By Khanh Chau Le
Half I. Linear conception: 1. unmarried oscillator -- 2. Coupled oscillators -- three. non-stop oscillators -- four. Linear waves -- half II. Nonlinear idea: five. self reliant unmarried oscillator -- 6. Non-autonomous unmarried oscillator -- 7. Coupled oscillators -- Nonlinear waves
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3) in the form ϕ1 = ϕˆ 1 est , ϕ2 = ϕˆ 2 est , where ϕˆ 1 and ϕˆ 2 are unknown constants. 3) we obtain [s2 ϕˆ 1 + ω02 ϕˆ 1 − α (ϕˆ2 − ϕˆ 1 )]est = 0, [s2 ϕˆ 2 + ω02 ϕˆ 2 + α (ϕˆ 2 − ϕˆ 1 )]est = 0. Since est is not equal to zero, the expressions in the square brackets must vanish. We may present these equations in the matrix form as follows s2 + ω02 + α −α −α s2 + ω02 + α ϕˆ 1 0 = . 4) exist if its determinant vanishes s2 + ω02 + α −α = (s2 + ω02 + α )2 − α 2 = 0. 5), quadratic with respect to s2 , yields s21 = −ω02 , s22 = −(ω02 + 2α ).
4) exist if its determinant vanishes s2 + ω02 + α −α = (s2 + ω02 + α )2 − α 2 = 0. 5), quadratic with respect to s2 , yields s21 = −ω02 , s22 = −(ω02 + 2α ). 6) with ω1 = ω0 and ω2 = ω02 + 2α being called the eigenfrequencies. Note that the amplitudes ϕˆ 1 and ϕˆ 2 cannot be arbitrary. 4) implies that ϕˆ 1 = ϕˆ 2 , or, in the vector form, ϕˆ = ϕˆ 1 ϕˆ 2 = C1 q1 , 1 1 q1 = √ . 2 1 Thus, the vector ϕˆ is proportional to the eigenvector q1 which is normalized to have the length 1. 4) ϕ1 = −ϕ2 , or ϕˆ = C2 q2 , 1 −1 q2 = √ .
22 Magnification factor M versus frequency ratio η at different damping ratio δ (case a). The equation for the magnification factor M can still be reduced to the form independent of the phase ψ . 5 h 2 Fig. 23 Magnification factor M versus frequency ratio η at different damping ratio δ (case b). (1 − η 2) cos ψ + 2δ η sin ψ = 2δ η (sin ψ + cos ψ 2δ η )= . 35) we can easily express sin ψ as sin ψ = 2δ η (1 − η 2)2 + 4δ 2η 2 Thus, M= α (1 − η 2)2 + 4δ 2 η 2 . 36) . 37) The plots of magnification factor M versus the frequency ratio η for different values of damping ratio δ are shown in Figs.