Year 11 SQA Engineering: Quick Reference Formula Handbook | SQA 工程：公式定理速查手册

📚 Year 11 SQA Engineering: Quick Reference Formula Handbook | SQA 工程：公式定理速查手册

This handbook provides a condensed yet comprehensive summary of the fundamental formulas and theorems required for Year 11 SQA Engineering Science. It is designed as a quick revision tool to help you master the essential quantitative relationships in mechanics, electronics, structures, and energy systems. Each formula is paired with clear definitions and standard units to support effective problem-solving.

本手册精简扼要地总结了 Year 11 SQA 工程科学所需的基础公式与定理，旨在作为快速复习工具，帮助您掌握力学、电子、结构与能源系统中的重要数量关系。每条公式都配有清晰的定义和标准单位，以助高效解题。

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1. Basic Mechanics: Force and Motion | 基础力学：力与运动

The relationship between unbalanced force, mass and acceleration is given by Newton's second law, which is a cornerstone of dynamics. It allows us to calculate any one of the three variables when the other two are known.

牛顿第二定律描述了非平衡力、质量与加速度之间的关系，是动力学的基石。利用它可以由已知的两个变量求出第三个变量。

F = m × a

F = force (N), m = mass (kg), a ector equation indicates that acceleration occurs in the direction of the resultant force.

F = 力 (N)，m = 质量 (kg)，a = 加速度 (m/s²)。该矢量方程表明加速度方向与合外力方向一致。

Weight is a special case of force caused by gravity. It is vital for calculating loads in structures and lifting systems.

重量是重力引起的特殊力，对于计算结构和起重系统中的荷载至关重要。

W = m × g

W = weight (N), g m/s² on Earth, often taken as 9.8 N/kg). Mass remains constant regardless of location, while weight changes with gravity.

W = 重量 (N)，g = 重力场强度 (地球取 9.8 N/kg)。质量不随位置改变，重量则随重力变化。

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2. Moments and Simple Machines | 力矩与简单机械

The turning effect of a force about a pivot is called the moment. It depends on both the magnitude of the force and the perpendicular distance from its line of action to the pivot.

力绕支点的转动效应称为力矩，其大小取决于力的大小与力作用线到支点的垂直距离。

M = F × d

M = moment (Nm), F = force (N), d (m). Clockwise and anticlockwise moments are considered opposite signs in equilibrium calculations.

M = 力矩 (Nm)，F = 力 (N)，d = 支点到力作用线的垂直距离 (m)。在平衡计算中，顺时针和逆时针力矩符号相反。

For a lever to be in equilibrium, the sum of clockwise moments must equal the sum of anticlockwise moments about any point. This principle underpins all simple machines.

杠杆平衡时，对任一点的顺时针力矩总和等于逆时针力矩总和。这一原理是所有简单机械的基础。

ΣM_clockwise = ΣM_{anticlockwise}

Mechanical advantage indicates how much a machine multiplies the effort force. Velocity ratio compares the distance moved by effort to that moved by load.

机械增益表示机器将作用力放大的倍数。速比则比较施力点与载荷的移动距离。

MA = Load / Effort

VR = Distance moved by effort / Distance moved by load

Efficiency can also be determined from these ratios: η = MA / VR.

效率也可由此求得：η = MA / VR。

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3. Stress, Strain and Young's Modulus | 应力、应变与杨氏模量

When a material is subjected to axial forces, stress measures the internal resistance and strain measures the deformation. They form the basis of material selection in engineering design.

材料承受轴向力时，应力表征内部抗力，应变表征形变程度，它们是工程设计选材的基础。

σ = F / A

σ = stress (Pa or N/m²), F = applied force (N), A ). Tensile stress tends to stretch, while compressive stress tends to squash the material.

σ = 应力 (Pa 或 N/m²)，F = 作用力 (N)，A = 横截面积 (m²)。拉应力使材料伸长，压应力使其缩短。

ε = ΔL / L₀

ε = strain (no units, dimensionless), ΔL = change in length (m), L₀ original length (m). Strain is often expressed as a percentage in practical applications.

ε = 应变 (无量纲)，ΔL = 长度变化 (m)，L₀ = 原始长度 (m)。实际应用中常以百分比表示。

Young's modulus is the ratio of stress to strain within the elastic limit and indicates a material's stiffness. A higher value means greater rigidity.

杨氏模量是弹性极限内应力与应变之比，反映材料的刚度。数值越大，材料越不易变形。

E = σ / ε

E = Young' . This constant is valid only within the linear elastic region of the stress-strain curve.

E = 杨氏模量 (Pa)。该常数值仅适用于应力-应变曲线的线弹性区域。

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4. Work, Energy and Power | 功、能与功率

Work is done when a force moves its point of application through a distance. It represents energy transferred mechanically.

力使其作用点移动一定距离时即做功，表示机械能传递的量。

W = F × d

W = work done (J), F = force (N), d on of force (m). If force is not parallel to displacement, the resolved component is used.

W = 功 (J)，F = 力 (N)，d = 力方向上的位移 (m)。若力与位移不平行，需使用分力计算。

Gravitational potential energy and kinetic energy are the two primary forms of mechanical energy. Their expressions are essential for energy conservation problems.

重力势能与动能是两种主要的机械能形式，其表达式是能量守恒问题的基础。

E_p = m g h

E_k = ½ m v²

E_p = gravitational potential energy (J), h = height (m). E_k = kinetic energy (J), v = speed (m/s). In absence of friction, E_p lost equals E_k gained.

E_p = 重力势能 (J)，h = 高度 (m)；E_k = 动能 (J)，v = 速率 (m/s)。无摩擦时，减少的势能等于增加的动能。

Power is the rate at which work is done or energy is transferred. It is also the product of force and velocity for an object moving at constant speed.

功率是做功或能量传递的速率，也可表示匀速运动物体的力与速度之积。

P = W / t or P = F × v

P = power (W), t = time (s), v = constant velocity (m/s). These two forms are equivalent for steady motion.

P = 功率 (W)，t = 时间 (s)，v = 匀速速度 (m/s)。对于匀速运动两式等价。

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5. Mechanical Efficiency and Drive Systems | 机械效率与传动系统

Efficiency compares useful output to total input, quantifying energy losses due to friction, heat, or sound. It is always less than 1 (or 100%) in real machines.

效率比较有用输出与总输入，量化因摩擦、热、声造成的能量损耗。实际机器中效率始终小于 1 (或 100%)。

η = P_out / P_in

η (%) = (P_out / P_in) × 100

η = efficiency (no units), P_out = useful output power (W), P_in For a series of components, overall efficiency is the product of individual efficiencies.

η = 效率 (无单位)，P_out = 有用输出功率 (W)，P_in = 总输入功率 (W)。多个组件串联时，总效率为各级效率的乘积。

In drive systems such as belts and gears, the speed ratio determines how rotational speed is altered between driving and driven shafts. It is the inverse of the gear ratio.

在带传动与齿轮传动系统中，速比决定主动轴与从动轴转速的变化关系，是传动比的反比。

Speed Ratio = Input speed / Output speed = N_driven / N_driver

Torque transmitted is related to power and angular speed. For an ideal (100% efficient) transmission, power remains constant, so torque changes inversely with speed.

传递的扭矩与功率和角速度相关。理想传动中功率恒定，故扭矩与转速成反比变化。

P = T × ω

T = torque (Nm), ω = angular velocity (rad/s). When speed increases, torque reduces if power is constant.

T = 力矩 (Nm)，ω = 角速度 (rad/s)。功率一定时，转速升高则扭矩下降。

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6. Electrical Fundamentals: Ohm’s Law | 电工基础：欧姆定律

Ohm's law relates voltage, current and resistance in a linear circuit element at constant temperature. It is the most used formula in circuit analysis.

欧姆定律描述了恒温下线性元件中电压、电流与电阻的关系，是电路分析中应用最多的公式。

V = I × R

V = potential difference or voltage (V), I = current (A), R e (Ω). The law assumes ohmic conductors where the graph of V against I is a straight line.

V = 电压或电势差 (V)，I = 电流 (A)，R = 电阻 (Ω)。欧姆定律适用于欧姆导体，即 V-I 图呈直线的元件。

Resistance depends on the material, length and cross-sectional area of a conductor. This explains why thicker wires have lower resistance.

导体的电阻取决于材料、长度和截面积，这也解释了为何粗导线电阻较小。

R = ρ × (L / A)

ρ = resistivity (Ω·m), L = length (m), A ross-sectional area (m²). Resistivity is a property of the material, independent of shape.

ρ = 电阻率 (Ω·m)，L = 长度 (m)，A = 横截面积 (m²)。电阻率是材料属性，与形状无关。

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7. Resistors in Series and Parallel | 电阻的串联与并联

Combining resistors correctly is essential for designing circuits with desired resistance values. Series circuits share the same current, while parallel circuits share the same voltage.

正确组合电阻是设计所需阻值电路的关键。串联电路电流相同，并联电路电压相同。

Series: R_total = R₁ + R₂ + R₃ + …

In a series arrangement, the total resistance is the sum of individual resistances. The same current flows through each component, and the supply voltage is divided among them.

串联时总电阻为各电阻之和，电流处处相等，电源电压在各电阻上分配。

Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …

For two parallel resistors, a simpler product-over-sum form can be used: R_total R₁ × R₂) / (R₁ + R₂). Adding more parallel branches always decreases the total resistance.

两个电阻并联时，可用简便公式：R_total = (R₁ × R₂) / (R₁ + R₂)。并联支路越多，总电阻越小。

Voltage divider and current divider rules allow quick calculation of intermediate values without solving full network equations.

分压与分流规则可快速求取中间变量，无需解完整网络方程。

V₁ = V_supply × (R₁ / (R₁ + R₂))

I₁ = I_total × (R₂ / (R₁ + R₂)) (for two parallel branches)

These formulas work only for resistive dividers and assume no significant loading at the output.

这些公式仅适用于电阻分压，并假设输出端无显著负载效应。

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8. Electrical Power and Energy | 电功率与电能

Electrical power is the rate at which electrical energy is converted into another form, such as heat, light or mechanical work. Multiple equivalent expressions exist.

电功率是电能转换为其他形式（热、光、机械功等）的速率，有多种等价表达式。

P = I × V

P = I² × R

P = V² / R

P eral for any component. The second form shows heating losses increase with current squared, which is why high-voltage transmission is used.

P = 功率 (W)。第一式为通用式；第二式表明发热损耗与电流平方成正比，由此可理解高压输电的意义。

Electrical energy consumed is power multiplied by time. This is the quantity measured by electricity meters in kilowatt-hours.

消耗的电能等于功率乘以时间，电表以千瓦时 (kWh) 度量。

E = P × t

E (J if using W and s, or kWh using kW and h). A 2 kW heater run for 3 hours consumes 6 kWh.

E = 电能 (若用 W 和 s 则得 J，用 kW 和 h 得 kWh)。例如 2 kW 的电热器工作 3 小时消耗 6 kWh。

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9. Statics and Structural Equilibrium | 静力学与结构平衡

A structure or component is in static equilibrium when it remains at rest. This requires both forces and moments to balance.

结构或部件保持静止时处于静力平衡状态，需同时满足力平衡与力矩平衡条件。

ΣF_x = 0, ΣF_y = 0

ΣM = 0

The sum of all horizontal forces and the sum of all vertical forces must each be zero. Additionally, the sum of moments about any point must be zero to prevent rotation.

所有水平分力之和与所有垂直分力之和均须为零。此外，对任一点力矩之和为零才能防止转动。

These conditions enable the analysis of trusses, beams and frames. Reactions at supports are calculated by applying equilibrium equations.

据此可分析桁架、梁与框架，通过平衡方程计算支座反力。

In a simple three-force member, if three non-parallel forces maintain equilibrium, their lines of action must intersect at a common point.

对于简单三力构件，若三个非平行力维持平衡，其作用线必汇交于一点。

Factor of safety is the ratio of ultimate stress to allowable working stress. It provides a margin against unexpected overloads.

安全因数是极限应力与许用工作应力之比，用以防范意外超载。

FS = Ultimate stress / Working stress

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10. Gear and Belt Drive Ratios | 齿轮与带传动比

Gear trains transmit rotary motion and torque between shafts. The velocity ratio depends on the number of teeth on each gear.

齿轮系在轴间传递旋转运动和扭矩，速比取决于各齿轮的齿数。

VR = N_driven / N_driver = T_driver / T_driven

N = rotational speed (rpm), T er turning a larger driven gear results in a reduction in speed and an increase in torque.

N = 转速 (rpm)，T = 齿数。小齿轮驱动大齿轮时转速降低、扭矩增大。

For compound gear trains, the overall velocity ratio is the product of individual stages. Idler gears change direction but not the magnitude of the ratio.

复合轮系的总速比为各级速比之积。惰轮只改变方向，不改变速比大小。

In belt and pulley systems, the diameter ratio replaces the tooth ratio. Slip may cause a slight difference between theoretical and actual output speed.

在带轮系统中，直径比取代齿数比。打滑可能导致理论与实际输出转速间存在轻微差异。

VR = D_driven / D_driver

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11. Hydraulics and Pneumatics: Pascal’s Principle | 液压与气动：帕斯卡原理

Pascal's principle states that pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid and the walls of the container. This enables force multiplication.

帕斯卡原理指出，施加于封闭流体上的压强会大小不变地传递至流体各部分及容器壁，由此可实现力的放大。

p = F₁ / A₁ = F₂ / A₂

Pressure p is constant throughout a hydraulic system, so a small force on a small piston can support a large load on a large piston.

整个液压系统中压强 p 处处相等，因此小活塞上的小力可支撑大活塞上的大负载。

F₂ = F₁ × (A₂ / A₁)

ortional to its area: d₁ / d₂ = A₂ / A₁. Work input equals work output in an ideal system.

活塞移动距离与面积成反比：d₁ / d₂ = A₂ / A₁。理想系统中输入功等于输出功。

For pneumatic systems using compressed air, the same principles apply, but the compressibility of gases must be considered in dynamic situations.

对于使用压缩空气的气动系统，原理相同，但在动态工况下需考虑气体的可压缩性。

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12. Material Properties and Selection | 材料性能与选用

Engineers select materials based on key mechanical properties such as strength, stiffness, ductility and toughness. Standard formulas help quantify these empirically.

工程师根据强度、刚度、延展性和韧性等关键力学性能选用材料，标准公式有助于将其量化。

• Tensile strength: maximum stress a material can withstand while being stretched before necking.

• 抗拉强度：材料在拉伸时颈缩前所能承受的最大应力。

• Hardness: resistance to indentation; often measured by Brinell, Vickers or Rockwell tests.

• 硬度：抵抗压痕的能力，常通过布氏、维氏或洛氏硬度测试衡量。

• mass / volume (kg/m³) influences weight and specific strength (strength-to-weight ratio).

• 密度 ρ = 质量 / 体积 (kg/m³)，影响重量及比强度（强度与重量比）。

Thermal expansion must be considered in assemblies. The linear expansion ΔL depends on temperature change ΔT, original length L₀ and the coefficient α.

装配时需考虑热膨胀。线性膨胀 ΔL 取决于温度变化 ΔT、原长 L₀ 和线膨胀系数 α。

ΔL = α × L₀ × ΔT

Conductivity (electrical or thermal) determines suitability for wiring, heat sinks or insulators. The selection matrix links these properties to application requirements.

导电性与导热性决定了是否适合用作导线、散热器或绝缘材料。选用矩阵将这些性能与应用需求联系起来。

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