Solving Harmonic Forcing Terms Using Complex Solutions | 利用复数解法求解谐波强迫项

📚 Solving Harmonic Forcing Terms Using Complex Solutions | 利用复数解法求解谐波强迫项

In the study of differential equations, one of the most important applications is modelling forced oscillations, such as a mass on a spring subjected to an external periodic force or an electrical RLC circuit driven by an alternating voltage. When the forcing term is a simple sine or cosine function, the traditional method of undetermined coefficients requires handling cumbersome trigonometric identities. A far more elegant approach is to replace the real trigonometric forcing term with a complex exponential, solve the resulting complex ODE, and then extract the real part of the solution. This technique, widely used in physics and engineering, simplifies calculations dramatically and provides deep insight into the amplitude and phase of the steady-state response. In this article, we will explore how to apply complex solutions to second‑order linear ordinary differential equations with harmonic forcing terms, covering the theory, worked examples, and the physical interpretation of resonance.

在研究微分方程时，最重要的应用之一是对受迫振动进行建模，例如受周期性外力作用的弹簧振子，或由交流电压驱动的RLC电路。当强迫项是简单的正弦或余弦函数时，传统的待定系数法需要处理繁琐的三角恒等式。一种更巧妙的方法是用复指数函数替换实三角强迫项，求解所得的复常微分方程，再从解中提取实部。这种广泛应用于物理和工程的技术不仅极大地简化了计算，还能让我们深刻洞察稳态响应的振幅与相位。本文将探讨如何利用复数解法处理具有谐波强迫项的二阶线性常微分方程，内容涵盖理论、实例以及共振的物理解释。

1. Introduction to Forced Oscillations | 受迫振动简介

Many physical systems can be modelled by a second‑ or F sin(ωt), the forcing term is said to be harmonic. The general solution of such an equation is the sum of the complementary function (the transient response) and a particular integral (the steady‑ he complementary function usually decays to zero if damping is present, leaving the steady‑ e frequency as the forcing term but with a possibly different amplitude and a phase shift.

许多物理系统可以用二阶线性常微分方程 y″ + a y′ + b y = f(t) 来建模，其中 a 和 b 是由阻尼和回复力决定的常数，f(t) 表示外部驱动力。当 f(t) = F cos(ωt) 或 F sin(ωt) 时，强迫项被称为谐波的。此类方程的通解是余函数（瞬态响应）与特积分（稳态响应）之和。若存在阻尼，足够长时间后余函数通常衰减为零，剩下稳态振荡，其频率与强迫项相同，但振幅和相位可能不同。

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2. The Harmonic Forcing Term | 谐波强迫项

e cosine function can be written as the real part of a complex exponential, cos(ωt) = Re(e^iωt), we can replace the real forcing term R cos(ωt) with the complex forcing term R e^iωt. This is the crucial step that allows us to work entirely in the complex domain. Once a complex particular solution z_p is found, the real particular solution is simply y_p = Re(z_p).

考虑标准形式 y″ + p y′ + q y = R cos(ωt)，其中 p, q 和 R 为实常数，ω > 0 是驱动力的角频率。由于余弦函数可以写成复指数函数的实部，即 cos(ωt) = Re(e^iωt)，我们可以将实强迫项 R cos(ωt) 替换为复强迫项 R e^iωt。这是允许我们在复数域中完整求解的关键步骤。一旦求出复数特解 z_p，实数特解就是 y_p = Re(z_p)。

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3. Complex Numbers and the Exponential Form | 复数与指数形式

Recall Euler's formula, e^iθ = cosθ + i sinθ. From this, we have cosθ = (e^iθ + e^−iθ)/2 and sinθ = (e^iθ − e^−iθ ith a single harmonic forcing term, it is much simpler to use the fact that cos(ωt) = Re(e^iωt). If we extend the real differential equation to a complex one by substituting the forcing term R e^iωt ator L[y] = y″ + p y′ + q y are linear and have real coefficients, so L[Re(z)] = Re(L[z]).

回顾欧拉公式 e^iθ = cosθ + i sinθ。由此可得 cosθ = (e^iθ + e^−iθ)/2 以及 sinθ = (e^iθ − e^−iθ)/(2i)。但是，对于求解单个谐波强迫项的常微分方程，更简单的方式是利用 cos(ωt) = Re(e^iωt)。如果将强迫项替换为 R e^iωt，从而将实微分方程延拓为复微分方程，那么微分算子的线性性质保证了复数解的实部必然满足原来的实方程。这是因为微分运算和算子 L[y] = y″ + p y′ + q y 是线性的且系数为实数，所以 L[Re(z)] = Re(L[z])。

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4. Converting the Real ODE to a Complex ODE | 将实ODE转化为复ODE

Starting from y″ + p y′ + q y = R cos(ωt), we consider the associated complex differential equation

z″ + p z′ + q z = R e^iωt.

Here, z(t) is a complex‑valued function of the real variable t. The great advantage is that the forcing term is now a pure exponential, which is an eigenfunction of the differential operator. This means we can guess a particular solution of the form z_p = C e^iωt, where C is a complex constant to be determined. The complementary function, which solves the homogeneous equation, can be found separately, but our focus is on the steady‑state response.

从 y″ + p y′ + q y = R cos(ωt) 出发，我们考虑相应的复微分方程
z″ + p z′ + q z = R e^iωt.
这里 z(t) 是实变量 t 的复值函数。最大的好处是强迫项现在是一个纯指数函数，而指数函数是微分算子的特征函数。这意味着我们可以假设一个形如 z_p = C e^iωt 的特解，其中 C 是一个待定的复常数。齐次方程的解（余函数）可以单独求出，但我们关注的重点是稳态响应。

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5. Solving the Complex ODE for a Particular Solution | 求解复ODE的特解

Substitute z_p = C e^iωt into the complex ODE. Since z_p′ = iω C e^iωt and z_p″ = (iω)² C e^iωt = −ω² C e^iωt, we obtain:

(−ω² + i p ω + q) C e^iωt = R e^iωt.

Cancelling the non‑zero factor e^iωt gives the complex algebraic equation:

(q − ω² + i p ω) C = R.

Therefore, the complex amplitude C is

C = R / (q − ω² + i p ω).

This is a complex number whose modulus and argument will determine the amplitude and phase shift of the real steady‑state solution.

将 z_p = C e^iωt 代入复常微分方程。由于 z_p′ = iω C e^iωt 且 z_p″ = (iω)² C e^iωt = −ω² C e^iωt，我们得到： (−ω² + i p ω + q) C e^iωt = R e^iωt. 消去非零因子 e^iωt 后得到复数代数方程： (q − ω² + i p ω) C = R. 因此，复振幅 C 为 C = R / (q − ω² + i p ω). 这个复数的模和辐角将决定实数稳态解的振幅和相位偏移。

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6. Extracting the Real Solution | 提取实数解

The complex particular solution is z_p = C e^iωt. To obtain the real particular solution y_p we take the real part:

y_p = Re(C e^iωt).

If we express C in polar form, C = |C| e^iφ, where |C| = √( Re(C)² + Im(C)² ) and φ = arg(C) = atan2(Im(C), Re(C)), then

C e^iωt = |C| e^(i(ωt + φ)).

Its real part is |C| cos(ωt + φ). However, engineers often prefer to write the steady‑state response as y_p = D cos(ωt − δ), where D = |C| (since φ is usually negative). The exact value of δ is obtained from the complex number C.

复数特解为 z_p = C e^iωt。通过取实部，我们得到实数特解 y_p： y_p = Re(C e^iωt). 如果将 C 写成极坐标形式 C = |C| e^iφ，其中 |C| = √(Re(C)² + Im(C)²)，φ = arg(C) = atan2(Im(C), Re(C))，那么 C e^iωt = |C| e^(i(ωt + φ)). 其实部为 |C| cos(ωt + φ)。不过，工程师通常更喜欢将稳态响应写作 y_p = D cos(ωt − δ)，其中 D = |C| 是振幅，δ = −φ 是相位滞后（因为 φ 通常为负）。δ 的精确值可以从复数 C 得出。

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7. Amplitude and Phase Shift | 振幅和相移

the amplitude and phase explicity. The denominator is a complex number with real part q − ω² and imaginary part p ω. Its modulus squared is (q − ω²)² + (p ω)², so

D = |C| = R / √( (q − ω²)² + (p ω)² ).

The phase angle φ of C is given by

φ = arg( R ) − arg(q − ω² + i p ω) = 0 − arctan( p ω / (q − ω²) ).

Therefore, φ = − arctan( p ω / (q − ω²) ), and the phase lag δ is

δ = −φ = arctan( p ω / (q − ω²) ).

Care must be taken with the quadrant when q − ω² is negative; the arctan function must be interpreted appropriately (using the atan2 function). In practice, for q > ω² the phase lag is between 0 and π/2; for q < ω² it is between π/2 and π.

由 C = R / (q − ω² + i p ω)，我们可以显式地计算振幅和相位。分母是一个复数，实部为 q − ω²，虚部为 p ω。其模的平方为 (q − ω²)² + (p ω)²，因此 D = |C| = R / √( (q − ω²)² + (p ω)² ). C 的辐角 φ 为 φ = arg(R) − arg(q − ω² + i p ω) = 0 − arctan( p ω / (q − ω²) ). 因此 φ = − arctan( p ω / (q − ω²) )，相位滞后 δ 为 δ = −φ = arctan( p ω / (q − ω²) ). 当 q − ω² 为负时，需要注意象限；必须正确解释 arctan 函数（使用 atan2 函数）。在实际中，当 q > ω² 时相位滞后在 0 到 π/2 之间；当 q < ω² 时，在 π/2 到 π 之间。

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8. Resonance and Damping | 共振与阻尼

) shows a resonance phenomenon. For a given set of parameters, the amplitude is maximal when the denominator is minimal. If damping p is small, the amplitude peaks when ω is close to √q, the natural frequency of the undamped system. The exact resonance frequency is ω_r = √q − p²/2, derived by minimising the square of the denominator. At resonance, the steady‑ nds to infinity, although physically this cannot occur indefinitely because the linear model breaks down or energy losses limit the amplitude. The complex method naturally captures this behaviour through the magnitude of the complex denominator.

振幅 D = R / √( (q − ω²)² + (p ω)² ) 展现出共振现象。对于给定的一组参数，当分母最小时振幅最大。如果阻尼 p 很小，振幅在 ω 接近 √q（无阻尼系统的固有频率）时达到峰值。精确的共振频率为 ω_r = √q − p²/2，这是通过最小化分母的平方得出的。在共振点，稳态振幅可以变得非常大；在无阻尼情形 (p = 0) 下，振幅趋向无穷大，尽管实际上这不会无限发生，因为线性模型会失效或者能量损失会限制振幅。复数方法通过复分母的大小自然地捕捉到了这种行为。

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9. Worked Example | 例题

y = 10 cos(2t) using complex numbers. The associated complex ODE is z″ + 2 z′ + 5 z = 10 e^{2 i t}. Assume z_p = C e^{2 i t} gate 1 − 4i: C = 10(1 − 4i) / (1 + 16) = (10 − 40i) / 17 = 10/17 − i 40/17. The modulus is |C| = √( (10/17)² + (40/17)² ) = √( (100 + 1600) / 289 ) = √1700/289 = √100×17 / 289 = (10/17)√17 = 10/√17 (10/17) ) = arctan(−4) ≈ −1.3258 rad (or −75.96°). Thus the real particular solution is y_p = Re(C e^{2 i t}) = (10/√17) cos(2t − 1.3258). The amplitude is 10/√17 ≈ 2.425, and the phase lag is about 1.33 rad.

让我们用复数方法求解 y″ + 2 y′ + 5 y = 10 cos(2t)。对应的复常微分方程为 z″ + 2 z′ + 5 z = 10 e^{2 i t}。设 z_p = C e^{2 i t}。代入后得 (−4 + 4i + 5) C = 10 → (1 + 4i) C = 10。因此 C = 10 / (1 + 4i)。分子分母同乘以共轭复数 1 − 4i： C = 10(1 − 4i) / (1 + 16) = (10 − 40i) / 17 = 10/17 − i 40/17。模为 |C| = √( (10/17)² + (40/17)² ) = √( (100 + 1600) / 289 ) = √1700/289 = √100×17 / 289 = (10/17)√17 = 10/√17。辐角 φ = arg(C) = arctan( (−40/17) / (10/17) ) = arctan(−4) ≈ −1.3258 rad（或 −75.96°）。因此实数特解为 y_p = Re(C e^{2 i t}) = (10/√17) cos(2t − 1.3258)。振幅约为 10/√17 ≈ 2.425，相位滞后约为 1.33 rad。

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10. Advantages of the Complex Method | 复数法的优势

Compared with the method of undetermined coefficients using real trigonometric trial solutions, the complex exponential approach offers several striking benefits. First, it avoids the need to recall or derive identities like cos(ωt + φ) expansions. Second, a single complex trial function C e^iωt replaces the trial combination A cos(ωt) + B sin(ωt), and the constants A and B would otherwise require solving a system of two equations. Third, the phase and amplitude emerge directly from the polar form of the complex constant, giving an immediate physical interpretation. Fourth, the method extends elegantly to any forcing term that is a linear combination of sines and cosines, using the principle of superposition. These advantages make the complex method the standard tool in analysing AC circuits, mechanical vibrations, and control systems.

与使用实数三角试探解的待定系数法相比，复指数方法具有几个显著的优点。首先，它无需记忆或推导诸如 cos(ωt + φ) 展开的恒等式。其次，单一的复试探函数 C e^iωt 取代了试探组合 A cos(ωt) + B sin(ωt)，而原本需要求解含两个方程的方程组来确定 A 和 B。第三，相位和振幅直接从复常数的极坐标形式得出，可立即给出物理解释。第四，该方法可以优雅地扩展到任何由正弦和余弦线性组合而成的强迫项，只需使用叠加原理。这些优点使复数方法成为分析交流电路、机械振动和控制系统中的标准工具。

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11. Summary | 总结

We have seen how a harmonic forcing term in a second‑order linear ODE can be handled efficiently by switching to a complex ODE. By taking the real part of a complex exponential forcing term, the particular integral reduces to algebraic manipulation of complex numbers. The key formulas are:

• Complex ODE: z″ + p z′ + q z = R e^iωt
• Complex particular solution: z_p = C e^iωt with C = R / (q − ω² + i p ω)
• Real steady‑state solution: y_p = |C| cos(ωt + φ) = D cos(ωt − δ)
• Amplitude: D = R / √( (q − ω²)² + (p ω)² )
• Phase lag: δ = arctan( p ω / (q − ω²) ) (adjusted for quadrant).

This technique is not only a computational shortcut but also deepens our understanding of how systems respond to sinusoidal driving forces. Mastering it is essential for tackling problems in IB Higher Level Mathematics and beyond, especially in physics and engineering applications.

我们已经看到，通过转化为复常微分方程，可以高效地处理二阶线性常微分方程中的谐波强迫项。借助复指数强迫项的实部，特积分简化为复数的代数运算。核心公式为：

• 复ODE： z″ + p z′ + q z = R e^iωt
• 复特解： z_p = C e^iωt，其中 C = R / (q − ω² + i p ω)
• 实稳态解： y_p = |C| cos(ωt + φ) = D cos(ωt − δ)
• 振幅： D = R / √( (q − ω²)² + (p ω)² )
• 相位滞后： δ = arctan( p ω / (q − ω²) )（需根据象限调整）。

这一技术不仅是计算上的捷径，还加深了我们对系统如何响应正弦驱动力的理解。掌握它对解决IB高级数学乃至物理与工程应用中的问题至关重要。

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