📚 Free and Forced Oscillations | 自由振荡与受迫振荡
When an object moves back and forth repeatedly about an equilibrium position, it performs an oscillation. In CIE A-Level Physics, oscillations are separated into two main types: free oscillations, where a system oscillates after an initial displacement without an external periodic force, and forced oscillations, where a periodic external driving force continuously supplies energy to the system. Understanding this distinction is essential for explaining damping, resonance, phase relationships and many real-world systems such as musical instruments, clocks, bridges and suspension systems.
当一个物体围绕平衡位置反复来回运动时，它就在进行振荡。在 CIE A-Level 物理中，振荡主要分为两类：自由振荡是系统在初始位移后、没有外部周期性驱动力作用下的振荡；受迫振荡则是周期性外部驱动力持续向系统提供能量而产生的振荡。理解这一区别对于解释阻尼、共振、相位关系以及乐器、钟表、桥梁和悬挂系统等许多实际系统都至关重要。
1. Oscillation Basics | 振荡基础
An oscillation is characterised by displacement x, amplitude A, period T and frequency f. The displacement is the distance from the equilibrium position at any instant, while the amplitude is the maximum displacement from equilibrium. The period T is the time for one complete oscillation, and the frequency f is the number of complete oscillations per unit time. Frequency and period are related by f = 1/T. Angular frequency ω is often used because it simplifies sinusoidal descriptions: ω = 2πf.
振荡由位移 x、振幅 A、周期 T 和频率 f 来表征。位移是任意时刻离开平衡位置的距离，而振幅是离开平衡位置的最大位移。周期 T 是完成一次完整振荡所需的时间，频率 f 是单位时间内完成完整振荡的次数。频率与周期之间的关系为 f = 1/T。角频率 ω 常用于简谐运动的描述，因为其能使正弦表达式更简洁：ω = 2πf。
f = 1/T and ω = 2πf
For example, a mass attached to a spring or a simple pendulum can both perform oscillations. The equilibrium position is where the net force is zero; if the system is displaced from this position, a restoring force acts to bring it back, causing repeated motion.
例如，连接在弹簧上的质量块或单摆都可以进行振荡。平衡位置是合力为零的位置；如果系统偏离该位置，就会受到一个指向平衡位置的回复力，从而产生反复运动。
2. Simple Harmonic Motion (SHM) | 简谐运动
Simple harmonic motion is a special type of free oscillation. In SHM, the restoring force F is directly proportional to the displacement x from equilibrium and always acts towards the equilibrium position. Mathematically, F = -kx, where k is the force constant. The minus sign shows that the force is always opposite to the displacement.
简谐运动是一种特殊的自由振荡。在简谐运动中，回复力 F 与离开平衡位置的位移 x 成正比，并且始终指向平衡位置。数学上可写为 F = -kx，其中 k 是力常数。负号表示力的方向始终与位移方向相反。
F = -kx
Using Newton’s second law, the acceleration in SHM is given by a = -ω²x. This condition is the defining feature of SHM: acceleration is proportional to displacement and directed opposite to it. The displacement can be described by x = A sin(ωt) or x = A cos(ωt), depending on the starting point.
根据牛顿第二定律，简谐运动中的加速度可写为 a = -ω²x。这一条件正是简谐运动的定义特征：加速度与位移成正比且方向相反。位移可以表示为 x = A sin(ωt) 或 x = A cos(ωt)，具体形式取决于初始时刻的位置。
a = -ω²x
Important derived results include the maximum speed vmax = ωA at the equilibrium position, the maximum acceleration amax = ω²A at the extremes, and the speed at an arbitrary displacement v = ±ω√(A² – x²). For a mass-spring system, ω = √(k/m), so T = 2π√(m/k). For a simple pendulum, ω = √(g/L), so T = 2π√(L/g).
重要的推导结果包括平衡位置处的最大速度 vmax = ωA、端点处的最大加速度 amax = ω²A，以及任意位移处的速度 v = ±ω√(A² – x²)。对于弹簧-质量系统，ω = √(k/m)，因此 T = 2π√(m/k)。对于单摆，ω = √(g/L)，因此 T = 2π√(L/g)。
3. Free Oscillations | 自由振荡
A free oscillation occurs when a system is displaced from its equilibrium position and then released, so it oscillates without any external periodic driving force. In an ideal free oscillation with no energy losses, the amplitude remains constant and the system oscillates forever at its natural frequency. In practice, resistive forces always remove some energy, so the amplitude gradually decreases unless energy is supplied.
自由振荡是指系统偏离平衡位置后释放，在没有外部周期性驱动力的情况下发生的振荡。在理想的无能量损耗自由振荡中，振幅保持不变，系统以固有频率永远振荡下去。实际上，阻力总是会带走一部分能量，因此如果不补充能量，振幅会逐渐减小。
The frequency of free oscillation is determined only by the physical properties of the system, such as mass, spring constant, length or gravitational field strength. Examples include a pendulum released from a small angle, a mass on a spring after being pulled and released, and a tuning fork after being struck.
自由振荡的频率仅由系统本身的物理性质决定，例如质量、弹簧劲度系数、摆长或重力场强度等。常见例子包括从一个小角度释放的单摆、被拉后释放的弹簧振子，以及被敲击后的音叉。
4. Natural Frequency | 固有频率
The natural frequency f₀ is the frequency at which a system freely oscillates when it is displaced from equilibrium and released. It is the system’s preferred frequency, determined by the balance between inertia and the restoring force. For a mass-spring system:
固有频率 f₀ 是系统离开平衡位置后自由振荡的频率。它是系统最倾向于保持的振荡频率，由惯性与回复力之间的平衡决定。对于弹簧-质量系统：
f₀ = (1/2π)√(k/m)
For a simple pendulum, the natural frequency is independent of the mass and depends on the length L and gravitational field strength g:
对于单摆，固有频率与质量无关，而取决于摆长 L 和重力场强度 g：
f₀ = (1/2π)√(g/L)
Every oscillating system has one or more natural frequencies. A guitar string, for instance, has a natural frequency determined by its length, tension and mass per unit length. When a system is forced to vibrate at a frequency other than f₀, the response is usually smaller than at f₀.
每个振荡系统都有一个或多个固有频率。例如，吉他的琴弦具有由其长度、张力和单位长度质量决定的固有频率。当系统被迫以不同于 f₀ 的频率振动时，其响应通常比在 f₀ 处要小。
5. Energy in Free Oscillations | 自由振荡中的能量
In an undamped free oscillation, total mechanical energy remains constant. Energy is continuously converted between kinetic energy and potential energy. At the equilibrium position, speed is maximum and kinetic energy is maximum, while potential energy is minimum. At the extreme positions, speed is zero and all the energy is stored as potential energy.
在无阻尼自由振荡中，总机械能保持不变。能量在动能和势能之间不断转化。在平衡位置，速度最大，动能最大，而势能最小。在最大位移处，速度为零，所有能量都以势能形式储存。
For a mass-spring system, the maximum potential energy is ½ kA² and the maximum kinetic energy is ½ mvmax². Since vmax = ωA and ω² = k/m, both give the same total energy:
对于弹簧-质量系统，最大势能为 ½ kA²，最大动能为 ½ mvmax²。由于 vmax = ωA 且 ω² = k/m，两种表达式给出的总能量相同：
Etotal = ½ kA²
This result shows that the total energy of a free oscillator is proportional to the square of the amplitude. If damping is present, energy is lost to the surroundings and the amplitude decreases with time, so the total energy also decreases.
这一结果表明，自由振荡器的总能量与振幅的平方成正比。如果存在阻尼，能量会散失到周围环境中，振幅随时间减小，因此总能量也会随之减小。
6. Damping | 阻尼
Damping is the removal of energy from an oscillating system, usually by friction or air resistance. Damping affects the amplitude of oscillations but, for light damping, it has little effect on the frequency. There are three main types of damping: light damping, critical damping and heavy damping.
阻尼是振荡系统能量的耗散，通常由摩擦或空气阻力引起。阻尼会影响振荡的振幅，但在轻度阻尼下，它对频率影响很小。阻尼主要有三种类型：轻度阻尼、临界阻尼和重度阻尼。
| Damping type | Typical behaviour |
|---|---|
| Light damping | Amplitude decreases gradually; system still completes many oscillations; period remains almost unchanged. |
| Critical damping | The system returns to equilibrium in the shortest possible time without oscillating. |
| Heavy damping | The system returns to equilibrium very slowly without oscillating. |
Light damping produces an exponential decrease in amplitude. Critical damping is important in car suspension and moving-coil meters, where the pointer should settle quickly without bouncing. Heavy damping may be useful in heavy doors or safety systems where oscillation must be suppressed completely.
轻度阻尼会使振幅呈指数式减小。临界阻尼在汽车悬挂和动圈式仪表中非常重要，因为指针需要快速稳定且不发生来回摆动。重度阻尼可用于重型门或安全系统等必须完全抑制振荡的场合。
7. Forced Oscillations | 受迫振荡
A forced oscillation occurs when a periodic external driving force acts on an oscillating system. In the steady state, the system oscillates at the driving frequency, not at its natural frequency. The amplitude of the forced oscillation depends on the size of the driving force, the amount of damping, and how close the driving frequency is to the natural frequency.
受迫振荡是指周期性外部驱动力作用在振荡系统上所产生的振荡。在稳定状态下，系统以驱动力的频率振荡，而不是以其固有频率振荡。受迫振荡的振幅取决于驱动力的大小、阻尼的大小以及驱动频率与固有频率的接近程度。
When a driving force is first applied, the system may briefly show transient motion involving both the driving frequency and its natural frequency. After a short time, the transient motion dies away due to damping, leaving only steady-state motion at the driving frequency.
当驱动力刚刚施加时，系统可能会出现短暂的瞬态运动，其中既包含驱动频率也包含固有频率。经过很短时间后，瞬态运动因阻尼而消失，只留下以驱动频率进行的稳态运动。
A useful mechanical model is a mass-spring system subjected to a sinusoidal external force. The mass initially resists the driving frequency, but soon settles into a steady oscillation whose frequency equals that of the driving force. The amplitude can be large or small depending on the frequency match.
一个有用的力学模型是受到正弦外力作用的弹簧-质量系统。质量块起初对驱动频率有排斥作用，但很快就会进入频率等于驱动力频率的稳定振荡状态。振幅大小取决于频率的匹配程度。
8. Resonance | 共振
Resonance is a special case of forced oscillations. It occurs when the driving frequency equals the natural frequency of the system, fdrive = f₀. At resonance, the system absorbs energy from the driving force most efficiently, and the amplitude of oscillation becomes much larger than at other driving frequencies.
共振是受迫振荡的一种特殊情况。当驱动频率等于系统的固有频率 fdrive = f₀ 时，就会发生共振。在共振时，系统从驱动力中吸收能量的效率最高，振荡振幅会远大于其他驱动频率下的振幅。
fdrive = f₀ → maximum amplitude
The sharpness of the resonance peak depends on damping. With light damping, the resonance peak is very high and narrow. With heavier damping, the peak is lower and broader, and the maximum amplitude occurs at a frequency slightly below the natural frequency in real damped systems. For CIE problems, the shift is often small and may be ignored unless stated.
共振峰的尖锐程度取决于阻尼的大小。轻度阻尼下，共振峰非常高且狭窄。阻尼较大时，共振峰较低且较宽，实际阻尼系统中共振频率会略低于固有频率。对于 CIE 问题，这一偏移通常很小，除非题目特别说明，否则常常可以忽略。
9. Phase Relationship in Forced Oscillations | 受迫振荡中的相位关系
In a forced oscillation, the way the displacement of the oscillator relates to the driving force depends on how the driving frequency compares with the natural frequency. At very low driving frequencies below f₀, the displacement is almost in phase with the driving force. The system moves slowly, and the force and displacement reach their peaks at nearly the same time.
在受迫振荡中，振子的位移与驱动力之间的相位关系取决于驱动频率与固有频率的比较。在远低于 f₀ 的驱动频率下，位移几乎与驱动力同相。系统运动较慢，驱动力和位移几乎同时达到最大值。
At resonance, the displacement lags behind the driving force by approximately π/2 radians (90°). This phase difference means that the driving force is in phase with the velocity of the oscillator, allowing maximum energy transfer per cycle and hence maximum amplitude.
在共振时，位移落后于驱动力约 π/2 弧度（90°）。这一相位差意味着驱动力与振子的速度同相，使得每个周期内能量传递最大，从而获得最大振幅。
At very high driving frequencies above f₀, the displacement approaches a phase difference of π radians (180°) with respect to the driving force. In this situation, the system cannot follow the fast driving force, and the force and displacement are almost opposite in phase.
在远高于 f₀ 的驱动频率下，位移与驱动力之间的相位差接近 π 弧度（180°）。在这种情况下，系统无法跟上快速变化的驱动力，力与位移几乎反相。
10. Applications and Safety | 应用与安全
Resonance is widely used in everyday technology. Musical instruments use resonance to amplify sound: the air column in a flute or the body of a guitar reinforces certain natural frequencies. Radio receivers use electrical resonance to select one station by adjusting a circuit so its natural frequency matches the carrier frequency. Microwave ovens use resonance of water molecules to heat food.
共振在日常技术中被广泛应用。乐器利用共振来放大声音：长笛中的空气柱或吉他的琴体增强了某些固有频率。无线电接收器利用电共振来选择一个电台，通过调节电路使其固有频率与载波频率匹配。微波炉则利用水分子的共振来加热食物。
However, resonance can also be dangerous if large amplitudes cause mechanical failure. The collapse of the Tacoma Narrows Bridge is often discussed as a warning example where wind-driven oscillations led to destructive resonance-like motion. Engineers therefore include damping or design structures so that their natural frequencies are far from possible driving frequencies such as wind loads, seismic waves or foot traffic.
然而，如果共振产生大幅振荡并引发机械破坏，就可能造成危险。塔科马海峡大桥的垮塌常被作为警示案例，当时风力驱动的振荡导致了破坏性的类共振运动。因此，工程师会加入阻尼，或将结构设计得使其固有频率远离风荷载、地震波或行人脚步等可能的驱动频率。
11. Graphical Analysis and Exam Tips | 图形分析与考试技巧
CIE questions often ask you to sketch a graph of amplitude against driving frequency for a forced oscillator. The graph shows a sharp peak at the natural frequency f₀ when damping is light. With increased damping, the peak shifts slightly to lower frequency, its height decreases, and the curve becomes broader.
CIE 考题经常要求画出受迫振荡器的振幅-驱动频率关系图。轻度阻尼时，该图在固有频率 f₀ 处显示出一个尖峰。阻尼增大时，峰值略微向低频方向移动，峰高降低，曲线变宽。
When interpreting such graphs, remember that the maximum amplitude does not occur at zero frequency; it occurs near f₀. Also remember that a forced oscillator does not oscillate at its natural frequency in the steady state. It oscillates at the driving frequency. The natural frequency only determines the condition for the largest response.
解读这类图形时，要记住最大振幅并不出现在零频率处，而是出现在 f₀ 附近。还要记住，受迫振荡器在稳定状态下并不以固有频率振荡，而是以驱动频率振荡。固有频率只决定最大响应的条件。
Exam questions may also ask you to identify natural frequency from a graph, compare damping levels, or explain how resonance can be reduced in a structure. Always refer to energy transfer: at resonance, energy is transferred most efficiently, and damping broadens the response because energy is removed more quickly.
考题还可能要求你从图中找出固有频率、比较阻尼大小，或解释如何减小结构中的共振。回答时始终要联系能量传递：共振时能量传递效率最高，而阻尼会因更快地耗散能量而使响应曲线变宽。
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