1 Ky Words. ω. ω.3 osω snω.4 ω snω ω osω.5

2 .6 ω osω snω.7 ω ω ( sn( ω φ.7 ( ω os( ω φ.8 ω ( ω sn( ω φ.9 ω anφ / ω ω φ ω T ω T s π T π. ω Hz ω. T π π rad/s π ω π T. T ω φ

3 T. T.5 3. ω hω 3. hω ω 3.3 λ λ λ λ λ λ ( hωλ ω λ 3.7 λ λ h ωλ ω 3.8 λ λ λ hω ± ω h 3.9 3

4 3. λ λ h - h λ λ h -< h< λ hω ω h 3. λ hω ω h a b λ λ 3. a hω 3.3 b ω h 3.4 λ a b 3.5 λ a b 3.6 ( a b ( a b b b ( 3.7 a os sn 3.8 os sn a {( osb ( sn b} ( 3 osb 4 sn b {( b a b ( b a sn b} a a 4 3 os b 4 ( a sn b 3.3 b a a ( osb sn b b ( a / b a sn( b φ 3.4 a ( ( a / b { bos( b φ asn( b φ } 4

5 a b b ( b φ a sn 3.5 a hω b ω h b an φ a / h 3.6 h h h 3.8 b a b a T. T.5 5

6 snω ω 4. snω 4. ω snω 4.3 ω snω 4.4 hω ω snω 4.5 ω h ω 4.6 s ( ( ( 4.7 s h ( ( os h sn h ω ω ω 4.8 ( osω snω ( ω snω ω osω ( ω osω ω ω 4.9 s 4. s 4. s sn { hωω } osω { hωω } snω snω h ( ω hωω 4. ωω 4.3 ω 4.4 6

7 hωω ( hωω ( hωω ( { hωω osω ( ω ω snω } 4.7 s ( ω ω ( hωω ( ω sn s ( hωω ( ω φ ( ω φ sn 4.8 ω 4.9 ω ω hωω ( ( hω / ω anφ ω / ω ω snω ( s ( ( ( a ( os b sn b sn( ω φ ( a {( b a osb ( b a sn b} ω ( ω φ os ω h a hω b ω h ω ω ( hωω aω an φ 4.4 ω ω aω ω ω ( aω ω b ( ( ω ( aω ( a ω ω ( aω 3 ( aω a a os b b ( a ω ω sn b sn( ω φ {( a ω ω os b ab sn b} ω os( ω φ ω a

8 ( ( aω { ab os b ( a ω ω sn b} ω sn( ω φ ω b a ω h a hω b ω h ω ω ( hωω aω an φ 4.3 ω ω s 8

9 T T. T s.. -. s

10 ω snω ω 5.3 ω ω 5.4 ω ω ω ω 5.5 ω h ω h ω 5.6 s s ω ( osω sn ω ω ( ω ω ( ω 5.7 s 5.8 s 5.9 s R I R I θ ( osθ snθ R I an θ I R ω hωω ω ω ω hωω ω θ ( hωω hωω an ω ω

11 anθ anφ 5.4 θ 5.5 ( ω θ ω s ( hωω ( ω θ ( ω θ ( ω θ os sn 5.6 osω sn ω / / θ π/ ω ω ω hωω ω θ ω ( ω ω ( hωω ω ( hωω θ 5.7 ω / ω ( ω / ω ( hω / ω θ 5.8 hω / ω anθ ω / ω ω π / ( / ( h / θ 5.9

12 h anθ / / / T T / T ( T / T ( ht / T θ 5. ht / T anθ T / T. h. h. 8 h.5.8 h.5 / 6 4 h. h. h φ.6.4 h. φ/π h h. /. 3 ω /ω ω /ω hω ω 5. hω ω 5. ω ( hωω ω ( hωω ω ω ( hωω ω ω ω ω hωω hωω ω 5.3 ω ω hωω hωω ω ω ω hωω h ( ω / ω anθ ω ( hω / ω ( ω / ω ( hω / ω / ω 3 ( 4h θ 5.4

13 ω π h ( / anθ ( h / ( / ( h / / 3 ( 4h θ 5.5 / T h ( T / T anθ T ( ht / T ( T / T ( ht / T / T 3 ( 4h θ 5.6 h.. h. 8 h.5.8 h.5 / 6 4 h. h. h φ.6.4 h. φ/π h. / 3 ω /ω. h ω /ω ω ω hω ω ω ω ω hωω H (ω ω ω ω hωω ( ω ω H 5.9 H ( ω 5.3 ω ω hωω 3

14 Duhal s Nwar-β ( ω hω 6.3 hω ω 6.4 Duhal s Duhal s { [ ]} hω ( τ ( ( τ sn ω h ( τ ω h dτ 6.5 Nwar-β Nwar-β F 6.8 4

15 Nwar-β ( 6.9 β β 6. β β β ( β β β β 4β ( β β 4β β β β β 4β β β ( F ( F β β β 4β β β ( β β β 4β 6.8 β β 4β β 6.9 β β β β 4β (

16 β β 4β β β β < a END Nwar-β β/8 β/6 β/4 β β/8 β/6 β/4 β β β/

17 β Nwar-β β. T.7.7 Nwar-b β/4 /6 /8 Nwar-β T.7 T.7 β / β/ β/

18 Nwar-β Nwar-β β/4... T.7 Nwar-β

19 R DYNAE y Asn B os A B A B sn os A B A B A osα B snα A B A B A B ( sn osα os snα y A B B A B sn snα an α os α ( α B A α A R I R I θ ( osθ snθ R I I an θ R os sn os sn 9

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1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π . 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()

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