Forced Oscillations and Transient Phenomena | 受迫振荡与瞬态现象

📚 Forced Oscillations and Transient Phenomena | 受迫振荡与瞬态现象

In IB Mathematics, the study of forced oscillations and transient phenomena brings together differential equations, damping, and periodic forcing. This article explores the mathematical structure of such systems, with emphasis on the transient solution and the steady-state solution.

在IB数学中，受迫振荡与瞬态现象的研究将微分方程、阻尼和周期强迫项结合在一起。本文将深入探讨这类系统的数学结构，重点分析瞬态解与稳态解。

1. The Differential Equation of Forced Oscillations | 受迫振荡的微分方程

Consider a damped mass-spring system subject to an external periodic force \(F_0\cos(\omega t)\) or \(F_0\sin(\omega t)\). The equation of motion is:

考虑一个受到外部周期力 \(F_0\cos(\omega t)\) 或 \(F_0\sin(\omega t)\) 作用的阻尼弹簧质量系统，其运动方程为：

m x″ + c x′ + k x = F₀ cos(ωt)

Here \(m\) is mass, \(c\) the damping coefficient, \(k\) the spring constant, and \(F_0\) the amplitude of the external force. Dividing by \(m\) gives the standard form:

其中 \(m\) 为质量，\(c\) 为阻尼系数，\(k\) 为弹簧刚度，\(F_0\) 为外力的振幅。两边除以 \(m\) 得到标准形式：

x″ + 2γ x′ + ωₙ² x = f₀ cos(ωt)

with \(\gamma = c/(2m)\), \(\omega_n = \sqrt{k/m}\), and \(f_0 = F_0/m\). The natural frequency \(\omega_n\) is the frequency with which the undamped, unforced system would oscillate freely.

其中 \(\gamma = c/(2m)\)，\(\omega_n = \sqrt{k/m}\)，\(f_0 = F_0/m\)。自然频率 \(\omega_n\) 是无阻尼、无外力时系统自由振荡的频率。

2. Homogeneous Solution and Transient Term | 齐次解与瞬态项

The homogeneous equation \(x″ + 2γ x′ + ωₙ² x = 0\) has solutions determined by the characteristic equation \(r² + 2γ r + ωₙ² = 0\). The roots are:

齐次方程 \(x″ + 2γ x′ + ωₙ² x = 0\) 的解由特征方程 \(r² + 2γ r + ωₙ² = 0\) 决定，其根为：

r = −γ ± √(γ² − ωₙ²)

Depending on the discriminant, the system is underdamped, critically damped, or overdamped. In the underdamped case, \(\gamma < \omega_n\), the homogeneous solution is:

根据判别式的取值，系统分为欠阻尼、临界阻尼或过阻尼。在欠阻尼情形（\(\gamma < \omega_n\)）下，齐次解为：

x_h(t) = e^(−γt) [ A cos(√(ωₙ² − γ²) t) + B sin(√(ωₙ² − γ²) t) ]

Because of the factor \(e^{-\gamma t}\), this part decays to zero as \(t \to \infty\). It is called the transient solution, since it represents the initial disturbance that dies out over time.

由于因子 \(e^{-\gamma t}\) 的存在，这部分随 \(t \to \infty\) 衰减至零，称为瞬态解，它代表随时间消逝的初始扰动。

Damping Condition	Roots	Homogeneous Solution Behaviour
Underdamped \(\gamma < \omega_n\)	Complex pair	Decaying oscillation
Critically damped \(\gamma = \omega_n\)	Repeated real root	Decays without oscillation
Overdamped \(\gamma > \omega_n\)	Distinct real roots	Slow exponential decay

In all damped cases, the homogeneous solution is transient. The constants \(A\) and \(B\) are determined by the initial displacement and velocity.

在所有阻尼情形下，齐次解都是瞬态的。常数 \(A\) 和 \(B\) 由初始位移和初始速度确定。

3. Particular Solution and Steady State | 特解与稳态

For a sinusoidal forcing function, we look for a particular solution with the same frequency \(\omega\):

对于正弦型强迫函数，我们寻找具有相同频率 \(\omega\) 的特解：

x_p(t) = C cos(ωt) + D sin(ωt)

Substituting into the differential equation and equating coefficients of \(\cos(\omega t)\) and \(\sin(\omega t)\) gives a pair of linear equations for \(C\) and \(D\). This particular solution is also called the steady-state solution, because it persists after the transient has died away.

将特解代入微分方程并比较 \(\cos(\omega t)\) 与 \(\sin(\omega t)\) 的系数，得到关于 \(C\) 和 \(D\) 的线性方程组。这一特解也称为稳态解，因为它在瞬态消失后仍然持续存在。

It is often convenient to write the steady-state solution in amplitude-phase form:

通常更方便将稳态解写成振幅-相位形式：

x_p(t) = R cos(ωt − φ)

where \(R\) is the steady amplitude and \(\phi\) is the phase lag between the forcing and the response.

其中 \(R\) 是稳态振幅，\(\phi\) 是强迫与响应之间的相位滞后。

4. Amplitude and Phase of the Steady Response | 稳态响应的振幅与相位

Using the equations from substitution, the amplitude \(R\) is found to be:

利用代入得到的方程，可求得振幅 \(R\)：

R = F₀ / √( (k − mω²)² + (cω)² )

Equivalently, in the standard form:

等价地，用标准形式表示：

R = f₀ / √( (ωₙ² − ω²)² + (2γω)² )

The phase lag \(\phi\) satisfies:

相位滞后 \(\phi\) 满足：

tan φ = cω / (k − mω²)

The quadrant of \(\phi\) must be chosen carefully: if \(k > m\omega^2\), then \(0 < \phi < \pi/2\); if \(k < m\omega^2\), then \(\pi/2 < \phi < \pi\). In the undamped case, \(\phi = 0\) for \(\omega < \omega_n\) and \(\phi = \pi\) for \(\omega > \omega_n\).

必须仔细确定 \(\phi\) 的象限：若 \(k > m\omega^2\)，则 \(0 < \phi < \pi/2\)；若 \(k < m\omega^2\)，则 \(\pi/2 < \phi < \pi\)。在无阻尼情形下，当 \(\omega < \omega_n\) 时 \(\phi = 0\)，当 \(\omega > \omega_n\) 时 \(\phi = \pi\)。

5. Transient vs Steady-State: The Total Solution | 瞬态与稳态：完整解

The general solution of a forced damped oscillator is the sum of the homogeneous and particular solutions:

受迫阻尼振荡器的通解是齐次解与特解之和：

x(t) = x_h(t) + x_p(t)

Because \(x_h(t)\) decays to zero as \(t\to\infty\), the long-term behaviour is completely described by \(x_p(t)\). The transient part carries the memory of the initial conditions; once it disappears, the motion becomes purely periodic at the driving frequency.

由于 \(x_h(t)\) 在 \(t\to\infty\) 时趋于零，系统的长期行为完全由 \(x_p(t)\) 描述。瞬态部分保留了初始条件的信息；一旦它消失，运动即变成完全以驱动频率周期运动的稳态。

In exam problems, students are often asked to determine both parts and then use initial conditions to find the arbitrary constants in the transient solution. The steady-state solution is independent of initial conditions.

在考试题目中，通常要求同时求出两部分，然后利用初始条件确定瞬态解中的任意常数。稳态解与初始条件无关。

6. Resonance and Practical Implications | 共振与实际意义

When the damping coefficient \(c\) is small, the amplitude \(R\) becomes very large as the driving frequency \(\omega\) approaches the natural frequency \(\omega_n\). This phenomenon is called resonance.

当阻尼系数 \(c\) 很小时，驱动频率 \(\omega\) 接近自然频率 \(\omega_n\)，振幅 \(R\) 会变得非常大，这种现象称为共振。

For a damped oscillator, the maximum amplitude occurs at a frequency slightly lower than \(\omega_n\). Differentiating \(R\) with respect to \(\omega\) gives:

对于阻尼振荡器，最大振幅出现在略低于 \(\omega_n\) 的频率处。对 \(R\) 关于 \(\omega\) 求导可得：

ω_res = √(ωₙ² − 2γ²)

provided \(\gamma < \omega_n/\sqrt{2}\). At exact resonance, the amplitude is \(R = F_0/(c\,\omega_n)\), which shows why small damping can lead to dangerously large oscillations.

前提是 \(\gamma < \omega_n/\sqrt{2}\)。在严格共振时，振幅为 \(R = F_0/(c\,\omega_n)\)，这说明了为什么小阻尼可能导致危险的大幅振荡。

This explains why engineers design structures and bridges to avoid resonance with environmental vibrations. The study of transient phenomena is also crucial in control systems and electrical circuits.

这解释了为什么工程师设计建筑和桥梁时要避免与环境振动发生共振。瞬态现象的研究在控制系统和电路中同样至关重要。

7. Beat Phenomenon and Transient Behaviour in Undamped Case | 无阻尼情况下的拍频与瞬态行为

If there is no damping (\(c=0\)) and the driving frequency is not equal to the natural frequency, the homogeneous part does not decay. For initial conditions \(x(0)=0\), \(x'(0)=0\), the solution is:

如果无阻尼（\(c=0\)）且驱动频率不等于自然频率，齐次部分不会衰减。对于初始条件 \(x(0)=0\)，\(x'(0)=0\)，解为：

x(t) = [F₀ / (m(ωₙ² − ω²))] [cos(ωt) − cos(ωₙ t)]

Using the trigonometric identity for the difference of cosines, this becomes:

利用余弦差的三角恒等式，可写成：

x(t) = [2F₀ / (m(ωₙ² − ω²))] sin((ωₙ+ω)t/2) sin((ωₙ−ω)t/2)

When \(\omega\) is close to \(\omega_n\), the \(\sin((\omega_n-\omega)t/2)\) factor acts as a slowly varying envelope, producing beats. Since there is no damping, this behaviour is not strictly transient; it persists forever. In the exact resonance case \(\omega = \omega_n\), the amplitude grows linearly with time:

当 \(\omega\) 接近 \(\omega_n\) 时，\(\sin((\omega_n-\omega)t/2)\) 因子作为缓慢变化的包络，产生拍频。由于没有阻尼，这种表现并非严格的瞬态，而是永久持续。在精确共振 \(\omega = \omega_n\) 时，振幅随时间线性增长：

x(t) = [F₀ / (2mωₙ)] t sin(ωₙ t)

8. Analysing a Worked Example | 例题解析

Consider the forced oscillator described by

考虑如下受迫振荡系统

x″ + 4x′ + 100x = 5 cos(6t), with x(0)=0 and x′(0)=0.

Here \(m=1\), \(c=4\), \(k=100\), \(F_0=5\), \(\omega=6\). The natural frequency is \(\omega_n=\sqrt{100}=10\), and \(\gamma=c/(2m)=2\). Thus \(\mu=\sqrt{\omega_n^2-\gamma^2}=\sqrt{96}=4\sqrt{6}\).

这里 \(m=1\)，\(c=4\)，\(k=100\)，\(F_0=5\)，\(\omega=6\)。自然频率 \(\omega_n=\sqrt{100}=10\)，且 \(\gamma=c

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Forced Oscillations and Transient Phenomena | 受迫振荡与瞬态现象