📚 SQA Engineering Year 12 Formula & Theorems Quick Reference | 苏格兰资格认证局工程学12年级公式定理速查手册
This quick reference guide brings together the essential formulas, theorems and key relationships required for the SQA Year 12 Engineering course. Each section presents a core topic with paired English and Chinese explanations to support bilingual learners in mastering the fundamentals, from mechanics and materials to electrical and fluid systems.
本速查手册汇集了 SQA 工程学 12 年级课程所需的基本公式、定理和关键关系。每个小节都以核心主题为单元，提供配对的英文和中文解释，帮助双语学习者掌握从力学、材料到电气与流体系统的基础知识。
1. Stress and Strain | 应力与应变
Stress (σ) is the force applied per unit area, and strain (ε) is the resulting deformation relative to original dimensions. For a prismatic member under axial load, direct stress and strain are given by:
应力（σ）是施加在单位面积上的力，而应变（ε）是相对于原始尺寸的变形。对于轴向载荷作用下的等截面构件，正应力与正应变由下式给出：
σ = F / A and ε = ΔL / L₀
where F is the force, A the cross‑sectional area, ΔL the change in length, and L₀ the original length. Strain is dimensionless; stress is quoted in pascals (Pa) or N/m².
其中 F 是力，A 是横截面积，ΔL 是长度变化量，L₀ 是原始长度。应变无量纲；应力单位为帕斯卡（Pa）或 N/m²。
Engineers work within the material’s elastic limit, where stress and strain remain proportional. The proportional limit and yield point are critical in design to avoid permanent deformation.
工程师工作在材料的弹性极限内，此时应力与应变仍保持正比关系。比例极限和屈服点对于设计至关，能避免永久变形。
2. Young’s Modulus | 杨氏模量
Young’s modulus (E) is a measure of a material’s stiffness, defined as the ratio of normal stress to normal strain within the elastic region.
杨氏模量（E）是衡量材料刚度的指标，定义为弹性范围内正应力与正应变的比值。
E = σ / ε = (F / A) / (ΔL / L₀)
This relationship is Hooke’s law in its material‑science form. A high E value means the material is stiff and resists deformation; steel has E ≈ 200 GPa, while polymers are orders of magnitude lower.
此关系即材料学形式的胡克定律。高 E 值表示材料刚度大且抵抗变形；钢的 E 约为 200 GPa，而聚合物则低几个数量级。
When interpreting tensile test data, Young’s modulus is the slope of the initial linear portion of the stress‑strain curve. It is an essential parameter in beam bending and column buckling calculations.
在解读拉伸试验数据时，杨氏模量是应力－应变曲线初始线性段的斜率。它是梁弯曲和压杆屈曲计算中的关键参数。
3. Factor of Safety | 安全系数
Factor of safety (FoS) is a design margin that accounts for uncertainties in load, material properties and manufacturing. It is the ratio of a material’s ultimate (or yield) strength to the allowable working stress.
安全系数（FoS）是为应对载荷、材料性能与制造中的不确定性而设置的设计裕度。它是材料极限（或屈服）强度与许用工作应力之比。
FoS = Ultimate Strength / Allowable Stress
Typical FoS values range from 1.5 for ductile materials under well‑known loads to 10 or more for structural components in critical applications. A FoS ≤ 1 indicates imminent failure.
典型的安全系数值从已知载荷下韧性材料的 1.5 到关键应用中结构件的 10 或更高。FoS ≤ 1 表示即将失效。
When using yield stress instead of ultimate stress, the term ‘factor of safety on yield’ is often used. SQA problems frequently ask learners to calculate allowable load from given material properties and a specified FoS.
当用屈服应力代替极限应力时，常称“基于屈服的安全系数”。SQA 题目常要求学习者根据给定材料性能和规定安全系数计算许用载荷。
4. Moments and Equilibrium | 力矩与平衡
A moment (M) is the turning effect produced by a force acting at a distance from a pivot. For a force perpendicular to a lever arm:
力矩（M）是力在距支点一定距离处产生的转动效应。对于垂直于力臂的力：
M = F × d
where d is the perpendicular distance from the line of action to the pivot. Moments are taken as positive in one sense (e.g., clockwise) and negative in the opposite sense by convention.
其中 d 是作用线到支点的垂直距离。习惯上规定一种转向（如顺时针）为正，相反转向为负。
For static equilibrium, two conditions must be met: the sum of all horizontal forces equals zero, the sum of all vertical forces equals zero, and the sum of moments about any point equals zero.
对于静力平衡，必须满足两个条件：所有水平分力之和为零，所有垂直分力之和为零，且对任意点的力矩之和为零。
The principle of moments is used to find unknown forces in beams, levers and frames. Beam reaction calculations are a staple of Year 12 Engineering analysis.
力矩原理用来求梁、杠杆和框架中的未知力。梁反力计算是 12 年级工程分析的主要内容。
5. Work, Energy and Power | 功、能与功率
Work done (W) by a constant force acting in the direction of motion, and the energy forms commonly encountered, are linked by the work–energy principle.
恒力沿运动方向所做的功（W）以及常见的能量形式通过功－能原理相联系。
Work: W = F × s (force × displacement in the direction of the force)
功： W = F × s （力 × 沿力方向的位移）
Gravitational potential energy: PE = m g h
重力势能： PE = m g h
Kinetic energy: KE = ½ m v²
动能： KE = ½ m v²
Power is the rate of doing work; in mechanical systems it can also be expressed as force × velocity.
功率是做功的速率；在机械系统中也可表示为力 × 速度。
P = W / t and P = F v
In the SQA course, learners combine these to analyse lifting gear, vehicles and machines, often including efficiency losses.
在 SQA 课程中，学习者结合这些原理分析起重装置、车辆和机械，通常包含效率损失。
6. Mechanical Advantage, Velocity Ratio and Efficiency | 机械优势、速比与效率
Simple machines provide a mechanical advantage (MA) that multiplies effort, but they trade distance for force. Three linked concepts define their performance.
简单机械提供放大力作用的机械优势（MA），但以距离换取力。三个相互关联的概念定义了其性能。
MA = Load / Effort
Velocity Ratio (VR) = Distance moved by effort / Distance moved by load
Efficiency (η) = (MA / VR) × 100%
VR depends purely on geometry (for ideal machines). Efficiency accounts for friction and other losses; real machines always have η < 100%.
速比（VR）完全取决于几何构型（对于理想机械）。效率计入了摩擦和其他损失；实际机械总有 η < 100%。
Typical classroom examples include lever systems, pulley blocks, screw jacks and hydraulic presses. Efficiency improvement is a core design goal.
典型的课堂示例包括杠杆系统、滑轮组、螺旋千斤顶和液压机。提高效率是设计的核心目标之一。
7. Gear Ratios | 齿轮比
Gear trains transmit rotary motion and torque. The gear ratio determines the speed and torque conversion between driver and driven shafts.
齿轮系传递旋转运动和扭矩。齿轮比决定了主动轴与从动轴之间的转速和扭矩转换。
Gear Ratio = Number of teeth on driven gear / Number of teeth on driver gear
Also, for two mating gears:
Gear Ratio = Speed of driver / Speed of driven
A ratio greater than 1 means speed reduction but torque multiplication (like in a car’s first gear). A ratio less than 1 is an overdrive. Torque ratio is ideally the inverse of the speed ratio if 100% efficient.
比值大于 1 表示减速但扭矩倍增（如汽车一挡）。比值小于 1 为增速传动。在 100% 效率的理想情况下，扭矩比是转速比的倒数。
Compound gear trains combine multiple pairs; the overall ratio is the product of individual stage ratios. SQA problems often require calculation of output speed and torque from given inputs.
复式齿轮系组合了多对齿轮；总传动比是各级传动比的乘积。SQA 题目常要求根据给定输入计算输出转速和扭矩。
8. Ohm’s Law and Resistance | 欧姆定律与电阻
Ohm’s law links potential difference (V), current (I) and resistance (R) in a metallic conductor at constant temperature. It is the starting point for all DC circuit calculations.
欧姆定律将恒定温度下金属导体两端的电势差（V）、电流（I）和电阻（R）联系起来。它是所有直流电路计算的出发点。
V = I × R
Resistance depends on material and geometry: for a uniform conductor of length L and cross‑sectional area A, resistivity ρ gives:
电阻取决于材料和几何尺寸：对于长度为 L、横截面积为 A 的均匀导体，电阻率 ρ 满足：
R = ρ × (L / A)
Resistivity has units Ω·m. Conductors like copper have low ρ, insulators high ρ. Temperature changes affect resistance; the temperature coefficient of resistance explains this variation.
电阻率单位为 Ω·m。铜等导体的 ρ 低，绝缘体 ρ 高。温度变化会影响电阻；电阻温度系数解释了这一变化。
9. Kirchhoff’s Laws and Circuit Analysis | 基尔霍夫定律与电路分析
Kirchhoff’s two laws extend Ohm’s law to complex networks with multiple branches and sources. They are essential for analysing series‑parallel circuits in Engineering.
基尔霍夫两条定律将欧姆定律扩展到多支路、多电源的复杂网络。它们是工程中分析串并联电路的基础。
Kirchhoff’s Current Law (KCL): The total current entering a junction equals the total current leaving it (conservation of charge).
基尔霍夫电流定律（KCL）： 进入某节点的总电流等于离开该节点的总电流（电荷守恒）。
Kirchhoff’s Voltage Law (KVL): The algebraic sum of all potential differences around any closed loop is zero (conservation of energy).
基尔霍夫电压定律（KVL）： 任意闭合回路中所有电势差的代数和为零（能量守恒）。
Applying KVL involves assigning polarities and creating loop equations. These equations are solved simultaneously to find branch currents. SQA problems typically involve a mix of series and parallel resistors with one or more power supplies.
应用 KVL 需要标定极性并建立回路方程。并联立求解这些方程可得支路电流。SQA 问题通常包含串并联电阻混合及一或多个电源的电路。
10. Power in Electrical Circuits | 电路功率
Electrical power (P) in a resistive circuit is the rate at which electrical energy is converted to heat or mechanical work. Three equivalent forms arise from Ohm’s law.
电阻电路中的电功率（P）是电能转化为热能或机械功的速率。由欧姆定律可得出三种等价形式。
P = V I = I² R = V² / R
The unit is the watt (W), where 1 W = 1 J/s. Power ratings of components (resistors, motors) must be respected to avoid overheating. Energy consumed over time is E = P × t, often expressed in kilowatt‑hours (kWh) for large quantities.
单位为瓦特（W），1 W = 1 J/s。必须遵守元件（电阻、电机）的额定功率，以防过热。随时间消耗的电能为 E = P × t，对大数值常以千瓦时（kWh）表示。
Understanding power efficiency is key: electrical input power to an actuator is partly lost as heat, with output mechanical power being the useful portion.
理解功率效率是关键：输入执行器的电功率部分以热形式损耗，输出的机械功率才是有效部分。
11. Fluid Systems: Pascal’s Principle | 流体系统：帕斯卡原理
Pascal’s principle states that pressure applied to an enclosed fluid is transmitted undiminished throughout the fluid. This underpins hydraulic press and brake systems.
帕斯卡原理指出，施加于封闭流体的压力会大小不变地传递到流体各处。这是液压机和液压制动系统的基础。
p = F₁ / A₁ = F₂ / A₂
Thus, a small force F₁ on a small piston area A₁ can support a large force F₂ on a large piston, giving force amplification. The compromise is that the small piston must move a proportionately larger distance.
因此，施加在小活塞面积 A₁ 上的小力 F₁ 可在大活塞上支撑起大力 F₂，实现力放大。代价是必须小活塞移动相应更大的距离。
The ideal mechanical advantage for a hydraulic system is the area ratio A₂/A₁. In practice, friction and fluid compressibility reduce this advantage slightly.
液压系统的理想机械优势为面积比 A₂/A₁。实际上，摩擦和流体可压缩性会略微降低这一优势。
12. Equations of Linear Motion | 直线运动方程
When an object moves with constant acceleration along a straight line, its motion is described by three kinematic equations (SUVAT). These link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t).
物体沿直线做匀加速运动时，其运动由三个运动学方程（SUVAT）描述，它们关联了位移（s）、初速度（u）、末速度（v）、加速度（a）和时间（t）。
v = u + a t
s = u t + ½ a t²
v² = u² + 2 a s
These equations assume constant acceleration and straight‑line motion. In engineering, they are applied to vehicle dynamics, lift mechanisms and projectile analysis where air resistance is neglected.
这些方程假设加速度恒定且为直线运动。在工程中，它们被应用于车辆动力学、升降机构以及忽略空气阻力的抛体分析。
Choosing the correct equation depends on the known quantities. For example, if time is not given, the third equation is most convenient. Careful sign conventions (positive direction) are essential when solving inclined plane problems.
选择正确方程取决于已知量。例如，若未给出时间，第三个方程最方便。在求解斜面问题时，务必注意符号约定（规定正方向）。
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