📚 Quick Reference Handbook of Formulas and Theorems | 公式定理速查手册
This handbook provides a concise summary of key formulas, laws, and theorems required for the CCEA Pre-U Engineering specification. It is designed as a quick revision aid, covering core topics from mechanics, materials, thermodynamics, fluid mechanics, electrical circuits, and digital logic. All symbols are defined in context, and principal equations are highlighted for rapid recall during problem-solving and examination preparation.
本手册简明汇总了 CCEA Pre-U 工程课程所必需的核心公式、定律与定理，旨在为考前快速复习提供便利。内容涵盖力学、材料学、热力学、流体力学、电路及数字逻辑等主要领域。所有符号均在上下文中给出说明，主要方程均突出显示，便于在解题和备考中迅速调用。
1. Fundamental Quantities and SI Units | 基本量与国际单位制
All engineering calculations rely on a consistent set of base units. The SI base units are metre (m) for length, kilogram (kg) for mass, second (s) for time, ampere (A) for electric current, kelvin (K) for thermodynamic temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. Derived units commonly used in engineering include the newton (N = kg·m/s²), pascal (Pa = N/m²), joule (J = N·m), watt (W = J/s), and hertz (Hz = s⁻¹).
所有工程计算都依赖一套统一的基本单位。国际单位制的基本单位是：米(m)表长度，千克(kg)表质量，秒(s)表时间，安培(A)表电流，开尔文(K)表热力学温度，摩尔(mol)表物质的量，坎德拉(cd)表发光强度。工程中常用的导出单位包括牛顿(N = kg·m/s²)、帕斯卡(Pa = N/m²)、焦耳(J = N·m)、瓦特(W = J/s)和赫兹(Hz = s⁻¹)。
F = m·a
P = F / A
W = F·d
Newton’s second law defines force as the product of mass and acceleration. Pressure is force distributed over area, work is the product of force and displacement in the direction of the force, and power is the rate of doing work (P = W / t). Dimensional analysis using these base quantities ensures the correctness of derived equations.
牛顿第二定律将力定义为质量与加速度的乘积。压力为力在面积上的分布，功为力与沿力方向的位移之积，功率则为做功的速率(P = W / t)。运用这些基本量进行量纲分析可以确保导出方程的正确性。
2. Statics: Equilibrium and Moments | 静力学：平衡与力矩
For a rigid body to be in static equilibrium, two conditions must be satisfied simultaneously: the vector sum of all external forces must be zero, and the algebraic sum of the moments of all forces about any point must also be zero.
刚体处于静力平衡时，必须同时满足两个条件：所有外力的矢量和为零，且所有力对任意点的力矩代数和也为零。
∑F = 0
∑M = 0
The moment of a force (torque) about a pivot is calculated as the product of the force magnitude and the perpendicular distance from the pivot to the line of action of the force. In coplanar systems, a sign convention (e.g. clockwise negative) must be used when summing moments. These principles are fundamental to the analysis of trusses, frames, and support reactions.
力对支点的力矩等于力的大小乘以支点到力作用线的垂直距离。在平面力系中，求合力矩时需要规定一个正负号惯例(如顺时针为负)。这些原理是分析桁架、框架和支反力的基础。
M = F × d⊥
3. Stress, Strain and Elastic Moduli | 应力、应变与弹性模量
When an axial force is applied to a member, the direct stress is the force per unit cross‑sectional area. Direct strain is the proportional change in length. For many engineering materials within the elastic limit, stress is directly proportional to strain – this is Hooke’s law.
当轴向力作用于构件时，正应力即单位截面积上的力。正应变为长度的相对变化量。在弹性极限内，许多工程材料的应力与应变成正比，这就是胡克定律。
σ = F / A
ε = ΔL / L₀
E = σ / ε
Young’s modulus E is a measure of a material’s stiffness. Shear stress τ and shear strain γ are related by the shear modulus G. Poisson’s ratio ν relates lateral strain to axial strain. These constants are essential for predicting deformation under load.
杨氏模量 E 衡量材料的刚度。剪切应力 τ 与剪切应变 γ 通过剪切模量 G 关联。泊松比 ν 将横向应变与轴向应变联系起来。这些常数对于预测载荷下的变形至关重要。
τ = Fshear / A
G = τ / γ
ν = –εlateral / εaxial
4. Bending of Beams | 梁的弯曲
The simple bending equation relates the bending stress in a beam to the applied bending moment, the distance from the neutral axis, and the geometric property of the cross‑section. For a symmetrical beam under pure bending, the neutral axis passes through the centroid.
简单弯曲方程将梁中的弯曲应力与作用的弯矩、到中性轴的距离以及截面几何性质联系起来。对于纯弯曲下的对称截面梁，中性轴通过形心。
M / I = σ / y = E / R
Here M is the bending moment, I is the second moment of area about the neutral axis, σ is the bending stress at a distance y from the neutral axis, E is Young’s modulus, and R is the radius of curvature. The section modulus Z = I / ymax is often used to determine the maximum stress in a beam.
其中 M 为弯矩，I 为截面对中性轴的面积的二次矩，σ 为距中性轴 y 处的弯曲应力，E 为杨氏模量，R 为曲率半径。截面模量 Z = I / ymax 常用于确定梁内的最大应力。
Deflection of a simply supported beam carrying a central point load is given by the standard formula below. Other loading and support configurations have analogous expressions.
简支梁承受中点集中载荷时的挠度由下式给出。其他载荷和支承情况也有类似的表达式。
δmax = FL³ / (48EI)
5. Linear and Rotational Kinematics | 直线与旋转运动学
For constant linear acceleration, the motion of a particle is described by three familiar equations – often called the SUVAT equations. In these, u is initial velocity, v is final velocity, a is constant acceleration, t is time, and s is displacement.
对于恒定的直线加速度，质点的运动由三个熟悉的方程描述，常称为 SUVAT 方程。其中 u 为初速度，v 为末速度，a 为恒定加速度，t 为时间，s 为位移。
v = u + at
s = ut + ½at²
v² = u² + 2as
The analogous relationships for rotation about a fixed axis use angular displacement θ, angular velocity ω, and angular acceleration α. The equations are structurally identical.
绕固定轴转动的类似关系使用角位移 θ、角速度 ω 和角加速度 α。方程结构完全相同。
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
The link between linear and rotational motion is through the radius r: tangential velocity v = ωr, and tangential acceleration a_t = αr. The torque T required to produce angular acceleration is T = Iα, where I is the moment of inertia about the axis of rotation.
直线运动与转动之间的联系通过半径 r 建立：切向速度 v = ωr，切向加速度 a_t = αr。产生角加速度所需转矩 T = Iα，其中 I 为绕转轴的转动惯量。
6. Work, Energy and Power | 功、能与功率
Kinetic energy of a translating body depends on its mass and speed, while gravitational potential energy depends on height above a datum. The work done by a constant force is the product of the force, displacement, and cosine of the angle between them.
平动物体的动能取决于其质量和速度，重力势能则取决于相对于基准面的高度。恒力所做的功等于力、位移以及二者夹角余弦的乘积。
KE = ½mv²
PE = mgh
W = F·d·cos θ
For a spring obeying Hooke’s law, the elastic potential energy stored is ½kx², where k is the spring stiffness and x is the extension or compression from the natural length. The principle of conservation of energy states that in an isolated system, total energy remains constant.
对于遵循胡克定律的弹簧，储存的弹性势能为 ½kx²，其中 k 为弹簧刚度，x 为相对于原长的拉伸或压缩量。能量守恒原理指出，在孤立系统中总能量保持不变。
Eelastic = ½kx²
Mechanical power is the rate of energy transfer. For a force moving at constant velocity in its own direction, power equals force times velocity. Efficiency η is the ratio of useful output power to input power, always less than 1 due to energy losses.
机械功率是能量传递的速率。当力沿自身方向以恒速运动时，功率等于力与速度之积。效率 η 为有用输出功率与输入功率之比，由于能量损失，该比值总是小于1。
P = F·v
η = Pout / Pin
7. Thermodynamics and Heat Transfer | 热力学与传热
Temperature change causes linear expansion in solids. The coefficient of linear expansion α is used to calculate the change in length. Heat conduction through a plane wall is governed by Fourier’s law, where Q/t is the rate of heat flow, k thermal conductivity, A area, ΔT temperature difference, and d wall thickness.
温度变化引起固体线膨胀。线膨胀系数 α 用于计算长度变化。通过平壁的热传导由傅里叶定律描述，其中 Q/t 为热流率，k 为导热系数，A 为面积，ΔT 为温差，d 为壁厚。
ΔL = α L₀ ΔT
Q / t = k A (ΔT / d)
For an ideal gas, its pressure p, volume V, and absolute temperature T are related by the equation pV = nRT, where n is the number of moles and R is the universal gas constant. In many engineering applications, specific forms such as pV/m = RT are used.
对于理想气体，其压力 p、体积 V 和绝对温度 T 之间的关系式为 pV = nRT，其中 n 为摩尔数，R 为普适气体常数。在许多工程应用中，会使用诸如 pV/m = RT 的特定形式。
pV = nRT
The maximum theoretical efficiency of a heat engine operating between a hot reservoir at temperature T₁ and a cold reservoir at T₂ is given by the Carnot efficiency. This provides an upper limit for real engine performance.
工作在热源温度 T₁ 和冷源温度 T₂ 之间的热机理论上能达到的最高效率由卡诺效率给出。这为实际发动机的性能提供了一个上限。
ηCarnot = 1 – (T₂ / T₁)
8. Fluid Mechanics | 流体力学
Pressure in a static fluid increases linearly with depth, independent of the shape of the container. Archimedes’ principle states that the buoyant force on an object equals the weight of the fluid displaced.
静止流体中的压力随深度线性增加，与容器形状无关。阿基米德原理指出，物体所受的浮力等于排开流体的重量。
P = ρgh
Fb = ρfluid Vdisp g
For a steady, incompressible flow, the mass flow rate must be conserved along a streamline, leading to the continuity equation. Bernoulli’s equation expresses the conservation of mechanical energy in a flowing fluid along a streamline, where P is static pressure, ρ fluid density, v flow speed, and h elevation.
对于定常、不可压缩的流动，沿流线的质量流量必须守恒，从而得到连续性方程。伯努利方程表达了沿流线流动流体中的机械能守恒，其中 P 为静压，ρ 为流体密度，v 为流速，h 为高度。
A₁v₁ = A₂v₂
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
These relationships are fundamental to the analysis of pipe flow, orifice flow, and the lift on aerofoils. In practice, energy losses due to friction are accounted for by empirical correction factors.
这些关系是分析管流、孔口流和翼型升力的基础。实际中，由摩擦引起的能量损失通过经验修正系数加以考虑。
9. Electric Circuits | 电路
Ohm’s law defines the relationship between voltage V, current I, and resistance R. The total resistance of resistors connected in series is the sum of individual resistances; for resistors in parallel, the reciprocal of total resistance equals the sum of reciproc
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