Damped and forced oscillations · 阻尼振动与受迫振动
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| driving force/ˈdraɪvɪŋ fɔːs/ | 驱动力 | qū dòng lì |
| damping/ˈdæmpɪŋ/ | 阻尼 | zǔ ní |
| forced oscillations/fɔːst ˌɒsɪˈleɪʃnz/ | 受迫振动 | shòu pò zhèn dòng |
| resonance/ˈrezənəns/ | 共振 | gòng zhèn |
| driving frequency/ˈdraɪvɪŋ ˈfriːkwənsi/ | 驱动频率 | qū dòng pín lǜ |
| natural frequency/ˈnætʃərəl ˈfriːkwənsi/ | 固有频率 | gù yǒu pín lǜ |
The bridge that had to be closed on its opening day
- London's Millennium Bridge opened in June 2000. Within hours it was swaying sideways so alarmingly that it was closed, and it stayed closed for nearly two years.
- Nobody had done anything wrong with the steel. The problem was that a small sideways sway made people adjust their step, that adjustment pushed the bridge at exactly the rhythm it liked, and the sway grew.
- A driving force at the right frequency, an oscillator with too little damping: those two ingredients build amplitude out of almost nothing.
- This lesson is damping 阻尼, forced oscillations 受迫振动, and resonance 共振.
开放当天就不得不关闭的桥
- 伦敦千禧桥于 2000 年 6 月开放。几小时之内它就侧向摇晃得令人不安,被迫关闭,而且一关将近两年。
- 没有人在钢材上做错什么。问题在于:一点点侧向摇晃让人们调整了步伐,那种调整恰好以桥喜欢的节奏推动它,于是摇晃变大。
- 频率恰当的驱动力,加上阻尼太小的振子:这两样东西能几乎从无到有地把振幅堆起来。
- 这一课讲阻尼(damping)、受迫振动(forced oscillations)和共振(resonance)。
Damping
- Damping is the removal of energy from an oscillator by a resistive force such as friction or air drag.
- The amplitude decreases with time, and the lost energy is transferred to thermal energy in the oscillator and its surroundings.
- The frequency is essentially unchanged for light damping, so the oscillation keeps its rhythm as it fades. That is a bell: same note, dying away.
The envelope shrinks; the spacing of the peaks does not
阻尼
- 阻尼是摩擦或空气阻力这样的耗散力把能量从振子中带走。
- 振幅随时间减小,而失去的能量转化为振子及其周围的热能。
- 轻阻尼下频率基本不变,所以振动在衰减时保持自己的节奏。那就是一口钟:同一个音,慢慢消失。
包络在缩小;而波峰的间距没有变
Damping of an oscillation is caused by: · 振动的阻尼是由什么引起的:
Friction or drag carries energy away as heat, so each swing is smaller than the last. · 摩擦或空气阻力以热的形式带走能量,所以每次摆动都比上一次小。
Damping reduces the frequency of an oscillation rather than its amplitude. · 阻尼减小的是振动的频率而不是振幅。
The amplitude falls as energy becomes thermal. For light damping the frequency is essentially unchanged, which is why a bell keeps its note as it fades. · 随着能量变成热能,振幅下降。轻阻尼下频率基本不变,这就是钟声在衰减时音高不变的原因。
The three degrees of damping
- Light damping: the amplitude decreases slowly over many oscillations. The system still oscillates.
- Critical damping: the system returns to equilibrium in the shortest possible time without oscillating or overshooting. This is what a car suspension and an analogue meter needle are designed for.
- Heavy damping: the system returns to equilibrium slowly and without oscillating, taking longer than critical. A heavy door closer is the example.
- The exam distinction is between critical and heavy: both avoid oscillating, but only critical does it in the shortest time.
Two curves that both avoid overshooting, one much slower than the other
三种阻尼程度
- 轻阻尼:振幅在许多次振动中缓慢减小。系统仍在振动。
- 临界阻尼:系统以最短的时间返回平衡且不振荡、不过冲。汽车悬挂和模拟仪表指针正是为此设计的。
- 重阻尼:系统缓慢地返回平衡且不振荡,用时比临界阻尼更长。沉重的闭门器是例子。
- 考试要区分的是临界与重阻尼:两者都不振荡,但只有临界是用最短时间做到的。
两条都不过冲的曲线,其中一条慢得多
Match each type of damping to its behaviour. · 把每种阻尼与它的行为配对。
Critical damping returns to equilibrium in the shortest time without overshooting; heavy damping is slower still. · 临界阻尼在最短时间内回到平衡而不过冲;重阻尼还要更慢。
Worked example: choose the damping
- A car's suspension is designed so that after hitting a bump the car returns to its normal height quickly and without bouncing. What damping is used, and why?
- Critical damping: it returns the system to equilibrium in the shortest time with no oscillation, so the car neither bounces (light damping) nor sags back slowly (heavy damping).
- An analogue ammeter needle must settle on a reading as quickly as possible. Critical damping again, and for the same reason: a needle that overshoots must be waited for, and one that is heavily damped is slow to read.
- Name the type, then give both properties: shortest time and no oscillation.
例题:选择阻尼
- 汽车的悬挂被设计成:碾过凸起后车身迅速回到正常高度且不上下颠簸。用的是什么阻尼,为什么?
- 临界阻尼:它以最短时间把系统带回平衡且不振荡,所以车既不会颠簸(轻阻尼),也不会慢慢沉回去(重阻尼)。
- 模拟安培计的指针必须尽快稳定在读数上。 还是临界阻尼,理由相同:过冲的指针要等,而阻尼过重的指针读起来太慢。
- 说出类型,再给出两条性质:最短时间并且不振荡。
A car suspension returns the car to its normal height quickly without bouncing. Which damping is this, and why? · 汽车悬挂让车身迅速回到正常高度且不颠簸。这是哪种阻尼,为什么?
Heavy damping also avoids oscillating but takes longer, so the car would sag back slowly. Give both properties: shortest time and no oscillation. · 重阻尼也不振荡,但用时更长,车会慢慢沉回去。两条性质都要给:最短时间和不振荡。
Forced oscillations
- A forced oscillation happens when a driving force 驱动力 of frequency $f_{\text{d}}$ is applied to a system.
- After the start-up settles, the system oscillates at the driving frequency 驱动频率, not at its own natural frequency.
- Its natural frequency 固有频率 $f_0$ is the frequency at which it oscillates when displaced and left alone, with no driving force.
- Both facts matter and are often confused: the driving frequency sets how fast it oscillates, and the natural frequency decides how big the response is.
受迫振动
- 当频率为 $f_{\text{d}}$ 的驱动力(driving force)施加到系统上时,就发生受迫振动。
- 起始过程平息之后,系统以驱动频率(driving frequency)振动,而不是以它自己的固有频率。
- 它的固有频率(natural frequency)$f_0$ 是它被拉离后不受驱动、自行振动时的频率。
- 两个事实都要紧而且常被混淆:驱动频率决定它振动得多快,固有频率决定响应多大。
A forced oscillator vibrates at: · 受迫振子以什么频率振动:
It follows the driver — it oscillates at the driving frequency $f_{\text{d}}$. · 它跟随驱动——它以驱动频率 $f_{\text{d}}$ 振动。
Once the start-up has settled, a forced oscillator vibrates at the ____ frequency. · 起始过程平息后,受迫振子以____频率振动。
The driving frequency sets how fast it oscillates; the natural frequency decides only how large the response is. · 驱动频率决定它振动得多快;固有频率只决定响应有多大。
Resonance
- Resonance occurs when the driving frequency equals the natural frequency of the system. The amplitude is then a maximum, and there is maximum transfer of energy from the driver to the system.
- The lighter the damping, the higher and sharper the resonance peak, and the closer the peak sits to $f_0$. Heavier damping gives a lower, broader peak.
- Everyday resonance: pushing a swing in time with it, a wine glass shattered by a sung note, a radio tuned to a station, and the Millennium Bridge.
Same natural frequency, very different responses
共振
- 当驱动频率等于系统的固有频率时发生共振。此时振幅最大,从驱动者到系统的能量传递也最大。
- 阻尼越轻,共振峰越高越尖,峰的位置也越靠近 $f_0$。阻尼越重,峰越低越宽。
- 日常的共振:合着节奏推秋千、被唱出的音震碎的酒杯、调到某个电台的收音机,以及千禧桥。
同样的固有频率,截然不同的响应
Resonance happens when the driving frequency equals the ____ frequency. · 当驱动频率等于 ____ 频率时发生共振。
At $f_{\text{d}} = f_0$ the driver feeds energy in most effectively and the amplitude peaks. · 在 $f_{\text{d}} = f_0$ 时,驱动最有效地输入能量,振幅达到峰值。
Lighter damping gives a sharper, higher resonance peak. · 阻尼越轻,共振峰越尖越高。
Less energy is lost each cycle, so the amplitude builds higher and the peak is narrower. · 每个周期损失的能量更少,所以振幅积累得更高,峰更窄。
Worked example: describe a resonance curve
- Sketch and describe how the amplitude of a lightly damped driven oscillator varies with driving frequency. [3]
- The amplitude is small at low driving frequency, rises to a sharp maximum when the driving frequency equals the natural frequency, and falls again at higher frequencies.
- With greater damping the peak is lower and broader, and occurs at a slightly lower frequency.
- The mark most often lost is failing to say where the peak is: at the natural frequency, which is the whole definition of resonance.
例题:描述共振曲线
- 画出并描述轻阻尼受迫振子的振幅怎样随驱动频率变化。[3]
- 低驱动频率时振幅很小,在驱动频率等于固有频率处升到一个尖锐的最大值,在更高频率处又落下。
- 阻尼更大时,峰更低更宽,而且出现在略低一点的频率处。
- 最常丢的一分是没说出峰在哪里:在固有频率处——而这正是共振的全部定义。
Resonance · 共振
Drive the swing at different frequencies. Far from its natural frequency it barely moves; tune them to match and the amplitude explodes — resonance, the same effect that can shake a bridge apart. · 以不同频率驱动摆动。远离其固有频率时，它几乎不动；调整至匹配时，振幅激增——这就是共振，同一种效应也能使桥梁解体。
Describing a resonance curve, which statements earn marks? Select all · 所有 that apply. · 描述共振曲线时,哪些陈述能得分?选出所有适用的。
Beyond the peak the amplitude falls again. Saying where the peak is, at the natural frequency, is the mark most often lost. · 越过峰之后振幅又下降。说出峰在哪里——在固有频率处——是最常丢的一分。
Wanted and unwanted
- Wanted: a radio or television tuned by matching a circuit's natural frequency to the broadcast; a microwave oven driving water molecules; magnetic resonance imaging; a musical instrument's air column or string.
- Unwanted: a bridge or building driven by wind, footsteps or an earthquake; a vibrating car panel at one particular engine speed.
- The engineering answer to unwanted resonance is to increase the damping, which lowers and broadens the peak, or to change the natural frequency so it no longer matches the driver.
需要的与不需要的
- 需要的:把电路的固有频率调到与广播相同来选台的收音机或电视;驱动水分子的微波炉;磁共振成像;乐器的气柱或琴弦。
- 不需要的:被风、脚步或地震驱动的桥或建筑;在某个特定发动机转速下振动的汽车面板。
- 对不需要的共振,工程上的答案是增大阻尼——让峰变低变宽——或者改变固有频率,使它不再与驱动者匹配。
Select all · 所有 the examples of resonance. · 选出所有共振的例子。
Each of the first three matches a driving frequency to a natural frequency. A rolling ball is not a driven oscillation. · 前三个都是把驱动频率匹配到固有频率。滚动的球不是受迫振动。
Marks that slip away
- A damped oscillator loses amplitude, not frequency. Say what decreases.
- Critical damping is the shortest time without oscillating; heavy damping also avoids oscillating but is slower. Give both properties.
- A forced oscillator vibrates at the driving frequency. Its natural frequency decides only how large the response is.
- Resonance is the driving frequency equal to the natural frequency, not merely close to it. Name where the peak is.
容易丢掉的分
- 有阻尼的振子失去的是振幅,不是频率。要说清减小的是什么。
- 临界阻尼是不振荡下用时最短;重阻尼也不振荡,但更慢。两条性质都要给。
- 受迫振子以驱动频率振动。它的固有频率只决定响应有多大。
- 共振是驱动频率等于固有频率,不只是接近。要说出峰在哪里。
An engineer must stop a footbridge resonating with pedestrians' footsteps. Put a sensible approach in order. · 一名工程师必须阻止人行桥与行人脚步共振。把一种合理的做法按顺序排列。
The two engineering answers to unwanted resonance are more damping or a different natural frequency. Both were used on the Millennium Bridge. · 对不需要的共振,工程上的两个答案是加大阻尼或改变固有频率。千禧桥两者都用了。
You've got it
- damping is a resistive force removing energy, so the amplitude falls and the energy becomes thermal; the rhythm is essentially unchanged
- light damping decays slowly and still oscillates; critical returns to equilibrium in the shortest time without oscillating; heavy also does not oscillate but is slower
- a forced oscillator vibrates at the driving frequency, whatever its natural frequency
- resonance is driving frequency equal to natural frequency, giving maximum amplitude and maximum energy transfer; lighter damping gives a higher, sharper peak
你掌握了
- 阻尼是耗散力带走能量,于是振幅下降、能量变成热能;节奏基本不变
- 轻阻尼缓慢衰减且仍在振荡;临界以最短时间不振荡地回到平衡;重阻尼也不振荡但更慢
- 受迫振子以驱动频率振动,无论它的固有频率是多少
- 共振是驱动频率等于固有频率,给出最大振幅和最大能量传递;阻尼越轻,峰越高越尖