Year 13 CAIE Engineering: Core Knowledge Review | Year 13 CAIE 工程：核心知识点梳理

📚 Year 13 CAIE Engineering: Core Knowledge Review | Year 13 CAIE 工程：核心知识点梳理

The second year of A Level Engineering under CAIE specification 9336 brings together advanced principles in mechanics, materials, thermodynamics, fluid dynamics, electronics and control. A deep understanding of these interconnected topics is essential for tackling Paper 1 and Paper 2, and for applying theory to practical design in the coursework project. This article summarises the core knowledge areas that every Year 13 student must master, with clear explanations and key formulas.

在CAIE 9336工程学科的第二年，学生需要整合力学、材料学、热力学、流体动力学、电子学和控制理论中的高级原理。透彻理解这些相互关联的主题对于应对卷一和卷二，以及在课程项目中把理论应用于实际设计至关重要。本文梳理了Year 13学生必须掌握的核心知识领域，配以清晰的解释和关键公式。

1. Statics and Equilibrium | 静力学与平衡

For a body in static equilibrium, both the resultant force and the resultant moment about any point must be zero. This fundamental condition is expressed as ΣF = 0 and ΣM = 0, and it underpins the analysis of forces in beams, trusses and frames.

对于处于静力平衡的物体，任何点上的合力和合力矩都必须为零。这一基本条件表示为 ΣF = 0 和 ΣM = 0，它是梁、桁架和框架中力分析的基础。

∑F = 0, ∑M = 0

In solving truss problems, you will apply the method of joints and method of sections, distinguishing between tension and compression members. A free-body diagram is always the first step, isolating the forces acting on a chosen section or node.

在解决桁架问题时，需要运用节点法和截面法，区分受拉和受压杆件。包括绘制受力分析图，隔离作用在选定截面或节点上的力，永远是第一步。

When a simply supported beam carries point loads or uniformly distributed loads (UDL), you calculate support reactions by taking moments about one support and then resolving vertically. For a beam of length L with a central point load W, the reactions are each W/2.

当简支梁承受集中荷载或均布荷载时，通过对某一支座取矩再竖向分解来计算支座反力。对于跨中承受集中荷载W、跨度为L的梁，每个支座反力为 W/2。

Rₐ = R₆ = W/2

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2. Material Properties and Stress-Strain | 材料性质与应力-应变

Engineering materials are characterised by their response to applied loads. Stress σ is defined as force per unit original cross-sectional area, and strain ε is the change in length divided by the original length.

工程材料通过其对施加荷载的响应来表征。应力 σ 定义为单位原始横截面积上的力，应变 ε 定义为长度变化量除以原始长度。

σ = F/A₀, ε = ΔL/L₀

Young’s modulus E = σ/ε describes the stiffness of a material within the linear elastic region. On a stress-strain graph, important features include the elastic limit, yield point (for ductile materials), ultimate tensile strength (UTS) and fracture point.

杨氏模量 E = σ/ε 描述材料在线弹性区内的刚度。在应力-应变曲线上，重要特征包括弹性极限、屈服点（对于延性材料）、极限抗拉强度 (UTS) 和断裂点。

Ductility, brittleness, toughness and hardness are key terms. Toughness is the energy absorbed up to fracture, given by the area under the stress-strain curve. Fatigue failure occurs under cyclic loading, even when stresses are below the UTS, and is analysed using S-N curves.

延性、脆性、韧性和硬度是关键术语。韧性是直至断裂所吸收的能量，由应力-应变曲线下的面积给出。疲劳破坏在循环荷载下发生，即使应力低于 UTS，可通过 S-N 曲线进行分析。

Property | 性质	Definition | 定义	Units | 单位
Young’s Modulus | 杨氏模量	σ/ε in linear region | 线性区内 σ/ε	Pa or GPa
UTS | 极限抗拉强度	Maximum stress before necking | 颈缩前的最大应力	Pa
Ductility | 延性	Percentage elongation or reduction in area | 延伸率或断面收缩率	%

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3. Kinematics and Projectile Motion | 运动学与抛体运动

Constant acceleration motion is described by the SUVAT equations. For an object moving along a straight line with uniform acceleration a, initial velocity u, final velocity v, displacement s, and time t:

匀加速运动由 SUVAT 方程描述。对于沿直线做匀加速运动的物体，加速度为 a，初速度为 u，末速度为 v，位移为 s，时间为 t：

v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t

These equations are applied in both horizontal and vertical components for projectile motion. The horizontal component of velocity remains constant (neglecting air resistance), while the vertical component is subject to gravitational acceleration g = 9.81 m/s².

这些方程应用于抛体运动的水平与竖直分量。水平方向速度保持不变（忽略空气阻力），而竖直方向受重力加速度 g = 9.81 m/s² 影响。

Key derived results include time of flight, maximum height and horizontal range. For a projectile launched with speed v₀ at angle θ to the horizontal, the range R = (v₀² sin 2θ)/g. Maximum range is achieved at θ = 45°.

关键推导结果包括飞行时间、最大高度和水平射程。对于以速度 v₀ 与水平方向夹角 θ 发射的抛体，射程 R = (v₀² sin 2θ)/g，当 θ = 45° 时射程最大。

R = (v₀² sin 2θ)/g, H_max = (v₀² sin²θ)/(2g)

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4. Dynamics and Newton’s Laws | 动力学与牛顿定律

Newton’s second law states that the resultant force acting on an object is equal to the rate of change of its momentum. For constant mass, this simplifies to F = ma. This law forms the basis for solving connected body problems (e.g. pulleys and lifts) and for analysing motion on inclined planes.

牛顿第二定律指出，作用在物体上的合外力等于其动量的变化率。对于恒定质量的情况，简化为 F = ma。这一定律是求解连接体问题（如滑轮和升降机）以及分析斜面上运动的基础。

F = ma

On a rough inclined plane, the force parallel to the slope is mg sin θ minus friction. Friction is given by F_fr = μN, where μ is the coefficient of friction and N is the normal reaction (mg cos θ). The body will accelerate if mg sin θ > μ mg cos θ.

在粗糙斜面上，沿斜面的分力为 mg sin θ 减去摩擦力。摩擦力 F_fr = μN，其中 μ 为摩擦系数，N 为法向反力 (mg cos θ)。若 mg sin θ > μ mg cos θ，物体将加速下滑。

Momentum p = mv is conserved in the absence of external forces. Impulse = FΔt = Δp. Elastic and inelastic collisions are distinguished by whether kinetic energy is conserved.

动量 p = mv 在无外力时守恒。冲量 Impulse = FΔt = Δp。弹性碰撞与非弹性碰撞的区别在于动能是否守恒。

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5. Work, Energy and Power | 功、能量与功率

Work done by a constant force F moving its point of application through displacement s in the direction of the force is W = F × s. When the force is not parallel, work = Fs cos θ. The area under a force-distance graph represents the work done.

恒力 F 沿力的方向使作用点位移 s 所做的功为 W = F × s。当力不平行时，功 = Fs cos θ。力-距离图下的面积表示做功大小。

Kinetic energy is ½mv² and gravitational potential energy is mgh. The principle of conservation of mechanical energy states that, in the absence of resistive forces, the sum of KE and PE remains constant.

动能为 ½mv²，重力势能为 mgh。机械能守恒定律指出，在没有阻力的条件下，动能与势能之和保持不变。

KE = ½mv², PE = mgh

Power is the rate of doing work: P = W/t = Fv (for constant force and velocity). Efficiency of a machine is (useful power output / total power input) × 100%.

功率是做功的速率：P = W/t = Fv（适用于恒力与恒速度）。机器的效率为（有用输出功率 / 总输入功率）× 100%。

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6. Thermodynamic Cycles and Efficiency | 热力学循环与效率

The First Law of Thermodynamics states that the increase in internal energy of a system equals the heat added to the system minus the work done by the system: ΔU = Q – W. For a complete cycle, ΔU = 0, hence net work = net heat transfer.

热力学第一定律指出，系统内能的增量等于加入系统的热量减去系统对外做的功：ΔU = Q – W。对于完整循环，ΔU = 0，因此净功等于净传热量。

ΔU = Q – W

Carnot’s theorem gives the maximum possible efficiency for a heat engine operating between a hot reservoir at temperature Tₕ and a cold sink at Tₗ (in kelvin): η_Carnot = 1 – Tₗ/Tₕ. No real engine can exceed this efficiency.

卡诺定理给出了在高温热源 Tₕ 与低温冷源 Tₗ（单位为开尔文）之间工作的热机所能达到的最高效率：η_Carnot = 1 – Tₗ/Tₕ。任何实际热机都无法超越这一效率。

η_Carnot = 1 – Tₗ/Tₕ

The Second Law implies that entropy of an isolated system always increases. In engineering, understanding the Rankine cycle (steam power) and Otto/Diesel cycles (internal combustion) is important for evaluating thermal efficiency and specific fuel consumption.

第二定律意味着孤立系统的熵总是增加。在工程中，理解朗肯循环（蒸汽动力）和奥托/狄塞尔循环（内燃机）对于评估热效率和比燃料消耗至关重要。

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7. Fluid Mechanics and Bernoulli’s Equation | 流体力学与伯努利方程

An ideal fluid is incompressible and inviscid. The continuity equation A₁v₁ = A₂v₂ states that mass flow rate is constant along a streamline. Bernoulli’s equation expresses conservation of energy in a fluid:

理想流体不可压缩且无粘性。连续性方程 A₁v₁ = A₂v₂ 表明沿一条流线的质量流量恒定。伯努利方程表达了流体中的能量守恒：

p₁ + ½ρv₁² + ρgh₁ = p₂ + ½ρv₂² + ρgh₂

This principle explains lift on an aerofoil, flow measurement using a Venturi meter, and the effect of fluid velocity on pressure. The term ½ρv² is dynamic pressure, and ρgh is the hydrostatic pressure component.

该原理解释了翼型的升力、文丘里流量计的应用以及流速对压力的影响。其中 ½ρv² 为动压，ρgh 为静压分量。

Reynolds number Re = ρvd/μ determines whether flow is laminar or turbulent. For a circular pipe, Re < 2300 is typically laminar, while Re > 4000 indicates turbulent flow. The Moody chart links Re, relative roughness, and friction factor for head loss calculations.

雷诺数 Re = ρvd/μ 决定流动是层流还是湍流。对于圆管，通常 Re < 2300 为层流，Re > 4000 为湍流。穆迪图用于关联 Re、相对粗糙度和摩擦系数，以计算水头损失。

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8. Electrical Circuit Analysis | 电路分析

Kirchhoff’s Current Law (KCL) states that the sum of currents entering a node equals the sum leaving it. Kirchhoff’s Voltage Law (KVL) states that the algebraic sum of voltages around any closed loop is zero. These laws are used to analyse complex DC circuits containing resistors, voltage sources and current sources.

基尔霍夫电流定律 (KCL) 指出，进入节点的电流之和等于离开节点的电流之和。基尔霍夫电压定律 (KVL) 指出，沿任意闭合回路的电压代数和为零。这些定律用于分析包含电阻、电压源和电流源的复杂直流电路。

For operational amplifier (op-amp) circuits, the ideal rules are: no current flows into the inputs, and the voltage at the inverting and non-inverting inputs are equal (virtual earth). The inverting amplifier gain is given by -R_f / R_in, and the non-inverting amplifier gain is 1 + R_f / R₁.

对于运算放大器电路，理想规则为：无电流流入输入端，且反相输入端和同相输入端的电压相等（虚地）。反相放大器的增益为 -R_f / R_in，同相放大器的增益为 1 + R_f / R₁。

A_v (inv) = -R_f / R_in

In AC circuits, phasor analysis deals with impedance Z (combining resistance, inductive reactance X_L = ωL, and capacitive reactance X_C = 1/(ωC)). Resonance in a series RLC circuit occurs when X_L = X_C, giving maximum current at resonant frequency f₀ = 1/(2π√(LC)).

在交流电路中，相量分析法使用阻抗 Z（综合电阻、感抗 X_L = ωL、容抗 X_C = 1/(ωC)）。串联 RLC 电路在 X_L = X_C 时发生谐振，谐振频率 f₀ = 1/(2π√(LC)) 时电流达到最大值。

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9. Digital Logic and Microcontrollers | 数字逻辑与微控制器

Combinational logic circuits use gates such as AND, OR, NOT, NAND and NOR to produce outputs determined solely by current inputs. Boolean algebra and Karnaugh maps allow simplification of logic expressions to minimise gate count.

组合逻辑电路使用与门、或门、非门、与非门和或非门等门电路，输出仅由当前输入决定。布尔代数和卡诺图可用于化简逻辑表达式，减少门数量。

Sequential circuits incorporate memory elements like flip-flops. A D-type flip-flop captures the input at the rising edge of the clock signal and holds it until the next edge. This is fundamental to registers and counters.

时序电路包含像触发器这样的存储元件。D 型触发器在时钟信号的上升沿捕获输入，并保持至下一个边沿。这对于寄存器和计数器至关重要。

Microcontrollers like the PIC or Arduino are programmed in C or assembly to read sensors, execute control algorithms, and drive actuators. Key features include digital I/O, analogue-to-digital converters (ADC), PWM outputs and serial communication (UART, SPI, I²C).

像 PIC 或 Arduino 这样的微控制器通过 C 语言或汇编语言编程，可读取传感器、执行控制算法并驱动执行器。关键特性包括数字 I/O、模数转换器 (ADC)、PWM 输出和串行通信 (UART, SPI, I²C)。

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10. Control Systems and Stability | 控制系统与稳定性

A control system aims to regulate a process variable (e.g. speed, temperature, position) by comparing feedback with a desired setpoint. The basic elements are sensor, controller, actuator and process. The transfer function G(s) represents the system’s input-output relationship in the Laplace domain.

控制系统旨在通过将反馈与期望设定值进行比较来调节过程变量（如速度、温度、位置）。基本要素包括传感器、控制器、执行器和过程。传递函数 G(s) 在拉普拉斯域表示系统的输入-输出关系。

PID (Proportional-Integral-Derivative) control is widely used. The proportional term gives an output proportional to the error, the integral term eliminates steady-state error, and the derivative term anticipates future error based on its rate of change.

PID（比例-积分-微分）控制应用广泛。比例项给出与误差成比例的输出，积分项消除稳态误差，微分项根据误差变化率预测未来误差。

Stability analysis uses root locus, Bode plots and Nyquist diagrams. A system is stable if all poles of its closed-loop transfer function lie in the left half of the s-plane. Gain margin and phase margin quantify relative stability.

稳定性分析运用根轨迹图、伯德图和奈奎斯特图。若闭环传递函数的所有极点均位于 s 平面左半部，则系统稳定。增益裕度和相位裕度用于量化相对稳定性。

Closed-loop T(s) = G(s)/(1+G(s)H(s))

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Published by TutorHao | Engineering Revision Series | aleveler.com

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