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2.固有频率和主振型多自由度系统的固有频率和主振型是从求解系统的无阻尼自由振动方程得到,设方程的解为{X}={A}e~(iwnt),则有=i~2ω_n~2{A}e~(iωnt),将它们代入公式(50),消去因子e~(iωnt)后得 ([K]-ω_n~2[M]){A}={O} (55)求解方程(55)就能得固有频率。由于振幅向量{A}的系数行列式等于零,即Δ(ω_n~2)=det([K]-ω_n~2[M])=0 (56)公式(56)称为特征方程或频率方程,将其展开后可得一个ω_n~2的n次代数方程ω_n~(2n)+a_1ω_n~(2(n-1))+a_2ω_n~(2(n-2))+……+a_(n-1)ω_n~2+ a_n=O解此方程,可得n个根:ω_(n1)~2,w_(2n)~2……ω_(nn)~2,称为特征值,将其开方即可得n个固有频率,按大小顺序排
2. The natural frequency and the main vibration mode The natural frequency and the main vibration mode of the system are obtained from the undamped free vibration equation of the system. Let the solution of the equation be {X} = {A} e ~ (iwnt), then (I) = i ~ 2ω_n ~ 2 {A} e ~ (iωnt), substituting them into equation (50) and eliminating factor e ~ (iωnt) O} (55) to solve equation (55) will be natural frequency. Since the coefficient determinant of amplitude vector {A} is equal to zero, Δ (ω_n ~ 2) = det ([K] -ω_n ~ 2 [M]) = 0 Equation (56) is called eigenvalue or frequency equation, We can get an n-order algebraic equation ω_n ~ (2n) + a_1ω_n ~ (2 (n-1)) + a_2ω_n ~ (2 (n-2)) + ...... + a_ (n- 1) ω_n ~ 2 + a_n = 0 By solving this equation, we can get n roots: ω_ (n1) ~ 2, w_ (2n) ~ 2 ...... ω_ (nn) You can get n natural frequency, according to the size of the row