📚 Features of Simple Harmonic Motion and the Oscillation Equation | 简谐运动特征与振动方程
Simple harmonic motion (SHM) is one of the most fundamental types of periodic motion in physics. It describes systems ranging from a mass on a spring to the vibration of atoms in a crystal. In this article, we examine its defining features, derive its kinematic equations, and explore the connections between displacement, velocity, acceleration and energy.
简谐运动（SHM）是物理学中最基本的周期运动类型之一，它描述从弹簧上的物体到晶体中原子的振动等各种系统。在本文中，我们将审视其定义特征，推导其运动学方程，并探索位移、速度、加速度与能量之间的联系。
1. Definition of Simple Harmonic Motion | 简谐运动的定义
A particle performs simple harmonic motion if its acceleration is directly proportional to its displacement from a fixed equilibrium position and is always directed towards that equilibrium position. Mathematically:
若一个质点的加速度与其相对于固定平衡位置的位移成正比，且始终指向平衡位置，则该质点做简谐运动。数学表达式为：
a = -ω²x
Here a is the acceleration, x is the displacement from equilibrium, and ω (omega) is the angular frequency. The negative sign indicates that acceleration always opposes displacement, ensuring that the motion is oscillatory rather than runaway.
其中 a 为加速度，x 为相对于平衡位置的位移，ω（欧米伽）为角频率。负号表示加速度始终与位移方向相反，从而保证运动是振荡性的而非发散的。
2. Key Characteristics of SHM | 简谐运动的关键特征
The defining features of SHM include:
简谐运动的定义性特征包括：
Restoring force is proportional to displacement: F = -kx, which is Hooke's law behaviour.
回复力与位移成正比：F = -kx，即胡克定律行为。
Acceleration is proportional to displacement and always opposes it, giving a = -ω²x.
加速度与位移成正比且方向始终相反，即 a = -ω²x。
Motion is periodic and oscillates symmetrically about a fixed equilibrium point.
运动是周期性的，并围绕固定平衡点对称振荡。
Displacement, velocity and acceleration are sinusoidal functions of time with the same frequency.
位移、速度和加速度均为同频率的时间正弦（或余弦）函数。
The total mechanical energy remains constant for an ideal undamped oscillator.
对于理想无阻尼振子，总机械能保持恒定。
3. Kinematic Quantities | 运动学量
The key quantities that describe SHM are amplitude, period, frequency and angular frequency.
描述简谐运动的关键物理量包括振幅、周期、频率和角频率。
Amplitude A: the maximum displacement from equilibrium; it determines the total energy of the oscillation. Amplitude is always positive and measured in metres.
振幅 A：离开平衡位置的最大位移，决定振动的总能量。振幅恒为正，单位为米。
Period T: the time taken for one complete oscillation, measured in seconds.
周期 T：完成一次全振动所需的时间，单位为秒。
Frequency f: the number of oscillations per second, f = 1/T, measured in hertz (Hz).
频率 f：每秒钟完成的振动次数，f = 1/T，单位为赫兹（Hz）。
Angular frequency ω: related to frequency and period by ω = 2πf = 2π/T, measured in rad s⁻¹.
角频率 ω：通过 ω = 2πf = 2π/T 与频率和周期相联系，单位为弧度每秒（rad s⁻¹）。
ω = 2πf = 2π/T
Angular frequency is particularly useful in SHM because the arguments of sine and cosine functions must be in radians, and ω naturally carries the unit rad s⁻¹.
角频率在简谐运动中尤为重要，因为正弦与余弦函数的自变量必须使用弧度，而 ω 天然带有 rad s⁻¹ 的单位。
4. The Displacement Equation | 位移方程
The displacement of an oscillator performing SHM can be expressed as a function of time:
做简谐运动的振子的位移可以表示为时间的函数：
x = A cos(ωt + φ)
m position moving in the positive direction, the equation may be written as x = A sin(ωt).
其中 x 为 t 时刻的位移，A 为振幅，ω 为角频率，φ 为相位常数（初相位），取决于运动的初始条件。若振子在 t = 0 时从最大位移处开始运动，则 φ = 0；若它从平衡位置沿正方向开始运动，则方程可写为 x = A sin(ωt)。
actly this sinusoidal form, which confirms that any system obeying the restoring-force condition will oscillate sinusoidally.
简谐运动微分方程 d²x/dt² = -ω²x 的通解正是这种正弦形式，这证实了任何满足回复力条件的系统都将做正弦振荡。
5. Velocity in SHM | 简谐运动中的速度
Differentiating the displacement equation with respect to time gives velocity:
将位移方程对时间求导，可得速度：
v = dx/dt = -ωA sin(ωt + φ)
The maximum speed is vmax A), the velocity is zero. There is also a useful relationship between velocity and displacement:
最大速度为 vmax = ωA，出现在振子经过平衡位置（x = 0）时。在端点（x = ±A）处，速度为零。速度与位移之间还有一个非常有用的关系式：
v = ±ω√A² - x²
This equation is obtained by eliminating time from the displacement and velocity equations, and it is extremely useful in energy calculations and exam problems where x and v are given.
该式通过消去位移方程与速度方程中的时间而得到，在能量计算以及给出 x 与 v 的考试题目中极为有用。
6. Acceleration in SHM | 简谐运动中的加速度
Differentiating the velocity equation gives acceleration:
对速度方程求导，可得加速度：
a = dv/dt = -ω²A cos(ωt + φ) = -ω²x
-ω²x. The acceleration is maximum at the extreme positions where displacement is maximum:
这证实了定义条件 a = -ω²x。在位移最大的端点处，加速度最大：
amax = ω²A
n is zero. The direction of acceleration is always towards the equilibrium position, and its magnitude is proportional to the instantaneous displacement.
在平衡位置 x = 0 处，加速度为零。加速度方向始终指向平衡位置，其大小与瞬时位移成正比。
7. Phase and Phase Difference | 相位与相位差
In SHM, the argument (ωt + φ) is called the phase. It determines the state of the oscillator at any instant. Two oscillators of the same frequency are said to be in phase if their phase difference is 0, and in antiphase if their phase difference is π (180°).
在简谐运动中，(ωt + φ) 这一参数称为相位，它决定振子在任一瞬时的运动状态。两个同频率的振子，若相位差为 0，则称为同相；若相位差为 π（180°），则称为反相。
Displacement and velocity differ in phase by π/2 (90°): velocity leads displacement by 90°.
位移与速度的相位差为 π/2（90°）：速度超前位移 90°。
Displacement and acceleration differ in phase by π (180°): acceleration is always opposite to displacement.
位移与加速度的相位差为 π（180°）：加速度始终与位移方向相反。
Velocity and acceleration differ in phase by π/2 (90°): acceleration leads velocity by 90°.
速度与加速度的相位差为 π/2（90°）：加速度超前速度 90°。
8. Energy in SHM | 简谐运动中的能量
During SHM, energy is continuously transformed between kinetic and potential forms. The total mechanical energy remains constant for an ideal (undamped) oscillator.
在简谐运动过程中，能量不断在动能与势能之间相互转化。对于理想（无阻尼）振子，总机械能保持恒定。
Etotal = ½kA² = ½mω²A²
The kinetic energy and potential energy at any displacement x are given by:
在任意位移 x 处，动能与势能分别为：
KE = ½mω²(A² - x²), PE = ½mω²x²
energy over one full cycle equals the average potential energy, each being half of the total energy.
在平衡位置（x = 0）处，全部能量为动能：KE = ½mω²A²。在端点（x = ±A）处，全部能量为势能：PE = ½mω²A²，动能为零。在一个完整周期内，平均动能等于平均势能，各占总能量的一半。
9. Graphical Representation | 图像表示
acement-time, velocity-time and acceleration-time graphs show the following relationships:
对于从最大位移处开始（φ = 0）的振子，位移-时间、速度-时间和加速度-时间图像表现出以下关系：
| Quantity 物理量 | Equation 方程 | Maximum value 最大值 | Value at x = 0 |
|---|---|---|---|
| Displacement 位移 | x = A cos(ωt) | A | 0 |
| Velocity 速度 | v = -ωA sin(ωt) | ωA | ±ωA |
| Acceleration 加速度 | a = -ω²A cos(ωt) | ω²A | 0 |
Note that velocity is 90° ahead of displacement, and acceleration is 180° out of phase with displacement (i.e. in antiphase). The slopes
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