PROBLEMA 1

EXCITACIÓN IMPULSIVA E INTEGRAL DE DUHAMEL

CASO 1: Para 0 ≤ t ≤ td/2

Las condiciones iniciales son:

Reemplazando las condiciones iniciales en la ecuación de DUHAMEL

Calculando la integral:

Haciendo:

Pero:

Reemplazando en ①:

𝑦(𝑡)=𝑦_𝑜 cos 𝑤𝑡+𝑣_𝑜/𝑤 𝑠𝑒𝑛 𝑤𝑡+1/𝑚𝑤 ∫24_𝑜^𝑡▒ 〖𝐹 (𝜏) 𝑠𝑒𝑛 𝑤(𝑡−𝜏)𝑑𝜏 〗

•〖 𝑦〗 _𝑜=0•〖 𝑉〗 _𝑜=0•〖 𝐹 ( )𝜏 =𝐹 〗 _𝑜

⇒ 𝑦(𝑡)=1/𝑚𝑤 ∫24_𝑜^(𝑡𝑑/2)▒ 〖𝐹 _𝑜 𝑠𝑒𝑛 𝑤(𝑡−𝜏)𝑑𝜏 〗 = _ / ∫_ ^(𝐹 𝑜 𝑚𝑤 𝑜 𝑡𝑑/2)▒ 〖 𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗

𝐴=∫▒ 〖𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 … ① 〗

𝑤( − )𝑡 𝜏 =𝑥⇒𝜏=𝑡−𝑥/𝑤⇒𝑑𝜏=−𝑑𝑥/𝑤𝐴=∫_ ^( /2)𝑜 𝑡𝑑 ▒ 〖𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗 =∫_ ^( /2)𝑜 𝑡𝑑 ▒ 〖𝑠𝑒𝑛 𝑥 (−𝑑𝑥/𝑤) 〗 =−1/𝑤 ∫_ ^(𝑜 𝑡𝑑/2)▒ 〖𝑠𝑒𝑛 𝑥 𝑑𝑥〗 =1/𝑤 cos 𝑥 |(𝑡𝑑/2)¦0┤

𝑥= ( − )𝑤 𝑡 𝜏⇒𝐴=1/ cos ( − ) |( /2)¦0𝑤 𝑤 𝑡 𝜏 𝑡𝑑 ┤

⇒ 𝑦(𝑡)= _ /𝐹 𝑜 𝑚𝑤∗1/ cos ( − ) |(𝑤 𝑤 𝑡 𝜏 𝑡𝑑/2)¦0 = _ /(┤ 𝐹 𝑜 𝑚𝑤^2 )∗[cos ( −𝑤 𝑡 𝑡𝑑/2)−cos 𝑤𝑡 ]⇒ 𝑦(𝑡)= _ /(𝐹 𝑜 𝑚𝑤^2 )∗[cos ( −𝑤 𝑡 𝑡𝑑/2)−cos 𝑤𝑡 ], 𝑝𝑎𝑟𝑎 0≤𝑡≤𝑡𝑑/2

CASO 2: Para td/2 ≤ t ≤ td

Las condiciones iniciales para t = td/2:

• Como:

• Como:

Reemplazando las condiciones iniciales en la ecuación de DUHAMEL, haciendo t = t - td/2

Calculando la integral:

Haciendo:

⇒ 𝑦(𝑡)= _ /(𝐹 𝑜 𝑚𝑤^2 )∗[cos ( −𝑤 𝑡 𝑡𝑑/2)−cos 𝑤𝑡 ], 𝑝𝑎𝑟𝑎 0≤𝑡≤𝑡𝑑/2

𝑦(𝑡)= _ /(𝐹 𝑜 𝑚𝑤^2 )∗[cos ( −𝑤 𝑡 𝑡𝑑/2)−cos 𝑤𝑡 ]⇒ 𝑦(𝑡𝑑/2)= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗[cos ( /2− /2)−cos 𝑤 𝑡𝑑 𝑡𝑑 〖 𝑤 𝑡𝑑 /2 〗 ]= _ /(𝐹 𝑜 𝑚𝑤^2 )∗[1−cos 〖𝑤 𝑡𝑑 /2 〗 ]

𝑣(𝑡)=𝑦′(𝑡)= _ /(𝐹 𝑜 𝑚𝑤^2 )∗[−𝑤 sen ( −𝑤 𝑡 𝑡𝑑/2)+𝑤 𝑠𝑒𝑛 𝑤𝑡]= _ /𝐹 𝑜 𝑚𝑤∗[−sen ( − /2)𝑤 𝑡 𝑡𝑑 + ]𝑠𝑒𝑛 𝑤𝑡⇒𝑣( /2)𝑡𝑑 =𝑦^′ ( /2)𝑡𝑑 = _ /𝐹 𝑜 𝑚𝑤∗[−sen ( /2−𝑤 𝑡𝑑 𝑡𝑑/2)+ 𝑠𝑒𝑛 𝑤 /2𝑡𝑑 ]= _ /𝐹 𝑜 𝑚𝑤∗ /2𝑠𝑒𝑛 𝑤 𝑡𝑑

•〖 𝐹 ( )𝜏 =𝐹 〗 _𝑜/2𝑦(𝑡)= ( /2) 𝑦 𝑡𝑑 〖∗ cos 〗𝑤 ( − /2)𝑡 𝑡𝑑 +1/𝑤∗ ( /2)𝑣 𝑡𝑑 ∗ ( − /2)𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 +1/𝑚𝑤 ∫24_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝐹 _ /𝑜 2 𝑠𝑒𝑛 𝑤(𝑡−𝜏)𝑑𝜏 〗

A B

𝐴=𝐹_𝑜/(𝑚𝑤^2 )∗[1−cos 〖𝑤 𝑡𝑑 /2 〗 ] 〖∗ cos 〗𝑤 (𝑡−𝑡𝑑/2)+1/𝑤∗𝐹_𝑜/𝑚𝑤∗𝑠𝑒𝑛 𝑤 𝑡𝑑/2∗𝑠𝑒𝑛 𝑤(𝑡−𝑡𝑑/2)𝐴= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗[cos ( − /2)𝑤 𝑡 𝑡𝑑 −cos 〖𝑤 𝑡𝑑 /2 〗∗ cos ( − /2)𝑤 𝑡 𝑡𝑑 + /2 ( − /2)𝑠𝑒𝑛 𝑤 𝑡𝑑 ∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 ]

〖− cos 〗〖 (𝛼+𝛽)= 〖− cos 〗 〖𝛼∗ cos 𝛽+𝑠𝑒𝑛 𝛼∗𝑠𝑒𝑛 𝛽 〗 〗𝐴= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗[cos ( − /2)𝑤 𝑡 𝑡𝑑 −cos (𝑤 𝑡𝑑/2+ − /2)𝑡 𝑡𝑑 ]= _ /( ^2 ) [cos ( − /2)−cos 𝐹 𝑜 𝑚𝑤 ∗ 𝑤 𝑡 𝑡𝑑 𝑤𝑡 ]

𝐵=1/𝑚𝑤 ∫24_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝐹 _ /𝑜 2 𝑠𝑒𝑛 𝑤(𝑡−𝜏)𝑑𝜏 〗 = _ /𝐹 𝑜 2 ∫_(𝑚𝑤 𝑡𝑑/2)^𝑡𝑑▒ 〖 𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗

𝐶=∫▒ 〖𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 … ② 〗

𝑤( − )𝑡 𝜏 =𝑥⇒𝜏=𝑡−𝑥/𝑤⇒𝑑𝜏=−𝑑𝑥/𝑤

Pero:

Reemplazando A y B:

CASO 3:

Las condiciones iniciales para t = td:

• Como:

• Como:

Reemplazando las condiciones iniciales en la ecuación de DUHAMEL, haciendo t = t - td

Reemplazando en ②:

Para t ≥ td

𝐶=∫_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗 =∫_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝑠𝑒𝑛 𝑥 (−𝑑𝑥/𝑤) 〗 =−1/𝑤 ∫_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝑠𝑒𝑛 𝑥 𝑑𝑥〗 =1/𝑤 cos 𝑥 |𝑡𝑑¦(𝑡𝑑/2)┤𝑥= ( − )𝑤 𝑡 𝜏⇒𝐶=1/𝑤 cos 𝑤(𝑡−𝜏) |𝑡𝑑¦(𝑡𝑑/2)┤

⇒𝐵= _ /𝐹 𝑜 2𝑚𝑤∗1/ cos ( − ) |𝑤 𝑤 𝑡 𝜏 𝑡𝑑¦(𝑡𝑑/2) = _ /(┤ 𝐹 𝑜 2𝑚𝑤^2 )∗[cos ( −𝑤 𝑡 𝑡𝑑)−cos ( − /2)𝑤 𝑡 𝑡𝑑 ]

𝑦(𝑡)= _ /( ^2 ) [cos ( − /2)−cos ]𝐹 𝑜 𝑚𝑤 ∗ 𝑤 𝑡 𝑡𝑑 𝑤𝑡 + _ /(2 ^2 ) [cos ( − )−cos ( − /2) ]𝐹 𝑜 𝑚𝑤 ∗ 𝑤 𝑡 𝑡𝑑 𝑤 𝑡 𝑡𝑑𝑦(𝑡)= _ /( ^2 ) [cos ( − /2)−cos 𝐹 𝑜 𝑚𝑤 ∗ 𝑤 𝑡 𝑡𝑑 𝑤𝑡+1/2∗cos ( − )𝑤 𝑡 𝑡𝑑 −1/2 cos ( − /2)∗ 𝑤 𝑡 𝑡𝑑 ]

𝑦(𝑡)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ 1/2∗cos ( − /2)−cos 𝑤 𝑡 𝑡𝑑 𝑤𝑡+1/2∗cos ( − )𝑤 𝑡 𝑡𝑑 ], /2≤ ≤𝑝𝑎𝑟𝑎 𝑡𝑑 𝑡 𝑡𝑑

𝑦(𝑡)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ 1/2∗cos ( − /2)−cos 𝑤 𝑡 𝑡𝑑 𝑤𝑡+1/2∗cos ( − )𝑤 𝑡 𝑡𝑑 ]

⇒𝑦(𝑡𝑑)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ 1/2∗cos (𝑤 𝑡𝑑− /2)−cos 𝑡𝑑 𝑤𝑡𝑑+1/2∗cos (𝑤 𝑡𝑑− )𝑡𝑑 ]

⇒𝑦(𝑡𝑑)= _ /( ^2 ) [1/2 cos ( /2)−cos +1/2]𝐹 𝑜 𝑚𝑤 ∗ ∗ 𝑤 𝑡𝑑 𝑤𝑡𝑑〖𝑣 (𝑦)=𝑦 〗 ^′ (𝑡)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ −𝑤/2∗𝑠𝑒𝑛 ( − /2)𝑤 𝑡 𝑡𝑑 +𝑤∗s𝑒𝑛 𝑤𝑡−𝑤/2∗𝑠𝑒𝑛 ( − )𝑤 𝑡 𝑡𝑑 ]

⇒𝑣(𝑡𝑑)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ −𝑤/2∗𝑠𝑒𝑛 (𝑤 𝑡𝑑− /2)𝑡𝑑 +𝑤∗s𝑒𝑛 𝑤𝑡𝑑−𝑤/2∗𝑠𝑒𝑛 (𝑤 𝑡𝑑− )𝑡𝑑 ]

⇒𝑣(𝑡𝑑)= _ /𝐹 𝑜 𝑚𝑤 [∗ −1/2∗𝑠𝑒𝑛 ( /2)𝑤 𝑡𝑑 +s𝑒𝑛 𝑤𝑡𝑑 ]

•𝐹(𝜏)=0

𝑦(𝑡)= (𝑦 𝑡𝑑) 〖∗ cos 〗𝑤 ( −𝑡 𝑡𝑑)+1/𝑤∗ (𝑣 𝑡𝑑)∗ ( −𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑)

𝑦(𝑡)= _ /( ^2 ) [1/2 cos ( /2)−cos +1/2] 𝐹 𝑜 𝑚𝑤 ∗ ∗ 𝑤 𝑡𝑑 𝑤𝑡𝑑 〖∗ cos 〗𝑤 ( −𝑡 𝑡𝑑)+1/𝑤∗ _ / [−1/2 ( /2)+s ]𝐹 𝑜 𝑚𝑤∗ ∗𝑠𝑒𝑛𝑤 𝑡𝑑 𝑒𝑛𝑤𝑡𝑑 ∗ ( −𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑)

𝑦(𝑡)= _ /( ^2 ) [1/2 cos ( /2)−cos +1/2] 𝐹 𝑜 𝑚𝑤 ∗ ∗ 𝑤 𝑡𝑑 𝑤𝑡𝑑 〖∗ cos 〗𝑤 ( −𝑡 𝑡𝑑)+1/𝑤∗ _ / [−1/2 ( /2)+s ]𝐹 𝑜 𝑚𝑤∗ ∗𝑠𝑒𝑛𝑤 𝑡𝑑 𝑒𝑛𝑤𝑡𝑑 ∗ ( −𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑)

𝑦(𝑡)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ █( 1/2 cos ( /2) ∗ 𝑤 𝑡𝑑 ∗〖 cos 〗𝑤 ( − )−1/2 ( /2) ( − )𝑡 𝑡𝑑 ∗𝑠𝑒𝑛𝑤 𝑡𝑑 ∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 @ 〖− cos ∗〗 〖𝑤𝑡𝑑 cos 〗𝑤 ( − )+s ( − )+1/2 𝑡 𝑡𝑑 𝑒𝑛𝑤𝑡𝑑∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 ∗〖 cos 〗〖𝑤 ( − )𝑡 𝑡𝑑 〗 )]

−〖 cos 〗〖 (𝛼+𝛽)= 〖− cos 〗 〖𝛼∗ cos 𝛽+𝑠𝑒𝑛 𝛼∗𝑠𝑒𝑛 𝛽 〗 〗

cos 〖 (𝛼+𝛽)=cos 〖𝛼∗ cos 𝛽−𝑠𝑒𝑛 𝛼∗𝑠𝑒𝑛 𝛽 〗 〗

𝑦(𝑡)= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗{ ( 1/2 [cos ( /2) █ ∗ 𝑤 𝑡𝑑 ∗〖 cos 〗𝑤 ( − )− ( /2) ( − )]@𝑡 𝑡𝑑 𝑠𝑒𝑛𝑤 𝑡𝑑 ∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 −〖 cos ∗〗 〖𝑤𝑡𝑑 cos 〗𝑤 ( − )+s ( − )+1/2 𝑡 𝑡𝑑 𝑒𝑛𝑤𝑡𝑑∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 ∗〖 cos 〗〖𝑤 ( − ) 𝑡 𝑡𝑑 〗 )}

𝑦(𝑡)= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗[1/2∗cos ( /2𝑤 𝑡𝑑 +𝑡−𝑡𝑑)−cos (𝑤 𝑡𝑑+ − )+1/2 𝑡 𝑡𝑑 ∗〖 cos 〗〖𝑤 ( − ) 𝑡 𝑡𝑑 〗 ]𝑦(𝑡)= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗[1/2∗cos (𝑤 𝑡− /2)𝑡𝑑 −cos 𝑤𝑡+1/2 ∗〖 cos 〗〖𝑤 ( − ) 𝑡 𝑡𝑑 〗 ], 𝑝𝑎𝑟𝑎 𝑡≥𝑡𝑑

⇒ 𝑦(𝑡)=1/𝑚𝑤 ∫24_𝑜^(𝑡𝑑/2)▒ 〖𝐹 _𝑜 𝑠𝑒𝑛 𝑤(𝑡−𝜏)𝑑𝜏 〗 = _ / ∫_ ^(𝐹 𝑜 𝑚𝑤 𝑜 𝑡𝑑/2)▒ 〖 𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗

𝐴=∫_ ^( /2)𝑜 𝑡𝑑 ▒ 〖𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗 =∫_ ^( /2)𝑜 𝑡𝑑 ▒ 〖𝑠𝑒𝑛 𝑥 (−𝑑𝑥/𝑤) 〗 =−1/𝑤 ∫_ ^(𝑜 𝑡𝑑/2)▒ 〖𝑠𝑒𝑛 𝑥 𝑑𝑥〗 =1/𝑤 cos 𝑥 |(𝑡𝑑/2)¦0┤

𝑦(𝑡)= ( /2) 𝑦 𝑡𝑑 〖∗ cos 〗𝑤 ( − /2)𝑡 𝑡𝑑 +1/𝑤∗ ( /2)𝑣 𝑡𝑑 ∗ ( − /2)𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 +1/𝑚𝑤 ∫24_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝐹 _ /𝑜 2 𝑠𝑒𝑛 𝑤(𝑡−𝜏)𝑑𝜏 〗

𝐴=𝐹_𝑜/(𝑚𝑤^2 )∗[1−cos 〖𝑤 𝑡𝑑 /2 〗 ] 〖∗ cos 〗𝑤 (𝑡−𝑡𝑑/2)+1/𝑤∗𝐹_𝑜/𝑚𝑤∗𝑠𝑒𝑛 𝑤 𝑡𝑑/2∗𝑠𝑒𝑛 𝑤(𝑡−𝑡𝑑/2)

𝐶=∫_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝑠𝑒𝑛 𝑤 ( − )𝑡 𝜏 𝑑𝜏 〗 =∫_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝑠𝑒𝑛 𝑥 (−𝑑𝑥/𝑤) 〗 =−1/𝑤 ∫_(𝑡𝑑/2)^𝑡𝑑▒ 〖𝑠𝑒𝑛 𝑥 𝑑𝑥〗 =1/𝑤 cos 𝑥 |𝑡𝑑¦(𝑡𝑑/2)┤

〖𝑣 (𝑦)=𝑦 〗 ^′ (𝑡)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ −𝑤/2∗𝑠𝑒𝑛 ( − /2)𝑤 𝑡 𝑡𝑑 +𝑤∗s𝑒𝑛 𝑤𝑡−𝑤/2∗𝑠𝑒𝑛 ( − )𝑤 𝑡 𝑡𝑑 ]

𝑦(𝑡)= _ /( ^2 ) [𝐹 𝑜 𝑚𝑤 ∗ █( 1/2 cos ( /2) ∗ 𝑤 𝑡𝑑 ∗〖 cos 〗𝑤 ( − )−1/2 ( /2) ( − )𝑡 𝑡𝑑 ∗𝑠𝑒𝑛𝑤 𝑡𝑑 ∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 @ 〖− cos ∗〗 〖𝑤𝑡𝑑 cos 〗𝑤 ( − )+s ( − )+1/2 𝑡 𝑡𝑑 𝑒𝑛𝑤𝑡𝑑∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 ∗〖 cos 〗〖𝑤 ( − )𝑡 𝑡𝑑 〗 )]

𝑦(𝑡)= _ /( ^2 )𝐹 𝑜 𝑚𝑤 ∗{ ( 1/2 [cos ( /2) █ ∗ 𝑤 𝑡𝑑 ∗〖 cos 〗𝑤 ( − )− ( /2) ( − )]@𝑡 𝑡𝑑 𝑠𝑒𝑛𝑤 𝑡𝑑 ∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 −〖 cos ∗〗 〖𝑤𝑡𝑑 cos 〗𝑤 ( − )+s ( − )+1/2 𝑡 𝑡𝑑 𝑒𝑛𝑤𝑡𝑑∗𝑠𝑒𝑛 𝑤 𝑡 𝑡𝑑 ∗〖 cos 〗〖𝑤 ( − ) 𝑡 𝑡𝑑 〗 )}