正弦波のスケーリング、ストレッチ、シフト
この記事では、標準的な正弦波 sin(x) の振幅、周波数、位相シフトを調整する方法を簡潔に解説する。具体的には、一般関数 s(x) = A・sin(w・x + θ) において、パラメータ A、w、θ を変更した際にグラフがどのように変化するかを説明する。
この記事では、標準的な正弦波 sin(x) の振幅、周波数、位相シフトを調整する方法を簡潔に解説する。具体的には、一般関数 s(x) = A・sin(w・x + θ) において、パラメータ A、w、θ を変更した際にグラフがどのように変化するかを説明する。
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The article presents the exact closed-form solution for the nonlinear (large-angle) pendulum using Jacobi elliptic functions, contrasting it with the common small-angle approximation that uses sine functions. It explains how the exact solution involves elliptic integrals and Jacobi sn, cn, and dn functions, providing formulas for the angular displacement and period.
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The article discusses how the nonlinear behavior of a pendulum deviates from the simple harmonic oscillator approximation, particularly at large amplitudes where the period depends on the amplitude. It explains that this nonlinearity means the small-angle approximation (sinθ ≈ θ) breaks down, leading to longer periods than predicted by the linearized model.
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