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Enunciado

Mecânica e Resistência dos Materiais - Rao - Vibrações mecânicas- Ed: 4º - Capítulo 4.Problemas - Ex. 53

Derive as Equações 4.68 e 4.71.

Passo 1

Eq 4.68

Diferenciando Eq 4.66

overset(˙)(x) ( t ) = (F_j)/(k) e^(- zeta omega_n (t - t_j - 1)) omega_d (1 + (zeta^2 phi_n^2)/(omega_d^2)) s i n omega_d (t - t_j - 1) + omega_d e^(- zeta omega_n (t - t_j - 1))

{- x_j - 1 s i n omega_d (t - t_j - 1) + (overset(˙)(x)_j - 1)/(omega_d) c o s omega_alpha (t - t_j - 1))

(- (zeta omega_n)/(omega_(alpha)^2) (overset(˙)(x)_j - 1 + zeta omega_n x_j - 1) s i n omega_alpha (t - t_j - 1)}

Passo 2

Para t = t_j ,

overset(˙)(x)_j = (F_j)/(k) e^(- zeta omega_n Delta t_j) omega_d (1 + (zeta^2 omega_n^2)/(omega_d^2)) s i n omega_d Delta t_j + ( omega_d e^(- zeta omega_n - x_j - 1) s i n omega_d (+ (overset(˙)(x)_j - 1)/(omega_d) c o s omega_d Delta t_j - (zeta omega_n)/(omega_d^2) (overset(˙)(x)_j - 1 + zeta omega_n x_j - 1) s i n omega_d Delta t_j}

Eq 4.71

overset(˙)(x) ( t ) = (Delta F_j)/(k Delta t_i) [1 - zeta omega_n e^(- zeta omega_n (t - t_j - 1)) (2 zeta)/(omega_n) c o s omega_d (t - t_j - 1) - e^(- zeta omega_n (t - t_j - 1)))

(2 zeta)/(omega_n) * omega_d * s i n omega_d (t - t_j - 1) + zeta omega_n e^(- zeta Q_n (t - t_j - 1)) ((omega_d^2 - zeta^2 omega_n^2)/(omega_n^2 omega_d)) s i n omega_d (t - t_j - 1)

(- e^(- zeta omega_n (t - t_j - 1)) ((omega_d^2 - r^2 omega_n^2)/(omega_n^2 omega_d)) omega_d c o s omega_d (t - t_j - 1)] + (F_j - 1)/(k) [zeta omega_n e^(- zeta omega_n (t - t_j - 1)))

c o s omega_d (t - t_j - 1) + e^(- zeta omega_n (t - t_j - 1)) omega_alpha s i n omega_d (t - t_j - 1) + T omega_n e^(- zeta omega_n (t - t_j - 1))

(((zeta omega_n)/(omega_d)) * s i n omega_d (t - t_j - 1) - e^(- zeta omega_n (t - t_j - 1)) ((T omega_n)/(omega_d)) omega_alpha * c o s omega_d (t - t_j - 1)]

s i n omega_d (t - t_j - 1) - zeta omega_n e^(- zeta omega_n (t - t_j - 1)) ((overset(˙)(x)_j - 1 + zeta omega_n x_j - 1)/(omega_d)) s i n omega_d (t - t_j - 1)

+ e^(- zeta omega_n (t - t_j - 1)) ((overset(˙)(x)_j - 1 + zeta omega_n x_j - 1)/(omega_d)) omega_alpha c o s omega_d (t - t_j - 1)

Passo 3

Simplificação desta equação dá t = t_j

overset(˙)(x)_j = (Delta F_j)/(x Delta t_j) [1 - e^(- zeta omega_n Delta t_j) {c o s omega_d Delta t_j + (zeta omega_n)/(omega_d) s i n omega_alpha Delta t_j}]

+ (F_j - 1)/(k) e^(- zeta omega_n Delta t_j) (omega_n^2)/(omega_alpha) s i n omega_d Delta t_j .

+ e^(- zeta omega_n Delta t_j) {overset(˙)(x)_j - 1 c o s omega_d Delta t_j - (zeta omega_n)/(Delta_j) (overset(˙)(x)_j - 1 + (omega_n)/(zeta) x_j - 1) s i n omega_d Delta t_j}

Resposta

overset(˙)(x)_j = (Delta F_j)/(x Delta t_j) [1 - e^(- zeta omega_n Delta t_j) {c o s omega_d Delta t_j + (zeta omega_n)/(omega_d) s i n omega_alpha Delta t_j}]

+ (F_j - 1)/(k) e^(- zeta omega_n Delta t_j) (omega_n^2)/(omega_alpha) s i n omega_d Delta t_j .

+ e^(- zeta omega_n Delta t_j) {overset(˙)(x)_j - 1 c o s omega_d Delta t_j - (zeta omega_n)/(Delta_j) (overset(˙)(x)_j - 1 + (omega_n)/(zeta) x_j - 1) s i n omega_d Delta t_j}

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