Simple Harmonic Motion (SHM): A Complete Guide — 简谐运动（SHM）：完整指南

📚 Simple Harmonic Motion | 简谐运动

Simple Harmonic Motion (SHM) is one of the most fundamental concepts in A-Level Physics. It describes a special type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and always acts towards the equilibrium position. From the oscillation of a mass on a spring to the swinging of a simple pendulum, SHM provides the mathematical framework for understanding countless natural and engineered systems.

简谐运动（SHM）是A-Level物理中最基础的概念之一。它描述了一种特殊的周期性运动：回复力与偏离平衡位置的位移成正比，并且始终指向平衡位置。从弹簧振子的振动到单摆的摆动，简谐运动为理解无数自然和工程系统提供了数学框架。

1. Defining Simple Harmonic Motion | 定义简谐运动

An object undergoes simple harmonic motion when two conditions are satisfied. First, the acceleration of the object must be directly proportional to its displacement from the equilibrium position. Second, the acceleration must always be directed towards the equilibrium position — that is, opposite in direction to the displacement. Mathematically, this is expressed as a = -ω²x, where a is acceleration, x is displacement, and ω is the angular frequency of the oscillation.

当满足两个条件时，物体做简谐运动。第一，物体的加速度必须与其偏离平衡位置的位移成正比。第二，加速度必须始终指向平衡位置——即与位移方向相反。数学上表达为 a = -ω²x，其中 a 是加速度，x 是位移，ω 是振动的角频率。

The negative sign in the defining equation is not merely a mathematical formality — it captures the essential physics of SHM. Because the restoring force always points towards equilibrium, the acceleration and displacement always have opposite signs. When the object is displaced to the right of equilibrium, the restoring force (and therefore acceleration) points to the left, driving the object back towards the centre.

定义方程中的负号不仅是数学形式——它捕捉了简谐运动的核心物理本质。由于回复力始终指向平衡位置，加速度和位移总是符号相反。当物体偏离到平衡位置右侧时，回复力（因此加速度）指向左侧，推动物体返回中心。

2. The Mathematics of SHM | 简谐运动的数学

The displacement of an object in SHM can be described by sinusoidal functions of time. The two most common forms are x(t) = A cos(ωt) and x(t) = A sin(ωt), where A is the amplitude (maximum displacement) and ω is the angular frequency. The choice between sine and cosine depends on the initial conditions — where the object starts at t = 0. If the object starts at maximum displacement (x = A), use cosine. If it starts at equilibrium moving in the positive direction, use sine.

简谐运动中物体的位移可以用时间的正弦函数来描述。两种最常见的形式是 x(t) = A cos(ωt) 和 x(t) = A sin(ωt)，其中 A 是振幅（最大位移），ω 是角频率。选择正弦还是余弦取决于初始条件——物体在 t = 0 时的位置。如果物体从最大位移处开始（x = A），使用余弦。如果从平衡位置朝正方向开始运动，使用正弦。

The angular frequency ω is related to the period T and the ordinary frequency f by the equations ω = 2π/T and ω = 2πf. The period is the time taken for one complete oscillation, and the frequency is the number of oscillations per unit time. These relationships are universal for all SHM systems, regardless of whether we are dealing with springs, pendulums, or floating objects.

角频率 ω 与周期 T 和普通频率 f 的关系为 ω = 2π/T 和 ω = 2πf。周期是完成一次完整振动所需的时间，频率是单位时间内的振动次数。这些关系对所有的简谐运动系统都是通用的，无论我们处理的是弹簧、摆还是浮体。

Differentiating the displacement equation gives us the velocity: v(t) = -Aω sin(ωt) for the cosine form, or v(t) = Aω cos(ωt) for the sine form. The maximum velocity occurs when the object passes through the equilibrium position and is given by v_max = Aω. The velocity is zero at the extreme positions where the object momentarily stops before reversing direction.

对位移方程求导得到速度：对于余弦形式 v(t) = -Aω sin(ωt)，对于正弦形式 v(t) = Aω cos(ωt)。最大速度出现在物体通过平衡位置时，由 v_max = Aω 给出。在极端位置速度为零，物体在反转方向前瞬间停止。

Differentiating again gives the acceleration: a(t) = -Aω² cos(ωt) = -ω²x(t), which is exactly the defining equation of SHM. The maximum acceleration occurs at the extreme positions and is a_max = Aω². This is where the restoring force is strongest, and it is often where students make errors in exam calculations — forgetting to square the angular frequency is a common mistake.

再次求导得到加速度：a(t) = -Aω² cos(ωt) = -ω²x(t)，这正是简谐运动的定义方程。最大加速度出现在极端位置，a_max = Aω²。这是回复力最强的地方，也是学生在考试计算中经常出错的地方——忘记对角频率平方是一个常见错误。

3. Energy in Simple Harmonic Motion | 简谐运动中的能量

One of the most elegant aspects of SHM is the continuous interchange between kinetic energy and potential energy. At the equilibrium position, all the energy is kinetic and the speed is maximum. At the extreme positions, all the energy is potential and the speed is zero. Throughout the oscillation, the total mechanical energy remains constant (assuming no damping), given by E_total = (1/2)mω²A².

简谐运动最优美的方面之一是动能和势能之间的持续转换。在平衡位置，所有能量为动能，速度最大。在极端位置，所有能量为势能，速度为零。在整个振动过程中，总机械能保持不变（假设无阻尼），由 E_total = (1/2)mω²A² 给出。

The kinetic energy at any point is KE = (1/2)mv² = (1/2)mω²(A² – x²). The potential energy is PE = (1/2)mω²x². Notice that both expressions contain mω²/2, making it easy to verify that KE + PE = (1/2)mω²A² at every position. This conservation of energy is a powerful tool for solving SHM problems — often simpler than working through the full differential equations.

任意点的动能为 KE = (1/2)mv² = (1/2)mω²(A² – x²)。势能为 PE = (1/2)mω²x²。注意两个表达式都包含 mω²/2，这使得验证每一位置 KE + PE = (1/2)mω²A² 变得容易。能量守恒是解决简谐运动问题的有力工具——通常比解完整微分方程更简单。

4. The Mass-Spring System | 弹簧振子系统

The mass-spring system is the canonical example of SHM. For a mass m attached to a spring of spring constant k, the restoring force is given by Hooke’s Law: F = -kx. Applying Newton’s Second Law (F = ma) yields ma = -kx, which rearranges to a = -(k/m)x. Comparing with the defining equation a = -ω²x, we find that ω² = k/m, so the angular frequency is ω = sqrt(k/m) and the period is T = 2π sqrt(m/k).

弹簧振子系统是简谐运动的典型例子。对于连接在劲度系数为 k 的弹簧上的质量 m，回复力由胡克定律给出：F = -kx。应用牛顿第二定律 F = ma 得到 ma = -kx，整理为 a = -(k/m)x。与定义方程 a = -ω²x 比较，我们得到 ω² = k/m，因此角频率为 ω = sqrt(k/m)，周期为 T = 2π sqrt(m/k)。

This result tells us something important: the period of a mass-spring system depends only on the mass and the spring constant, not on the amplitude of oscillation. Doubling the mass increases the period by a factor of sqrt(2). Doubling the spring constant decreases the period by a factor of 1/sqrt(2). This amplitude independence is a defining characteristic of SHM and is known as isochronism.

这个结果告诉我们一个重要的结论：弹簧振子系统的周期仅取决于质量和劲度系数，而与振幅无关。将质量加倍会使周期增加 sqrt(2) 倍。将劲度系数加倍会使周期减小到 1/sqrt(2) 倍。这种与振幅无关的特性是简谐运动的定义性特征，被称为等时性。

In exam problems, students must also consider the effective spring constant when springs are combined. For springs in parallel, k_eff = k1 + k2. For springs in series, 1/k_eff = 1/k1 + 1/k2. These combinations appear regularly in A-Level questions, particularly when a mass is suspended between two springs.

在考试题目中，学生还需要考虑弹簧组合时的等效劲度系数。并联弹簧：k_eff = k1 + k2。串联弹簧：1/k_eff = 1/k1 + 1/k2。这些组合在A-Level题目中经常出现，特别是当质量悬挂在两个弹簧之间时。

5. The Simple Pendulum | 单摆

The simple pendulum consists of a point mass (the bob) suspended from a fixed point by a light, inextensible string. When displaced by a small angle, the bob executes SHM. The restoring force is the component of weight tangential to the arc: F = -mg sin θ. For small angles (θ less than about 10 degrees), sin θ ≈ θ, and with the arc length s = Lθ, the equation reduces to a = -(g/L)s, giving ω = sqrt(g/L) and T = 2π sqrt(L/g).

单摆由一个悬挂在固定点上的质点（摆锤）和一根轻质、不可伸长的绳子组成。当偏离一个小角度时，摆锤做简谐运动。回复力是重力沿弧线切向的分量：F = -mg sin θ。对于小角度（θ 小于约10度），sin θ ≈ θ，结合弧长 s = Lθ，方程简化为 a = -(g/L)s，得到 ω = sqrt(g/L) 和 T = 2π sqrt(L/g)。

Note the critical small-angle approximation. Without it, the pendulum’s motion is not simple harmonic — it becomes a nonlinear oscillator whose period depends on amplitude. For angles beyond about 10 degrees, the period increases with amplitude, and the simple formula T = 2π sqrt(L/g) becomes increasingly inaccurate. This is why grandfather clocks use small-amplitude pendulums.

注意关键的小角度近似。没有它，摆的运动就不是简谐运动——它变成了非线性振动，其周期取决于振幅。对于超过约10度的角度，周期随振幅增加而增加，简单的公式 T = 2π sqrt(L/g) 变得越来越不准确。这就是落地钟使用小振幅摆的原因。

The independence of period from mass is a striking feature of the simple pendulum. Unlike the mass-spring system where T ∝ sqrt(m), the pendulum period depends only on length and gravitational field strength. Galileo reportedly discovered this while watching a swinging chandelier in Pisa Cathedral, using his pulse to time the oscillations.

周期与质量无关是单摆的一个显著特征。与弹簧振子系统中 T ∝ sqrt(m) 不同，摆的周期仅取决于长度和重力场强度。据传伽利略在比萨大教堂观察吊灯摆动时发现了这一点，用自己的脉搏来计时摆动。

6. Graphical Representations | 图形表示

Understanding the displacement-time, velocity-time, and acceleration-time graphs is essential for A-Level Physics exams. For an oscillator starting at maximum positive displacement (x = A at t = 0), the displacement graph is a cosine curve starting at +A. The velocity graph is a negative sine curve — it starts at zero, becomes negative (moving towards equilibrium), reaches its most negative value at equilibrium, and then returns to zero at the negative extreme.

理解位移-时间、速度-时间和加速度-时间图对A-Level物理考试至关重要。对于从最大正位移开始的振子（t = 0 时 x = A），位移图是从 +A 开始的余弦曲线。速度图是负正弦曲线——从零开始，变为负值（向平衡位置运动），在平衡位置达到最负值，然后在负极端恢复到零。

The acceleration graph is simply the displacement graph scaled by -ω² and reflected across the time axis. When displacement is at its maximum positive value, acceleration is at its maximum negative value. When displacement is zero (at equilibrium), acceleration is also zero. This phase relationship — acceleration always opposite in sign to displacement — is the graphical signature of SHM and a common exam question topic.

加速度图就是位移图乘以 -ω² 并关于时间轴翻转。当位移处于最大正值时，加速度处于最大负值。当位移为零（在平衡位置）时，加速度也为零。这种相位关系——加速度始终与位移符号相反——是简谐运动的图形标志，也是常见的考题主题。

The phase difference between displacement and velocity is π/2 (90 degrees). Velocity leads displacement by a quarter of a cycle. The phase difference between displacement and acceleration is π (180 degrees) — they are in antiphase. Between velocity and acceleration, the phase difference is also π/2. Mastering these phase relationships allows students to sketch any SHM graph from any starting condition.

位移和速度之间的相位差是 π/2（90度）。速度超前位移四分之一个周期。位移和加速度之间的相位差是 π（180度）——它们是反相的。速度与加速度之间的相位差同样为 π/2。掌握这些相位关系使学生能够从任何初始条件绘制任何简谐运动图像。

7. Damped Harmonic Motion | 阻尼简谐运动

In real physical systems, energy is gradually lost to the surroundings through friction, air resistance, or internal forces. This energy dissipation causes the amplitude of oscillation to decrease over time — a phenomenon called damping. The mathematical treatment introduces a damping force proportional to velocity: F_damping = -bv, where b is the damping coefficient.

在真实物理系统中，能量通过摩擦、空气阻力或内力逐渐散失到环境中。这种能量耗散导致振幅随时间减小——这种现象称为阻尼。数学处理中引入了一个与速度成正比的阻尼力：F_damping = -bv，其中 b 是阻尼系数。

There are three regimes of damping. Light damping (underdamping) occurs when the system oscillates with a gradually decreasing amplitude — the most common case in everyday life, such as a car suspension bouncing over speed bumps. Critical damping occurs when the system returns to equilibrium in the shortest possible time without oscillating — this is the design goal for car shock absorbers and door-closing mechanisms. Heavy damping (overdamping) occurs when the system returns to equilibrium very slowly without oscillating, as in a heavily oiled piston.

阻尼有三种状态。轻阻尼（欠阻尼）发生在系统以逐渐减小的振幅振动时——这是日常生活中最常见的情况，如汽车悬挂通过减速带时的弹跳。临界阻尼发生在系统以最短时间返回平衡位置而不发生振荡时——这是汽车减震器和关门机构的设计目标。重阻尼（过阻尼）发生在系统非常缓慢地返回平衡位置而不发生振荡时，如重油润滑的活塞。

A-Level students should be able to distinguish between these regimes from displacement-time graphs and understand the practical applications. Critical damping is particularly important in engineering because it provides the fastest return to equilibrium without the wear and discomfort of continued oscillation.

A-Level学生应能够从位移-时间图中区分这些状态并理解其实际应用。临界阻尼在工程中尤为重要，因为它提供了最快的返回平衡且没有持续振荡带来的磨损和不适。

8. Forced Oscillations and Resonance | 受迫振动和共振

When an oscillating system is subjected to a periodic external driving force, it undergoes forced oscillations. The system vibrates at the frequency of the driving force, not its natural frequency — though the amplitude of the response depends strongly on how close the driving frequency is to the natural frequency. This leads to the phenomenon of resonance, one of the most dramatic and practically important aspects of oscillatory physics.

当一个振动系统受到周期性的外部驱动力作用时，它进行受迫振动。系统以驱动力的频率振动，而不是其固有频率——尽管响应的振幅在很大程度上取决于驱动频率与固有频率的接近程度。这导致了共振现象，这是振动物理学中最引人注目且具有重要实际意义的方面之一。

Resonance occurs when the driving frequency equals the natural frequency of the system. At resonance, the amplitude of oscillation becomes dramatically large — theoretically infinite in an undamped system, though damping always limits it in practice. The sharpness of the resonance peak depends on the degree of damping: light damping produces a sharp, tall peak, while heavy damping produces a broad, low peak.

当驱动频率等于系统固有频率时发生共振。在共振状态下，振幅变得非常巨大——在无阻尼系统中理论上为无穷大，尽管实际中阻尼总有限制。共振峰的尖锐程度取决于阻尼程度：轻阻尼产生尖锐的高峰，而重阻尼产生宽而低的峰。

The Tacoma Narrows Bridge collapse in 1940 is the classic cautionary tale of resonance. Wind-induced vibrations matched the bridge’s natural frequency, causing torsional oscillations that grew until the structure failed. On the constructive side, resonance is exploited in microwave ovens (water molecules resonate at microwave frequencies), MRI scanners (nuclear magnetic resonance), and musical instruments (air columns and strings resonating at specific frequencies).

1940年塔科马海峡大桥的坍塌是关于共振的经典警示故事。风致振动与桥梁的固有频率相匹配，导致扭转振动不断增大直至结构失效。在建设性方面，微波炉（水分子在微波频率下共振）、核磁共振扫描仪和乐器（空气柱和弦以特定频率共振）都利用了共振原理。

9. Experimental Determination of g Using a Pendulum | 利用单摆实验测定重力加速度 g

A classic A-Level practical investigation uses the simple pendulum to determine the acceleration due to gravity, g. The method is elegant in its simplicity: measure the period T for different pendulum lengths L, then plot T² against L. Since T² = (4π²/g)L, the graph should be a straight line through the origin with gradient 4π²/g. From the gradient, g = 4π²/gradient.

一个经典的A-Level实验研究利用单摆测定重力加速度 g。方法优雅而简洁：测量不同摆长 L 下的周期 T，然后绘制 T² 对 L 的图像。由于 T² = (4π²/g)L，图像应该是一条过原点的直线，斜率为 4π²/g。由斜率可得 g = 4π²/斜率。

Key experimental techniques include: timing multiple oscillations (typically 20) and dividing to reduce timing uncertainty; measuring the pendulum length from the point of suspension to the centre of the bob; keeping the angular amplitude small (less than 10 degrees) to satisfy the small-angle approximation; and repeating measurements to identify and reduce random errors. Systematic errors such as an incorrectly measured length produce a non-zero intercept, which can be discussed in the evaluation.

关键实验技巧包括：计时多次振动（通常20次）然后除以次数以减少计时不确定度；从悬点到摆锤中心测量摆长；保持小角度振幅（小于10度）以满足小角度近似；重复测量以识别和减少随机误差。系统误差（如长度测量不正确）会产生非零截距，这可以在评估中讨论。

Typical results should give a value for g within about 5% of the accepted value of 9.81 m/s². Sources of uncertainty include reaction time in starting and stopping the stopwatch, difficulty in judging the exact centre of oscillation, and air resistance causing slight damping. Students should be able to calculate percentage uncertainty and compare their result with the accepted value.

典型结果得出的 g 值应在公认值 9.81 m/s² 的约5%以内。不确定度的来源包括启动和停止秒表的反应时间、判断确切振动中心的困难以及空气阻力引起的轻微阻尼。学生应能够计算百分比不确定度并将结果与公认值进行比较。

10. Exam Tips and Common Mistakes | 考试技巧和常见错误

In A-Level Physics examinations, SHM questions frequently combine several concepts in a single problem. Students must be comfortable switching between the defining equation a = -ω²x, the displacement equations x = A cos(ωt) or x = A sin(ωt), and the energy equations KE = (1/2)mω²(A² – x²). The most common mistake is mixing up sine and cosine — always check the initial conditions before writing the displacement equation.

在A-Level物理考试中，简谐运动题目经常将多个概念结合在一个问题中。学生必须能够熟练地在定义方程 a = -ω²x、位移方程 x = A cos(ωt) 或 x = A sin(ωt) 以及能量方程 KE = (1/2)mω²(A² – x²) 之间切换。最常见的错误是混淆正弦和余弦——在写位移方程前一定要检查初始条件。

Another frequent error involves the period formulas. Students often confuse T = 2π sqrt(m/k) for mass-spring systems with T = 2π sqrt(L/g) for pendulums. Remember: mass-spring depends on m and k; pendulum depends on L and g. A useful mnemonic: “spring has mass in the formula” and “pendulum has length”. Writing the wrong formula is a costly error that loses marks even when all subsequent working is correct.

另一个常见错误涉及周期公式。学生经常混淆弹簧振子的 T = 2π sqrt(m/k) 和单摆的 T = 2π sqrt(L/g)。记住：弹簧振子取决于 m 和 k；单摆取决于 L 和 g。一个有用的记忆法：”弹簧公式中有质量”，”摆的公式中有长度”。写错公式是一个代价高昂的错误，即使后续所有计算正确也会扣分。

When tackling multi-step problems involving energy conservation, always start by writing the total energy E = (1/2)mω²A². Then determine what fraction of this energy is kinetic or potential at the given position. Many problems ask “at what displacement is the kinetic energy equal to the potential energy?” — the answer is x = A/√2, derived from setting (1/2)mω²(A² – x²) = (1/2)mω²x².

在处理涉及能量守恒的多步问题时，始终从写出总能量 E = (1/2)mω²A² 开始。然后确定在给定位置该能量中有多少是动能或势能。许多问题问”在什么位移处动能等于势能？”——答案是 x = A/√2，由令 (1/2)mω²(A² – x²) = (1/2)mω²x² 得出。

Finally, practice drawing and interpreting graphs. Be able to sketch displacement, velocity, and acceleration against time for both sine and cosine starting conditions. Label key values: amplitude A for displacement, Aω for velocity, and Aω² for acceleration. Mark period T clearly on the time axis. These graphs are almost guaranteed to appear in A-Level Physics Paper 2 or Paper 4.

最后，练习绘制和解读图像。能够为正弦和余弦起始条件分别绘制位移、速度和加速度随时间变化的图。标注关键值：位移的振幅 A，速度的 Aω，加速度的 Aω²。在时间轴上清楚标注周期 T。这些图像几乎肯定会在A-Level物理卷二或卷四中出现。

11. Summary | 总结

Simple harmonic motion is defined by the relationship a = -ω²x, where acceleration is proportional to and opposite in direction to displacement. The mathematics of SHM is described by sinusoidal functions: x = A cos(ωt) or x = A sin(ωt), depending on initial conditions. The key parameters are amplitude A, angular frequency ω, period T = 2π/ω, and frequency f = 1/T = ω/2π.

简谐运动由关系式 a = -ω²x 定义，其中加速度与位移成正比且方向相反。简谐运动的数学由正弦函数描述：x = A cos(ωt) 或 x = A sin(ωt)，取决于初始条件。关键参数是振幅 A、角频率 ω、周期 T = 2π/ω 和频率 f = 1/T = ω/2π。

Energy in SHM continuously converts between kinetic and potential forms, with total energy E = (1/2)mω²A² conserved in the absence of damping. Real systems experience damping, which reduces amplitude over time, and forced oscillations can lead to resonance when the driving frequency matches the natural frequency. Mastering these concepts not only prepares students for examination success but also builds the foundation for understanding wave phenomena, alternating current circuits, and quantum mechanical oscillators at higher levels of study.

简谐运动中的能量在动能和势能之间持续转换，总能量 E = (1/2)mω²A² 在无阻尼时守恒。真实系统经历阻尼，使振幅随时间减小，受迫振动在驱动频率与固有频率匹配时导致共振。掌握这些概念不仅为学生的考试成功做准备，也为更高层次学习中理解波动现象、交流电路和量子力学谐振子打下基础。