📚 Year 13 CIE Engineering: Formula & Theorem Quick Reference | CIE 工程 A2 公式定理速查手册
This quick reference handbook covers the essential formulas, theorems and principles required for the CIE Year 13 Engineering syllabus. Each section groups related concepts, presents key equations in a clear format and provides paired English–Chinese explanations to support bilingual learning and revision.
本速查手册涵盖 CIE 工程 Year 13 课程所需的核心公式、定理与原理。各小节将关联概念分组，以清晰格式呈现关键方程，并提供英中对照解释，帮助双语学习与复习。
1. Stress and Strain | 应力与应变
σ = F / A
Engineering stress σ (sigma) is the applied force F divided by the original cross‑sectional area A. It is measured in pascals (Pa) or newtons per square metre (N m⁻²). Tensile stress is positive, compressive stress negative by convention.
工程应力 σ（西格玛）等于施加的力 F 除以原始横截面积 A，单位为帕斯卡 (Pa) 或牛每平方米 (N m⁻²)。通常拉伸应力为正，压缩应力为负。
ε = ΔL / L₀
Engineering strain ε (epsilon) is the change in length ΔL divided by the original gauge length L₀. Strain is dimensionless and often expressed as a percentage.
工程应变 ε（伊普西隆）等于长度变化 ΔL 除以原始标距 L₀。应变无量纲，常以百分比表示。
σ = E ε
Hooke’s Law: within the proportional limit, stress is directly proportional to strain. The constant of proportionality E is Young’s modulus, a measure of material stiffness. Unit: Pa.
胡克定律：在比例极限内，应力与应变成正比。比例常数 E 为杨氏模量，衡量材料刚度。单位：Pa。
Yield stress and ultimate tensile stress (UTS) are determined from the stress–strain curve. Ductile materials exhibit a distinct yield point and plastic region; brittle materials fracture with little plastic deformation.
屈服应力和极限拉伸强度 (UTS) 由应力‑应变曲线确定。延性材料具有明显的屈服点和塑性区域；脆性材料几乎无塑性变形即断裂。
2. Elastic Constants and Poisson’s Ratio | 弹性常数与泊松比
ν = –ε_lateral / ε_axial
Poisson’s ratio ν (nu) is the negative ratio of transverse strain to axial strain in a uniaxially loaded material. For most engineering metals ν ≈ 0.25–0.35.
泊松比 ν（纽）为单轴受力时横向应变与轴向应变的负比值。大多数工程金属的 ν ≈ 0.25–0.35。
Relationship between elastic constants for isotropic materials (theoretical reference):
各向同性材料弹性常数间的关系（理论参考）：
G = E / [2(1 + ν)]
where G is the shear modulus (modulus of rigidity). This relates shear stress and shear strain: τ = G γ.
其中 G 为剪切模量（刚性模量），关联剪应力与剪应变：τ = G γ。
K = E / [3(1 – 2ν)]
where K is the bulk modulus, linking hydrostatic pressure to volumetric strain.
K 为体积模量，关联静水压力与体积应变。
3. Beam Bending Theory | 梁弯曲理论
M / I = σ / y = E / R
The simple bending equation for a beam in pure bending: M is the applied bending moment, I the second moment of area of the cross‑section, σ the normal stress at a distance y from the neutral axis, E Young’s modulus and R the radius of curvature. The neutral axis passes through the centroid of the cross‑section.
纯弯曲梁的简明弯曲公式：M 为施加的弯矩，I 为横截面的二次矩，σ 为距中性轴距离 y 处的正应力，E 为杨氏模量，R 为曲率半径。中性轴通过截面形心。
Common beam deflection formulas (elastic, small deflections):
常见梁挠度公式（弹性、小变形）：
| Loading & Support | Maximum Deflection δ_max |
|---|---|
| Simply supported, centre point load F | δ_max = FL³ / (48 EI) |
| Cantilever, end point load F | δ_max = FL³ / (3 EI) |
| Simply supported, uniformly distributed load w (total W=wL) | δ_max = 5 wL⁴ / (384 EI) = 5 WL³ / (384 EI) |
In all cases, L is beam length, E Young’s modulus, I the relevant second moment of area. Superposition can combine effects of multiple loads for linear elastic behaviour.
以上各公式中 L 为梁长，E 为杨氏模量，I 为相应的截面二次矩。对于线弹性行为，可利用叠加法组合多载荷效应。
4. Torsion of Circular Shafts | 圆轴扭转
τ / r = T / J = G θ / L
The torsion equation for a solid or hollow circular shaft: T is the applied torque, J the polar second moment of area, τ the shear stress at radius r, G the shear modulus, θ the angle of twist (radians) over length L. For a solid shaft, J = π d⁴ / 32; for a hollow shaft, J = π (d_o⁴ – d_i⁴) / 32.
实心或空心圆轴的扭转公式：T 为施加的扭矩，J 为极二次矩，τ 为半径 r 处的剪应力，G 为剪切模量，θ 为长度 L 上的扭转角（弧度）。实心轴 J = π d⁴ / 32；空心轴 J = π (d_o⁴ – d_i⁴) / 32。
Power transmitted by a rotating shaft: P = T ω, where ω is angular velocity (rad s⁻¹). With speed N in rpm, P = (2π N T) / 60.
旋转轴传递的功率：P = T ω，ω 为角速度 (rad s⁻¹)。若转速 N 单位 rpm，则 P = (2π N T) / 60。
5. Buckling – Euler’s Formula | 压杆屈曲 – 欧拉公式
P_cr = π² E I / L_e²
Euler’s critical load for a slender column: P_cr is the axial load at which buckling occurs, E Young’s modulus, I the minimum second moment of area of the cross‑section, and L_e the effective length. Effective length depends on end fixity: both ends pinned, L_e = L; both ends fixed, L_e = 0.5 L; one end fixed, one free, L_e = 2 L; one end fixed, one pinned, L_e ≈ 0.7 L.
细长柱的欧拉临界载荷：P_cr 为发生屈曲时的轴向载荷，E 为杨氏模量，I 为截面最小二次矩，L_e 为有效长度。有效长度取决于端部约束：两端铰支 L_e = L；两端固定 L_e = 0.5 L；一端固定、一端自由 L_e = 2 L；一端固定、一端铰支 L_e ≈ 0.7 L。
Euler’s formula is valid only for long columns where the slenderness ratio L_e/r (r = √(I/A)) is greater than the critical value for the material. Short columns fail by yielding.
欧拉公式仅适用于长柱，其长细比 L_e/r（r = √(I/A)）高于材料临界值。短柱因屈服破坏。
6. Strain Energy | 应变能
U = (1/2) F δ
For a linearly elastic system, the strain energy U stored is half the product of the gradually applied load F and the corresponding displacement δ in the direction of the load.
对于线弹性系统，储存的应变能 U 等于逐渐施加的载荷 F 与沿载荷方向相应位移 δ 乘积的一半。
In a uniform bar under axial load: U = F²L / (2AE) = σ² AL / (2E)
In a beam under pure bending: U = ∫₀ᴸ (M² / (2EI)) dx. For a shaft under torsion: U = T²L / (2GJ).
均匀直杆受轴向载荷：U = F²L / (2AE) = σ² AL / (2E)。梁受纯弯：U = ∫₀ᴸ (M² / (2EI)) dx。轴受扭：U = T²L / (2GJ)。
Castigliano’s theorem: the partial derivative of the total strain energy with respect to a load gives the displacement at that load: δ = ∂U/∂F. It is a powerful energy method for finding deflections in structures.
卡氏定理：总应变能对某一载荷的偏导数等于该载荷作用点沿其方向的位移：δ = ∂U/∂F。这是求解结构挠度的强大能量法。
7. Fluid Mechanics – Bernoulli and Reynolds | 流体力学 – 伯努利与雷诺数
p₁ + ½ ρ v₁² + ρ g h₁ = p₂ + ½ ρ v₂² + ρ g h₂
Bernoulli’s equation for steady, incompressible, inviscid flow along a streamline. It expresses conservation of energy per unit volume: p is static pressure, ρ density, v velocity, g gravitational acceleration, h elevation.
伯努利方程适用于沿流线稳定、不可压缩、无粘流动，表示单位体积能量守恒：p 为静压，ρ 密度，v 流速，g 重力加速度，h 高程。
Re = ρ v d / μ = v d / ν
Reynolds number characterises flow regime: Re < 2300 typically laminar; Re > 4000 turbulent. Here d is characteristic length (e.g. pipe diameter), μ dynamic viscosity, ν = μ/ρ kinematic viscosity.
雷诺数表征流态：Re < 2300 常为层流；Re > 4000 为湍流。d 为特征长度（如管径），μ 为动力粘度，ν = μ/ρ 为运动粘度。
Mass flow rate: ṁ = ρ A v. Continuity for incompressible flow: A₁ v₁ = A₂ v₂.
质量流量：ṁ = ρ A v。不可压缩流动的连续性方程：A₁ v₁ = A₂ v₂。
8. Thermodynamics – Laws and Cycles | 热力学 – 定律与循环
First Law for a closed system: ΔU = Q – W, where ΔU is change in internal energy, Q heat added to system, W work done by system.
封闭系统第一定律：ΔU = Q – W，ΔU 为内能变化，Q 为系统吸热，W 为系统对外做功。
η_Carnot = 1 – T_C / T_H
The maximum theoretical efficiency of a heat engine operating between a high‑temperature reservoir T_H and a low‑temperature reservoir T_C (temperatures in kelvin). No engine can exceed this efficiency.
工作在高温热源 T_H 与低温热源 T_C（开尔文温标）间的热机最大理论效率。任何热机效率不可能超过此值。
Otto cycle efficiency (ideal spark‑ignition): η = 1 – 1 / r^(γ–1), where r is compression ratio, γ = c_p / c_v (ratio of specific heats).
奥托循环效率（理想点燃式）：η = 1 – 1 / r^(γ–1)，r 为压缩比，γ = c_p / c_v（比热比）。
For a constant pressure process: W = p ΔV. For an ideal gas: pV = nRT, and specific heat relations c_p – c_v = R.
等压过程功：W = p ΔV。理想气体状态方程：pV = nRT，比热关系 c_p – c_v = R。
9. Rotational Dynamics | 转动动力学
τ = I α
Newton’s second law for rotation: the net torque τ acting on a body equals the product of its moment of inertia I and angular acceleration α (rad s⁻²). Analogous to F = ma.
转动牛顿第二定律：作用在物体上的净扭矩 τ 等于其转动惯量 I 与角加速度 α 的乘积，类比 F = ma。
KE_rot = ½ I ω²
Rotational kinetic energy, where ω is angular velocity. Total KE of a rolling object = ½ m v² + ½ I ω².
转动动能，ω 为角速度。滚动体总动能 = ½ m v² + ½ I ω²。
Angular momentum: L = I ω. Conservation of angular momentum applies when external torque is zero.
角动量：L = I ω。当外扭矩为零时角动量守恒。
Moment of inertia for common shapes about centroidal axis: solid cylinder/disk I = ½ m r²; hoop/ring I = m r²; slender rod (centre) I = (1/12) m L².
常见形状绕质心轴的转动惯量：实心圆柱/圆盘 I = ½ m r²；圆环/薄壁环 I = m r²；细杆（绕中心）I = (1/12) m L²。
10. Electronics: Operational Amplifiers | 电子学：运算放大器
Ideal op‑amp assumptions: infinite open‑loop gain, infinite input impedance, zero output impedance. In linear mode, with negative feedback, the two input terminals are at virtually the same voltage (virtual short) and draw negligible current.
理想运放假设：开环增益无穷大、输入阻抗无穷大、输出阻抗为零。线性模式下，引入负反馈，两输入端电压近似相等（虚短），且输入电流可忽略。
A_v = – R_f / R_i
Inverting amplifier: output is inverted. Gain is set solely by the feedback resistor R_f and input resistor R_i.
反相放大器：输出倒相。增益仅由反馈电阻 R_f 和输入电阻 R_i 决定。
A_v = 1 + R_f / R₁
Non‑inverting amplifier: output in phase with input. R₁ is the resistor from the inverting input to ground, R_f is the feedback resistor.
同相放大器：输出与输入同相。R₁ 为反相端到地电阻，R_f 为反馈电阻。
Summing amplifier (inverting): V_out = – R_f (V₁/R₁ + V₂/R₂ + …). Difference amplifier: V_out = (R_f/R₁)(V₂ – V₁) when resistor ratios match.
加法器（反相）：V_out = – R_f (V₁/R₁ + V₂/R₂ + …)。差分放大器：当电阻比值匹配时 V_out = (R_f/R₁)(V₂ – V₁)。
11. Control Systems – Transfer Functions | 控制系统 – 传递函数
T(s) = G(s) / (1 + G(s) H(s))
For a negative‑feedback system with forward‑path transfer function G(s) and feedback‑path transfer function H(s), the closed‑loop transfer function T(s) = C(s)/R(s) is given by the above formula. The product G(s)H(s) is the open‑loop transfer function.
对于前向通道传递函数 G(s)、反馈通道传递函数 H(s) 的负反馈系统，闭环传递函数 T(s) = C(s)/R(s) 由上式给出。G(s)H(s) 为开环传递函数。
Steady‑state error for a unity‑feedback system (H(s)=1) can be found from the final‑value theorem: e_ss = lim_{t→∞} e(t) = lim_{s→0} s E(s).
单位反馈系统 (H(s)=1) 的稳态误差可由终值定理求得：e_ss = lim_{t→∞} e(t) = lim_{s→0} s E(s)。
Common controller actions: Proportional (P) u = K_p e; Integral (I) u = K_i ∫ e dt; Derivative (D) u = K_d de/dt. PID control combines all three to improve transient response and eliminate steady‑state error.
常见控制器作用：比例 (P) u = K_p e；积分 (I) u = K_i ∫ e dt；微分 (D) u = K_d de/dt。PID 控制三者结合，改善瞬态响应并消除稳态误差。
Stability: the characteristic equation is 1 + G(s)H(s) = 0. Routh–Hurwitz criterion or root‑locus can be used to assess stability without solving for roots explicitly.
稳定性：特征方程为 1 + G(s)H(s) = 0。可采用劳斯‑赫尔维茨判据或根轨迹法，无需显式求解根即可评估稳定性。
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