Free Vibration vs Forced Vibration | 自由振动与受迫振动的区别

📚 Free Vibration vs Forced Vibration | 自由振动与受迫振动的区别

In A-Level Physics, the distinction between free vibration and forced vibration is essential for understanding oscillatory systems, damping and resonance. A free vibration occurs when a system oscillates on its own after being displaced, with no continuous external driving force. A forced vibration occurs when an external periodic force repeatedly supplies energy to the system.

在 A-Level 物理中，区分自由振动与受迫振动是理解振荡系统、阻尼和共振的关键。自由振动是指系统在受到初始位移或冲量后，在没有持续外部驱动力的情况下自行振荡；受迫振动则是指系统在外界周期性驱动力反复输入能量下发生的振动。

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1. What Is Free Vibration? | 什么是自由振动？

A free vibration is the oscillation of a system under the action of its own restoring forces only, after it has been displaced from equilibrium. Once the initial displacement or impulse is given, no further external energy is supplied by an outside agent. The system therefore vibrates at a frequency determined entirely by its own physical properties, known as the natural frequency.

自由振动是指系统在偏离平衡位置后，仅依靠自身回复力进行的振荡。在给予初始位移或瞬时冲击之后，外界不再向系统输入额外能量。因此，系统振动的频率完全由其自身的物理性质决定，这个频率被称为固有频率。

For example, a pendulum pulled to one side and released, or a mass on a spring stretched and then let go, both perform free vibrations. In the ideal case with no energy loss, the amplitude would remain constant for ever. In real systems, air resistance and internal friction remove energy, so the amplitude gradually decreases.

例如，把单摆拉到一侧后释放，或者把弹簧上的质量块拉伸后松手，它们都在做自由振动。在无能量损失的理想情况下，振幅将永远保持不变。但在实际系统中，空气阻力和内摩擦会不断消耗能量，使振幅逐渐减小。

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2. Natural Frequency and Examples | 固有频率与实例

The natural frequency is the frequency at which a system would oscillate if there were no damping and no external driving force. It depends on the restoring force and the inertia of the system. For a mass-spring system with mass m and spring constant k, the natural frequency is given by

固有频率是系统在无阻尼、无外部驱动力时自身振动的频率。它取决于系统的回复力和惯性。对于质量 m、劲度系数 k 的弹簧振子，其固有频率为

f₀ = (1/2π)√(k/m)

For a simple pendulum of length L, the natural frequency is approximately

对于摆长为 L 的单摆，其固有频率近似为

f₀ = (1/2π)√(g/L)

Common examples include a guitar string plucked and then left to ring, a tuning fork struck with a hammer, and a child on a swing initially pushed once and then allowed to move without further pushes. In each case, the system oscillates at its own natural frequency, not at the frequency of any repeated external force.

常见实例如下：拨动后任其发声的吉他弦、用音锤敲击后的音叉，以及只被推一次后靠惯性摆动的秋千。在上述每种情况下，系统都按自身固有频率振动，而不是按任何外界周期性驱动力的频率振动。

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3. Damping in Free Vibration | 自由振动中的阻尼

Damping is the removal of energy from an oscillating system. A system undergoing free vibration can still be damped, because friction and air resistance are always present in real situations. The damping force is not a periodic driving force; it only opposes motion and dissipates mechanical energy.

阻尼是指振荡系统能量不断耗散的过程。发生自由振动的系统仍然可以存在阻尼，因为在真实环境中摩擦和空气阻力总是存在。阻尼力不是周期性的驱动力，它只是阻碍运动并耗散机械能。

Depending on the amount of damping, three cases are often distinguished:

根据阻尼大小，通常区分三种情况：

• Light damping: the system continues to oscillate with a gradually decreasing amplitude; the period remains almost constant.

  轻阻尼：系统继续振动，但振幅逐渐减小，周期几乎保持不变。

• Critical damping: the system returns to equilibrium in the shortest possible time without oscillating.

  临界阻尼：系统以最短时间回到平衡位置，且不发生振荡。

• Heavy damping: the system returns to equilibrium very slowly and never oscillates.

  过阻尼：系统缓慢回到平衡位置，且完全不振荡。

In free vibration, damping only changes the amplitude over time; it does not change the natural frequency in light damping. This distinction becomes important when comparing with forced vibration, where the driving frequency, not the natural frequency, controls the steady oscillation.

在自由振动中，阻尼只改变振幅随时间的变化；在轻阻尼情况下，它并不会改变固有频率。这一区别在比较受迫振动时非常重要：受迫振动最终由驱动频率控制，而非固有频率。

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4. What Is Forced Vibration? | 什么是受迫振动？

A forced vibration occurs when an external periodic force, known as the driving force, is applied continuously to a system. The system is compelled to oscillate at the frequency of the driving force, even if that frequency is different from the system’s natural frequency.

受迫振动是指一个外部周期性力，即驱动力，持续作用在系统上时发生的振动。系统被迫按照驱动力的频率振荡，即使该频率与系统固有频率不同。

During forced vibration, the driving force transfers energy into the system on every cycle. The amplitude of the steady oscillation depends on the balance between the energy supplied by the driving force and the energy lost through damping. Examples include a building swaying in a steady wind, a washing machine drum vibrating during the spin cycle, and the cone of a loudspeaker driven by an alternating current.

在受迫振动中，驱动力在每一个周期都会向系统输入能量。稳态振幅取决于驱动力输入能量与阻尼消耗能量之间的平衡。常见例子包括：稳定风吹拂下的高楼轻微摆动、洗衣机在脱水时滚筒的振动，以及由交流电驱动的扬声器纸盆的振动。

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5. The Steady State and Amplitude | 稳态与振幅

When a forced vibration first starts, the motion contains a transient component at the natural frequency. After a short time, this transient dies away because of damping, and the system settles into a steady state in which it oscillates purely at the driving frequency.

受迫振动刚开始时，运动中包含一个以固有频率振动的瞬态分量。由于阻尼的存在，这个瞬态分量很快衰减，系统随后进入稳态，只按驱动频率稳定振荡。

In the steady state, the amplitude is constant as long as the driving force and damping remain unchanged. The phase difference between the displacement and the driving force also depends on the driving frequency. At low driving frequencies, displacement is nearly in phase with the driving force; at very high frequencies, displacement is nearly half a cycle out of phase with the driving force.

在稳态中，只要驱动力和阻尼不变，振幅就保持恒定。位移与驱动力之间的相位差也取决于驱动频率。在低驱动频率时，位移与驱动力几乎同相；在很高频率时，位移与驱动力几乎相差半个周期。

The amplitude is largest when the driving frequency is close to the natural frequency. If damping is very small, the amplitude can become extremely large. This condition is called resonance.

当驱动频率接近固有频率时，振幅达到最大值。如果阻尼很小，振幅会变得非常大。这种现象称为共振。

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6. Resonance | 共振

Resonance is the condition in which the driving frequency equals the natural frequency of the system. At resonance, energy is transferred from the driving force to the system in the most efficient way, so the amplitude reaches a maximum.

共振是指驱动频率恰好等于系统固有频率时发生的现象。在共振时，驱动力向系统传递能量的效率最高，因此振幅达到最大值。

The graph of amplitude against driving frequency shows a peak at f₀. The sharpness of the peak depends on damping:

振幅随驱动频率变化的图像在 f₀ 处出现峰值。峰的尖锐程度取决于阻尼：

Damping | 阻尼	Amplitude at resonance | 共振振幅	Sharpness of peak | 峰值尖锐程度
Small damping | 小阻尼	Very large | 非常大	Very sharp | 非常尖锐
Large damping | 大阻尼	Small | 较小	Broad | 平缓

At resonance, the driving force is always acting in the same direction as the velocity, so positive work is done on the system each cycle. If damping is absent and the force keeps acting, the amplitude would grow without limit. In real systems, damping always limits the maximum amplitude.

在共振时，驱动力始终与速度方向相同，因此每个周期都对外做正功。如果完全没有阻尼且驱动力持续作用，振幅将无限增大。实际系统中，阻尼总会限制最大振幅。

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7. Useful and Harmful Effects of Resonance | 共振的利与弊

Resonance can be very useful. Musical instruments rely on resonance to amplify certain frequencies. A radio receiver uses electrical resonance to select a particular station from many signals. In medicine, magnetic resonance imaging uses resonant excitation of nuclei to create detailed images of the human body.

共振非常有用。乐器依靠共振来放大特定频率的声音。无线电接收器利用电路谐振从众多信号中选出特定电台。医学中的磁共振成像则是利用原子核的共振激发来生成人体精细图像。

Resonance can also be destructive. A tuning fork or wine glass can be shattered by a driving force at its natural frequency. Bridges and tall buildings can suffer severe damage if wind or earthquakes drive them at a resonant frequency. The famous Tacoma Narrows Bridge collapse is often cited as a dramatic example of vibration caused by a periodic forcing effect.

共振也可能造成破坏。音叉或酒杯在其固有频率的驱动力作用下可能破裂。如果风或地震使桥梁和高层建筑发生共振，就可能造成严重损坏。著名的塔科马海峡大桥坍塌常被引为例证。

To avoid destructive resonance, engineers change the natural frequency of a structure by altering its mass or stiffness, and they often add damping devices to absorb vibrational energy. This is why suspension bridges have tuned masses and tall buildings contain tuned liquid dampers.

为了避免破坏性共振，工程师通过改变结构的质量或刚度来改变其固有频率，并经常添加阻尼装置来吸收振动能量。这就是悬索桥使用调谐质量块、高层建筑安装调谐液体阻尼器的原因。

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8. Comparing Free and Forced Vibration | 自由振动与受迫振动的对比

The table below summarises the main differences between free vibration and forced vibration. You should be able to state these clearly in an exam question.

下表总结了自由振动与受迫振动的主要区别。你应该能够在考试中清晰表述这些内容。

Feature | 特征	Free vibration | 自由振动	Forced vibration | 受迫振动
Energy supply | 能量来源	Only initial energy given | 仅获得初始能量	Continuous energy from driving force | 驱动力持续输入能量
Oscillation frequency | 振动频率	Natural frequency f₀ | 固有频率 f₀	Driving frequency f | 驱动频率 f
Amplitude | 振幅	Usually decreases if damped | 若有阻尼，通常逐渐减小	Constant in steady state | 稳态时保持恒定
Resonance possibility | 共振可能性	Resonance not defined | 不讨论共振	Resonance occurs when f = f₀ | 当 f = f₀ 时发生共振
Examples | 实例	Plucked guitar string, released pendulum | 拨动后的吉他弦、释放后的单摆	Loudspeaker cone, vibrating building | 扬声器纸盆、振动中的建筑

Notice that even in forced vibration, the natural frequency remains a property of the system. It does not disappear, but it no longer controls the steady oscillation frequency. Instead, it controls the conditions under which resonance will occur.

注意，即使在受迫振动中，固有频率仍然是系统自身的一种属性，它不会消失，只是不再控制稳态振荡频率，而是控制共振发生的条件。

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9. Common Misconceptions and Exam Tips | 常见误区与考试要点

One common misconception is that free vibration always means there is no damping. This is incorrect. Free vibration refers to the absence of an external driving force, not the absence of energy loss. A damped free vibration still vibrates at its natural frequency, with decreasing amplitude.

一个常见误区是认为自由振动总是意味着没有阻尼。这是错误的。自由振动指的是没有外部驱动力，而不是没有能量损耗。有阻尼的自由振动仍然以固有频率振动，只是振幅逐渐减小。

Another misconception is that resonance can only happen if there is no damping. In reality, resonance happens with any amount of damping, but the resonant amplitude is smaller and the peak is broader when damping is stronger.

另一个误区是认为只有无阻尼才能发生共振。实际上，任何阻尼情况下都能发生共振，只是阻尼越大，共振振幅越小，峰值越平缓。

In exam questions, always check whether the problem asks about the frequency of vibration. If the forcing is continuous, the answer is the driving frequency. If the system is released and left alone, the answer is the natural frequency. Also remember to quote the condition for resonance as driving frequency equals natural frequency.

在考试题中，务必检查题目问的是哪种频率。如果驱动力持续作用，答案为驱动频率；如果系统被释放后不再受外力，答案为固有频率。同时要记住共振条件是驱动频率等于固有频率。

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10. Worked Example | 例题演练

A mass of 0.20 kg is attached to a spring with spring constant 80 N m⁻¹. The mass is pulled down and released, and then the system is driven by a motor whose frequency can be adjusted.

一个质量 0.20 kg 的物体连接在劲度系数 80 N m⁻¹ 的弹簧上。将物体下拉后释放，之后用频率可调的电机驱动该系统。

First, calculate the natural frequency of the system. Using the formula

首先，计算系统的固有频率。使用公式

f₀ = (1/2π)√(k/m) = (1/2π)√(80/0.20) = (1/2π)√400 = 20/2π ≈ 3.18 Hz

The natural frequency is approximately 3.2 Hz. If the motor is adjusted until the driving frequency is exactly 3.2 Hz, the system will resonate. The amplitude at resonance becomes much larger because the motor supplies energy most efficiently when it matches the natural frequency.

固有频率约为 3.2 Hz。如果电机频率被精确调节到 3.2 Hz，系统将发生共振。共振时振幅变得大得多，因为当电机频率等于固有频率时，能量输入效率最高。

If damping were increased, the resonant amplitude would become smaller and the resonance peak would become broader, but the resonance would still occur at approximately the same driving frequency.

如果增大阻尼，共振振幅会变小，共振峰变宽，但共振仍大致发生在同样的驱动频率处。

Published by TutorHao | Physics Revision Series | aleveler.com

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