📚 Simple Harmonic Motion: Characteristics and Equations | 简谐运动的特征与方程
Simple harmonic motion (SHM) is one of the most fundamental models in physics, describing any periodic oscillation where the restoring force is directly proportional to the displacement from equilibrium. From pendulums to springs, from sound waves to alternating current, SHM provides a mathematical bridge between mechanical systems and wave phenomena.
简谐运动（SHM）是物理学中最基本的模型之一，它描述了恢复力与偏离平衡位置的位移成正比的周期性振荡。从单摆到弹簧，从声波到交流电，简谐运动为机械系统和波动现象之间架起了一座数学桥梁。
1. Definition of Simple Harmonic Motion | 简谐运动的定义
For a system to exhibit simple harmonic motion, the net restoring force F acting on the object must be proportional to its displacement x from the equilibrium position and directed opposite to that displacement. Mathematically, this condition is expressed as:
一个系统要呈现简谐运动，作用在物体上的净恢复力 F 必须与物体偏离平衡位置的位移 x 成正比，且方向与位移相反。这一条件用数学表达式可写为：
F = −kx
where k is the force constant (units: N·m⁻¹). The negative sign indicates that the force always acts toward the equilibrium position. This equation is known as the restoring force condition for SHM.
其中 k 为力常数（单位：N·m⁻¹）。负号表示力始终指向平衡位置。该方程被称为简谐运动的恢复力条件。
An object is said to be executing SHM if its acceleration is proportional to its displacement and always directed toward the equilibrium position. Since F = ma, we can write:
若物体的加速度与位移成正比且始终指向平衡位置，则该物体在做简谐运动。由 F = ma，可写出：
a = −(k/m)x = −ω²x
where ω = √(k/m) is the angular frequency. This acceleration condition is the defining equation of SHM.
其中 ω = √(k/m) 为角频率。这个加速度条件就是简谐运动的定义方程。
2. Key Physical Quantities | 关键物理量
To fully describe simple harmonic motion, several kinematic and dynamic quantities must be defined precisely:
要完整地描述简谐运动，我们必须精确定义若干运动学和动力学物理量：
- Displacement (x) — the distance from the equilibrium position at any instant. It is a vector quantity, ranging from −A to +A.
- 位移（x） —— 任意时刻离开平衡位置的距离。它是矢量，取值范围为 −A 到 +A。
- Amplitude (A) — the maximum displacement from equilibrium. It is always positive and measured in metres.
- 振幅（A） —— 离开平衡位置的最大位移。它恒为正值，单位为米。
- Period (T) — the time taken for one complete oscillation. Its SI unit is seconds. For a mass–spring system, T = 2π√(m/k).
- 周期（T） —— 完成一次全振动所需的时间。其国际单位制单位为秒。对弹簧振子系统，T = 2π√(m/k)。
- Frequency (f) — the number of complete oscillations per second, given by f = 1/T. Its SI unit is hertz (Hz).
- 频率（f） —— 每秒内完成的全振动次数，由 f = 1/T 给出。其国际单位制单位为赫兹（Hz）。
- Angular Frequency (ω) — the rate of change of phase, ω = 2πf = 2π/T. It is measured in rad·s⁻¹.
- 角频率（ω） —— 相位的变化率，ω = 2πf = 2π/T。其单位为 rad·s⁻¹。
- Phase (φ) — the argument of the sine or cosine function that specifies the state of oscillation. The initial phase (φ₀) is the phase at t = 0.
- 相位（φ） —— 正弦或余弦函数中的自变量，用于确定振动状态。初相位（φ₀）是 t = 0 时刻的相位。
3. The SHM Displacement Equation | 简谐运动的位移方程
The solution to the differential equation a = −ω²x depends on the initial conditions, giving rise to either a sine or cosine function. The general form of the displacement equation is:
微分方程 a = −ω²x 的解依赖于初始条件，从而产生正弦或余弦函数。位移方程的一般形式为：
x = A sin(ωt + φ₀)
where A is the amplitude, ω is the angular frequency, t is the time elapsed, and φ₀ is the initial phase constant. The term (ωt + φ₀) is called the phase of the motion.
其中 A 是振幅，ω 是角频率，t 是经过的时间，φ₀ 是初相位常数。表达式 (ωt + φ₀) 被称为运动的相位。
If the object starts from the equilibrium position moving in the positive direction, the initial phase is zero and the displacement is given by x = A sin(ωt). If the object starts at maximum displacement, the displacement is described by x = A cos(ωt).
如果物体从平衡位置沿正方向开始运动，初相位为零，位移由 x = A sin(ωt) 给出。如果物体从最大位移处开始运动，则位移由 x = A cos(ωt) 描述。
4. Velocity and Acceleration Equations | 速度与加速度方程
Taking the time derivative of the displacement equation yields the velocity as a function of time:
对位移方程求时间导数，得到速度随时间变化的函数：
v = dx/dt = Aω cos(ωt + φ₀)
The maximum velocity occurs when the object passes through equilibrium (x = 0) and is given by vₘₐₓ = Aω. Note that at the extreme positions, where x = ±A, the velocity is zero.
最大速度出现在物体经过平衡位置（x = 0）时，其值为 vₘₐₓ = Aω。注意在极端位置 x = ±A 处，速度为零。
Differentiating velocity once more gives the acceleration:
对速度再求一次导数，得到加速度：
a = dv/dt = −Aω² sin(ωt + φ₀) = −ω²x
This confirms that acceleration is proportional to negative displacement — a hallmark of SHM. The maximum acceleration is aₘₐₓ = Aω², occurring at the turning points.
这证实了加速度与负位移成正比——这是简谐运动的标志。最大加速度为 aₘₐₓ = Aω²，出现在转向点处。
5. Relationships Between x, v, and a | 位移、速度与加速度之间的关系
One of the most elegant features of SHM is the phase relationships among the three kinematic quantities. While displacement and acceleration are in anti-phase (180° apart), velocity leads displacement by 90°.
简谐运动最优雅的特征之一就是三个运动学量之间的相位关系。位移与加速度反相（相差180°），而速度领先位移90°。
At equilibrium (x = 0): velocity is maximum, acceleration is zero.
在平衡位置（x = 0）：速度最大，加速度为零。
At maximum displacement (x = ±A): velocity is zero, acceleration is maximum in magnitude.
在最大位移处（x = ±A）：速度为零，加速度大小最大。
| Position | x | v | a |
| Equilibrium | 0 | ±Aω (maximum) | 0 |
| Maximum displacement | ±A | 0 | ∓Aω² (maximum) |
位置关系表
Simple harmonic motion thus exhibits a constant interchange between kinetic and potential energy, which we will explore in the next section.
简谐运动因此表现出动能与势能之间的持续相互转换，我们将在下一节探讨这一点。
6. Energy in Simple Harmonic Motion | 简谐运动中的能量
In an ideal SHM system with no damping, mechanical energy is conserved. The total energy E is the sum of kinetic energy (Eₖ) and potential energy (Eₚ).
在没有阻尼的理想简谐运动系统中，机械能守恒。总能量 E 是动能（Eₖ）与势能（Eₚ）之和。
The kinetic energy is given by:
动能由下式给出：
Eₖ = ½mv² = ½m(Aω)²cos²(ωt + φ₀)
The potential energy stored in the system is:
存储在系统中的势能为：
Eₚ = ½kx² = ½kA²sin²(ωt + φ₀)
Since k = mω², the total energy becomes:
由于 k = mω²，总能量变为：
E = Eₖ + Eₚ = ½kA² = ½mω²A²
This expression shows that total energy is proportional to the square of the amplitude. When the oscillator reaches equilibrium, all energy is kinetic; at maximum displacement, all energy is potential.
该表达式表明总能量与振幅的平方成正比。当振子到达平衡位置时，所有能量均为动能；在最大位移处，所有能量均为势能。
7. Simple Harmonic Motion and Circular Motion | 简谐运动与圆周运动
A powerful way to visualise SHM is through its connection to uniform circular motion. If a particle moves in a circle of radius A with constant angular speed ω, then the projection of its position onto a diameter performs perfect SHM.
通过简谐运动与匀速圆周运动的联系，可以非常直观地理解简谐运动。若一个质点以恒定角速度 ω 在半径为 A 的圆上运动，则它在直径上的投影所做的是完美的简谐运动。
Consider a particle at angular position θ = ωt + φ₀ from the x-axis. Its x-coordinate is:
考虑一个从 x 轴起算角度位置为 θ = ωt + φ₀ 的质点。它的 x 坐标为：
x = A cos(ωt + φ₀)
which is the standard SHM displacement equation. Similarly, the y-component of the particle’s velocity gives the SHM velocity, and the centripetal acceleration’s component provides the SHM acceleration. This equivalence is often called the “reference circle” method.
这正是标准的简谐运动位移方程。类似地，质点速度的 y 分量给出简谐运动的速度，向心加速度的分量给出简谐运动的加速度。这种等价关系常被称为“参考圆”方法。
8. Worked Example: Mass–Spring System | 示例：弹簧振子系统
A block of mass 0.50 kg is attached to a light spring with force constant 200 N·m⁻¹ and set into SHM with an amplitude of 0.080 m. Determine: (a) the angular frequency, (b) the period, (c) the maximum speed, (d) the total energy.
一个质量为 0.50 kg 的物块连接在劲度系数为 200 N·m⁻¹ 的轻弹簧上，以振幅 0.080 m 做简谐运动。求：(a) 角频率；(b) 周期；(c) 最大速度；(d) 总能量。
(a) Angular frequency:
(a) 角频率：
ω = √(k/m) = √(200/0.50) = √400 = 20 rad·s⁻¹
(b) Period:
(b) 周期：
T = 2π/ω = 2π/20 = 0.314 s
(c) Maximum speed:
(c) 最大速度：
vₘₐₓ = Aω = 0.080 × 20 = 1.6 m·s⁻¹
(d) Total energy:
(d) 总能量：
E = ½kA² = ½ × 200 × (0.080)² = 0.64 J
These results illustrate how the defining parameters of SHM are interconnected. Changing the mass or the spring constant alters both the period and the energy characteristics of the system.
这些结果表明简谐运动各基本参数之间的相互联系。改变质量或弹簧常数会同时影响系统的周期和能量特性。
9. Common Misconceptions | 常见误区
Many students confuse velocity and acceleration in SHM. A common error is to assume that maximum acceleration occurs when velocity is maximum. In fact, the opposite is true: acceleration is maximum at the extremes where velocity vanishes.
许多学生容易混淆简谐运动中的速度与加速度。一个常见错误是认为最大加速度与最大速度同时出现。事实上恰恰相反：加速度在速度为零的极端位置达到最大。
Another misconception is that period depends on amplitude. For an ideal spring, the period T = 2π√(m/k) is independent of amplitude — a property known as isochronism. The same applies to small-angle oscillations of a pendulum.
另一个误区是认为周期与振幅有关。对理想弹簧而言，周期 T = 2π√(m/k) 与振幅无关——该性质称为等时性。单摆的小角度摆动同样如此。
Finally, students often forget the role of initial conditions. The full description of SHM requires both the amplitude and the initial phase, which are determined by how the motion is started.
最后，学生常忽略初始条件的作用。完整描述简谐运动需要同时知道振幅和初相位，这两个参数取决于运动是如何开始的。
10. SHM in Real-World Systems | 简谐运动在现实系统中的应用
While ideal SHM assumes no energy loss, real oscillators are subject to damping. In addition, many systems approximate SHM for small amplitudes: buildings swaying in wind, quartz crystals in watches, electric circuits with inductors and capacitors, and even the vibrations of atoms in a crystal lattice.
尽管理想简谐运动假设没有能量损失，实际振子总会受到阻尼。此外，许多系统在小振幅条件下近似做简谐运动：风中摇摆的建筑物、手表中的石英晶体、含有电感和电容的电路，甚至晶格中原子的振动。
The pendulum is a particularly instructive example. For small angular displacement (θ < 10°), sin θ ≈ θ, and the pendulum's motion becomes approximately simple harmonic with period:
单摆是一个特别有启发性的例子。在小角度位移条件下（θ < 10°），sin θ ≈ θ，摆的运动近似简谐运动，其周期为：
T = 2π√(L/g)
where L is the pendulum length and g is the gravitational acceleration. Notably, the period does not depend on the mass of the bob.
其中 L 是摆长，g 是重力加速度。值得注意的是，周期与摆锤的质量无关。
Published by TutorHao | 物理 Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导