Mechanical Watchmaking

These are my running notes on how a mechanical watch actually works — and on the harder question underneath it: where does craftsmanship live in a piece of metal? Rather than scatter the topic across many short posts, I keep everything here as one growing document. Each major idea is a section; each moving part is a subsection with its own interactive figure you can play with.

Start by exploring the movement itself. Click any part below to see what it is, what it does, and which subsection explains it.

这是我关于机械表究竟如何运作的随手笔记——以及它底下那个更难的问题：工艺究竟栖身于一块金属的何处？ 与其把话题拆成许多篇短文，我把所有内容都汇集在这一篇不断生长的长文里。每个大主题是一个章节（section）；每个运动部件是一个子节（subsection），并配有一个可以亲手把玩的交互图。

先来探索机芯本身。点击下方任意部件，看看它是什么、有什么功效、以及在哪个子节里讲解。

A mechanical watch has no battery, no quartz crystal, no microchip. It is a purely mechanical machine that takes the energy you store by winding a spring, releases it in tiny, precisely-timed increments, and counts those increments on a dial. Everything else — the decoration, the brand, the price — is built on top of this one feat: turning stored energy into a steady, countable rhythm.

Every mechanical movement, from a $50 Seiko to a $500,000 Patek Philippe, is organized around the same five subsystems, plus the finishing that holds them together:

Subsystem	English	Function	Subsection
Energy source	Mainspring + barrel	Stores winding energy, releases torque	§1.1
Gear train	Wheel train	Steps the torque down and the speed up	§1.2
Escapement	Escapement	Releases energy in discrete “ticks”	§1.3
Oscillator	Balance + hairspring	Sets the rhythm — the timekeeping organ	§1.4
Display	Motion works + hands	Counts and shows the result	§1.5
Finishing	Bevels, polish, jewels	Where craft meets function	§1.6

The subsections below walk through each one, with an interactive figure for the key idea, and end with the question this whole document is really about: where does craftsmanship actually live?

机械表没有电池、没有石英晶体、也没有芯片。它是一台纯粹的机械装置：把你上链时储存进发条的能量，以极小而精确定时的增量释放出来，再在表盘上把这些增量数出来。其余的一切——打磨、品牌、价格——都建立在这一项本领之上：把储存的能量转化为稳定、可计数的节奏。

从 50 美元的精工到 50 万美元的百达翡丽，每一枚机械机芯都围绕同样的五个子系统组织，外加把它们维系在一起的修饰工艺：

子系统	名称	功能	子节
能量源	发条 + 条盒	储存上链能量，释放扭矩	§1.1
传动轮系	轮系	降低扭矩、提升转速	§1.2
擒纵机构	擒纵	以离散的”滴答”释放能量	§1.3
振荡器	摆轮 + 游丝	设定节奏——计时器官	§1.4
显示	显示轮系 + 指针	计数并呈现结果	§1.5
修饰	倒角、抛光、宝石	工艺与功能相遇之处	§1.6

下面的各子节将逐一讲解，并为每个关键概念配一个交互图，最后回到贯穿全文的真正问题：工艺究竟栖身于何处？

The mainspring is a long, thin ribbon of hardened steel alloy coiled inside a drum called the barrel. Winding the crown tightens the spring; as it unwinds, it delivers torque to the gear train.

The trouble is that a coiled spring does not deliver constant torque. To a first approximation it behaves like a torsional spring,

\[\tau = -k\,\theta,\]

so the torque \( \tau \) falls roughly linearly as the spring unwinds through angle \( \theta \). A fully wound watch therefore pushes harder than a nearly run-down one — and as we’ll see in §1.4, an oscillator’s accuracy is sensitive to the push it receives: more torque drives a wider balance swing, and once the torque drops below a certain minimum the swing collapses, so the watch loses time or stops outright. Two centuries of watchmaking are, in large part, a war against this single fact:

• The going barrel uses only the middle portion of the spring’s torque curve, where it is flattest.
• A stopwork (Maltese cross) mechanically forbids the extreme over-wound and run-down turns.
• A fusée-and-chain — the most beautiful solution — wraps a chain around a cone-shaped pulley so that as torque falls, leverage rises, holding the delivered force nearly constant.

A typical modern mainspring stores enough energy for 40–80 hours of running. “Power reserve” on a spec sheet is just how long the barrel can feed the escapement before torque drops below the threshold the oscillator needs — and, as the figure shows, that trustworthy window is often far shorter than the spring’s full life.

发条是一条又长又薄、由淬硬钢合金制成的带子，盘绕在一个叫做条盒的鼓状容器里。旋转表冠上链会拧紧发条；当它松开时，便向轮系输出扭矩。

麻烦在于，盘绕的弹簧并不输出恒定扭矩。一阶近似下它表现为一个扭转弹簧，

\[\tau = -k\,\theta,\]

因此随着发条松开转过角度 \( \theta \)，扭矩 \( \tau \) 大致线性下降。满链的表因而比快走完时推得更猛——而正如我们将在 §1.4 看到的，振荡器的精度对它所受到的推力很敏感：扭矩越大，摆轮摆幅越大；一旦扭矩跌破某个最小值，摆幅便会塌缩，表就走慢、甚至直接停摆。两个世纪的制表，很大程度上就是在与这一个事实作战：

• 走时条盒（going barrel）只使用发条扭矩曲线中段最平坦的部分。
• 停止装置（马耳他十字）从机械上禁止过度上链和走完链的极端圈数。
• 宝塔轮链条（fusée-and-chain）——最优雅的解法——把一条链子缠在锥形滑轮上，使扭矩下降时杠杆臂增大，从而把输出力保持得近乎恒定。

一条典型的现代发条储存的能量足够运行 40–80 小时。规格表上的”动力储备”，无非是条盒在扭矩跌破振荡器所需阈值之前，能给擒纵供能多久——而正如下图所示，这个可信赖的窗口往往远短于发条的整个寿命。

How to read this figure. The horizontal axis is hours since you last fully wound the watch; the vertical axis is the torque the barrel is delivering, as a percentage of a fully-wound spring. Each element answers one question:

• Steel-blue curve — a bare going barrel. Torque starts at 100% and sags as the spring unwinds.
• Red dashed line — the escapement threshold: the least torque the balance needs to keep swinging widely enough to keep time. Below it, the swing collapses (the link we just drew above).
• Green band — the usable window: the stretch where the blue curve sits above the threshold. This — not the spring’s full life — is the part of the run you can actually trust.
• Moving dot — the watch right now, at the time-elapsed value you pick. It reads RUNNING (green, above the line) or STOPPED (red, below).
• Gold curve (toggle the checkbox) — a fusée-equalized barrel, held nearly flat so almost the entire reserve stays inside the green window.

Now drive it: raise power reserve and the blue curve stretches to the right; raise the escapement threshold (a hungrier movement) and the green window shrinks; drag time elapsed to watch the dot cross from green to red. The takeaway: a bare barrel’s trustworthy window is much shorter than its spec-sheet reserve — which is exactly why the stopwork and the fusée exist.

如何读这张图。 横轴是自上次满链以来经过的小时数；纵轴是条盒此刻输出的扭矩，以满链发条为 100%。每个图元各回答一个问题：

• 钢蓝色曲线——裸走时条盒。扭矩从 100% 起步，随发条松开而下垂。
• 红色虚线——擒纵阈值：摆轮维持足够大摆幅、从而准确走时所需的最小扭矩。低于它，摆幅就会塌缩（即上文刚建立的那条联系）。
• 绿色色带——可用窗口：蓝色曲线高于阈值的那一段。这一段——而非发条的整个寿命——才是你真正可以信赖的走时区间。
• 移动圆点——此刻的表，位于你设定的已用时间处。它显示 运行中（绿色，在线之上）或 已停止（红色，在线之下）。
• 金色曲线（勾选复选框显示）——宝塔轮均衡后的条盒，被拉得近乎平直，使几乎整个动储都落在绿色窗口内。

动手试试：调大动力储备，蓝色曲线就向右伸长；调高擒纵阈值（更”耗力”的机芯），绿色窗口随之缩小；拖动已用时间，看圆点从绿色越过红线变成红色。结论：裸条盒真正可信赖的窗口，远短于它规格表上的动储——这正是停止装置与宝塔轮存在的理由。

Between the barrel and the escapement sits the going train: a cascade of wheels and pinions (a wheel is the large toothed gear, a pinion the small one it drives). Each meshing pair multiplies speed and divides torque by its tooth ratio. The classic train is four stages:

\[\text{barrel} \;\to\; \text{centre wheel} \;\to\; \text{third wheel} \;\to\; \text{fourth wheel} \;\to\; \text{escape wheel}\]

The ratios are chosen so that the centre wheel turns exactly once per hour (it carries the minute hand) and the fourth wheel once per minute (often carrying the seconds hand). The barrel, by contrast, might turn only a handful of times per day. The train converts the barrel’s slow, powerful rotation into the escape wheel’s fast, feeble one — feeble because by the time torque reaches the escapement it has been divided down to a few micronewton-metres, which is exactly what a delicate oscillator can tolerate. Speed up the animation below to watch the cascade run.

在条盒与擒纵之间是走时轮系：一连串轮与齿（轮是大的带齿齿轮，齿是它带动的小齿轮）。每一对啮合都按齿数比放大转速、缩小扭矩。经典轮系分四级：

\[\text{条盒} \;\to\; \text{中心轮} \;\to\; \text{三轮} \;\to\; \text{四轮} \;\to\; \text{擒纵轮}\]

齿比的选择，使得中心轮恰好每小时转一圈（它带动分针），而四轮每分钟转一圈（常带动秒针）。相比之下，条盒可能一天才转区区几圈。轮系把条盒缓慢而有力的转动，转化为擒纵轮快速而微弱的转动——之所以微弱，是因为扭矩传到擒纵时已被分到只剩几微牛·米，而这恰是娇弱的振荡器所能承受的量级。把下方动画加速，看这条级联运转起来。

If the watch has a heart, it is the escapement. It does two jobs at once, and that double duty is why it is the hardest part to make well:

1. It locks the gear train, stopping the whole cascade of energy from spinning out at once.
2. It releases the train in tiny, equal steps, and on each release it gives the oscillator a small push to keep it swinging.

The dominant design for 200+ years is the Swiss lever escapement. A pallet fork with two jewelled “pallets” rocks between the teeth of the escape wheel. Each beat of the balance unlocks one tooth, the wheel rotates a step (the drop), delivers an impulse through the fork to the balance, then re-locks on the next tooth. That familiar tick-tick-tick is the sound of the escape wheel stepping forward, one tooth per half-swing. Slow the figure right down to watch one full lock → unlock → impulse → drop cycle.

The escapement is where the most delicate craftsmanship concentrates:

• The pallet stones are tiny blocks of synthetic ruby, angled to a few arc-minutes, polished so their locking and impulse faces are optically flat.
• The escape-wheel teeth are pointed and finished to minimise the energy lost on impact.
• A poorly finished escapement wastes the mainspring’s energy as friction and shock, robbing the balance of the clean impulse it needs to keep good time.

This is the first place where “finishing” stops being decoration and becomes function: a mirror-polished pallet locking face genuinely loses less energy than a rough one (we return to this in §1.6).

如果说表有一颗心脏，那就是擒纵机构。它同时承担两项工作，而正是这种双重职责使它成为最难做好的部件：

1. 它锁住轮系，阻止整串能量一次性失控释放。
2. 它以极小而相等的步进释放轮系，并在每次释放时给振荡器一个小推力，维持其摆动。

200 多年来占主导地位的设计是瑞士杠杆式擒纵（Swiss lever）。带有两枚红宝石”擒纵叉瓦”的擒纵叉，在擒纵轮的轮齿间往复摆动。摆轮的每一拍解锁一枚轮齿，擒纵轮前进一步（跳动 drop），通过叉子向摆轮传递一次冲量 impulse，随后在下一枚齿上重新锁定。那熟悉的滴答滴答，正是擒纵轮一拍前进一齿的声音。把下方图放到最慢，看清一个完整的锁定 → 解锁 → 冲量 → 跳动循环。

擒纵机构是最精细工艺最集中之处：

• 叉瓦是极小的合成红宝石块，研磨到只差几角分的角度，锁面与冲面被抛光到光学平整。
• 擒纵轮齿被做成尖形并精修，以最小化撞击时损失的能量。
• 打磨拙劣的擒纵会把发条的能量以摩擦和冲击的形式浪费掉，使摆轮失去维持走时所需的干净冲量。

这是”打磨”第一次不再是装饰、而成为功能的地方：一个镜面抛光的叉瓦锁面，确实比粗糙的损失更少能量（我们将在 §1.6 回到这一点）。

The escapement paces the train, but it does not decide the pace. That is the job of the oscillator: a balance wheel coupled to a fine spiral hairspring. Together they form a torsional pendulum that swings back and forth at a frequency set by the wheel’s inertia and the spring’s stiffness:

\[T = 2\pi \sqrt{\frac{I}{k}},\]

where \( I \) is the balance’s moment of inertia and \( k \) the hairspring’s torque constant. The escapement nudges it once per swing to replace energy lost to friction — exactly like pushing a child on a swing at the right moment — but the natural period \( T \) is what the watch counts. In the figure below, change \( I \) and \( k \) and watch the beat rate move.

Beat rates are quoted in vibrations per hour (vph), counting half-swings:

Beat rate	Frequency	Feel
18,000 vph	2.5 Hz	Slow, vintage; visible “stepping” seconds hand
21,600 vph	3 Hz	Common in robust, long-reserve movements
28,800 vph	4 Hz	The modern standard; smooth sweep
36,000 vph	5 Hz	High-beat; finer resolution, harder on lubrication

The deep design goal is isochronism: the period \( T \) should stay constant whether the balance swings through a wide arc (fully wound) or a narrow one (nearly run down), and whether the watch lies flat or stands on edge. Two facts fight isochronism — the falling mainspring torque from §1.1, and gravity pulling on the balance in vertical positions. The craft of adjusting a movement to keep good time in multiple positions, and the use of an overcoil (Breguet) hairspring whose terminal curve breathes concentrically, are direct attacks on those two enemies.

擒纵调节轮系的节拍，但它并不决定节拍。那是振荡器的工作：一个摆轮与一根细密的螺旋游丝耦合。二者构成一个扭摆，以由摆轮转动惯量与游丝刚度决定的频率往复摆动：

\[T = 2\pi \sqrt{\frac{I}{k}},\]

其中 \( I \) 是摆轮的转动惯量，\( k \) 是游丝的扭矩常数。擒纵每摆动一次轻推它一下，以补偿被摩擦消耗的能量——正如在恰当时机推一个荡秋千的孩子——但表所计数的，是那个固有周期 \( T \)。在下方图中，改变 \( I \) 与 \( k \)，看摆频如何变化。

摆频以每小时振动次数（vph）表示，按半摆计：

摆频	频率	体感
18,000 vph	2.5 Hz	慢、复古；秒针有可见的”跳步”
21,600 vph	3 Hz	常见于坚固、长动储的机芯
28,800 vph	4 Hz	现代标准；扫秒顺滑
36,000 vph	5 Hz	高频；分辨率更高，但对润滑要求更苛刻

深层的设计目标是等时性（isochronism）：无论摆轮以大摆幅（满链）还是小摆幅（将走完）摆动，无论表平放还是竖立，周期 \( T \) 都应保持恒定。两个事实与等时性作对——§1.1 提到的下降的发条扭矩，以及竖直方位时重力对摆轮的牵拉。让机芯在多个方位都走时准确的校准（adjusting）工艺，以及采用末端曲线同心”呼吸”的宝玑上绕游丝（overcoil），正是针对这两个敌人的直接进攻。

The last subsystem is almost an afterthought mechanically, but it is what you actually look at. The motion works is a small set of gears under the dial that takes the once-per-hour rotation of the centre wheel and produces the 12:1 reduction needed to drive the hour hand from the minute hand:

\[\omega_{\text{hour}} = \frac{1}{12}\,\omega_{\text{minute}}.\]

A friction-fitted cannon pinion lets you set the time by turning the hands without forcing the whole train backwards. From here, complications — date, chronograph, moonphase, minute repeater — are extra gear trains tapped off this same rotation. The figure below lets you speed up time and watch the hour hand crawl at exactly one-twelfth the minute hand’s pace.

最后一个子系统在机械上几乎像是事后补充，却正是你真正在看的东西。显示轮系（motion works）是表盘下的一小组齿轮，把中心轮每小时一圈的转动，做成驱动时针所需的、相对分针的 12:1 减速：

\[\omega_{\text{时}} = \frac{1}{12}\,\omega_{\text{分}}.\]

一个以摩擦配合安装的跨轮（cannon pinion）让你能通过转动指针来校时，而不必强行倒转整个轮系。由此出发，各种复杂功能——日历、计时码表、月相、三问报时——都是从这同一转动上引出的额外轮系。下方图让你把时间加速，看时针如何恰好以分针十二分之一的步速缓缓爬行。

Now we can answer the question. A skeptic looks at a hand-finished movement and a machine-made one keeping the same time and asks: why does the finishing matter? The honest answer has two layers — and the figure below lets you watch them pull apart.

The functional layer. Some finishing is not cosmetic at all. Flat, polished pivots and bearing surfaces lower friction; well-shaped escape teeth and pallet faces lose less energy on impact; properly poised balances and adjusted hairsprings hold isochronism across positions. These directly affect amplitude, rate stability, and how long the watch keeps good time as the mainspring runs down. Here, craft is performance.

The cultural layer. The rest — anglage (hand-bevelled, mirror-polished bridge edges), Côtes de Genève stripes, perlage swirls, blued screws, engraved balance cocks — is largely invisible once the watch is cased and does nothing for accuracy. But it is the visible signature of human time and skill spent on surfaces no one was forced to touch. The bevel on an inward corner is the classic tell: a machine cannot cut a sharp interior angle, so a crisp inner anglage is proof a hand held a file there.

That, finally, is what these notes mean by craftsmanship: the place where a constraint of physics (lower friction, better isochronism) and a gesture of human care (a polished surface no function required) happen to meet on the same piece of metal. As this document grows, later sections will take individual subsystems — escapements, hairsprings, finishing techniques, complications — and look at exactly where, and how, that meeting happens.

现在我们可以回答那个问题了。怀疑者看着一枚手工修饰的机芯和一枚机器量产的机芯走出同样的时间，问道：打磨究竟有什么意义？ 诚实的答案有两层——下方的图让你亲眼看着这两层分道扬镳。

功能层。 有些修饰根本不是为了好看。平整、抛光的轴尖与轴承面降低摩擦；形状良好的擒纵齿与叉瓦面在撞击时损失更少能量；经过校平（poise）的摆轮与调校过的游丝，能在各方位保持等时性。这些直接影响摆幅、走时稳定性，以及发条放松过程中表能准走多久。在这里，工艺就是性能。

文化层。 其余的——倒角（anglage，手工削出、镜面抛光的夹板边缘）、日内瓦纹（Côtes de Genève）、珍珠纹（perlage）旋涡、烧蓝螺丝、雕花摆轮夹板——一旦表壳合上便大多不可见，对精度也毫无贡献。但它们是人投入在无人强迫他去触碰的表面上的时间与技艺的可见签名。内凹角上的倒角是经典的判据：机器无法切出锐利的内角，所以一道利落的内倒角，便是有一只手在那里执锉的证据。

而这，正是这份笔记所说的工艺：物理约束（更低的摩擦、更好的等时性）与人的用心姿态（一个功能并不需要的抛光表面）恰好在同一块金属上相遇的地方。随着本文的生长，后续章节将逐一拿起各个子系统——擒纵、游丝、修饰技法、复杂功能——去细看这场相遇究竟在何处、以何种方式发生。

Section 1 treated the mainspring as a black box that “stores energy and releases torque.” This section opens the box. Everything a barrel can do — how much energy it holds, how flat its torque is, how many hours it runs — falls out of one piece of beam mechanics applied to a thin steel strip, plus the geometry of packing that strip into a drum.

第 1 节把发条当作一个”储存能量、释放扭矩”的黑箱。本节把箱子打开。条盒所能做的一切——能储多少能量、扭矩多平、能走多少小时——都源自把一段梁力学施加到一条薄钢带上，再加上把这条钢带塞进鼓里的几何。

A mainspring is a flat ribbon: thickness \( e \) (the dimension that bends), width \( h \) (the height that fits the barrel), length \( L \). Coiling it bends it to some radius of curvature \( \rho \). For a rectangular cross-section the second moment of area is

\[I = \frac{h\,e^{3}}{12},\]

and elementary beam theory gives the bending moment needed to hold that curvature, together with the peak (surface) stress:

\[M = \frac{E\,I}{\rho} = \frac{E\,h\,e^{3}}{12\,\rho}, \qquad \sigma = \frac{M\,(e/2)}{I} = \frac{E\,e}{2\,\rho}.\]

Two design facts fall straight out. First, torque scales as \( e^{3} \) — a 20% thicker spring delivers \( 1.2^{3}\approx 1.7\times \) the torque, which is why thickness is the watchmaker’s coarsest power knob. Second, the stress \( \sigma=E e/2\rho \) must stay below the alloy’s yield strength \( \sigma_y \) or the spring takes a permanent “set” and loses force. That caps how tightly it can be wound: \( \rho \ge E e / 2\sigma_y \).

The stored energy is the integral of moment over the angle wound. A strip bent uniformly to curvature \( 1/\rho \) along its whole length stores

\[U = \tfrac{1}{2}\,\frac{M^{2} L}{E I} = \frac{\sigma^{2}}{6E}\,V, \qquad V = L\,h\,e,\]

so the useful figure of merit for a spring material is the energy density \( \sigma_y^{2}/E \): high yield strength, modest stiffness. This single ratio is why modern springs are cobalt alloys (Nivaflex) rather than ordinary hardened carbon steel — same \( E \), far higher \( \sigma_y \).

发条是一条扁带：厚度 \( e \)（弯曲的方向）、宽度 \( h \)（装进条盒的高度）、长度 \( L \)。盘绕使它弯成某个曲率半径 \( \rho \)。对矩形截面，截面二次矩为

\[I = \frac{h\,e^{3}}{12},\]

初等梁理论给出维持该曲率所需的弯矩，以及表面峰值应力：

\[M = \frac{E\,I}{\rho} = \frac{E\,h\,e^{3}}{12\,\rho}, \qquad \sigma = \frac{M\,(e/2)}{I} = \frac{E\,e}{2\,\rho}.\]

两条设计事实立刻浮现。其一，扭矩正比于 \( e^{3} \)——厚 20% 的发条输出 \( 1.2^{3}\approx 1.7 \) 倍的扭矩，这就是为什么厚度是制表师最粗的功率旋钮。其二，应力 \( \sigma=E e/2\rho \) 必须低于合金的屈服强度 \( \sigma_y \)，否则发条会产生永久”变形（set）”并失去力量。这限制了它能被拧多紧：\( \rho \ge E e / 2\sigma_y \)。

储能是弯矩对上链角度的积分。一条沿全长均匀弯到曲率 \( 1/\rho \) 的带子，储能为

\[U = \tfrac{1}{2}\,\frac{M^{2} L}{E I} = \frac{\sigma^{2}}{6E}\,V, \qquad V = L\,h\,e,\]

因此发条材料的优值是能量密度 \( \sigma_y^{2}/E \)：高屈服强度、适中的刚度。正是这一个比值，使现代发条采用钴合金（Nivaflex）而非普通淬火碳钢——\( E \) 相近，\( \sigma_y \) 高得多。

Now pack the strip into a barrel of inner wall radius \( R \) around an arbor of radius \( r \). A spiral whose successive coils are spaced by the thickness \( e \) and runs from radius \( a \) to radius \( b \) has length

\[L = \frac{\pi}{e}\,\bigl(b^{2}-a^{2}\bigr),\]

so the steel cross-section area \( L e = \pi(b^{2}-a^{2}) \) is just the annulus the coils occupy. The fraction of the free space the spring fills is

\[f = \frac{L\,e}{\pi\,(R^{2}-r^{2})}.\]

The classic result — derivable by maximizing the number of developed turns subject to fixed barrel size — is that a spring delivers the most turns of reserve when it half-fills the barrel, \( f\approx\tfrac12 \). Too thick and short, and it runs out of turns; too thin and long, and the coils bind before it can develop. When fully wound against the arbor the spring’s tightest coil sits at \( \rho=r \), so the peak stress is \( \sigma_{\max}=E e/2r \): a small arbor is brutal on the spring. The energy is bounded above by

\[U_{\max} = \frac{\sigma_{\max}^{2}}{6E}\,V = \frac{E\,e^{2}}{24\,r^{2}}\,L\,h\,e,\]

and the running time is just the developed turns of the barrel divided by how fast the train unwinds it — a going barrel turns only a handful of times per day, so 6–8 working turns becomes a 40–80 h reserve. The figure below is a live barrel: set \( e \), \( L \), \( r \), \( R \) and watch the spiral, the fill fraction, the peak stress against yield, and the resulting torque curve all move together.

现在把钢带塞进内壁半径为 \( R \)、绕着半径为 \( r \) 的条轴的条盒里。相邻圈以厚度 \( e \) 间隔、从半径 \( a \) 盘到半径 \( b \) 的螺旋，其长度为

\[L = \frac{\pi}{e}\,\bigl(b^{2}-a^{2}\bigr),\]

因此钢材截面积 \( L e = \pi(b^{2}-a^{2}) \) 恰好是这些圈所占的环面。发条填充自由空间的比例为

\[f = \frac{L\,e}{\pi\,(R^{2}-r^{2})}.\]

经典结论——可由在固定条盒尺寸下最大化可发展圈数推得——是：当发条填满一半条盒（\( f\approx\tfrac12 \)）时，可发展的动储圈数最多。太厚太短则圈数不够；太薄太长则圈与圈在能发展之前就贴死。完全上满、紧贴条轴时，最紧的一圈位于 \( \rho=r \)，故峰值应力为 \( \sigma_{\max}=E e/2r \)：小条轴对发条非常残酷。储能的上界为

\[U_{\max} = \frac{\sigma_{\max}^{2}}{6E}\,V = \frac{E\,e^{2}}{24\,r^{2}}\,L\,h\,e,\]

而走时时长，就是条盒可发展的圈数除以轮系放松它的快慢——走时条盒一天才转几圈，于是 6–8 个工作圈就成了 40–80 小时的动储。下图是一只活的条盒：设定 \( e \)、\( L \)、\( r \)、\( R \)，看螺旋、填充比例、峰值应力对屈服强度、以及由此得到的扭矩曲线如何一同变化。

The torque curve from §2.2 sags as the spring unwinds — bad for the oscillator (§1.4). The fusée cancels the sag with pure geometry. A chain links the barrel (acting on its drum of radius \( \rho_b \)) to a cone-shaped pulley, the fusée, of variable radius \( R_f \). The chain tension is set by the spring:

\[T = \frac{\tau_s(\theta)}{\rho_b},\]

and the torque the fusée delivers to the train is that tension times the radius at which the chain currently pulls:

\[\tau_{\text{out}} = T\,R_f(\theta) = \tau_s(\theta)\,\frac{R_f(\theta)}{\rho_b}.\]

For a constant output \( \tau_{\text{out}} \) we need the cone radius to be the exact inverse of the spring torque:

\[R_f(\theta) = \frac{\tau_{\text{out}}\,\rho_b}{\tau_s(\theta)}.\]

When the spring is full and strong, the chain pulls from the narrow top of the cone (small \( R_f \), small leverage); as it runs down and weakens, the chain has unwound to the wide base (large \( R_f \), large leverage). Rising leverage exactly compensates falling force. The cone’s profile is therefore nothing but the spring’s torque curve turned upside-down — which is why a fusée must be cut to match its specific spring. It is the most mechanically honest solution to the mainspring problem, and also the most expensive: a miniature chain of dozens of riveted links, a cone, and the space for both.

§2.2 的扭矩曲线随发条松开而下垂——这对振荡器（§1.4）不利。宝塔轮（fusée）用纯几何抵消这种下垂。一条链子把条盒（作用在其半径为 \( \rho_b \) 的鼓上）连到一个锥形滑轮——宝塔轮——其半径 \( R_f \) 可变。链条张力由发条决定：

\[T = \frac{\tau_s(\theta)}{\rho_b},\]

而宝塔轮传给轮系的扭矩，是该张力乘以链条当前拉拽处的半径：

\[\tau_{\text{out}} = T\,R_f(\theta) = \tau_s(\theta)\,\frac{R_f(\theta)}{\rho_b}.\]

要得到恒定输出 \( \tau_{\text{out}} \)，就需要锥半径恰好是发条扭矩的倒数：

\[R_f(\theta) = \frac{\tau_{\text{out}}\,\rho_b}{\tau_s(\theta)}.\]

发条满而强时，链条从锥的窄顶拉拽（\( R_f \) 小、杠杆小）；随着放松变弱，链条已退绕到宽底（\( R_f \) 大、杠杆大）。上升的杠杆恰好补偿下降的力。于是锥的轮廓不过是发条扭矩曲线倒过来——这正是为什么宝塔轮必须按它特定的发条来切削。它是发条问题在机械上最诚实的解法，也是最昂贵的：一条由数十枚铆接链节组成的微型链、一个锥体、以及容纳两者的空间。

The fusée flattens torque the hard way. There is a cheaper, cruder option: don’t flatten the curve — just hide its bad ends. A stopwork (the Maltese cross on the barrel arbor) is a mechanical end-stop that forbids the first turn or two of full wind and the last turn or two of run-down, confining the watch to the flat middle of the curve. You give up reserve to buy torque consistency.

The modern answer abandons both. A long, thin spring of a high-\( \sigma_y \) cobalt alloy in a large barrel already has a torque curve flat enough over its working range that, paired with a good escapement and a hairspring adjusted for isochronism (§5), a fusée buys too little to justify its cost and bulk. Automatics add a slipping bridle: the spring’s outer end grips the barrel wall by friction and slips once torque exceeds a set point, which both prevents over-winding and clamps the top of the curve for free. The figure below lets you compare all four regimes on the same axes — raw barrel, fusée-equalized, stopwork-clipped, and modern flat-alloy — and read off the trade between usable reserve and torque flatness.

宝塔轮以”硬办法”压平扭矩。还有一个更便宜、更粗暴的选择：不压平曲线——只把它糟糕的两端藏起来。 停止装置（条轴上的马耳他十字）是一个机械限位，禁止满链的头一两圈和走完链的末一两圈，把表约束在曲线平坦的中段。你以动储为代价，换取扭矩的一致性。

现代的答案两者都不要。一条又长又薄、采用高 \( \sigma_y \) 钴合金、装在大条盒里的发条，其工作区间内的扭矩曲线本就足够平；配上好的擒纵与为等时性调校过的游丝（§5），宝塔轮带来的好处太少，不值它的成本与体积。自动表再加一道打滑制动（slipping bridle）：发条外端靠摩擦咬住条盒壁，一旦扭矩超过设定值就打滑，既防止上链过满，又顺带把曲线顶端钳平。下图让你在同一坐标上比较全部四种方案——裸条盒、宝塔轮均衡、停止装置截断、现代平合金——并读出可用动储与扭矩平整度之间的取舍。

The train is the bridge from barrel to escapement, and it is defined by two questions. What tooth shape lets one wheel drive the next at a perfectly steady ratio? And what tooth counts turn the barrel’s once-a-day crawl into the escape wheel’s sixteen-turns-a-minute blur? The first is a problem in differential geometry; the second is pure integer arithmetic. Both are visible on the dial.

轮系是从条盒到擒纵的桥梁，由两个问题定义。什么样的齿形能让一只轮以完全平稳的比率带动下一只？又是什么样的齿数把条盒一天一圈的蜗行，变成擒纵轮每分钟十六圈的飞旋？前者是微分几何问题，后者是纯整数算术。两者都写在表盘上。

For the seconds hand to sweep evenly, every mesh must transmit a constant angular-velocity ratio — no flutter as each tooth rolls through. The condition is the fundamental law of gearing: at the contact point, the common normal to the two tooth surfaces must pass through a fixed point \( P \) on the line joining the two centres. That point divides the centre distance in the inverse ratio of the speeds,

\[\frac{\omega_1}{\omega_2} = \frac{O_2 P}{O_1 P} = \frac{z_2}{z_1},\]

so the ratio is set by the tooth counts \( z_1, z_2 \) alone. Two profiles satisfy this law. The involute (the unwinding of a string off a base circle) makes the line of contact a straight line at a fixed pressure angle, and — its great virtue — keeps the ratio exact even if the centre distance is slightly wrong. It rules general engineering. But horology mostly uses the older cycloidal profile (epicycloidal tips, radial-ish flanks). The reason is the watch’s extreme gearing: driven pinions have only 6–12 leaves. At such low counts an involute pinion would have to be undercut at the root, weakening it; a cycloidal pinion does not, and it sheds less energy to sliding friction near the pitch line — exactly the priority when the torque is a few micronewton-metres and every microjoule must reach the balance. Modern profiles (the NIHS/ogival standards) are tuned descendants of the cycloid.

要让秒针均匀扫动，每一处啮合都必须传递恒定的角速度比——不能在每颗齿滚过时抖动。其条件是齿轮啮合基本定律：在接触点，两齿面的公法线必须通过两中心连线上的一个定点 \( P \)。该点按转速的反比分割中心距，

\[\frac{\omega_1}{\omega_2} = \frac{O_2 P}{O_1 P} = \frac{z_2}{z_1},\]

因此传动比仅由齿数 \( z_1, z_2 \) 决定。有两种齿形满足此定律。渐开线（一根线从基圆上展开的轨迹）使接触线成为一条固定压力角下的直线，并且——这是它的大优点——即便中心距略有偏差，比率仍精确。它统治着一般工程。但钟表多用更古老的摆线齿形（外摆线齿顶、近径向齿根）。原因在于表的极端传动比：被动的小齿（pinion）只有 6–12 个齿叶（leaves）。在如此低的齿数下，渐开线小齿不得不在齿根根切（undercut）而被削弱；摆线小齿则无需，且在节点附近因滑动摩擦损失的能量更少——当扭矩只有几微牛·米、每一微焦都必须送到摆轮时，这正是首要考量。现代齿形（NIHS／ogival 标准）是摆线调校后的后裔。

Each mesh is a step-up: a large wheel of \( z_w \) teeth drives a small pinion of \( z_p \) leaves, multiplying speed by \( z_w/z_p \). Chaining the going train multiplies these factors. The balance frequency \( f \) (in Hz) fixes the whole thing through the beat rate. Counting vibrations (half-swings), there are \( 2f \) per second, so

\[\text{vph} = 2f \times 3600 = 7200\,f,\]

and since the lever escapement advances the escape wheel by one tooth per full oscillation (two vibrations), an escape wheel of \( Z \) teeth must turn at

\[n_{\text{esc}} = \frac{\text{vph}}{2Z} \quad[\text{rev/hr}].\]

For \( f=4\,\)Hz and \( Z=15 \): vph \( =28{,}800 \) and \( n_{\text{esc}}=960 \) rev/hr \( =16 \) rev/min. The centre wheel turns once per hour (it carries the minute hand), so the train from centre to escape must multiply by 960. A canonical solution that also puts the seconds hand on the fourth wheel (60 rev/hr) is

\[\underbrace{\frac{75}{10}}_{7.5}\;\cdot\;\underbrace{\frac{80}{10}}_{8}\;=\;60\ (\text{fourth wheel}),\qquad \underbrace{\frac{96}{6}}_{16}\;\Rightarrow\;960\ (\text{escape}).\]

The figure below is a live beat-rate calculator: set the frequency and escape-tooth count, watch the meshing pair hold its exact ratio, and read off the train.

每一处啮合都是一次升速：一只 \( z_w \) 齿的大轮带动一只 \( z_p \) 叶的小齿，转速放大 \( z_w/z_p \) 倍。把走时轮系串起来，这些因子相乘。摆轮频率 \( f \)（赫兹）通过摆频锁定全局。按振动（半摆）计，每秒有 \( 2f \) 次，故

\[\text{vph} = 2f \times 3600 = 7200\,f,\]

又因杠杆擒纵每整次摆动（两次振动）使擒纵轮前进一齿，一只 \( Z \) 齿的擒纵轮须以

\[n_{\text{esc}} = \frac{\text{vph}}{2Z} \quad[\text{转/小时}]\]

旋转。取 \( f=4\,\)Hz、\( Z=15 \)：vph \( =28{,}800 \)，\( n_{\text{esc}}=960 \) 转/时 \( =16 \) 转/分。中心轮每小时一圈（带动分针），故从中心到擒纵的轮系须放大 960 倍。一个同时把秒针放在四轮（60 转/时）上的经典解是

\[\underbrace{\frac{75}{10}}_{7.5}\;\cdot\;\underbrace{\frac{80}{10}}_{8}\;=\;60\ (\text{四轮}),\qquad \underbrace{\frac{96}{6}}_{16}\;\Rightarrow\;960\ (\text{擒纵}).\]

下图是一只活的摆频计算器：设定频率与擒纵齿数，看那对啮合轮维持其精确比率，并读出整条轮系。

A mesh is never lossless. Teeth slide as well as roll, and the sliding dissipates energy as friction; the loss is smallest when the contact is exactly at the pitch point and grows with the depth of engagement and any error in centre distance. Setting that distance correctly — depthing — is done with a dedicated tool and is one of the quiet skills of movement-making: too shallow and the teeth skip or “butt”; too deep and friction and wear climb. A well-cut, well-depthed mesh transmits with efficiency \( \eta \) around 0.97–0.99, but efficiency compounds: across a four-mesh train,

\[\eta_{\text{train}} = \prod_{i} \eta_i \approx \eta^{4},\]

so even \( \eta=0.97 \) per mesh leaves \( 0.97^{4}\approx 0.89 \) — more than a tenth of the barrel’s torque gone before it reaches the escapement. That lost fraction is exactly the margin §1.1 was fighting for: every percent saved in the train is a percent more amplitude at the balance, and amplitude is timekeeping.

啮合从无无损。齿在滚动之外还有滑动，滑动以摩擦耗散能量；当接触恰在节点时损失最小，并随啮合深度与中心距误差而增大。把这个距离设对——定深（depthing）——要用专门的工具完成，是制芯里一项不张扬的功夫：太浅则齿打滑或”顶齿”；太深则摩擦与磨损攀升。一处切削良好、定深得当的啮合，传动效率 \( \eta \) 约 0.97–0.99，但效率会连乘：经过四处啮合的轮系，

\[\eta_{\text{train}} = \prod_{i} \eta_i \approx \eta^{4},\]

故即便每处 \( \eta=0.97 \)，也只剩 \( 0.97^{4}\approx 0.89 \)——条盒扭矩还没到擒纵就丢了一成多。这丢掉的一份，正是 §1.1 在力争的余量：轮系里每省下一个百分点，摆轮就多一个百分点的摆幅，而摆幅就是走时。

The other place torque leaks away is the pivots — the thin ends of each arbor turning in its bearing. A pivot of radius \( a \) carrying normal load \( N \) at friction coefficient \( \mu \) costs a retarding torque

\[M_f = \mu\,N\,a,\]

so you fight friction by making pivots thin (small \( a \)) and \( \mu \) low. Thin steel pivots running in soft brass would wear and seize; the fix, dating to 1704, is to run them in jewels — synthetic ruby/sapphire (corundum) bushings, hard, polished, and dimensionally stable, holding a tiny reservoir of oil. Whether that oil actually separates the surfaces is the Stribeck story: at very low speed the bearing runs in the boundary regime (metal-to-jewel contact, high \( \mu \)); as speed rises a hydrodynamic film lifts the pivot and \( \mu \) drops, then climbs again with viscous drag. A watch pivot lives near the bottom of that curve, which is why oil choice and the polished, slightly-domed pivot end (running against a flat capstone) matter so much. This is the first appearance of a theme §7 makes explicit: a polished surface is not vanity here — it is lower \( \mu \), more amplitude, better rate. The figure below traces the barrel’s torque as it bleeds through meshes and pivots, with jewels on or off.

扭矩泄漏的另一处是轴尖——每根轴在轴承里转动的细端。一个半径为 \( a \)、承受法向载荷 \( N \)、摩擦系数为 \( \mu \) 的轴尖，带来阻滞扭矩

\[M_f = \mu\,N\,a,\]

故对抗摩擦的办法是把轴尖做细（\( a \) 小）、让 \( \mu \) 低。细钢轴尖在软黄铜里转动会磨损、卡死；自 1704 年起的解法，是让它们在宝石里转动——合成红宝石/蓝宝石（刚玉）轴承，硬、抛光、尺寸稳定，并存住一小汪油。这汪油究竟有没有把两面分开，是Stribeck（斯特里贝克）的故事：极低速时轴承处于边界润滑（金属与宝石接触，\( \mu \) 高）；速度升高，流体动压油膜把轴尖托起，\( \mu \) 下降，随后又因黏性阻力回升。表的轴尖就活在这条曲线的谷底附近，这正是为什么选油、以及抛光、略带球面的轴尖端（顶在平的顶石上）如此要紧。这里第一次出现一个 §7 将明说的主题：此处一个抛光的表面绝非虚饰——它意味着更低的 \( \mu \)、更大的摆幅、更好的走时。下图追踪条盒扭矩如何在啮合与轴尖间渗漏，可切换宝石的有无。

Section 1 called the escapement the heart and left it there. It is also the single component where the most force, the most friction, and the most cleverness concentrate in the smallest space. Everything upstream exists to deliver a trickle of torque here; everything downstream exists to count what happens here. This section dissects the Swiss lever action by action, puts numbers on its angles, and asks why — after two centuries — anyone still builds the alternatives.

第 1 节把擒纵称作心脏便搁下了。它也是整枚表里，最大的力、最多的摩擦、最巧的机心都挤进最小空间的那个部件。上游的一切，都是为了把一缕扭矩送到这里；下游的一切，都是为了数清这里发生了什么。本节把瑞士杠杆式擒纵逐个动作拆开，为它的各个角度标上数字，并追问：两个世纪之后，为什么还有人去造那些替代方案？

The cast of parts is small: an escape wheel with pointed “club” teeth; a pallet fork carrying two ruby stones — the entry and exit pallets — pivoting on its own staff; and on the balance staff a roller bearing a single ruby impulse pin, plus a guard pin and banking pins for safety. One beat runs through five actions:

1. Lock. A tooth rests against the locking face of a pallet. The wheel cannot turn; the whole train is frozen (this is the held-back torque of §1.3 — the pallet withstands the tiny torque left after the train’s step-down, not the barrel’s).
2. Draw. The locking face is cut at a small negative angle so the wheel’s torque pulls the fork harder into its banking pin. Draw is what makes the lock safe against shock — the lever will not bounce open.
3. Unlock. The returning balance drives its impulse pin into the fork slot and rotates the lever, sliding the tooth off the locking face against draw. This briefly costs the balance energy.
4. Impulse (lift). The tooth, now on the impulse face, pushes the pallet → fork → impulse pin, returning energy to the balance — the one push per beat that keeps it alive.
5. Drop. The tooth falls off the pallet tip; the wheel “drops” freely until the next tooth locks on the other pallet. Drop is unavoidable lost motion — pure wasted energy, minimised but never zero.

Across one full oscillation the wheel advances one tooth — half as the entry pallet works, half as the exit pallet does — so a \( Z \)-tooth wheel makes \( 2Z \) vibrations per turn, exactly the \( \text{vph}=2Zn \) of §3.2.

角色不多：一只带尖”棒齿（club）”的擒纵轮；一只携两枚红宝石——进瓦与出瓦——并绕自身轴摆动的擒纵叉；以及装在摆轴上、带一枚红宝石冲击钉（impulse pin）的圆盘（roller），外加保安用的保险钉（guard pin）与限位钉（banking pins）。一拍要走过五个动作：

1. 锁定（Lock）。 一颗轮齿抵在叉瓦的锁面上。轮无法转动，整条轮系被冻结（这就是 §1.3 那被顶住的扭矩——叉瓦承受的是轮系降速后剩下的极小扭矩，而非条盒的扭矩）。
2. 拉回（Draw）。 锁面以一个小负角切削，使轮的扭矩把叉更紧地拉向限位钉。拉回正是让锁定能抗冲击的原因——叉不会被震开。
3. 解锁（Unlock）。 返回的摆轮把冲击钉送入叉口、转动杠杆，使齿逆着拉回滑离锁面。这一步短暂地消耗摆轮的能量。
4. 冲量（Impulse／lift）。 齿此刻落到冲面上，推动叉瓦 → 叉 → 冲击钉，把能量送回摆轮——这是每拍唯一的、维持它存活的一推。
5. 跳动（Drop）。 齿从瓦尖滑落，轮”空跳”到下一颗齿在另一枚瓦上锁定为止。跳动是不可避免的损耗动作——纯粹被浪费的能量，可减小却永不为零。

整整一次摆动，轮前进一齿——进瓦走一半、出瓦走一半——故一只 \( Z \) 齿的轮每转产生 \( 2Z \) 次振动，恰是 §3.2 的 \( \text{vph}=2Zn \)。

The fork’s total swing is split into named angles. The lift angle is the slice of the balance’s rotation during which the impulse pin is engaged in the fork — unlocking plus impulse — typically about 30–60° of balance rotation (a common value is 52°). The lock and the drop are small angles of the wheel (a degree or two each), and the draw angle on the locking face is around 10–15°. The efficiency follows directly from this budget: useful work is the impulse; the unlocking work and the drop are losses. A rough efficiency is

\[\eta_{\text{esc}} \approx \frac{\theta_{\text{impulse}}}{\theta_{\text{impulse}} + \theta_{\text{unlock}} + \theta_{\text{drop}}},\]

and a good lever escapement lands near 0.3–0.4 — most of the energy that reaches it is lost, which is why it must be fed by a torque already squeezed down through the train (§3) and why every microjoule saved upstream matters. The figure below steps through one beat: scrub the slider and watch a tooth move through lock → draw → unlock → impulse → drop, with the live phase and the balance’s lift arc called out.

叉的总摆动被切成几个有名字的角。升角（lift angle）是冲击钉嵌在叉中、摆轮转过的那一段——解锁加冲量——通常约为摆轮的 30–60°（常见值 52°）。锁定与跳动是轮的小角度（各一两度），锁面上的拉回角约 10–15°。效率直接由这本账得出：有用功是冲量；解锁功与跳动都是损失。一个粗略的效率为

\[\eta_{\text{esc}} \approx \frac{\theta_{\text{impulse}}}{\theta_{\text{impulse}} + \theta_{\text{unlock}} + \theta_{\text{drop}}},\]

一只好的杠杆擒纵落在 0.3–0.4 附近——到达它的能量大半被浪费，这正是为什么它必须由一个已经过轮系（§3）层层降下的扭矩来喂养，也是为什么上游每省一微焦都要紧。下图分步走过一拍：拖动滑块，看一颗齿历经 锁定 → 拉回 → 解锁 → 冲量 → 跳动，并实时标出当前相位与摆轮的升弧。

Here is the deep idea that made the lever great. The impulse and unlocking happen only within the lift angle — a narrow window at the bottom of the swing, where the balance is moving fastest. For the rest of its arc — the supplementary arc, often 200°+ on each side — the balance is detached: it swings entirely free, touching nothing, governed only by its own \( T=2\pi\sqrt{I/k} \). The escapement is called a detached escapement because it interferes with the oscillator as little, and as briefly, as possible.

Why at the bottom? Because that is where a disturbance does least harm. A push delivered exactly at the balance’s equilibrium point, symmetric on both half-swings, shifts the rate far less than a push delivered out near the turning points. The residual disturbance — the unavoidable cost of unlocking against draw, plus any asymmetry in the impulse — is the escapement error, and it is the part of a watch’s rate that depends on amplitude. A guard pin and the roller’s safety crescent make sure that during the long detached arc a knock cannot let the fork flip to the wrong banking. The takeaway: the larger and cleaner the supplementary arc, the smaller the fraction of each swing the escapement touches, and the better the rate.

这里是让杠杆式封王的深层思想。冲量与解锁只发生在升角之内——摆动最低点附近一个狭窄的窗口，那里摆轮速度最快。在其余的弧段——补充弧（supplementary arc），每侧常达 200° 以上——摆轮是分离的：它完全自由地摆动，不触碰任何东西，只服从自己的 \( T=2\pi\sqrt{I/k} \)。之所以叫分离式（detached）擒纵，正因为它对振荡器的干预尽可能地小、尽可能地短。

为什么在最低点？因为那里的扰动危害最小。一次恰在摆轮平衡点、且两个半摆对称的推力，对走时的偏移远小于靠近折返点处的推力。剩下的扰动——逆着拉回解锁不可避免的代价，加上冲量的任何不对称——就是擒纵误差（escapement error），它正是表的走时中依赖摆幅的那一部分。保险钉与圆盘上的保安月牙，确保在漫长的分离弧中，一次磕碰不会让叉翻到错误的限位上。结论是：补充弧越大、越干净，擒纵每摆触碰的比例就越小，走时也就越好。

The lever wins on robustness: it is self-starting, tolerant of shock, and forgiving to make. Its sin is sliding friction — the tooth slides along the pallet face during impulse, so the escapement depends critically on lubrication, and its rate drifts as the oil ages. Two rivals attack that sin:

• The detent (chronometer) escapement gives a single impulse per oscillation, delivered almost radially with no sliding on the impulse — the most efficient escapement ever made, and the choice for marine chronometers. But it is not self-starting (a stopped chronometer must be swung to life) and is fragile to shock, so it never suited the wristwatch.
• The co-axial escapement (George Daniels, 1974/1999) replaces sliding impulse with nearly radial pushing on two levels of wheel, so it can run with far less lubricant and holds its rate longer between services. The price is complexity, thickness, and tight tolerances.

So the modern lever survives not because it is the best escapement but because it is the best compromise — and because the lubrication and finishing crafts (§3.4, §7) have been pushed far enough to tame its one real flaw. The figure below plots where each design delivers its impulse across the swing, and the efficiency that buys.

杠杆式胜在皮实：它能自启动、耐冲击、也好做。它的原罪是滑动摩擦——冲量阶段齿沿瓦面滑过，故擒纵高度依赖润滑，且走时会随油的老化而漂移。两个对手专攻此罪：

• 天文台（detent，又称冲击式）擒纵每次摆动只给一次冲量，几乎沿径向送出、冲量阶段无滑动——是有史以来效率最高的擒纵，也是航海天文钟的选择。但它不能自启动（停了的天文钟须摇晃才能复活），且不耐冲击，故从不适合腕表。
• 同轴擒纵（co-axial，George Daniels，1974/1999）以两层轮上近乎径向的推动取代滑动冲量，从而能在远少的润滑下运转，两次保养之间走时更持久。代价是复杂、增厚与严苛的公差。

所以现代杠杆式得以存活，不是因为它是最好的擒纵，而是因为它是最好的折中——并且因为润滑与修饰之工（§3.4、§7）已被推进到足以驯服它那唯一的真缺陷。下图画出每种设计在摆动的何处送出冲量，以及由此换来的效率。

The escapement paces; the oscillator decides. Everything else in the watch is in service of letting one spring-and-wheel resonator swing as freely, as steadily, and as indifferently to its surroundings as possible. This section is the physics of that resonator — its quality factor, why its period must not depend on how hard it swings, and why for two centuries the hardest battle in watchmaking was against a thermometer.

擒纵定节拍，振荡器作决定。表里其余的一切，都是为了让这一只弹簧–轮谐振子，尽可能自由、尽可能稳定、尽可能不理会周遭地摆动。本节即是这只谐振子的物理——它的品质因数、它的周期为何不可依赖于摆得多用力，以及为什么两个世纪以来制表最难的一仗，是对着一支温度计打的。

The balance + hairspring obey the equation of a torsional oscillator with damping \( c \) and the escapement’s once-per-beat drive \( M(t) \):

\[I\,\ddot\theta + c\,\dot\theta + k\,\theta = M(t), \qquad \omega_0 = \sqrt{\frac{k}{I}},\quad T = 2\pi\sqrt{\frac{I}{k}}.\]

Left alone, the swing would die away; the escapement’s small impulse exactly replaces the energy lost to \( c \) each cycle, holding a steady amplitude. The single number that governs quality is the quality factor

\[Q = \frac{\omega_0 I}{c} = 2\pi\,\frac{\text{energy stored}}{\text{energy lost per cycle}},\]

a watch balance running around \( Q\sim 100\text{–}300 \) (a quartz tuning fork is \( 10^4\text{–}10^5 \); an atomic standard far beyond). Q matters because the fractional pull the escapement can exert on the rate scales as \( 1/Q \): a high-Q resonator is stubborn, holding its own frequency against the very impulse keeping it alive. So the craft goal is to lose less per cycle — light balances, clean pivots (§3.4), large amplitude — which is the same goal as the whole rest of the movement, now stated as one ratio. The figure animates this: lower Q and watch the free swing decay faster and the rate grow more sensitive; the escapement’s top-up is what arrests the decay.

摆轮 + 游丝遵循带阻尼 \( c \) 与擒纵每拍一次的驱动 \( M(t) \) 的扭转振子方程：

\[I\,\ddot\theta + c\,\dot\theta + k\,\theta = M(t), \qquad \omega_0 = \sqrt{\frac{k}{I}},\quad T = 2\pi\sqrt{\frac{I}{k}}.\]

若放任不管，摆动会衰减殆尽；擒纵那一小份冲量恰好补回每周期被 \( c \) 消耗的能量，维持稳定的摆幅。主宰品质的那个数，是品质因数

\[Q = \frac{\omega_0 I}{c} = 2\pi\,\frac{\text{每周期储存的能量}}{\text{每周期损失的能量}},\]

表的摆轮 Q 约 \( 100\text{–}300 \)（石英音叉为 \( 10^4\text{–}10^5 \)；原子频标更远超之）。Q 之所以要紧，是因为擒纵能对走时施加的相对拉扯正比于 \( 1/Q \)：高 Q 的谐振子很固执，能顶住那个维持它存活的冲量、守住自己的频率。于是工艺目标就是每周期损失更少——轻摆轮、干净轴尖（§3.4）、大摆幅——这与机芯其余部分的目标完全一致，如今被表述为一个比值。下图把这点动画化：调低 Q，看自由摆动衰减更快、走时更敏感；而擒纵的补给正是止住衰减的那只手。

The spring constant \( k \) comes from a hairspring only a few hundredths of a millimetre thick, wound into a flat spiral of a dozen-odd turns. For a strip of width \( b \), thickness \( t \), length \( \ell \), the torsional stiffness is

\[k = \frac{E\,b\,t^{3}}{12\,\ell},\]

— again the \( t^{3} \) law, so the spring’s thickness is the final, finest adjustment of rate. But a flat spiral has a flaw: as it breathes in and out it does not stay concentric, its centre of gravity wanders off-axis, and that wander makes the rate depend on both amplitude and position. Breguet’s fix (1795) is the overcoil: lift the outer end up and curve it back over the body on a computed terminal curve (the Phillips curves give the ideal shape), so the spring develops and contracts concentrically, its CG pinned at the centre. A good overcoil is one of the clearest cases of geometry bought with hand-labour: it is bent and shaped by eye and feel, and it directly buys isochronism (§5.3). The figure’s overcoil toggle shows the difference between a flat spiral’s lopsided breathing and an overcoil’s concentric one.

弹性系数 \( k \) 来自一根只有几百分之一毫米厚的游丝，盘成十余圈的扁平螺旋。对宽 \( b \)、厚 \( t \)、长 \( \ell \) 的带子，其扭转刚度为

\[k = \frac{E\,b\,t^{3}}{12\,\ell},\]

——又是 \( t^{3} \) 律，故游丝的厚度是走时最后、最精细的调整。但扁平螺旋有一个缺陷：它一呼一吸地伸缩时并不保持同心，其质心会偏离轴线游走，而这游走使走时同时依赖摆幅与方位。宝玑的解法（1795）是上绕游丝（overcoil）：把外端抬起、循一条计算好的末端曲线（Phillips 曲线给出理想形状）弯回到游丝主体之上，使游丝同心地展开与收缩，质心被钉在中心。一条好的上绕游丝，是以手工换取几何最清晰的例子之一：它靠眼力与手感弯制成形，并直接换来等时性（§5.3）。图中的 overcoil 开关展示了扁平螺旋偏心的呼吸与上绕游丝同心呼吸之间的差别。

A perfect harmonic oscillator is isochronous: its period does not depend on amplitude. A real watch is not, because the hairspring is slightly non-linear, the escapement adds its amplitude-dependent error (§4.3), and the spring’s CG wanders (§5.2). The result is an isochronism defect — a rate that drifts as amplitude changes, which is exactly what happens as the mainspring runs down (§1.1). Plotting rate against amplitude shows a curve; the whole art of adjusting is to flatten it.

On top of that sits positional error. Lay the watch dial-up and the balance pivots ride on their flat ends; stand it on edge and gravity loads them sideways and, if the balance is not perfectly poised (mass-balanced about its axis), pulls the rate one way on each vertical position. Fine watches are adjusted in up to six positions (dial up/down, crown up/down/left/right); a chronometer certificate is in large part a promise about how little the rate varies across them. The figure below lets you turn an overcoil and good poise on and off and watch the rate-vs-amplitude curve flatten.

一个完美的谐振子是等时的：周期不依赖于摆幅。真实的表并非如此，因为游丝略有非线性、擒纵叠加了它依赖摆幅的误差（§4.3）、且游丝质心会游走（§5.2）。结果是等时性缺陷——走时随摆幅变化而漂移，而这恰恰就是发条放松时发生的事（§1.1）。把走时对摆幅作图会得到一条曲线；调校的全部艺术，就是把它压平。

在此之上还有方位误差。表盘朝上平放时，摆轮轴尖立在其平端上；竖起来时，重力从侧面加载它们，并且若摆轮没有被完美校平（poise，绕轴质量平衡），便会在每个竖直方位把走时拉向一侧。好表会被校准至多六个方位（盘上/下、冠上/下/左/右）；一纸天文台证书，很大程度上就是关于走时在这些方位间变化有多小的承诺。下图让你开关上绕游丝与良好校平，看走时–摆幅曲线如何变平。

The largest enemy of rate, historically, was heat. Two effects stack: the balance expands with temperature, raising its inertia \( I \); and far worse, the hairspring’s elastic modulus \( E \) falls as it warms, lowering \( k \). Both lengthen \( T \), so an uncompensated watch runs slow when warm — by minutes per day across a season. The fractional rate shift is

\[\frac{1}{T}\frac{dT}{d\vartheta} \approx \tfrac12\!\left(\frac{1}{I}\frac{dI}{d\vartheta} - \frac{1}{k}\frac{dk}{d\vartheta}\right),\]

and the \( dk/d\vartheta \) (thermoelastic) term dominates. The 19th-century fix was the bimetallic cut compensation balance: a split rim of brass fused to steel that curls inward as it heats, shrinking \( I \) to cancel the softening spring. The 20th-century fix made the balance obsolete: alloys — Elinvar, then Nivarox, Guillaume’s Invar — whose modulus is nearly temperature-independent (a near-zero thermoelastic coefficient), paired with a low-expansion Glucydur balance. The thermometer was finally beaten not by a clever mechanism but by metallurgy. The figure’s compensation toggle collapses the rate-vs-temperature line toward flat.

历史上走时最大的敌人是热。两种效应叠加：摆轮随温度膨胀，抬高其转动惯量 \( I \)；而更糟的是，游丝的弹性模量 \( E \) 随升温而下降，压低 \( k \)。两者都拉长 \( T \)，故未补偿的表遇热走慢——一季之内可达每天数分钟。其相对走时偏移为

\[\frac{1}{T}\frac{dT}{d\vartheta} \approx \tfrac12\!\left(\frac{1}{I}\frac{dI}{d\vartheta} - \frac{1}{k}\frac{dk}{d\vartheta}\right),\]

其中 \( dk/d\vartheta \)（热弹性）项占主导。19 世纪的解法是双金属切口补偿摆轮：黄铜熔合于钢的开口轮缘，受热向内卷曲，缩小 \( I \) 以抵消变软的游丝。20 世纪的解法让这种摆轮过时了：一类合金——Elinvar，继而 Nivarox，以及 Guillaume 的 Invar——其模量几乎不随温度变化（近零热弹性系数），再配一只低膨胀的 Glucydur 摆轮。温度计最终被击败，靠的不是某个精巧机构，而是冶金学。图中的补偿开关，会把走时–温度的直线压向水平。

Mechanically the display is the humblest subsystem, but it hides two neat tricks: a gear train that produces an exact irrational-looking ratio from small integers, and a clutch that lets you fight the whole train backwards with two fingers and lose gracefully. This is also where every complication plugs in.

论机械，显示是最朴素的子系统，却藏着两手漂亮活：一组用小整数凑出精确比率的齿轮，以及一个让你能用两根手指逆着整条轮系较劲、还能体面落败的离合器。这里也是每一项复杂功能的接口。

The centre wheel turns once per hour and carries the minute hand (via the cannon pinion). To drive the hour hand you need a 12:1 reduction, built under the dial from two meshes — cannon pinion \( z_1 \) → minute wheel \( z_2 \), and minute-wheel pinion \( z_3 \) → hour wheel \( z_4 \):

\[\frac{\omega_{\text{minute}}}{\omega_{\text{hour}}} = \frac{z_2}{z_1}\cdot\frac{z_4}{z_3} = 12.\]

A classic integer solution is \( (40/10)\cdot(36/12) = 4\times 3 = 12 \). Note the minute wheel turns the “wrong” way and the second mesh turns it back, so both hands run clockwise. The figure lets you set the four counts and shows instantly whether the product lands on exactly 12 — miss it and the hour hand simply lies.

中心轮每小时一圈，带动分针（经跨轮 cannon pinion）。要驱动时针，需要 12:1 的减速，在表盘下由两处啮合搭成——跨轮 \( z_1 \) → 分轮 \( z_2 \)，分轮小齿 \( z_3 \) → 时轮 \( z_4 \)：

\[\frac{\omega_{\text{分}}}{\omega_{\text{时}}} = \frac{z_2}{z_1}\cdot\frac{z_4}{z_3} = 12.\]

一个经典整数解是 \( (40/10)\cdot(36/12) = 4\times 3 = 12 \)。注意分轮转向”反了”，第二处啮合又把它转回来，故两针都顺时针走。图中可设定这四个齿数，并即时显示乘积是否恰好落在 12——错过它，时针就只是在撒谎。

The cannon pinion does not bite the centre arbor with teeth — it grips by friction, a tube sprung onto the arbor. Normally that grip is more than enough to carry the featherweight motion works, so the hands track the train. But pull the crown to set the time and you turn the cannon pinion directly; its friction fit slips over the still-locked train, so you reposition the hands without forcing the escapement backwards. The holding torque must sit in a narrow window: enough to drive the hands and any date load, little enough to slip under finger pressure.

Everything beyond hours and minutes is a complication tapped off this same slow rotation. A 24-hour wheel (one turn per day) carries a finger that, once each midnight, pushes a date ring of 31 teeth forward by one tooth; a moonphase divides further; a chronograph couples a separate train through clutches. None of it changes the going train — it only reads from it. The figure animates the 12:1 set and a date ring stepping once per simulated day.

跨轮并不用齿咬住中心轴——它靠摩擦夹持，是一段弹性套在轴上的管。平时这点夹持远够带动轻飘飘的显示轮系，故指针跟随轮系。但当你拔出表冠校时，便直接转动跨轮；它的摩擦配合在仍被锁住的轮系上打滑，于是你能重置指针而不必逆推擒纵。这夹持力矩须落在一个窄窗里：大到能带动指针与日历负载，小到在手指压力下打滑。

时与分之外的一切，都是从这同一缓慢转动上引出的复杂功能。一只 24 小时轮（每天一圈）带一根拨指，每到午夜把一圈 31 齿的日历环推进一齿；月相再作细分；计时码表则经离合耦合一条独立轮系。它们都不改变走时轮系——只从中读数。图中动画化了这个 12:1 组合，以及一圈每”天”步进一次的日历环。

Section 1.6 made the claim; this section proves half of it with numbers and lets the other half stand as what it is. Finishing splits cleanly into two layers — one that the equations of the previous sections demand, and one that no equation asks for at all.

§1.6 立下了那个论断；本节用数字证明它的一半，并让另一半坦然地做它本来的样子。修饰干净地分成两层——一层是前几节的方程所要求的，另一层则没有任何方程会去过问。

Every loss term in this document traces back to two surfaces rubbing: teeth at a mesh (§3.3), a pivot in its jewel (§3.4), a tooth on a pallet (§4.2). The friction at such a contact rises with its surface roughness \( R_a \) — asperities interlock and plough — so a rougher finish means a higher effective \( \mu \), more energy bled per cycle, lower balance amplitude, and (through the escapement error of §4.3) worse rate. The chain is direct:

\[R_a \uparrow \;\Rightarrow\; \mu \uparrow \;\Rightarrow\; \text{loss/cycle}\uparrow \;\Rightarrow\; Q_{\text{eff}}\downarrow,\ \text{amplitude}\downarrow.\]

This is why the finest finishing is reserved for the surfaces that do mechanical work: pivot ends and their capstones, pallet faces, escape teeth, the contact flanks of the wheels. The extreme is black polish (poli noir) — a surface lapped so flat that it scatters no light and shows as pure black or pure mirror depending on the angle. It is not for looks (though it is beautiful); a specular surface has the lowest friction and the best resistance to wear and corrosion a steel part can have. Here, polishing is engineering. The figure makes the chain live: drag the roughness down and watch friction, per-cycle loss, and amplitude respond.

本文里每一个损耗项，都能追到两个表面的摩擦：啮合处的齿（§3.3）、宝石里的轴尖（§3.4）、叉瓦上的齿（§4.2）。这类接触的摩擦随其表面粗糙度 \( R_a \) 升高——微凸体相互咬合、犁削——故更粗的表面意味着更高的有效 \( \mu \)、每周期渗漏更多能量、摆幅更低，并（经 §4.3 的擒纵误差）走时更差。这条链很直接：

\[R_a \uparrow \;\Rightarrow\; \mu \uparrow \;\Rightarrow\; \text{每周期损耗}\uparrow \;\Rightarrow\; Q_{\text{有效}}\downarrow,\ \text{摆幅}\downarrow.\]

这正是为什么最精的修饰，留给那些真正做机械功的表面：轴尖端及其顶石、叉瓦面、擒纵齿、轮的接触齿侧。极致是黑抛光（poli noir）——研磨到极平、不散射光、随角度呈纯黑或纯镜面的表面。它不为好看（尽管确实美）；镜面是钢件所能拥有的最低摩擦、最佳耐磨与抗蚀的表面。在这里，抛光就是工程。图把这条链变活：把粗糙度拖低，看摩擦、每周期损耗与摆幅如何回应。

The rest of finishing answers no equation. Côtes de Genève stripes, perlage swirls, blued screws heated to the precise straw-to-cornflower temperature, engraved balance cocks, and above all anglage — the hand-bevelled, mirror-polished bevels along every bridge edge — sit mostly on surfaces sealed inside the case, doing nothing for amplitude, rate, or reserve. They are the visible record of human hours spent where no function required them.

The connoisseur’s proof is the inward angle. A polished bevel that turns a re-entrant (inward) corner cannot be produced by a rotating tool — a wheel or lap leaves a small radius there. A crisp, sharp interior anglage can only be cut and polished by a hand working with a wooden stick and abrasive, by eye. So a single sharp inner corner certifies that a human spent the time, the way a brushstroke certifies a painting. That is the honest statement of what these notes have been circling: a watch is the rare object where a hard physical optimum (lower friction, flatter torque, better isochronism) and a purely human gesture (a polished surface no law of motion asked for) are worked into the same piece of metal, by the same hands, often in the same motion. The functional layer is why the watch keeps time; the cultural layer is why someone cared whether it kept time beautifully. The whole document has been one long answer to §1.6’s question — and the answer is: in the meeting of those two.

修饰的其余部分不回答任何方程。日内瓦纹、珍珠纹旋涡、加热到恰到好处的稻黄至矢车菊蓝的烧蓝螺丝、雕花摆轮夹板，尤以倒角（anglage）为最——沿每道夹板边缘手工削出、镜面抛光的斜面——大多落在封进表壳内的表面上，对摆幅、走时、动储毫无贡献。它们是人把时光花在功能并不要求之处的可见记录。

行家的判据是那道内角。一道转过内凹角的抛光倒角，无法由旋转刀具做出——轮或砂盘会在那里留下一个小圆角。一道利落锐利的内倒角，只能由一只手执木棒蘸磨料、凭眼力切磨而成。于是单单一个锐利的内角，就证明了有人花过那段时间，正如一道笔触证明一幅画。这正是这份笔记一直在绕的那句老实话：表是这样一种罕见之物——一个坚硬的物理最优（更低的摩擦、更平的扭矩、更好的等时性）与一个纯粹属人的姿态（一个运动定律从未要求的抛光表面）被做进同一块金属、由同一双手、常常在同一个动作里。功能层是表为何能走时；文化层是为何有人在乎它走得漂亮。整篇文章都是对 §1.6 那个问题的一个长长的回答——而答案是：就在这两者相遇之处。

• Daniels, G. (2011). Watchmaking (Updated ed.). Philip Wilson Publishers. — The canonical modern treatise; builds a complete watch from first principles, escapement included.
• Reymondin, C.-A., et al. (1999). The Theory of Horology. Federation of the Swiss Watch Industry (FH). — The standard Swiss watchmaking-school textbook; rigorous on trains, escapements, and oscillator physics.
• Headrick, M. V. (2002). Origin and evolution of the anchor clock escapement. IEEE Control Systems Magazine, 22(2), 41–52. — A control-theory reading of why the escapement is a self-sustaining oscillator.
• Rawlings, A. L. (1993). The Science of Clocks and Watches (3rd ed.). British Horological Institute. — Classic, physics-forward treatment of isochronism and the mainspring problem.
• Phillips, É. (1861). Mémoire sur le spiral réglant des chronomètres et des montres. — The mathematics of the balance-spring terminal curve behind the Breguet overcoil (§5.2).
• Guillaume, C.-É. (1920). Invar and Elinvar (Nobel lecture, Physics). — The metallurgical defeat of temperature error (§5.4).
• Daniels, G. (1994). The Practical Watch Escapement. — The co-axial escapement and the case against sliding-friction impulse (§4.4).

• Daniels, G. (2011). Watchmaking（更新版）. Philip Wilson Publishers. — 现代制表的经典论著；从第一性原理出发造出一整枚表，含擒纵。
• Reymondin, C.-A., 等 (1999). The Theory of Horology. 瑞士钟表工业联合会（FH）. — 瑞士制表学校的标准教科书；对轮系、擒纵与振荡器物理的论述严谨。
• Headrick, M. V. (2002). Origin and evolution of the anchor clock escapement. IEEE Control Systems Magazine, 22(2), 41–52. — 用控制论解读擒纵为何是一个自维持振荡器。
• Rawlings, A. L. (1993). The Science of Clocks and Watches（第 3 版）. British Horological Institute. — 关于等时性与发条问题的经典、偏物理的论述。
• Phillips, É. (1861). Mémoire sur le spiral réglant des chronomètres et des montres. — 宝玑上绕游丝背后、游丝末端曲线的数学（§5.2）。
• Guillaume, C.-É. (1920). Invar and Elinvar（诺贝尔物理学奖演讲）. — 用冶金学击败温度误差（§5.4）。
• Daniels, G. (1994). The Practical Watch Escapement. — 同轴擒纵，以及反对滑动摩擦冲量的论证（§4.4）。