Solución a un sistema con 9 grados de libertad y sus gráficas

1. UNVIERSIDAD DEL QUINDIO
ONDAS
EJERCICIO OSCILACIONES CON NUEVE GRADOS DE LIBERTAD
DIAGRAMA
Sistema simetrico de cuerda tensa con 9 grados de libertad con masas (m) y distancia
entre ellas (a)
HALLAMOS LAS FRECUENCIAS NORMALES
𝜔𝑝 = 2𝛺2
(1 − 𝑐𝑜𝑠 (
𝑝𝜋
𝑁+1
))
Donde:
𝛺2
=
𝐹
𝑚𝑎
𝑝 = 𝑁𝑢𝑚𝑒𝑟𝑜 𝑑𝑒𝑙 𝑚𝑜𝑑𝑜
𝑁 = 9
𝜔𝑝 =
2𝐹
𝑚𝑎
(1 − 𝑐𝑜𝑠 (
𝑝𝜋
10
))
(𝜔𝐼)2
=
𝐹
𝑚𝑎
(0.0979)
𝑟𝑎𝑑
𝑠
(𝜔𝐼𝐼)2
=
𝐹
𝑚𝑎
(0.3819)
𝑟𝑎𝑑
𝑠
(𝜔𝐼𝐼𝐼)2
=
𝐹
𝑚𝑎
(0.8244)
𝑟𝑎𝑑
𝑠
(𝜔𝐼𝑉)2
=
𝐹
𝑚𝑎
(1.3819)
𝑟𝑎𝑑
𝑠
(𝜔𝑉)2
=
𝐹
𝑚𝑎
(2)
𝑟𝑎𝑑
𝑠
(𝜔𝑉𝐼)2
=
𝐹
𝑚𝑎
(2.618)
𝑟𝑎𝑑
𝑠
(𝜔𝑉𝐼𝐼)2
=
𝐹
𝑚𝑎
(3.1756)
𝑟𝑎𝑑
𝑠
(𝜔𝑉𝐼𝐼𝐼)2
=
𝐹
𝑚𝑎
(3.618)
𝑟𝑎𝑑
𝑠

2. (𝜔𝐼𝑋)2
=
𝐹
𝑚𝑎
(3.9021)
𝑟𝑎𝑑
𝑠
Hallar los coeficientes de reparto
𝑆𝑝(𝑥𝑛, 𝑡) = 𝑠𝑜𝑝
𝑠𝑖𝑛(𝑘𝑝𝑥𝑛) 𝑠𝑖𝑛(𝜔𝑝𝑡 + 𝜑𝑝)
Sea 𝑠𝑜𝑝
y 𝜑𝑝 Las condiciones iniciales que se determinan arbitrariamente
Donde
𝑥𝑛 𝑒𝑠 𝑙𝑎 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑖𝑎 𝑒𝑛 𝑎
𝑘𝑝 =
(𝑝𝜋)
𝑎(10)
𝑝 = 𝑚𝑜𝑑𝑜 𝑜𝑠𝑐𝑖𝑙𝑎𝑐𝑖ó𝑛
𝑆𝑝(𝑥𝑛, 𝑡) = 𝑠𝑜𝑝
𝑠𝑒𝑛 (
(𝑝𝜋)
𝑎(10)
𝑥𝑛) 𝑠𝑒𝑛(𝜔𝑝𝑡 + 𝜑𝑝)
𝐸𝑙 𝑐𝑜𝑒𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑒 𝑑𝑒 𝑟𝑒𝑝𝑎𝑟𝑡𝑜 (𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑟𝑜𝑝𝑖𝑜) 𝑒𝑠: 𝑠𝑒𝑛 (
(𝑝𝜋)
𝑎(10)
𝑥𝑛)
COEFICIENTES DE REPARTO
FRECUENCIAS
NORMALES
1 2 3 4 5 6 7 8 9
I 0,3090 0,5878 0,8090 0,9511 1,0000 0,9511 0,8090 0,5878 0,3090
II 0,5878 0,9511 0,9511 0,5878 0,0000 -0,5878 -0,9511 -0,9511 -0,5878
III 0,8090 0,9511 0,3090 -0,5878 -1,0000 -0,5878 0,3090 0,9511 0,8090
VI 0,9511 0,5878 -0,5878 -0,9511 0,0000 0,9511 0,5878 -0,5878 -0,9511
V 1,0000 0,0000 -1,0000 0,0000 1,0000 0,0000 -1,0000 0,0000 1,0000
VI 0,9511 -0,5878 -0,5878 0,9511 0,0000 -0,9511 0,5878 0,5878 -0,9511
VII 0,8090 -0,9511 0,3090 0,5878 -1,0000 0,5878 0,3090 -0,9511 0,8090
VIII 0,5878 -0,9511 0,9511 -0,5878 0,0000 0,5878 -0,9511 0,9511 -0,5878
IX 0,3090 -0,5878 0,8090 -0,9511 1,0000 -0,9511 0,8090 -0,5878 0,3090
GRÁFICAS
Posición de las masas en cada uno de los modos normales de con:
𝑠𝑜𝑝
= 1,
𝐹
𝑚𝑎
= 1, en 𝑡 = 0

3. I)
II)
III)
IV)

4. V)
VI)
VII)
VIII)

5. IX)
CONJUNTOS DE ECUACIONES
Primer modo:
𝑆𝐼(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.309) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥2, 𝑡) = 𝑠𝑜𝑝
0.5878𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥3, 𝑡) = 𝑠𝑜𝑝
0.809 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥4, 𝑡) = 𝑠𝑜𝑝
0.951𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥5, 𝑡) = 𝑠𝑜𝑝
𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥6, 𝑡) = 𝑠𝑜𝑝
0.951𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥7, 𝑡) = 𝑠𝑜𝑝
0.809 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥8, 𝑡) = 𝑠𝑜𝑝
0.5878𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
𝑆𝐼(𝑥9, 𝑡) = 𝑠𝑜𝑝
0.309 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.0979)𝑡 + 𝜑𝐼)
Segundo modo:
𝑆𝐼𝐼(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥2, 𝑡) = 𝑠𝑜𝑝
0.9511𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)

6. 𝑆𝐼𝐼(𝑥3, 𝑡) = 𝑠𝑜𝑝
0.9511 𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥4, 𝑡) = 𝑠𝑜𝑝
0.5878𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥5, 𝑡) = 𝑠𝑜𝑝
0 𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥6, 𝑡) = 𝑠𝑜𝑝
−0.5878𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥7, 𝑡) = 𝑠𝑜𝑝
− 0.9511 𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥8, 𝑡) = 𝑠𝑜𝑝
−0.9511𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
𝑆𝐼𝐼(𝑥9, 𝑡) = 𝑠𝑜𝑝
− 0.5878 𝑠𝑒𝑛 (
𝐹
𝑚𝑎
(0.3819)𝑡 + 𝜑𝐼𝐼)
Tercer modo:
𝑆𝐼𝐼𝐼(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.8090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥2, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥3, 𝑡) = 𝑠𝑜𝑝
(0.3090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥4, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥5, 𝑡) = 𝑠𝑜𝑝
(−1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥6, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥7, 𝑡) = 𝑠𝑜𝑝
(0.3090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥8, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
𝑆𝐼𝐼𝐼(𝑥9, 𝑡) = 𝑠𝑜𝑝
(0.8090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(0.8244)𝑡 + 𝜑𝐼𝐼𝐼)
Cuarto modo
𝑆𝐼𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥2, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥3, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)

7. 𝑆𝐼𝑉(𝑥4, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥5, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥6, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥7, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥8, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
𝑆𝐼𝑉(𝑥9, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(1.3819)𝑡 + 𝜑𝐼𝑉)
Quinto modo
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(−1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(−1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
𝑆𝑉(𝑥1, 𝑡) = 𝑠𝑜𝑝
(1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2)𝑡 + 𝜑𝑉)
Sexto modo
𝑆𝑉𝐼(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)

8. 𝑆𝑉𝐼(𝑥2, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥3, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥4, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥5, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥6, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥7, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥8, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
𝑆𝑉𝐼(𝑥9, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(2.618)𝑡 + 𝜑𝑉𝐼)
Séptimo modo
𝑆𝑉𝐼𝐼(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.8090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥2, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥3, 𝑡) = 𝑠𝑜𝑝
(0.3090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥4, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥5, 𝑡) = 𝑠𝑜𝑝
(−1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥6, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥7, 𝑡) = 𝑠𝑜𝑝
(0.3090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥8, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
𝑆𝑉𝐼𝐼(𝑥9, 𝑡) = 𝑠𝑜𝑝
(0.8090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.1756)𝑡 + 𝜑𝑉𝐼𝐼)
Octavo modo

9. 𝑆𝑉𝐼𝐼𝐼(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥2, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥3, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥4, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥5, 𝑡) = 𝑠𝑜𝑝
(0) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥6, 𝑡) = 𝑠𝑜𝑝
(0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥7, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥8, 𝑡) = 𝑠𝑜𝑝
(0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
𝑆𝑉𝐼𝐼𝐼(𝑥9, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.618)𝑡 + 𝜑𝑉𝐼𝐼𝐼)
Noveno modo
𝑆𝐼𝑋(𝑥1, 𝑡) = 𝑠𝑜𝑝
(0.3090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥2, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥3, 𝑡) = 𝑠𝑜𝑝
(0.8090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥4, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥5, 𝑡) = 𝑠𝑜𝑝
(1) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥6, 𝑡) = 𝑠𝑜𝑝
(−0.9511) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥7, 𝑡) = 𝑠𝑜𝑝
(0.8090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
𝑆𝐼𝑋(𝑥8, 𝑡) = 𝑠𝑜𝑝
(−0.5878) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)

10. 𝑆𝐼𝑋(𝑥9, 𝑡) = 𝑠𝑜𝑝
(0.3090) 𝑠𝑒𝑛 (√
𝐹
𝑚𝑎
(3.9021)𝑡 + 𝜑𝐼𝑋)
ECUACION GENERAL
𝑆𝑝(𝑥𝑛, 𝑡) = ∑ 𝑠𝑜𝑝
𝑠𝑖𝑛 (
𝑝𝜋
(𝑁+1)
𝑛) 𝑠𝑖𝑛(𝜔𝑝𝑡 + 𝜑𝑝)
9
1
MILTON GUILLERMO CORSO USSA