Introduction to Mechanical Systems | A3.3

All movement in a mechanical system can be described using four fundamental motion types:

1. Linear motion — Movement in a straight line from one point to another. The direction may be horizontal, vertical, or at any angle, but the path is straight. Examples: A drawer opening and closing; a lift car travelling up and down; a bullet leaving a barrel; a rack in a rack-and-pinion system.
2. Rotary motion — Continuous rotation around a fixed axis. The most common motion type in mechanical systems. Examples: A gear shaft spinning; a wheel turning; a motor output shaft; a clock hand.
3. Oscillating motion — A back-and-forth arc of movement around a fixed pivot point. Unlike reciprocating motion (which is linear), oscillating motion follows a curved path. Examples: A pendulum swinging; windscreen wipers sweeping; a metronome; a child's swing.
4. Reciprocating motion — A back-and-forth movement along a straight line. Reciprocating motion is linear but reverses direction periodically. Examples: A piston moving up and down inside a cylinder; a saw blade cutting; a sewing machine needle; a bicycle pump.

Converting between motion types is one of the core tasks of mechanical design. A petrol engine converts reciprocating piston motion into rotary crankshaft motion. A rack and pinion converts rotary into linear. A cam converts rotary into reciprocating. Understanding which mechanism achieves which conversion is essential for designing functional products.

机械系统中所有运动都可以用四种基本运动类型描述：

1. 直线运动——从一点到另一点沿直线移动。方向可以是水平、垂直或任意角度，但路径是直线。例子：抽屉开关；电梯上下运行；子弹离开枪管；齿条齿轮系统中的齿条。
2. 旋转运动——绕固定轴连续旋转。机械系统中最常见的运动类型。例子：齿轮轴旋转；车轮转动；电机输出轴；钟表指针。
3. 摆动运动——绕固定支点来回弧形运动。与往复运动（直线）不同，摆动运动遵循弧形路径。例子：摆锤摆动；雨刷扫动；节拍器；秋千。
4. 往复运动——沿直线来回运动。往复运动是直线的但周期性反向。例子：活塞在气缸内上下运动；锯条切割；缝纫机针；自行车打气筒。

运动类型转换是机械设计的核心任务之一。汽油发动机将活塞往复运动转换为曲轴旋转运动。齿条齿轮将旋转转换为直线。凸轮将旋转转换为往复。理解哪种机构实现哪种转换对设计功能性产品至关重要。

Every mechanical system can be described using an input–process–output (IPO) model:

• Input: The energy or force applied to the system by the user or a power source. This may be a push, pull, twist, or rotational force.
• Process: The mechanism that transforms the input — changing its magnitude, direction, speed, or type of motion.
• Output: The resulting motion or force delivered to do useful work.

Examples across product types:

The IPO model helps designers identify which mechanism to choose. If an input is rotary and an output must be linear, the process must include a rotary-to-linear converter (rack and pinion, cam, crank and slider). If force magnitude must change, a lever, pulley, or gear ratio provides the process step.

Simple machines are the primitive building blocks: inclined plane, wedge, lever, pulley, wheel and axle, and screw. Every complex mechanism can be traced back to combinations of these six.

每个机械系统都可以用输入-过程-输出（IPO）模型描述：

• 输入：用户或动力源施加到系统的能量或力。可以是推、拉、扭或旋转力。
• 过程：转换输入的机构——改变其大小、方向、速度或运动类型。
• 输出：传递用于做有用功的运动或力。

跨产品类型的例子：

IPO模型帮助设计师确定选择哪种机构。如果输入是旋转而输出必须是直线，则过程必须包括旋转到直线转换器（齿条齿轮、凸轮、曲柄滑块）。如果力的大小必须改变，则杠杆、滑轮或齿轮比提供过程步骤。

简单机械是基本构建模块：斜面、楔子、杠杆、滑轮、轮轴和螺旋。每个复杂机构都可以追溯到这六种的组合。

Mechanical Advantage (MA) is the ratio of the output force produced by a machine to the input force applied by the user:

MA = Output force / Input force = F_out / F_in

MA > 1 means the machine multiplies force (you apply less force than the load requires). MA < 1 means you apply more force than the load, but gain speed or distance. The fundamental trade-off is: you cannot get more work out than you put in. Increasing force means the input must travel a greater distance; increasing speed means less force.

Ideal Mechanical Advantage (IMA) assumes a frictionless, perfect machine. It is calculated from geometry (distances), not actual forces:

IMA = Distance input travels / Distance output travels

Real machines have friction, so Actual Mechanical Advantage (AMA) is always less than IMA. Efficiency = AMA / IMA × 100%.

Inclined plane: Spreads the work of lifting over a longer sloped distance, reducing the force required at each instant.

IMA = Slope length (L) / Vertical height (h)

Example from the chapter: L = 1.46 m, h = 0.5 m → IMA = 1.46 / 0.5 = 2.92. The user applies 2.92 times less force than lifting straight up — but must push the load 1.46 m along the slope rather than 0.5 m straight up.

Wedge: Two inclined planes placed back-to-back. When driven forward, the wedge converts a forward force into two outward (splitting) forces perpendicular to the wedge faces. MA depends on the wedge angle — the thinner the wedge, the greater the MA, but the further it must be driven. Examples: Axe head, knife blade, wood chisel, door stop, ZIP fastener teeth.

Pulleys: A fixed pulley changes the direction of force but not its magnitude (MA = 1). A movable pulley provides MA = 2, because the load is supported by two rope segments. A block and tackle combines multiple movable pulleys; the MA equals the number of rope segments supporting the movable block. A 4-rope block and tackle gives MA = 4 (you lift a 400 N load with 100 N, but must pull 4× the lifting distance).

Wheel and axle: Applies a large effort at the rim (wheel) to produce a larger force at the axle — or vice versa. MA = radius of wheel / radius of axle. Examples: Steering wheel (large wheel radius → small force needed), screwdriver handle (wide handle → large torque at tip), doorknob, winch.

机械优势（MA）是机器产生的输出力与用户施加的输入力之比：

MA = 输出力 / 输入力 = F_out / F_in

MA > 1意味着机器放大力（施加的力小于负载所需的力）。MA < 1意味着施加的力大于负载，但获得了更大的速度或距离。基本权衡是：输出的功不能多于输入的功。增加力意味着输入必须行进更大距离；增加速度意味着力更小。

理想机械优势（IMA）假设无摩擦的完美机器。它由几何（距离）而非实际力计算：

IMA = 输入行进距离 / 输出行进距离

真实机器有摩擦，所以实际机械优势（AMA）总是小于IMA。效率 = AMA / IMA × 100%。

斜面：将提升工作分散到更长的斜坡距离上，减少每时刻所需的力。

IMA = 斜面长度（L）/ 垂直高度（h）

章节示例：L = 1.46 m，h = 0.5 m → IMA = 1.46 / 0.5 = 2.92。用户施加比垂直提升小2.92倍的力——但必须沿斜面推动负载1.46 m而非直接向上0.5 m。

楔子：两个背靠背的斜面。向前驱动时，楔子将向前的力转换为垂直于楔面的两个向外（劈开）的力。MA取决于楔子角度——楔子越薄，MA越大，但必须被驱动的距离越远。例子：斧头、刀刃、木凿、门挡、拉链齿。

滑轮：定滑轮改变力的方向但不改变大小（MA = 1）。动滑轮提供MA = 2，因为负载由两段绳子支撑。滑轮组结合多个动滑轮；MA等于支撑动滑轮的绳段数量。4绳滑轮组MA = 4（用100 N提起400 N的负载，但必须拉4倍的提升距离）。

轮轴：在轮缘（轮）施加较大的力产生轴处较大的力——或反之。MA = 轮半径 / 轴半径。例子：方向盘（大轮半径→所需力小）、螺丝刀柄（宽手柄→尖端扭矩大）、门把手、绞盘。

Mechanical systems are grouped into five broad families by how they transmit and transform motion and force:

Speed and torque are always traded against each other (assuming constant power). Gearing up (small gear driving large gear) increases torque but reduces speed. Gearing down (large driving small) increases speed but reduces torque. This is the same principle as the inclined plane: gain force, lose distance — or gain speed, lose force.

Direction changes are achieved by bevel gears (90° shaft angle), reverse linkages (opposite linear direction), rack and pinion (rotary ↔ linear), bell-crank linkages (90° direction change), and idler gears (same shaft alignment, reversed rotation).

机械系统按传递和转换运动及力的方式分为五大类：

速度和扭矩总是相互权衡的（假设功率恒定）。升速（小齿轮驱动大齿轮）增加扭矩但降低速度。降速（大驱动小）增加速度但降低扭矩。这与斜面原理相同：获得力，失去距离——或获得速度，失去力。

方向改变通过锥齿轮（90°轴角）、反向连杆（相反直线方向）、齿条齿轮（旋转↔直线）、钟形曲柄连杆（90°方向改变）和惰轮（相同轴对齐，旋转方向相反）实现。

Simple machines become complex mechanical systems when two or more are combined in series or in parallel, each stage transforming the motion or force before passing it to the next. The output of one stage becomes the input of the next.

Everyday complex system examples:

• Can opener: Combines a lever (the handle provides MA to grip the can), a wedge (the cutting wheel pierces and shears the lid), and a wheel and axle (the turning handle drives the cutting wheel around the rim). Three simple machines, one seamless action.
• Car jack (scissor jack): Combines a screw (turning the handle raises the lead screw), a lever (the handle multiplies the turning force), and linkages (the X-shaped scissors convert the horizontal screw motion into vertical lift).
• Bicycle: Combines a lever (pedal arm), chain drive (sprocket to wheel), wheel and axle (road wheel), and on multi-speed versions, a derailleur linkage and multiple gear ratios.
• Petrol engine: Combines reciprocating motion (pistons), a crank and connecting rod (reciprocating → rotary), a camshaft (rotary → reciprocating for valves), and gears/chains (timing, gearbox).

Historical context — Georgius Agricola, De Re Metallica (1556): Agricola's landmark mining engineering treatise documented waterwheel-powered ore-crushing machines. His woodcut illustrations show waterwheels (rotary), linked via gear trains to hammers (reciprocating), with cams on the shaft converting rotary to the repeated hammer blows needed to crush ore. This is essentially the same system as a modern stamping press — the technology is half a millennium old.

Design principle: When analysing a complex mechanism, identify the chain from input to output. At each stage ask: what type of motion enters? What type must leave? What mechanism performs that conversion? This systematic decomposition is the foundation of mechanism design.

当两个或多个简单机械串联或并联组合时，每个阶段在传递给下一个之前转换运动或力，就形成了复杂机械系统。一个阶段的输出成为下一阶段的输入。

日常复杂系统例子：

• 开罐器：结合了杠杆（手柄提供MA夹住罐头）、楔子（切割轮穿透并剪切盖子）和轮轴（转动手柄驱动切割轮绕边缘运动）。三种简单机械，一个无缝动作。
• 汽车千斤顶（剪式千斤顶）：结合了螺旋（转动手柄提升丝杠）、杠杆（手柄放大转动力）和连杆机构（X形剪刀将水平螺旋运动转换为垂直提升）。
• 自行车：结合了杠杆（踏板臂）、链传动（链轮到车轮）、轮轴（后轮）以及多速版本上的拨链器连杆机构和多种齿轮比。
• 汽油发动机：结合了往复运动（活塞）、曲柄连杆（往复→旋转）、凸轮轴（旋转→气门往复）以及齿轮/链条（正时、变速箱）。

历史背景——格奥尔格乌斯·阿格里科拉，《论矿冶》（1556年）：阿格里科拉的里程碑式矿山工程著作记录了水轮驱动的矿石破碎机。他的木版画展示了水轮（旋转），通过齿轮系与锤子（往复）连接，轴上的凸轮将旋转转换为破碎矿石所需的重复锤击。这本质上与现代冲压机相同——这项技术已有五百年历史。

设计原则：分析复杂机构时，识别从输入到输出的链条。在每个阶段问：什么类型的运动进入？什么类型必须离开？什么机构执行该转换？这种系统分解是机构设计的基础。

Gears transmit rotary motion and torque between shafts. When two gears mesh, their teeth interlock — one tooth of the driving gear pushes one tooth of the driven gear. The gear ratio governs the speed and torque relationship:

Gear ratio = Teeth on driven gear / Teeth on driving gear = Speed of driving gear / Speed of driven gear

A 40-tooth driven gear meshing with a 20-tooth driving gear has a ratio of 2:1 — the driven gear turns at half the speed but with double the torque.

The main gear types are:

1. Spur gears: Straight teeth cut parallel to the shaft axis. The simplest, most common type. Teeth engage suddenly, which can cause noise at high speeds. Applications: Washing machines, clocks, simple gear trains, electric screwdrivers.
2. Bevel gears: Conical-shaped gears that transmit motion between shafts that intersect at an angle, typically 90°. The cone angle determines the shaft angle. Applications: Automotive differentials (distribute engine power to both rear wheels while allowing speed difference on corners); hand drills.
3. Hypoid gears: Similar to spiral bevel gears but with non-intersecting offset shafts — the pinion shaft sits below the ring gear centreline. This allows a lower vehicle floor, a larger pinion for greater strength, and quieter operation due to the spiral tooth engagement. Applications: Rear-wheel drive automobile axles.
4. Rack and pinion: A cylindrical pinion gear meshes with a flat linear rack. Converts rotary motion to linear motion and vice versa. Applications: Car steering (pinion turns, rack moves tie rods laterally); drill press spindle; railway switches.
5. Worm gears (screw gears): A helical screw (worm) meshes with a toothed wheel. One full rotation of the worm advances the wheel by just one tooth. This produces high gear reduction in a compact space. The system is usually self-locking — the wheel cannot back-drive the worm. Applications: Lifting equipment (screw jacks, hoists), automotive steering boxes, conveyor systems, tuning pegs on guitars.
6. Ratchet and pawl: A toothed wheel (ratchet) engages a spring-loaded latch (pawl) that allows rotation in one direction only. In the forward direction, the pawl clicks over the teeth. In the reverse direction, the pawl catches and locks. Applications: Socket wrenches (ratchet spanners), mechanical clocks, cable ties, climbing equipment.
7. Idler gears: A gear placed between two other gears to reverse the direction of rotation of the output gear without changing the speed ratio. An odd number of idlers gives the same rotation direction as the input; an even number gives the opposite. Idlers also allow the driving and driven shafts to be further apart. Applications: Reversing direction in a clock mechanism; print rollers.
8. Compound gears: Two or more gears fixed on the same shaft, rotating together at the same speed. Because the gears on the same shaft have different tooth counts, each pair in the compound train can multiply the gear ratio. Large speed/torque changes can be achieved in a compact arrangement. Applications: Mechanical watches, multi-speed gearboxes, can opener drives.

齿轮在轴之间传递旋转运动和扭矩。当两个齿轮啮合时，它们的齿相互咬合——主动齿轮的一个齿推动从动齿轮的一个齿。齿轮比控制速度和扭矩关系：

齿轮比 = 从动齿轮齿数 / 主动齿轮齿数 = 主动齿轮速度 / 从动齿轮速度

40齿从动齿轮与20齿主动齿轮啮合，齿轮比为2:1——从动齿轮以一半速度旋转但扭矩加倍。

主要齿轮类型：

1. 直齿圆柱齿轮：平行于轴线切割的直齿。最简单、最常见的类型。齿突然啮合，高速时可能产生噪音。应用：洗衣机、钟表、简单齿轮系、电动螺丝刀。
2. 锥齿轮：圆锥形齿轮，在通常以90°相交的轴之间传递运动。锥角决定轴角。应用：汽车差速器（在弯道时允许速度差的情况下将发动机动力分配给两个后轮）；手摇钻。
3. 准双曲面齿轮：类似螺旋锥齿轮，但具有不相交的偏置轴——小齿轮轴位于环形齿轮中心线以下。这允许更低的车辆底板、更大的小齿轮以获得更大强度，以及由于螺旋齿啮合而更安静的运行。应用：后轮驱动汽车车桥。
4. 齿条齿轮：圆柱形小齿轮与平直线性齿条啮合。将旋转运动转换为直线运动，反之亦然。应用：汽车转向（小齿轮转动，齿条横向移动拉杆）；钻床主轴；铁路道岔。
5. 蜗轮蜗杆（螺旋齿轮）：螺旋形蜗杆与带齿的蜗轮啮合。蜗杆每转一圈只将蜗轮推进一个齿。这在紧凑空间内产生高齿轮减速比。系统通常自锁——蜗轮不能反向驱动蜗杆。应用：起重设备（螺旋千斤顶、提升机）、汽车转向箱、传送系统、吉他调音弦钮。
6. 棘轮棘爪：带齿的棘轮与弹簧加载的棘爪啮合，仅允许单方向旋转。向前方向时，棘爪在齿上发出咔哒声。反向时，棘爪卡住锁定。应用：套筒扳手（棘轮扳手）、机械钟表、扎带、登山设备。
7. 惰轮：放置在两个其他齿轮之间的齿轮，反转输出齿轮的旋转方向而不改变速比。奇数个惰轮给出与输入相同的旋转方向；偶数个给出相反方向。惰轮还允许主动轴和从动轴相距更远。应用：钟表机构中的方向反转；印刷滚轮。
8. 组合齿轮：固定在同一轴上的两个或多个齿轮，以相同速度一起旋转。由于同一轴上的齿轮具有不同的齿数，复合齿轮系中的每对都可以放大齿轮比。可以在紧凑的排列中实现大幅度的速度/扭矩变化。应用：机械手表、多速变速箱、开罐器驱动装置。

Belt and chain drives transmit rotary power between two or more shafts that are physically separated — unlike gears, which require direct tooth-to-tooth contact.

Belt drives use a continuous flexible belt looped over two or more pulleys. Power transmission relies on friction between the belt and pulley surface. The belt cross-section (flat, V, or toothed) determines how the friction is generated:

• Flat belts: Simple; lower friction; used where some slip is acceptable.
• V-belts: Wedge into a groove in the pulley, increasing normal force and friction. Most common type for industrial power transmission.
• Toothed (synchronous) belts: Have teeth that mesh with matching teeth on the pulley — no slip, precise timing. Used where synchronisation is critical: vehicle timing belts (keeping crankshaft and camshaft in phase), 3D printer drives.

Belt drive speed ratio: Speed ratio = diameter of driven pulley / diameter of driving pulley. A 200 mm driven pulley driven by a 100 mm pulley runs at half the speed of the driving pulley with double the torque.

Historical use: 19th-century factories used a single steam-driven lineshaft running the length of the factory ceiling. Individual machines were connected to the lineshaft by their own belts and pulleys, each with a different pulley ratio to run at the correct speed. The belt drive was the power distribution system of the Industrial Revolution.

Modern belt drive applications: Conveyor belts, vehicle alternator drives, lathe and drill press drives, treadmill decks, gym equipment.

Chain drives use a continuous loop of chain links engaging with the teeth of sprocket wheels. Power is transmitted by meshing (mechanical interlocking), not friction. This eliminates slip and allows precise speed ratios.

• Advantages over belts: No slip; handles heavier loads; more durable; efficient power transmission.
• Disadvantages: Noisier; requires lubrication to reduce wear; heavier; more expensive.

Applications: Bicycles (the classic example — sprocket on pedal crank drives sprocket on rear wheel via chain); motorcycles; industrial conveyors; timing chains in engines; agricultural machinery.

The bicycle chain drive is elegant in its simplicity: changing sprocket sizes (the derailleur system) changes the gear ratio without any gear contact or noise, and the chain is easily replaced when worn.

带传动和链传动在物理上分离的两个或多个轴之间传递旋转动力——不像齿轮，需要齿对齿直接接触。

带传动使用绕在两个或多个带轮上的连续柔性皮带。动力传递依赖于皮带和带轮表面之间的摩擦力。皮带横截面（平带、V带或齿形带）决定摩擦的产生方式：

• 平带：简单；摩擦力较低；用于允许一定打滑的场合。
• V带：楔入带轮的槽中，增加法向力和摩擦力。最常见的工业动力传递类型。
• 齿形（同步）带：具有与带轮上匹配齿啮合的齿——无打滑，精确计时。用于同步至关重要的场合：汽车正时皮带（保持曲轴和凸轮轴同相）、3D打印机驱动。

带传动速比：速比 = 从动带轮直径 / 主动带轮直径。200 mm从动带轮由100 mm带轮驱动时，以主动带轮一半速度运行，扭矩加倍。

历史用途：19世纪工厂使用单一蒸汽驱动的天轴沿工厂天花板延伸。每台机器通过自己的皮带和带轮连接到天轴，每个都有不同的带轮比以正确速度运行。带传动是工业革命的动力分配系统。

现代带传动应用：传送带、汽车交流发电机驱动、车床和钻床驱动、跑步机、健身设备。

链传动使用连续链环与链轮齿啮合。动力通过啮合（机械互锁）而非摩擦传递。这消除了打滑并允许精确速比。

• 相比皮带的优势：无打滑；处理更重荷载；更耐用；高效动力传递。
• 劣势：更噪杂；需要润滑以减少磨损；更重；更昂贵。

应用：自行车（经典例子——踏板曲柄上的链轮通过链条驱动后轮上的链轮）；摩托车；工业传送机；发动机正时链；农业机械。

A cam is a specially shaped plate or cylinder mounted on a rotating shaft. As the shaft turns, a follower — a rod or lever resting against the cam surface — traces the cam profile and converts the rotary motion into a controlled reciprocating or oscillating motion. The exact motion of the follower depends entirely on the cam's shape (profile).

Follower types: Knife-edge (precise, wears quickly), flat-faced (lower surface pressure, less wear), roller (reduced friction, common in engines), and spherical (for curved cam surfaces).

The six standard cam shapes and the motion each produces:

1. Pear cam: Shaped like a pear. The follower remains stationary (dwell) for most of the rotation as the circular portion passes, then rises quickly over the pointed lobe and returns to rest. Produces a short rapid rise-and-fall action. Use: Engine camshafts for valve opening; printing presses.
2. Circular cam (eccentric): A circular disc with its centre of rotation offset from its geometric centre. As it rotates, the follower rises and falls smoothly in a sinusoidal-like pattern. There is no dwell — the follower is always moving. Use: Simple reciprocating pumps; windscreen wiper drives.
3. Triangular cam: Approximately equilateral triangular cross-section with rounded corners. Produces three rise-and-fall cycles per revolution, making the follower oscillate three times for each turn of the shaft. Use: Where multiple strokes per revolution are needed; some loom mechanisms.
4. Eccentric cam: Sometimes used interchangeably with circular cam above. More precisely, any cam whose effective rotation centre is offset. Use: Jigsaws, reciprocating saws — the blade's up-and-down stroke is produced by an eccentric cam on the motor shaft.
5. Oval (elliptical) cam: Egg-shaped or elliptical cross-section. Produces two rise-and-fall cycles per revolution. Smoother than the triangular cam. Use: Some knitting machines; mechanical typewriters.
6. Snail cam (drop cam): Shaped like a snail shell — the follower rises gradually as the cam rotates, then drops suddenly at the "foot" of the snail. Produces a slow rise followed by a rapid drop (or sudden release). Use: Trip hammers; alarm clock striking mechanisms (the hour hand cam releases the striker).

Agricola's 1556 waterwheel-powered ore crushers used snail or lobe cams on a rotating shaft to repeatedly lift and drop heavy hammers. The same cam-and-follower principle powers the valves in every petrol and diesel engine today.

凸轮是安装在旋转轴上的特殊形状的板或圆柱。当轴旋转时，靠在凸轮表面上的从动件——杆或杠杆——描绘凸轮轮廓，将旋转运动转换为受控的往复或摆动运动。从动件的确切运动完全取决于凸轮的形状（轮廓）。

从动件类型：刀刃型（精确，磨损快）、平面型（表面压力低，磨损少）、滚轮型（摩擦减少，发动机中常见）和球形型（用于弯曲凸轮表面）。

六种标准凸轮形状及其产生的运动：

1. 梨形凸轮：形如梨子。当圆形部分经过时，从动件大部分时间保持静止（停留），然后在尖形凸起上快速上升并返回静止。产生短暂快速的升降动作。用途：发动机凸轮轴气门开启；印刷机。
2. 圆形凸轮（偏心凸轮）：旋转中心偏离几何中心的圆形盘。旋转时，从动件以正弦形式平滑地升降。没有停留——从动件总在运动。用途：简单往复泵；雨刷驱动。
3. 三角形凸轮：近似等边三角形截面，带圆角。每转产生三次升降循环，轴每转一圈从动件振荡三次。用途：需要每转多次行程的场合；某些织机机构。
4. 偏心凸轮：有时与上述圆形凸轮互换使用。更准确地说，任何有效旋转中心偏移的凸轮。用途：曲线锯、往复锯——刀片的上下行程由电机轴上的偏心凸轮产生。
5. 椭圆形凸轮：卵形或椭圆形截面。每转产生两次升降循环。比三角形凸轮更平滑。用途：某些编织机；机械打字机。
6. 蜗牛形凸轮（落凸轮）：形如蜗牛壳——凸轮旋转时从动件缓慢上升，然后在蜗牛"脚"处突然下落。产生缓慢上升后的快速下落（或突然释放）。用途：碎矿锤；闹钟打铃机构（时针凸轮释放打铃器）。

阿格里科拉1556年水轮驱动的矿石破碎机使用旋转轴上的蜗牛形或凸轮凸起反复提起并落下重锤。如今每台汽油和柴油发动机中，同样的凸轮从动件原理驱动气门。

A lever is a rigid beam that rotates about a fixed point called the fulcrum. A force called the effort is applied at one point to move a load at another. The lever's class depends on the relative positions of these three elements.

Mechanical advantage of a lever:

MA = Effort arm length / Load arm length

Where the effort arm is the distance from the fulcrum to where effort is applied, and the load arm is the distance from the fulcrum to the load.

First-class lever (F–L–E or E–F–L): The fulcrum is between the load and the effort. MA can be greater than, equal to, or less than 1 depending on the relative arm lengths.

• Tool examples: See-saw (fulcrum in centre), crowbar (fulcrum near load end, MA > 1), scissors (two first-class levers sharing a fulcrum), pliers.
• Human body: The head and neck. The atlanto-occipital joint (base of skull) is the fulcrum. The weight of the head is the load, and the neck muscles (at the back) are the effort. When the head is tilted forward, the load arm is longer than the effort arm — MA < 1. When tilted back, the effort arm is longer — MA > 1.

Second-class lever (F–L–E): The load is between the fulcrum and the effort. The effort arm is always longer than the load arm, so MA is always > 1. A smaller effort can always lift a larger load, but the effort point must travel further than the load.

• Tool examples: Wheelbarrow (wheel is fulcrum, load in the tray, hands provide effort at the handles), bottle opener, nutcracker.
• Human body: Standing on tiptoes (plantarflexion). The ball of the foot (metatarsophalangeal joint) is the fulcrum. Body weight acts at the ankle (load). The calf muscles pull upward via the Achilles tendon at the heel (effort). Since the heel is behind the ankle but the fulcrum is at the ball of the foot, the effort arm is longer than the load arm → MA > 1. A relatively small calf muscle force lifts the whole body.

Third-class lever (F–E–L): The effort is between the fulcrum and the load. The effort arm is always shorter than the load arm, so MA is always < 1. The user must apply more force than the load — but the load moves faster and travels further than the effort point. Third-class levers trade force for speed and range of motion.

• Tool examples: Tweezers (pivot at one end, fingers apply effort in the middle, grip at the far end), fishing rod, broom (lower hand is fulcrum, upper hand is effort, bristles are load).
• Human body: Bicep curl (elbow flexion). The elbow joint is the fulcrum. The hand and weight are the load. The bicep muscle attaches to the radius bone close to the elbow (effort). Because the bicep attachment is very close to the elbow, the effort arm is short relative to the load arm (hand to elbow) → MA < 1. The bicep muscle must produce much more force than the weight being held — but the hand moves through a large arc for a small contraction of the bicep.
• Human body: Knee extension (quadriceps). The knee joint is the fulcrum. The foot/lower leg is the load. The quadriceps tendon attaches to the tibia just below the knee (effort). Same principle as the bicep: MA < 1, but the foot swings quickly through a large range.

Third-class levers dominate in the human body because evolution prioritises speed and range of limb movement over force multiplication — most powerful muscles achieve force through size, not lever geometry.

杠杆是绕称为支点的固定点旋转的刚性横梁。称为用力的力施加在一点以在另一点移动负载。杠杆的类型取决于这三个元素的相对位置。

杠杆的机械优势：

MA = 用力臂长度 / 负载臂长度

其中用力臂是从支点到施力点的距离，负载臂是从支点到负载的距离。

第一类杠杆（支-负-力或力-支-负）：支点在负载和用力之间。根据臂长的相对关系，MA可以大于、等于或小于1。

• 工具例子：跷跷板（支点在中心）、撬棍（支点靠近负载端，MA > 1）、剪刀（两个共享支点的第一类杠杆）、钳子。
• 人体：头部和颈部。寰枕关节（颅底）是支点。头部重量是负载，颈后肌肉是用力。头部前倾时，负载臂比用力臂长——MA < 1。后仰时，用力臂更长——MA > 1。

第二类杠杆（支-负-力）：负载在支点和用力之间。用力臂总是比负载臂长，因此MA总是 > 1。较小的力总能提起较大的负载，但用力点必须比负载移动更远的距离。

• 工具例子：独轮车（车轮是支点，负载在车斗里，双手在把手处提供用力）、开瓶器、坚果夹。
• 人体：踮脚尖站立（跖屈）。脚掌（跖趾关节）是支点。体重作用在脚踝处（负载）。小腿肌肉通过跟腱在脚跟处向上拉（用力）。用力臂比负载臂长 → MA > 1。相对较小的小腿肌肉力量提起整个身体。

第三类杠杆（支-力-负）：用力在支点和负载之间。用力臂总是比负载臂短，所以MA总是 < 1。用户必须施加比负载更大的力——但负载移动速度更快、距离更远。第三类杠杆以力换取速度和运动范围。

• 工具例子：镊子（一端为支点，手指在中间施力，顶端夹持）、钓鱼竿、扫帚（下手是支点，上手是用力，刷毛是负载）。
• 人体：肱二头肌弯举（肘关节屈曲）。肘关节是支点。手和重物是负载。肱二头肌附着在靠近肘部的桡骨上（用力）。因为附着点非常靠近肘部，用力臂相对于负载臂（手到肘）很短 → MA < 1。肱二头肌必须产生比持有重量大得多的力——但手通过大弧度运动，而肱二头肌只需小幅收缩。
• 人体：膝关节伸展（股四头肌）。膝关节是支点。脚/小腿是负载。股四头肌腱附着在膝盖正下方的胫骨上（用力）。原理与肱二头肌相同：MA < 1，但脚在大范围内快速摆动。

第三类杠杆在人体中占主导地位，因为进化优先考虑肢体运动的速度和范围而非力的放大——大多数强壮的肌肉通过体积而非杠杆几何形状获得力量。

A linkage is a system of rigid bars (links) connected by pivots (pins or hinges). Unlike gears and belts, linkages do not transmit continuous rotary motion — they guide specific paths of movement, constrain relative motion between parts, or convert one motion type into another over a limited range.

1. Parallel linkage: All links remain parallel to each other throughout the motion. The output member moves in the same direction as the input member, but maintains the same orientation (it does not rotate).

• Mechanism: A four-bar linkage where the two side links (cranks) are equal in length. As one crank moves, the other crank moves identically. The connecting bar (coupler) translates without rotating.
• Applications: Scissor lifts (Mobile Elevated Work Platforms) — use a criss-cross X pattern of parallel links to raise a platform while keeping it horizontal. The X pattern uses fixed pivots at the centre of each X and moving pivot points that slide as the lift extends. Important: although the X pattern visually resembles an X, it is still a parallel linkage because the platform moves in the same direction as the input (upward when extended). Pantograph — scales drawings; used by engravers and in railway overhead wire systems. Desk lamps — the arm maintains the shade in any position. Trainset buffer couplings.

2. Reverse linkage: The output member moves in the opposite direction to the input. A central pivot point reverses the motion.

• Mechanism: Two links share a fixed central pivot. When one end moves in one direction, the other end moves in the opposite direction — like a see-saw.
• Applications: Bolt cutters — squeezing the handles together forces the cutting jaws together (handles and jaws move in opposite directions). Vice grips (locking pliers). Brake pedal linkages in vehicles (pushing the pedal forward applies the brakes rearward).

3. Bell-crank linkage: Changes the direction of motion or force by approximately 90°. An L-shaped lever (the bell crank) has its pivot at the corner, so a horizontal input becomes a vertical output, or vice versa.

• Mechanism: A rigid L-shaped or angled lever pivots at its elbow. A push rod connected to one arm creates a pull rod movement in the perpendicular arm.
• Applications: Bicycle rim brakes (caliper brakes) — squeezing the brake lever pulls a cable that connects to a bell-crank mechanism at the wheel, pressing the brake pads inward against the rim. The 90° direction change allows the cable to run from the handlebars down to the wheel axle area. Aircraft control systems — converting control cable movement to surface movement. Early telegraph signal relays.

Linkage design principle: The geometry of the pivot points and link lengths determines the path, speed, and force amplification of the output. Changing the ratio of the bell-crank arms changes the mechanical advantage; changing the length ratio in a parallel linkage changes the speed ratio. Linkages are versatile because they guide motion with no friction in the transmission itself (only at the pivot pins), unlike belt and gear systems.

连杆机构是由支点（销或铰链）连接的刚性杆（连杆）系统。与齿轮和皮带不同，连杆机构不传递连续旋转运动——它们引导特定的运动路径，约束部件之间的相对运动，或在有限范围内将一种运动类型转换为另一种。

1. 平行连杆机构：所有连杆在运动中始终保持相互平行。输出构件与输入构件沿相同方向运动，但保持相同方向（不旋转）。

• 机构：两侧连杆（曲柄）等长的四杆机构。当一个曲柄运动时，另一个曲柄做相同运动。连接杆（连接件）平移而不旋转。
• 应用：剪叉式升降机（高空作业平台）——使用十字形X平行连杆模式提升平台同时保持水平。X模式使用每个X中心的固定支点和升降延伸时滑动的移动支点。受电弓（pantograph）——按比例缩放图纸；用于雕刻师和铁路架空线系统。台灯——臂在任何位置保持灯罩不倾斜。

2. 反向连杆机构：输出构件沿与输入相反的方向运动。中央支点点反转运动。

• 机构：两个连杆共享固定中央支点。当一端沿一个方向运动时，另一端沿相反方向运动——像跷跷板。
• 应用：断线钳——将手柄捏合在一起迫使切割钳口合拢（手柄和钳口沿相反方向运动）。大力钳（锁定钳）。车辆中的制动踏板连杆（向前踩踏板向后施加制动）。

3. 钟形曲柄连杆机构：将运动或力的方向改变约90°。L形杠杆（钟形曲柄）的支点在转角处，因此水平输入变成垂直输出，或反之。

• 机构：刚性L形或有角度的杠杆在其肘部处旋转。连接到一个臂的推杆在垂直臂产生拉杆运动。
• 应用：自行车轮缘刹车（卡钳制动器）——捏合刹车拉杆拉动连接到车轮处钟形曲柄机构的钢缆，将刹车片向内压在轮缘上。90°方向改变允许钢缆从车把沿着到车轴区域延伸。飞机控制系统——将控制钢缆运动转换为控制面运动。

连杆设计原则：支点位置和连杆长度的几何形状决定输出的路径、速度和力放大。改变钟形曲柄臂的比例改变机械优势；改变平行连杆中的长度比改变速比。连杆机构用途广泛，因为它们在传动本身中没有摩擦（只在支点销处有），不像皮带和齿轮系统。