📚 Resonance: The Mathematics of Forced Vibrations | 共振：强制振动的数学
Resonance is a phenomenon that occurs when a system is driven by an external periodic force whose frequency matches the system's natural frequency, leading to a dramatic increase in amplitude. In IB Mathematics, resonance provides a rich context for applying second-order linear differential equations, trigonometric identities, and amplitude–frequency analysis. Through mathematical modelling, we can predict the conditions under which resonance occurs and understand how damping affects the response.
共振是当系统受到频率与其固有频率相匹配的外界周期性驱动力作用时，振幅急剧增大的现象。在IB数学中，共振为应用二阶线性微分方程、三角恒等式以及振幅–频率分析提供了丰富的背景。通过数学建模，我们可以预测共振发生的条件，并理解阻尼如何影响系统的响应。
1. Introduction to Oscillations | 振动简介
Oscillatory motion is found in pendulums, springs, electrical circuits, and even in the swaying of bridges. Mathematically, such systems are often described by differential equations where the acceleration is proportional to the displacement and opposes it. Understanding free oscillations is the first step towards analysing forced oscillations and resonance.
振动运动在钟摆、弹簧、电路甚至桥梁的摇摆中都有体现。从数学上讲，这类系统通常用微分方程来描述，其中加速度与位移成正比且方向相反。理解自由振动是分析受迫振动和共振的第一步。
2. Simple Harmonic Motion (SHM) | 简谐运动
In simple harmonic motion, the restoring force follows Hooke's law. The governing equation for an undamped mass–spring system is m d2x/dt2 + k x = 0, which simplifies to d2x/dt2 + ω02 x = 0 where ω0 radic;(k/m) is the natural angular frequency. The general solution is x(t) = A cos(ω0t + φ) or a linear combination of sine and cosine, revealing a constant amplitude determined by initial conditions.
在简谐运动中，回复力遵循胡克定律。无阻尼质量–弹簧系统的控制方程为 m d2x/dt2 + k x = 0，可简化为 d2x/dt2 + ω02 x = 0，其中 ω0 = √(k/m) 是自然角频率。通解为 x(t) = A cos(ω0t + φ) 或正余弦的线性组合，显示出由初始条件决定的恒定振幅。
3. Forced Oscillations and the Governing Equation | 受迫振动与支配方程
When an external periodic force F(t) = F0 cos(ω t) acts on the system, the equation becomes non-homogeneous. Including damping, we have m d2x/dt2 + c dx/dt + k x = F0 cos(ω t). Dividing by m yields a standard form: d2x/dt2 + 2β dx/dt + ω02 x = A cos(ω t), where β = c/(2m) is the damping coefficient and A = F0/m. The interplay between the driving frequency ω, the natural frequency ω0, and the damping β governs the system's long-term behaviour.
当外部周期性力 F(t) = F0 cos(ω t) 作用在系统上时，方程变为非齐次形式。计入阻尼后，我们有 m d2x/dt2 + c dx/dt + k x = F0 cos(ω t)。两边除以 m 得到标准形式：d2x/dt2 + 2β dx/dt + ω02 x = A cos(ω t)，其中 β = c/(2m) 是阻尼系数，A = F0/m。驱动频率 ω、固有频率 ω0 和阻尼 β 之间的相互作用决定了系统的长期行为。
4. The Undamped Case and Resonance Catastrophe | 无阻尼情形与共振灾难
Consider the undamped scenario β = 0. The equation reduces to d2x/dt2 + ω02 x = A cos(ω t). Assuming a particular solution of the form xp = C cos(ω t), substitution gives C(ω02 – ω2) = A, so C = A/(ω02 – ω2). As the driving frequency ω approaches ω0, the denominator tends to zero, causing the amplitude to grow without bound. This unbounded response is known as resonance catastrophe and is why real structures must account for damping or avoid such frequencies.
考虑无阻尼情形 β = 0。方程简化为 d2x/dt2 + ω02 x = A cos(ω t)。假设特解具有形式 xp = C cos(ω t)，代入后得到 C(ω02 – ω2) = A，因此 C = A/(ω02 – ω2)。随着驱动频率 ω 趋近于 ω0，分母趋于零，振幅无限增大。这种无界响应被称为共振灾难，这也解释了为什么实际结构必须考虑阻尼或避开这样的频率。
5. Introducing Damping | 引入阻尼
Damping, represented by the term 2β dx/dt, dissipates energy and prevents infinite amplitudes. The homogeneous solution becomes a decaying exponential, leaving the steady-state particular solution as the dominant long-term response. For the underdamped case (β < ω0), the system can still exhibit a prominent resonance peak, but the amplitude remains finite. The steady-state solution is sought in the form xp = D cos(ω t – δ), where D is the amplitude and δ is the phase lag.
阻尼由项 2β dx/dt 表示，它会消耗能量并阻止振幅无限大。齐次解变为衰减的指数函数，使得稳态特解成为长期主导的响应。对于欠阻尼情况（β < ω0），系统仍能表现出显著的共振峰，但振幅保持有限。稳态解可设为 xp = D cos(ω t – δ)，其中 D 是振幅，δ 是相位滞后。
6. Steady-State Solution and Amplitude Response | 稳态解与振幅响应
Substituting xp = D cos(ω t – uation and comparing coefficients yields two equations. Solving these gives the amplitude:
D = A / √[( ω02 – ω2 )2 + (2βω)2]
and the phase angle tan δ = (2βω) / ( ω02 – ω2 ). This explicit function D(ω) allows us to study how the amplitude varies with the driving frequency.
将 xp = D cos(ω t – δ) 代入有阻尼的受迫方程并比较系数，得到两个方程。求解后得到振幅：
D = A / √[( ω02 – ω2 )2 + (2βω)2]
以及相位角 tan δ = (2βω) / ( ω02 – ω2 )。这一显式函数 D(ω) 使我们能够研究振幅如何随驱动频率变化。
7. Frequency Response Curve and Resonance Peak | 频率响应曲线与共振峰
Plotting D against ω produces a frequency response curve. For small β, the curve shows a sharp peak near ω0. The peak's height and width are controlled by the damping coefficient. As β increases, the peak lowers and broadens; eventually, if damping is sufficiently large, the peak disappears entirely. The maximum of D is not necessarily exactly at ω0 unless the damping is zero, a subtlety that reveals the mathematics of resonance.
绘制 D 关于 ω 的曲线可得到频率响应曲线。对于很小的 β，曲线在 ω0 附近出现尖锐的峰。峰的高度和宽度由阻尼系数控制。随着 β 增大，峰值降低并变宽；如果阻尼足够大，峰值最终完全消失。D 的最大值未必正好在 ω0 处，除非阻尼为零，这一微妙之处揭示了共振的数学本质。
| Damping level β (阻尼水平 β) | Peak behaviour (峰值行为) |
|---|---|
| β = 0 | Infinite at ω = ω₀ (在 ω = ω₀ 处无穷大) |
| 0 < β < ω₀/√2 | Finite peak at ωres > 0 (在 ωres > 0 处有限峰) |
| β ≥ ω₀/√2 | No peak, monotonically decreasing (无峰，单调递减) |
8. Derivation of the Resonant Frequency | 共振频率推导
is equivalent to minimising the denominator inside the square root, f(ω) = (ω02 – ω2)2 + (2βω)2. Setting f'(ω) = 0 gives –4ω(ω02 – ω2) + 8β2ω = 0, leading to ω(ω2 – ω02 + 2β2) = 0. The nontrivial solution yields the amplitude resonant frequency:
ωres = √( ω02 – 2β2 )
This expression is real only when β < ω0/√2. Notice that velocity resonance (maximum speed) occurs at ω = ω0 regardless of damping, which is a separate but related phenomenon.
为找到使振幅 D 最大化的频率，我们对 D 关于 ω 求导并令导数为零。这等价于最小化平方根内的分母 f(ω) = (ω02 – ω2)2 + (2βω)2。令 f'(ω) = 0 得到 –4ω(ω02 – ω2) + 8β2ω = 0，导出 ω(ω2 – ω02 + 2β2) = 0。非平凡解给出振幅共振频率：
ωres = √( ω02 – 2β2 )
该表达式仅在 β < ω0/√2 时为实数。注意速度共振（最大速度）发生在 ω = ω0 处，与阻尼无关，这是一个独立但相关的现象。
9. Quality Factor (Q-Factor) | 品质因子（Q因子）
The quality factor Q quantifies the sharpness of resonance. It is defined as Q = ω0 / (2β). A high Q indicates low damping and a narrow, tall resonance peak. In energy terms, Q is approximately the ratio of the total energy stored to the energy dissipated per cycle. The half-power bandwidth Δω is related by Δω = ω0 / Q = 2β. This makes Q a convenient parameter for characterising oscillatory systems in engineering and physics.
品质因子 Q 用于量化共振的尖锐程度，定义为 Q = ω0 / (2β)。高 Q 值表示低阻尼和窄而高的共振峰。从能量角度看，Q 近似等于系统总储能与每周期的能量耗散之比。半功率带宽 Δω 满足 Δω = ω0 / Q = 2β。这使得 Q 成为工程和物理中表征振动系统的一个方便参数。
10. Beats: Near-Resonance Phenomenon | 拍：近共振现象
When the driving frequency is close to the natural frequency with little damping, the solution can exhibit beats. For the undamped case with ω ≠ ω0, the complete solution (homogeneous + particular) can be re-expressed using trigonometric identities as:
x(t) = [2A/(ω02 – ω2)] sin( (ω0 – ω)t / 2 ) sin( (ω0 + ω)t / 2 )
The amplitude envelope oscillates at the beat frequency fbeat = |ω0 – ω| / (2π). This phenomenon is heard as a pulsing sound in acoustics and is a direct consequence of the linear superposition of two nearby frequencies.
当驱动频率接近固有频率且阻尼很小时，解可能呈现拍现象。对于无阻尼且 ω ≠ ω0 的情形，完整解（齐次解加特解）可通过三角恒等式重新表示为：
x(t) = [2A/(ω02 – ω2)] sin( (ω0 Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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