Oscillations | 振动 - aleveler.com

📚 Oscillations | 振动

Oscillations are repeated back-and-forth motions about a fixed equilibrium position. In CIE A Level Physics, simple harmonic motion (SHM) is the most important type of oscillation because it gives a clear mathematical model for systems such as pendulums, mass-spring systems and many real vibrating objects. Understanding displacement, velocity, acceleration, energy changes, damping and resonance is essential for tackling both theoretical and practical questions.

振动是物体围绕固定平衡位置所做的来回往复运动。在 CIE A Level 物理中，简谐运动（SHM）是最重要的振动类型，因为它为单摆、弹簧振子以及许多真实振动系统提供了清晰的数学模型。理解位移、速度、加速度、能量变化、阻尼和共振对于解决理论和实验问题都至关重要。

1. Simple Harmonic Motion (SHM) Definition | 简谐运动定义

Simple harmonic motion occurs when the acceleration of an object is directly proportional to its displacement from equilibrium and is always directed towards that equilibrium position. This can be written as a ∝ −x, which leads to the defining equation:

当物体的加速度与其偏离平衡位置的位移成正比，并且始终指向平衡位置时，物体就做简谐运动。这可以写成 a ∝ −x，由此得到定义方程：

a = −ω²x

Here ω is the angular frequency, measured in rad s⁻¹. The negative sign is important because it shows that acceleration and displacement act in opposite directions. In SHM the restoring force is also proportional to displacement, so F = −kx for a spring system.

式中 ω 是角频率，单位为 rad s⁻¹。负号很重要，因为它表明加速度与位移方向相反。在简谐运动中，回复力也与位移成正比，例如弹簧系统中 F = −kx。

For an oscillation to be truly simple harmonic, the acceleration–displacement graph must be a straight line through the origin with a negative gradient equal to −ω². This linear relationship distinguishes SHM from other back-and-forth motions.

对于真正的简谐运动，加速度-位移图像必须是过原点的直线，且斜率为负，等于 −ω²。这种线性关系将简谐运动与其他往复运动区分开来。

2. Key Quantities in SHM | 简谐运动的关键物理量

Several quantities are used to describe oscillations. Displacement x is the distance from equilibrium at any time. Amplitude x₀ is the maximum displacement. Period T is the time for one complete oscillation, and frequency f is the number of oscillations per second:

描述振动时会用到多个物理量。位移 x 是任意时刻物体偏离平衡位置的距离。振幅 x₀ 是最大位移。周期 T 是完成一次完整振动所需的时间，频率 f 是每秒振动的次数：

f = 1/T

The angular frequency ω is related to frequency and period by ω = 2πf = 2π/T. Phase difference φ describes how much one oscillation is ahead of or behind another, measured in radians.

角频率 ω 与频率和周期的关系为 ω = 2πf = 2π/T。相位差 φ 描述两个振动之间超前或落后的程度，单位为弧度。

Quantity	Symbol	SI unit
Displacement	x	m
Amplitude	x₀	m
Period	T	s
Frequency	f	Hz
Angular frequency	ω	rad s⁻¹
Phase difference	φ	rad

These definitions are tested frequently, so it is important to use precise wording in exam answers. For example, the period must be for one complete oscillation, not just one swing from left to right.

这些定义经常出现在考试中，因此答题时必须使用准确的表述。例如，周期是指一次完整振动的时间，而不仅仅是从左到右的一次摆动。

3. Equations of Motion for SHM | 简谐运动的运动方程

If timing starts when the object passes through equilibrium moving in the positive direction, the displacement can be written as:

如果从物体经过平衡位置并向正方向运动时开始计时，位移可以表示为：

x = x₀ sin(ωt)

The velocity is the derivative of displacement with respect to time:

速度是位移对时间的导数：

v = ωx₀ cos(ωt)

The acceleration is the derivative of velocity, giving a = −ω²x₀ sin(ωt), which simplifies to a = −ω²x. A very useful alternative equation links velocity to displacement without time:

加速度是速度对时间的导数，得到 a = −ω²x₀ sin(ωt)，简化为 a = −ω²x。另一个非常有用的方程将速度与位移联系起来，而不显含时间：

v = ±ω√(x₀² − x²)

This equation shows that speed is greatest when x = 0 and zero when x = ±x₀. The ± sign indicates direction of motion. The maximum speed occurs at equilibrium: v₀ = ωx₀.

该方程表明，当 x = 0 时速率最大，当 x = ±x₀ 时速度为零。± 号表示运动方向。最大速度出现在平衡位置：v₀ = ωx₀。

In terms of phase, velocity leads displacement by π/2 rad, and acceleration leads velocity by another π/2 rad. Therefore acceleration is π rad out of phase with displacement, which is why a and x always have opposite signs.

从相位上看，速度比位移超前 π/2 rad，加速度又比速度超前 π/2 rad。因此加速度与位移相位差为 π rad，这就是 a 与 x 总是反号的原因。

4. Graphical Representations | 图形表示

Displacement, velocity and acceleration can all be plotted against time. For an oscillation starting at equilibrium with x = x₀ sin(ωt), the displacement–time graph is a sine curve. The velocity–time graph is a cosine curve, reaching maxima when displacement is zero.

位移、速度和加速度都可以随时间作图。对于从平衡位置开始且 x = x₀ sin(ωt) 的振动，位移-时间图像是正弦曲线。速度-时间图像是余弦曲线，在位移为零时达到最大值。

The acceleration–time graph is an inverted sine curve because a = −ω²x. Its amplitude is a₀ = ω²x₀, and it is always opposite in sign to the displacement. On the same axes, acceleration and displacement are mirror images about the time axis.

加速度-时间图像是倒置的正弦曲线，因为 a = −ω²x。它的振幅为 a₀ = ω²x₀，并且始终与位移符号相反。在同一坐标轴上，加速度和位移的图像关于时间轴对称。

Graphs of kinetic energy, potential energy and total energy against time or displacement are also important. The total energy is constant, while kinetic and potential energies vary periodically. When plotted against displacement, the potential energy curve is a parabola given by Eₚ = ½mω²x², and the kinetic energy is an inverted parabola.

动能、势能和总能量随时间或位移变化的图像也很重要。总能量保持不变，而动能和势能呈周期性变化。当以位移为横轴作图时，势能曲线是由 Eₚ = ½mω²x² 给出的抛物线，动能则是倒置的抛物线。

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

In SHM there is a continuous interchange between kinetic energy and potential energy. At the equilibrium position, displacement is zero so all energy is kinetic. At the extreme positions, displacement equals amplitude so all energy is potential and the object is momentarily at rest.

在简谐运动中，动能和势能不断相互转化。在平衡位置，位移为零，因此所有能量都是动能。在极端位置，位移等于振幅，因此所有能量都是势能，物体瞬时静止。

The kinetic energy can be written as Eₖ = ½mv² = ½mω²(x₀² − x²). The potential energy is Eₚ = ½mω²x². Adding these gives the total energy:

动能可以写作 Eₖ = ½mv² = ½mω²(x₀² − x²)。势能为 Eₚ = ½mω²x²。将两者相加得到总能量：

E_total = ½mω²x₀²

The total energy is proportional to the square of the amplitude and the square of the angular frequency. If the amplitude doubles, the total energy increases by a factor of four because E_total ∝ x₀².

总能量与振幅的平方和角频率的平方成正比。如果振幅加倍，总能量将增加为原来的四倍，因为 E_total ∝ x₀²。

Energy graphs are useful for explaining damping. In lightly damped systems the total energy decreases gradually over many cycles, so the amplitude decreases. Since energy is proportional to amplitude squared, a small loss of energy at large amplitude causes a relatively small change in amplitude, while at small amplitude the same energy loss causes a larger fractional decrease.

能量图像有助于解释阻尼。在轻阻尼系统中，总能量在多个周期内逐渐减少，因此振幅减小。由于能量与振幅的平方成正比，在大振幅时损失少量能量对振幅影响较小，而在小振幅时相同能量损失会造成更大的相对变化。

6. The Simple Pendulum | 单摆

A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angular displacements, the restoring force is approximately proportional to displacement, so the motion is simple harmonic. The period is independent of the mass of the bob and the amplitude, provided the angle is small.

单摆由一个悬挂在轻质且不可伸长的细线上的质点组成。在小角度位移下，回复力近似与位移成正比，因此运动是简谐的。周期与摆球的质量和振幅无关，前提是角度很小。

The period of a simple pendulum is given by:

单摆的周期由下式给出：

T = 2π√(l/g)

Here l is the length of the pendulum and g is the acceleration of free fall. This equation is often used in experiments to determine g by measuring T for different lengths and plotting T² against l. The gradient of the graph is 4π²/g.

式中 l 是摆长，g 是重力加速度。该方程常用于实验中，通过测量不同摆长下的周期 T，并绘制 T² 对 l 的图像来测定 g。图像的斜率为 4π²/g。

The simple pendulum is only approximately SHM because the restoring force is proportional to sin θ rather than θ. For angles less than about 10° the approximation sin θ ≈ θ is valid, and the motion is close to simple harmonic.

单摆只是近似简谐运动，因为回复力与 sin θ 成正比，而不是与 θ 成正比。当角度小于约 10° 时，近似 sin θ ≈ θ 成立，运动非常接近简谐运动。

7. Mass-Spring System | 弹簧振子系统

A mass attached to a spring obeys Hooke’s law, F = −kx, where k is the spring constant and x is the extension from the natural length or from the new equilibrium position. This linear restoring force produces true SHM.

连接在弹簧上的质量遵循胡克定律 F = −kx，其中 k 是弹簧劲度系数，x 是相对自然长度或新平衡位置的伸长量。这种线性回复力产生真正的简谐运动。

For a mass m on a spring, the angular frequency is ω = √(k/m), so the period is:

对于弹簧上的质量 m，角频率为 ω = √(k/m)，因此周期为：

T = 2π√(m/k)

The period depends on mass and spring constant but not on amplitude. In a vertical mass-spring system, gravity shifts the equilibrium position but does not change the period because the net restoring force about the new equilibrium is still −kx.

周期取决于质量和弹簧劲度系数，而与振幅无关。在竖直弹簧振子系统中，重力会改变平衡位置，但不会改变周期，因为相对于新平衡位置的净回复力仍为 −kx。

This system is often used in laboratory work. By varying the mass and measuring the period, students can plot T² against m to obtain a straight line with gradient 4π²/k. The intercept should pass through the origin for an ideal spring.

该系统常用于实验。通过改变质量并测量周期，学生可以绘制 T² 对 m 的图像，得到一条斜率为 4π²/k 的直线。对于理想弹簧，图像应经过原点。

8. Free and Forced Oscillations | 自由振动与受迫振动

Free oscillations occur when a system is displaced and released without any external driving force. The system oscillates at its natural frequency f₀, determined by its physical properties. For example, a pendulum has a natural frequency that depends on length and g.

自由振动是指系统被移开并释放后，在没有任何外部驱动力的情况下发生的振动。系统以其固有频率 f₀ 振动，该频率由系统的物理性质决定。例如，单摆的固有频率取决于摆长和 g。

Forced oscillations occur when an external periodic driving force is applied to the system. After a short transient period, the system vibrates at the driving frequency, not necessarily its natural frequency. The amplitude depends on how close the driving frequency is to f₀.

受迫振动是指对系统施加周期性外力驱动时发生的振动。经过短暂的瞬态过程后，系统以驱动频率振动，而不一定是其固有频率。振幅取决于驱动频率与 f₀ 的接近程度。

The natural frequency is the frequency of free oscillation in the absence of driving or damping. If damping is small, the free oscillation frequency is approximately the same as the undamped natural frequency. Heavy damping can lower the observed frequency slightly.

固有频率是在没有驱动或阻尼时自由振动的频率。如果阻尼很小，自由振动频率近似等于无阻尼时的固有频率。重阻尼会使观测到的频率略微降低。

9. Damping | 阻尼

Damping is the removal of energy from an oscillating system by resistive forces such as air resistance, friction or eddy currents. The amplitude decreases over time because total energy is dissipated.

阻尼是指通过空气阻力、摩擦或涡流等阻力将能量从振动系统中耗散掉。由于总能量被耗散，振幅会随时间减小。

Light damping causes a gradual decrease in amplitude over many oscillations. The period remains almost unchanged. Critical damping brings the system to equilibrium in the shortest possible time without oscillating. Heavy damping also prevents oscillation but returns to equilibrium more slowly.

轻阻尼会在许多个振动周期内使振幅逐渐减小，周期几乎保持不变。临界阻尼使系统在不发生振动的情况下以最短时间回到平衡位置。重阻尼也阻止振动，但回到平衡位置的时间更长。

Graphs of displacement against time for damped oscillations show an exponential decay envelope. The decay is not linear. For light damping the envelope decreases slowly; for heavy damping the displacement may never cross equilibrium.

阻尼振动的位移-时间图像显示出指数衰减的包络线。衰减不是线性的。对于轻阻尼，包络线缓慢下降；对于重阻尼，位移可能根本不会越过平衡位置。

Damping is useful in car suspension systems, where critical damping gives a smooth ride, and in measuring instruments, where damping prevents the pointer from oscillating too much. However, damping also reduces the sharpness of resonance peaks.

阻尼在汽车悬挂系统中很有用，临界阻尼可以提供平稳的行驶体验；在测量仪表中，阻尼可以防止指针摆动过大。但阻尼也会降低共振峰的尖锐程度。

10. Resonance | 共振

Resonance occurs when a system is driven at a frequency close to its natural frequency. The system absorbs energy most efficiently, and the amplitude of forced oscillation becomes very large. At resonance the driving force is in phase with the velocity, so maximum power is transferred.

当系统被以接近其固有频率的频率驱动时，就会发生共振。系统最有效地吸收能量，受迫振动的振幅变得非常大。在共振时，驱动力与速度同相，因此传递的功率最大。

The resonance curve is a graph of amplitude against driving frequency. It shows a peak at the natural frequency. Increasing damping reduces the maximum amplitude and broadens the peak, while decreasing damping makes the peak sharper and higher.

共振曲线是振幅随驱动频率变化的图像。它在固有频率处出现峰值。增大阻尼会降低最大振幅并使峰变宽，而减小阻尼会使峰更尖锐、更高。

Examples of resonance include the sympathetic vibration of a tuning fork, the oscillation of a child’s swing when pushed at the right frequency, and the collapse of bridges driven by wind or marching soldiers. Resonance is also used in radio tuning circuits to select a particular frequency.

共振的例子包括音叉的共鸣、以正确频率推动的秋千，以及由风或齐步行进的士兵引起的桥梁坍塌。共振还用于无线电调谐电路，以选择特定频率。

In physics problems, remember that resonance is not necessarily about maximum displacement at the instant of maximum driving force; it is about the frequency match that produces maximum amplitude over many cycles.

在物理问题中，要记住共振并不一定意味着驱动力最大时位移最大；它是频率匹配的结果，经过许多周期后产生最大振幅。

11. Experimental Methods and Graphs | 实验方法与图像

Common experiments for SHM involve a simple pendulum, a mass-spring system, or a data logger with a motion sensor. To measure the period accurately, time at least 10 complete oscillations and divide by the number of oscillations. Start and stop timing at the equilibrium position, where the speed is greatest and the timing error is smallest.

常见的简谐运动实验涉及单摆、弹簧振子或带有运动传感器的数据记录仪。为了准确测量周期，应至少计时 10 次完整振动，再除以振动次数。应在平衡位置开始和停止计时，因为那里速度最大，计时误差最小。

For a pendulum, vary the length and plot T² against l to obtain a straight line. For a mass-spring system, vary the mass and plot T² against m. The gradients of these graphs give 4π²/g and 4π²/k respectively.

对于单摆，改变摆长并绘制 T² 对 l 的图像，得到一条直线。对于弹簧振子系统，改变质量并绘制 T² 对 m 的图像。这些图像的斜率分别给出 4π²/g 和 4π²/k。

When plotting energy or acceleration graphs, pay attention to the amplitude and period. The acceleration graph should have the same period as displacement but opposite phase. The potential energy graph has half the period of the displacement graph because energy depends on the square of displacement.

在绘制能量或加速度图像时，要注意振幅和周期。加速度图像的周期应与位移相同，但相位相反。势能图像的周期是位移图像周期的一半，因为能量与位移的平方有关。

To investigate damping, attach a card to a mass-spring system and observe how the amplitude decreases. Increasing the area of the card increases air resistance and therefore damping. The amplitude decay can be plotted against time to show an exponential decrease.

为了研究阻尼，可以在弹簧振子上附着一块卡片，观察振幅如何减小。增大卡片面积会增加空气阻力，从而增大阻尼。可以绘制振幅随时间的变化，以显示指数衰减。

12. Exam Tips and Common Mistakes | 考试技巧与常见错误

Always state the defining equation a = −ω²x when asked to prove that a system performs SHM. Show that acceleration is proportional to displacement and opposite in direction, rather than simply quoting the equation.

当要求证明一个系统做简谐运动时，一定要写出定义方程 a = −ω²x。要证明加速度与位移成正比且方向相反，而不是仅仅引用公式。

Do not confuse frequency f, angular frequency ω, and period T. Remember ω = 2πf and T = 1/f. Many calculation errors come from using f where ω is required or vice versa.

不要混淆频率 f、角频率 ω 和周期 T。记住 ω = 2πf 和 T = 1/f。许多计算错误来自在需要 ω 的地方使用了 f，或者相反。

For pendulum questions, the period is independent of mass and amplitude only for small angles. For large angles the motion is not simple harmonic and the period increases. Also, the length l must be measured from the pivot to the centre of mass of the bob.

对于单摆问题，只有在小角度时周期才与质量和振幅无关。对于大角度，运动不是简谐运动，周期会增大。此外，摆长 l 必须从悬挂点到摆球质心测量。

When drawing graphs, label axes with quantities and units, use sensible scales, and draw smooth curves. For damped oscillations, draw the exponential envelope clearly, not a linear decrease.

画图时要标注坐标轴的物理量和单位，使用合理的刻度，并绘制平滑曲线。对于阻尼振动，要清楚地画出指数衰减的包络线，而不是线性下降。

In resonance questions, remember that increasing damping lowers and broadens the resonance peak, but the natural frequency itself is almost unchanged. The maximum amplitude occurs when driving frequency equals natural frequency.

在共振问题中，记住增大阻尼会降低并拓宽共振峰，但固有频率本身几乎不变。当驱动频率等于固有频率时振幅最大。

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Oscillations | 振动