📚 Year 12 CAIE Engineering: Core Knowledge Review | Year 12 CAIE 工程：核心知识点梳理
The Year 12 CAIE Engineering syllabus introduces students to the fundamental principles of mechanical, electrical, thermal and materials engineering. This article provides a structured recap of the core knowledge areas, linking key concepts, essential equations and practical applications to help you consolidate your understanding for the AS examinations. Each section below pairs the English terminology you will meet in the question paper with a clear Chinese explanation, so that you can revise both the physics and the language of engineering at the same time.
Year 12 CAIE 工程教学大纲向学生介绍机械、电气、热学和材料工程的基本原理。本文对核心知识领域进行了系统的梳理，将关键概念、基本方程和实际应用联系起来，以帮助你巩固理解，为 AS 考试做好准备。以下每一节都将试卷中会出现的英文术语与清晰的中文解释配对呈现，让你在复习工程原理的同时，也同步掌握工程英语表达。
1. Forces and Equilibrium | 力与平衡
Forces are vector quantities that can be resolved into perpendicular components. A body is in static equilibrium when the net force in any direction is zero and the net moment about any point is zero. The principle of moments states that for a body in rotational equilibrium, the sum of clockwise moments equals the sum of anticlockwise moments. In examinations you are usually asked to draw a free body diagram first, label every reaction, weight and applied load, and only then substitute numbers into the equations.
力是矢量，可以分解为相互垂直的分量。当一个物体在所有方向上的合力都为零，且对任意点的合力矩也为零时，该物体处于静力平衡状态。力矩原理指出，对于处于转动平衡的物体，顺时针力矩之和等于逆时针力矩之和。在考试中，通常要求你先画受力图，标出每一个支座反力、重力和外加载荷，然后才把数值代入方程求解。
For a force F acting at an angle θ to the horizontal, the components are found with trigonometry, and the resultant of two perpendicular forces follows Pythagoras:
作用于某点、与水平方向成 θ 角的力 F，其分量可用三角函数求出，而两个相互垂直的力的合力则遵循勾股定理：
Fₓ = F cos θ Fᵧ = F sin θ F = √Fₓ² + Fᵧ² tan θ = Fᵧ / Fₓ
In many structures, support reactions are determined by applying the three equilibrium conditions for a coplanar force system:
在许多结构中，支座反力可以通过应用平面力系的三个平衡条件来确定：
ΣFₕ = 0 ΣFᵥ = 0 ΣM = 0
A simply supported beam carrying a central point load W of span L has two equal reactions of W / 2, whereas an overhanging or asymmetric beam requires you to take moments about one support to find the other reaction first. Always state the direction of each reaction in your answer, and check that the vertical reactions add up to the total downward load.
承受跨中集中载荷 W、跨度为 L 的简支梁，两端反力各为 W / 2；而对于外伸梁或非对称受载梁，则需要先对一侧支座取矩求出另一侧反力。答题时务必写明每个反力的方向，并检查竖直反力之和是否等于全部向下载荷之和。
2. Moments, Couples and Support Reactions | 力矩、力偶与支座反力
The moment of a force about a point equals the magnitude of the force multiplied by the perpendicular distance from the point to the line of action of the force:
力对某点的力矩等于力的大小乘上该点到力作用线的垂直距离：
M = F × d
The unit of moment is the newton metre (N m). Because a moment is a vector quantity in the full three-dimensional treatment, the direction of rotation must always be stated, and clockwise is conventionally taken as negative when you sum moments on one side of an equation.
力矩的单位是牛米（N m）。在完整的三维处理中力矩是矢量，因此必须说明转动方向；在方程一侧对力矩求和时，习惯上把顺时针取为负值。
A couple consists of two equal, opposite and parallel forces whose lines of action are separated by a perpendicular distance d. The resultant force of a couple is zero, but its turning effect, called the torque, is:
力偶由两个大小相等、方向相反且相互平行的力组成，它们的作用线之间相距垂直距离 d。力偶的合力为零，但其转动效应（称为转矩）为：
T = F × d
Couples are the natural way to describe the output of a motor, a steering wheel or a screwdriver. In a beam problem, a concentrated moment applied at a point does not appear in the ΣFₕ and ΣFᵥ equations at all; it appears only in the moment equation, which is a very common source of lost marks.
力偶是描述电动机、方向盘或螺丝刀输出作用的自然方式。在梁的问题中，施加于某点的集中力偶完全不出现在 ΣFₕ 和 ΣFᵥ 方程里，只出现在力矩方程中，这是非常常见的失分点。
3. Stress, Strain and Young Modulus | 应力、应变与杨氏模量
When a material is subjected to an external force it develops internal stress and deforms. Tensile stress σ is defined as the force applied per unit cross-sectional area:
当材料受到外力作用时，内部会产生应力并发生变形。拉应力 σ 定义为施加在单位截面积上的力：
σ = F / A
resses are usually quoted in megapascals (MPa, 10⁶ Pa) or gigapascals (GPa, 10⁹ Pa) because structural materials are extremely strong.
单位是帕斯卡（Pa），1 Pa = 1 N m⁻²。在工程实践中，应力通常用兆帕（MPa，10⁶ Pa）或吉帕（GPa，10⁹ Pa）表示，因为结构材料的强度极高。
Strain ε is the ratio of extension to original length and has no units:
应变 ε 是伸长量与原长之比，没有单位：
ε = ΔL / L₀
Young modulus E is a measure of the stiffness of a material and is defined within the linear elastic region as the ratio of stress to strain:
杨氏模量 E 是材料刚度的量度，在线弹性范围内定义为应力与应变之比：
E = σ / ε = (F × L₀) / (A × ΔL)
Hooke's Law states that the extension of a spring or a wire is directly proportional to the applied load, provided the elastic limit is not exceeded:
胡克定律指出，在不超过弹性极限的前提下，弹簧或金属丝的伸长量与所加载荷成正比：
F = k × x and k = E × A / L₀
The stiffness constant k is therefore not a material property: it depends on the Young modulus, the cross-sectional area and the original length of the particular specimen being tested.
因此，劲度系数 k 并不是材料本身的属性：它取决于被测样品的杨氏模量、横截面积和原长。
4. Stress–Strain Behaviour of Materials | 材料的应力—应变行为
A stress–strain graph for a ductile metal such as mild steel shows several distinct regions. The initial straight line is the linear elastic region, in which the gradient equals the Young modulus and the material returns to its original length when the load is removed. The limit of proportionality marks the end of the straight line, and the elastic limit marks the greatest stress from which the specimen can still recover completely.
低碳钢等韧性金属的应力—应变图会显示出几个明显不同的区域。最初的直线段是线弹性区，其斜率等于杨氏模量，卸载后材料能完全恢复到原长。比例极限标志着直线的终点，而弹性极限则标志着样品仍能完全恢复的最大应力。
Beyond the yield point, plastic deformation begins: the material deforms permanently and the curve flattens or even dips before rising again to the ultimate tensile strength (UTS). After the UTS the specimen begins to neck, the cross-sectional area falls rapidly, and fracture finally occurs at the breaking point. A brittle material such as cast iron or glass shows almost no plastic region: it breaks at or very near the limit of proportionality.
超过屈服点后进入塑性变形：材料发生永久变形，曲线变平甚至下凹，随后再次上升至抗拉强度极限（UTS）。达到 UTS 后样品开始出现缩颈，横截面积迅速减小，最终在断裂点处断裂。铸铁或玻璃等脆性材料几乎没有塑性区：它们几乎在比例极限处就直接断裂。
The area under the stress–strain curve up to a given point represents the energy absorbed per unit volume, which is why ductile materials are described as tough. In the elastic region the strain energy stored can be calculated in two equivalent ways:
应力—应变曲线在某一给定点之前下方的面积表示单位体积吸收的能量，这正是韧性材料被称为"韧"的原因。在弹性区，储存的应变能可用两种等价方式计算：
U = ½ × F × ΔL = ½ × k × x² = ½ × σ × ε × (A L₀)
Toughness, ductility, stiffness and strength are four different material properties and examination questions frequently test whether you can keep them apart. Stiffness is measured by E, strength by the UTS or yield stress, ductility by the permanent strain at fracture, and toughness by the total area under the curve.
韧性、延展性、刚度和强度是四种不同的材料性能，考试题经常考查你能否把它们区分开来。刚度由 E 衡量，强度由 UTS 或屈服应力衡量，延展性由断裂时的永久应变衡量，韧性则由曲线下总面积衡量。
5. Materials Selection and Engineering Properties | 材料选择与工程性能
Choosing a material for a component requires matching six or seven key properties to the service conditions: density, Young modulus, yield strength, ductility, thermal conductivity, electrical conductivity and cost. Metals are generally strong, stiff and ductile, polymers are light, corrosion resistant and electrically insulating, and composites such as carbon fibre reinforced polymer combine high strength with low mass.
为零件选择材料时，需要把六七项关键性能与服务条件匹配起来：密度、杨氏模量、屈服强度、延展性、导热率、导电率以及成本。金属通常强度高、刚度大且具有延展性；聚合物质轻、耐腐蚀且电绝缘；而碳纤维增强聚合物等复合材料则兼具高强度与低质量。
| Material | 材料 | Typical E / GPa | 典型 E | Typical UTS / MPa | 典型 UTS | Typical use | 典型用途 |
| Mild steel 低碳钢 | 200 | 400 | Beams, bolts 梁、螺栓 |
| Aluminium alloy 铝合金 | 70 | 300 | Aircraft, frames 飞机、框架 |
| Cast iron 铸铁 | 100 | 200 | Machine beds 机床床身 |
| Nylon 尼龙 | 3 | 70 | Gears, bushes 齿轮、轴套 |
| CFRP 碳纤维复合材料 | 150 | 1200 | Bicycle frames 自行车车架 |
Three failure mechanisms are worth memorising because they appear regularly in design questions. Fatigue is the growth of a crack under repeated cyclic loading well below the static UTS. Creep is slow, time-dependent deformation at high temperature under a constant load. Corrosion is electrochemical attack, usually prevented by painting, galvanising, alloying or cathodic protection.
有三种失效机理值得牢记，因为它们在设计题中经常出现。疲劳是在远低于静态 UTS 的反复循环载荷下裂纹的扩展；蠕变是在高温和恒定载荷下随时间缓慢发生的变形；腐蚀则是电化学侵蚀，通常通过涂漆、镀锌、合金化或阴极保护来防止。
6. Mechanical Systems: Levers, Gears and Cams | 机械系统：杠杆、齿轮与凸轮
A lever is a rigid bar pivoting about a fulcrum, and its mechanical advantage is the ratio of load to effort, while the velocity ratio is the ratio of the distance moved by the effort to the distance moved by the load. For an ideal frictionless machine, mechanical advantage equals velocity ratio, so:
杠杆是绕支点转动的刚性杆，其机械利益是载荷与作用力之比，而速度比则是作用力移动距离与载荷移动距离之比。对于理想的无力摩擦机械，机械利益等于速度比，因此：
MA = Load / Effort VR = d(effort) / d(load) η = MA / VR × 100%
Gear trains transmit rotation and change torque. The gear ratio depends on the numbers of teeth, and neighbouring gears rotate in opposite directions unless an idler is introduced:
齿轮系用于传递转动并改变转矩。传动比取决于齿数，相邻齿轮的转向相反，除非加入惰轮：
n(driven) / n(driver) = N(driver) / N(driven)
Because power is conserved in an ideal gear train, a reduction in speed produces a proportional increase in torque. Pulley systems follow the same logic: the velocity ratio equals the number of rope sections supporting the moving load. A single fixed pulley only changes the direction of the effort, whereas a block and tackle with four supporting sections has a velocity ratio of 4.
由于理想齿轮系中功率守恒，转速降低会带来转矩成比例增大。滑轮系统的逻辑相同：速度比等于承担动载荷的绳索段数。单个定滑轮只改变作用力方向，而具有四段承重绳的滑轮组速度比则为 4。
Cams convert uniform rotary motion into a prescribed reciprocating motion, and the shape of the cam profile determines the displacement diagram of the follower. Linkages and four-bar mechanisms are used to change the direction or the nature of motion, while bearings, lubrication and surface finish control friction losses.
凸轮把匀速转动转换为指定的往复运动，凸轮轮廓的形状决定了从动件的位移图。连杆机构和四杆机构用于改变运动方向或运动形式，而轴承、润滑与表面粗糙度则决定摩擦损失的大小。
7. Energy, Work, Power and Efficiency | 能量、功、功率与效率
Work is done when a force moves its point of application through a distance, and for a force at an angle to the displacement only the component along the displacement counts:
当力的作用点沿力的方向发生位移时，该力就做了功；若力与位移成一定夹角，则只有沿位移方向的分量作功：
W = F × d × cos θ KE = ½ m v² PE = m g h
Power is the rate of doing work or transferring energy, and for a rotating shaft the mechanical power output is the product of torque and angular velocity:
功率是作功或传递能量的速率；对于转轴，机械输出功率等于转矩与角速度的乘积：
P = W / t = F × v P = T × ω ω = 2π n (rad s⁻¹)
Efficiency compares the useful output with the total input and is always less than 100 % in a real machine because of friction, windage, electrical resistance and heat losses:
效率比较的是有用输出与总输入，由于摩擦、风阻、电阻和热损失，真实机器的效率总是小于 100 %：
η = (useful output energy / total input energy) × 100% = (useful output power / input power) × 100%
The principle of conservation of energy requires that every joule of input is accounted for, so an energy audit question usually asks you to list the losses in descending order of magnitude and suggest one practical improvement for the largest one.
能量守恒原理要求输入能量的每一焦耳都有去向，因此能量审计类题目通常要求你按从大到小的顺序列出各项损失，并针对最大的一项提出一条具体的改进措施。
8. Thermal Principles and Heat Transfer | 热学原理与传热
Heat is energy transferred because of a temperature difference. The quantity of heat needed to raise the temperature of a mass m by ΔT depends on the specific heat capacity c of the substance:
热量是由于温差而传递的能量。使质量为 m 的物质温度升高 ΔT 所需的热量取决于该物质的比热容 c：
Q = m × c × ΔT Q = m × L (change of state)
During a change of state the temperature remains constant while latent heat is absorbed or released, which is why the specific latent heat of fusion or vaporisation appears in the second equation. A mixture problem is solved by setting the heat lost by the hot body equal to the heat gained by the cold body, plus any heat absorbed by the container.
在物态变化过程中，温度保持不变，同时吸收或释放潜热，这就是第二个方程中出现熔化比潜热或汽化比潜热的原因。混合类问题可通过令高温物体放出的热量等于低温物体吸收的热量加上容器吸收的热量来求解。
Conduction through a solid slab of thickness L and cross-sectional area A obeys the steady-state relation below, where k is the thermal conductivity in W m⁻¹ K⁻¹. Convection transfers heat by the bulk movement of a fluid, and radiation transfers energy by electromagnetic waves without needing a medium.
通过厚度为 L、横截面积为 A 的固体板的传导遵循下面的稳态关系式，其中 k 为导热率，单位为 W m⁻¹ K⁻¹。对流通过流体的宏观运动传递热量，而辐射则通过电磁波传递能量，无需介质。
Q / t = k × A × ΔT / L ΔL = L₀ × α × ΔT
Thermal expansion must be designed for: railway rails need expansion gaps, bridges sit on bearings, and bimetallic strips convert a temperature change into mechanical displacement for thermostats. Insulation reduces the rate of heat loss by lowering k, reducing A or increasing L.
热膨胀必须在设计中加以考虑：铁轨需要伸缩缝，桥梁要放在支座上，而双金属片可把温度变化转化为机械位移用于恒温器。保温通过降低 k、减小 A 或增大 L 来降低散热速率。
9. Electrical Principles: Ohm's Law and Circuits | 电学原理：欧姆定律与电路
Ohm's Law states that the current through a metallic conductor is proportional to the potential difference across it, provided the temperature remains constant:
欧姆定律指出，在温度保持恒定的条件下，通过金属导体的电流与其两端电势差成正比：
V = I × R P = V × I = I² × R = V² / R
The resistance of a uniform conductor depends on its resistivity, length and cross-sectional area, which explains why long thin cables waste more energy as heat:
均匀导体的电阻取决于其电阻率、长度和横截面积，这解释了为什么又长又细的电缆会以热量形式浪费更多能量：
R = ρ × L / A
In a series circuit the current is the same everywhere and the potential differences add; in a parallel circuit the potential difference is the same across each branch and the currents add. Kirchhoff's two rules simply express conservation of charge and conservation of energy at a junction and around a loop.
在串联电路中各处电流相同，电势差相加；在并联电路中各支路电势差相同，电流相加。基尔霍夫两条定律不过是在节点和回路中表达电荷守恒与能量守恒。
A real source has internal resistance r, so the terminal potential difference falls as the current drawn increases, which is the reason a car battery struggles to crank a cold engine:
实际电源具有内阻 r，因此随着取用电流增大，端电压会下降，这正是汽车电瓶在寒冷天气下难以启动发动机的原因：
V = EMF − I × r I = EMF / (R + r)
The maximum power transfer theorem states that a load receives maximum power when its resistance equals the internal resistance of the source, a result that is used in matching loudspeakers and sensors to their amplifiers.
最大功率传输定理指出，当负载电阻等于电源内阻时负载获得最大功率，这一结果常用于扬声器与传感器同放大器的阻抗匹配。
10. Electromagnetism, Motors and Generators | 电磁学、电动机与发电机
A current-carrying conductor placed in a magnetic field experiences a force whose direction is given by Fleming's left-hand rule. The magnitude of the force depends on the flux density, the current and the length of conductor within the field:
置于磁场中的载流导体受到力的作用，其方向由弗莱明左手定则给出。力的大小取决于磁通密度、电流以及磁场中导体的长度：
F = B × I × L × sin θ F = B × q × v
ical d.c. motor adds a commutator to reverse the current every half revolution so that the rotation
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