Pre-U Edexcel Engineering: Quick Reference Formula & Theorem Handbook | Pre-U Edexcel 工程：公式定理速查手册

📚 Pre-U Edexcel Engineering: Quick Reference Formula & Theorem Handbook | Pre-U Edexcel 工程：公式定理速查手册

This handbook consolidates the essential formulae, theorems and key relationships required for the Pre-U Edexcel Engineering course. It serves as a rapid revision aid, covering statics, dynamics, materials, fluids, thermodynamics, electrical systems, control and core mathematics. Each section presents the most frequently examined results with paired English–Chinese explanations, ensuring both conceptual clarity and exam readiness.

本手册汇总了 Pre-U Edexcel 工程课程所需的核心公式、定理及重要关系式。它是一份快速复习工具，涵盖静力学、动力学、材料、流体、热力学、电气系统、控制以及核心数学。每一节都以中英双语配对讲解最常考的内容，帮助理清概念、备战考试。

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1. Statics and Equilibrium | 静力学与平衡

For a rigid body in static equilibrium, the vector sum of all external forces and the vector sum of all moments about any point must be zero. This ensures translational and rotational equilibrium simultaneously.

对于处于静态平衡的刚体，所有外力的矢量和以及关于任意点的所有力矩的矢量和必须为零。这同时保证了平动平衡和转动平衡。

ΣF = 0, ΣM = 0

When resolving forces on an inclined plane, the components parallel and perpendicular to the slope are given by mg sinθ and mg cosθ respectively, where m is mass, g is gravitational acceleration and θ is the angle of inclination.

在斜面上分解力时，平行和垂直于斜面的分力分别为 mg sinθ 与 mg cosθ，其中 m 为质量，g 为重力加速度，θ 为斜面倾角。

F_∥ = mg sinθ, F_⟂ = mg cosθ

The principle of moments states that for equilibrium the sum of clockwise moments equals the sum of anticlockwise moments about any pivot. This is used to find unknown reaction forces and distances in levers, beams and trusses.

力矩原理指出，平衡时绕任意支点的顺时针力矩之和等于逆时针力矩之和。这用于求解杠杆、梁和桁架中的未知反力和距离。

Σ M_cw = Σ M_acw

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2. Kinematics and Dynamics | 运动学与动力学

For uniform acceleration in one dimension, the four SUVAT equations link displacement s, initial velocity u, final velocity v, acceleration a and time t. They are fundamental for solving motion problems under constant net force.

对于一维匀加速运动，四个 SUVAT 方程关联了位移 s、初速度 u、末速度 v、加速度 a 和时间 t。它们是求解恒定合力下运动问题的基础。

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

Newton’s second law in linear form defines the net force as the product of mass and acceleration. When multiple forces act, the resultant force F_net determines the acceleration of the centre of mass.

牛顿第二定律的线性形式将合力定义为质量与加速度的乘积。当多个力作用时，合力 F_net 决定质心的加速度。

F = m a

For rotational motion, the analogous equation uses torque τ, moment of inertia I and angular acceleration α. Moment of inertia depends on mass distribution about the axis of rotation.

对于转动，类似方程使用扭矩 τ、转动惯量 I 和角加速度 α。转动惯量取决于质量绕转轴的分布。

τ = I α

Centripetal force required to keep an object moving in a circular path of radius r at speed v is directed towards the centre. It can also be expressed in terms of angular velocity ω.

使物体以速度 v 在半径为 r 的圆周上运动所需的向心力指向圆心。也可用角速度 ω 表示。

F_c = mv²/r = mω²r

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3. Stress, Strain and Elasticity | 应力、应变与弹性

Direct stress σ is defined as force F applied perpendicularly over cross-sectional area A. Engineering strain ε is the change in length ΔL divided by the original length L₀ under uniaxial loading.

正应力 σ 定义为垂直施加在截面积 A 上的力 F。工程应变 ε 是单轴加载下长度变化 ΔL 除以原始长度 L₀。

σ = F/A, ε = ΔL/L₀

Hooke’s law holds within the linear elastic region, relating stress and strain through Young’s modulus E. The material returns to its original shape upon unloading.

胡克定律在线弹性范围内成立，通过杨氏模量 E 关联应力与应变。卸载后材料恢复原状。

σ = E ε

Poisson’s ratio ν quantifies the lateral contraction relative to axial extension. It ranges typically from 0.25 to 0.35 for most engineering metals.

泊松比 ν 量化了横向收缩相对于轴向拉伸的程度。大多数工程金属的典型取值范围为 0.25 到 0.35。

ν = –ε_lateral/ε_axial

Shear stress τ and shear strain γ are linked by the shear modulus G. In isotropic materials, E, G and ν are related, allowing conversion between stiffness measures.

剪应力 τ 和剪应变 γ 通过剪切模量 G 关联。在各向同性材料中，E、G 与 ν 之间的关系使得不同刚度描述可以相互转换。

τ = G γ, G = E/[2(1+ν)]

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4. Bending and Torsion | 弯曲与扭转

The simple bending formula relates bending moment M to curvature through the flexural rigidity EI. Normal stress varies linearly across the beam depth, with neutral axis at the centroid.

纯弯曲公式将弯矩 M 与曲率通过抗弯刚度 EI 关联起来。正应力沿梁高线性变化，中性轴位于截面形心处。

σ/y = M/I = E/R

For symmetric beam cross-sections, maximum bending stress occurs at the extreme fibre distance y_max. The section modulus Z = I/y_max simplifies strength checks.

对于对称截面梁，最大弯曲正应力出现在最外层纤维距离 y_max 处。截面模量 Z = I/y_max 简化了强度校核。

σ_max = M/Z

In torsion of circular shafts, shear stress τ varies linearly with radius r. The torsion equation connects torque T, polar second moment of area J and the angle of twist per unit length θ/L.

在圆轴扭转中，剪应力 τ 随半径 r 线性变化。扭转方程将扭矩 T、极惯性矩 J 和单位长度扭转角 θ/L 联系起来。

τ/r = T/J = Gθ/L

Power transmitted by a rotating shaft is the product of torque and angular velocity. In engine design, this links mechanical power to shaft speed and stress levels.

转轴传递的功率是扭矩与角速度的乘积。在发动机设计中，该关系将机械功率与轴转速及应力水平联系起来。

P = T ω

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5. Energy Methods and Virtual Work | 能量法与虚功原理

The work done by a constant force F moving its point of application by distance d in the direction of the force is W = Fd. If the force acts at an angle, only the component along the displacement does work.

恒力 F 使其作用点沿力方向移动距离 d 所做的功为 W = Fd。若力与位移成角度，仅沿位移方向的分量做功。

W = F d cosθ

Elastic strain energy stored in a linearly elastic bar under axial load is U = (1/2)FΔL = (1/2)σ ε V, where V is the volume. This energy is recoverable upon load removal.

轴向受载的线弹性杆中储存的弹性应变能为 U = (1/2)FΔL = (1/2)σ ε V，其中 V 为体积。该能量在卸去载荷后可以恢复。

U_axial = ½ F ΔL = (σ²/2E) V

The principle of virtual work states that a system is in equilibrium if the total virtual work done by all forces during an arbitrary infinitesimal virtual displacement is zero. It is highly effective for analysing linkages and structures with geometric constraints.

虚功原理指出，若在任意微小虚位移下所有力所作的总虚功为零，则系统处于平衡。对于分析含几何约束的连杆机构和结构尤为有效。

Σ (F · δr) = 0, δW = 0

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6. Fluid Statics and Dynamics | 流体静力学与动力学

Hydrostatic pressure in a fluid at rest increases linearly with depth h. The gauge pressure p at a point is given by the specific weight ρg times depth, where ρ is fluid density.

静止流体中的静压力随深度 h 线性增加。某点的表压力 p 等于重度 ρg 乘以深度，其中 ρ 为流体密度。

p = ρ g h

Archimedes’ principle asserts that the buoyant force on a submerged body equals the weight of fluid displaced. This governs floatation and apparent weight in fluids.

阿基米德原理表明，浸没物体所受的浮力等于被排开流体的重量。它决定了浮力以及物体在流体中的表观重量。

F_B = ρ_fluid g V_displaced

The Bernoulli equation for steady, incompressible, inviscid flow along a streamline states that the sum of pressure, kinetic and potential energy per unit volume remains constant. It is widely applied in pipe flow, Venturi meters and aerofoil analysis.

对于沿流线的定常、不可压缩、无粘流动，伯努利方程指出单位体积的压力能、动能与势能之和保持恒定。它广泛应用于管道流动、文丘里流量计和翼型分析。

p + ½ρv² + ρgz = constant

Continuity equation for incompressible flow asserts that the mass flow rate is constant. For a streamtube, the product of area A and flow speed v is invariant.

不可压缩流动的连续性方程指出质量流量恒定。对于流管，截面积 A 与流速 v 的乘积保持不变。

A₁v₁ = A₂v₂

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7. Thermodynamic Laws and Cycles | 热力学定律与循环

The First Law of Thermodynamics is an energy conservation statement: the change in internal energy ΔU equals heat added Q minus work done W by the system. Sign conventions are critical in cycle analysis.

热力学第一定律是能量守恒表述：内能变化 ΔU 等于加入的热量 Q 减去系统对外做的功 W。循环分析中正负号约定至关重要。

ΔU = Q – W

For a reversible ideal gas process, several useful relations hold: isothermal (pV = constant), adiabatic (pV^γ = constant) and polytropic (pVⁿ = constant), where γ = c_p/c_v is the specific heat ratio.

对于可逆的理想气体过程，有若干有用关系：等温过程 pV = 常数，绝热过程 pV^γ = 常数，多变过程 pVⁿ = 常数，其中 γ = c_p/c_v 为比热比。

pV = nRT, pV^γ = const., pVⁿ = const.

Thermal efficiency of a heat engine is the ratio of net work output to heat input. For a Carnot cycle operating between temperatures T_H and T_C, the maximum possible efficiency is given by the temperature ratio.

热机效率是净输出功与输入热量之比。对于工作在温度 T_H 和 T_C 之间的卡诺循环，最大可能效率由温度比给出。

η = W_net/Q_in = 1 – T_C/T_H (Carnot)

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8. Electrical and Electronic Formulae | 电气与电子公式

Ohm’s law for a resistive circuit relates voltage V, current I and resistance R. Power dissipated in the resistor can be expressed in three equivalent forms.

电阻电路的欧姆定律联系了电压 V、电流 I 和电阻 R。电阻器消耗的功率可用三种等效形式表达。

V = IR, P = VI = I²R = V²/R

For a capacitor of capacitance C, charge stored Q is proportional to voltage. The energy stored in an electric field is a function of C and V.

对于电容 C，储存的电荷 Q 与电压成正比。电场中储存的能量是 C 与 V 的函数。

Q = CV, E = ½CV²

In an RL circuit, the time constant τ = L/R determines the rate of current rise or decay. Similarly, the RC time constant τ = RC governs capacitor charging and discharging.

在 RL 电路中，时间常数 τ = L/R 决定电流上升或衰减的速率。同样，RC 时间常数 τ = RC 支配电容的充放电过程。

τ = L/R (RL), τ = RC (RC)

Kirchhoff’s laws are essential for circuit analysis: the junction rule (Σ I = 0 at a node) and the loop rule (Σ V = 0 around a closed loop). They enable solution of complex networks.

基尔霍夫定律对电路分析至关重要：节点电流定律（节点处 Σ I = 0）和回路电压定律（闭合回路 Σ V = 0）。它们用于求解复杂网络。

Σ I_in = Σ I_out, Σ V_rises = Σ V_drops

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9. Control Systems and Transfer Functions | 控制系统与传递函数

A first-order system with time constant T has a transfer function G(s) = K/(Ts+1). Its step response reaches 63.2% of final value in one time constant, and steady-state error depends on loop gain.

时间常数为 T 的一阶系统具有传递函数 G(s) = K/(Ts+1)。其阶跃响应在一个时间常数后达到终值的 63.2%，稳态误差取决于回路增益。

G(s) = K/(Ts+1)

A second-order system is characterised by natural frequency ω_n and damping ratio ζ. The standard form reveals overshoot and settling behaviour critical for mechanical and electrical control.

二阶系统由固有频率 ω_n 和阻尼比 ζ 表征。标准形式揭示了超调量和调节时间，这对机电控制至关重要。

G(s) = ω_n²/(s² + 2ζω_ns + ω_n²)

The steady-state response to a sinusoidal input of frequency ω is given by the frequency response function G(jω). The gain in decibels is 20 log₁₀|G(jω)|, and phase shift is ∠G(jω).

频率为 ω 的正弦输入下的稳态响应由频率响应函数 G(jω) 给出。增益以分贝计为 20 log₁₀|G(jω)|，相移为 ∠G(jω)。

Gain (dB) = 20 log₁₀|G(jω)|, Phase = arg G(jω)

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10. Engineering Mathematics Essentials | 工程数学要点

Differentiation and integration are pervasive in engineering analysis. The derivative of a displacement function gives velocity, and the second derivative gives acceleration. Integration reverses these operations.

微分与积分在工程分析中无处不在。位移函数的导数给出速度，二阶导数给出加速度。积分则逆转这些运算。

v = ds/dt, a = d²s/dt²

Vector dot product and cross product are used to compute work (scalar) and moment (vector). Dot product: a·b = |a||b| cosθ; cross product: a×b yields a vector perpendicular to both with magnitude |a||b| sinθ.

向量点积和叉积用于计算功（标量）和力矩（矢量）。点积：a·b = |a||b| cosθ；叉积：a×b 产生一个垂直于两者的矢量，大小为 |a||b| sinθ。

W = F·d, M = r × F

Solving linear simultaneous equations often employs matrix methods. For a 2×2 system, the determinant Δ = ad – bc determines invertibility, and the solution is found via Cramer’s rule or inverse matrix.

求解线性联立方程组常使用矩阵方法。对于 2×2 系统，行列式 Δ = ad – bc 判断可逆性，解可通过克莱姆法则或逆矩阵求得。

[a b; c d][x; y] = [e; f] , x = (ed–bf)/Δ, y = (af–ec)/Δ (if Δ≠0)

The trapezoidal rule provides a numerical estimate of a definite integral by approximating the area under a curve as a series of trapezoids. The estimate improves as the number of strips n increases.

梯形法则通过将曲线下的面积近似为一系列梯形来给出定积分的数值估计。随着分割条数 n 增加，估计值改善。

∫_a^b f(x) dx ≈ (h/2)[f₀ + 2(f₁+…+f_n-1) + f_n], h = (b–a)/n

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