📚 Forced Oscillations and Transients | 强迫振荡与瞬态响应
Forced oscillations occur when an external driving force is applied to a vibrating system. In mathematics, they are modelled by non‑homogeneous second‑order linear differential equations. The complete solution consists of a transient part (the complementary function) that decays over time due to damping, and a steady‑state part (the particular integral) that persists as long as the forcing continues. Understanding these components is essential for analysing the long‑term behaviour of mechanical and electrical systems.
强迫振荡发生在振动系统受到外部驱动力时。在数学中，它们由非齐次二阶线性微分方程建模。完整解由随时间衰减的瞬态部分（余函数）和持续存在的稳态部分（特积分）组成，稳态部分在强迫持续期间一直存在。理解这些成分对于分析机械和电路系统的长期行为至关重要。
1. Modelling Oscillations with Differential Equations | 用微分方程建模振荡
Many physical systems, such as a mass on a spring or an RLC circuit, satisfy a second‑order differential equation of the form m x'' + c x' + k x = F(t) ) = 0, the system undergoes free oscillations; when F(t) ≠ 0, we have forced oscillations.
许多物理系统，如弹簧上的质量或RLC电路，满足形式的二阶微分方程 m x'' + c x' + k x = F(t)，其中 x(t) 是位移，m 是质量，c 是阻尼系数，k 是弹簧常数，F(t) 是外部驱动力。当 F(t) = 0 时，系统进行自由振动；当 F(t) ≠ 0 时，我们得到强迫振动。
For sinusoidal forcing, we write F(t) = F₀ cos(ω t) or F₀ sin(ω t), where ω is the driving angular frequency. The natural frequency of the undamped system is ω₀ = √k/m, and the damping factor is often written as β = c/(2m). The full equation then becomes x'' + 2β x' + ω₀² x = (F₀/m) cos(ω t).
对于正弦强迫，我们写作 F(t) = F₀ cos(ω t) 或 F₀ sin(ω t)，其中 ω 是驱动角频率。无阻尼系统的自然频率为 ω₀ = √k/m，阻尼因子常写成 β = c/(2m)。全方程便成为 x'' + 2β x' + ω₀² x = (F₀/m) cos(ω t)。
2. Homogeneous Solution and Transient Behaviour | 齐次解与瞬态行为
The homogeneous equation m x'' + c x' + k x = 0 governs the free motion. Its general solution depends on the discriminant of the characteristic equation m r² + c r + k = 0. Using the damping factor β = c/(2m) and the natural frequency ω₀ = √k/m in three cases: underdamped (β < ω₀), critically damped (β = ω₀), and overdamped (β > ω₀).
齐次方程 m x'' + c x' + k x = 0 控制自由运动。其通解取决于特征方程 m r² + c r + k = 0 的判别式。利用阻尼因子 β = c/(2m) 和自然频率 ω₀ = √k/m，我们得到三种情况：欠阻尼 (β < ω₀)、临界阻尼 (β = ω₀) 和过阻尼 (β > ω₀)。
In the underdamped case, the homogeneous solution is xh(t) = exp(−β t)[A cos(ω₁ t) + B sin(ω₁ t)], with the damped frequency ω₁ = √ω₀² − β² transient response. For critically damped and overdamped systems the homogeneous solution involves purely real exponentials, which also decay without oscillation.
在欠阻尼情况下，齐次解为 xh(t) = exp(−β t)[A cos(ω₁ t) + B sin(ω₁ t)]，其中阻尼频率 ω₁ = √ω₀² − β²。指数因子 exp(−β t) 迫使这部分解在 t → ∞ 时趋于零，因此它被称为瞬态响应。对于临界阻尼和过
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