📚 Case Study: Design Analysis of a Bicycle Crank Arm | 自行车曲柄设计案例分析
This case study presents a comprehensive, step-by-step engineering analysis of a bicycle crank arm. It aims to demonstrate how fundamental principles of mechanics, materials science, and manufacturing are integrated to solve a real-world design problem. The structured approach mirrors the style of extended response questions in CAIE AS-Level Engineering Paper 2, guiding students through the decision-making process from requirements to final recommendations.
本案例分析对自行车曲柄进行了全面、逐步的工程分析，旨在展示如何整合力学、材料科学和制造的基本原理来解决实际设计问题。这种结构化方法模拟了 CAIE AS 阶段工程试卷 2 中拓展回答题的风格，引导学生完成从需求到最终建议的决策过程。
1. Problem Statement and Design Requirements | 问题陈述与设计要求
A crank arm is the lever that connects the pedal to the bottom bracket spindle, converting the rider’s downward force into rotational torque. The design must satisfy the following key requirements:
曲柄是连接脚踏与中轴轴心的杠杆，将骑行者向下的力转化为旋转扭矩。设计必须满足以下关键要求：
- Crank length L = 170 mm (0.17 m), to suit standard adult bicycle geometry. | 曲柄长度 L = 170 mm (0.17 m)，适配标准成人自行车几何尺寸。
- Maximum static pedal force F = 1500 N, representing a heavy rider standing on the pedal. | 最大静态脚踏力 F = 1500 N，代表较重骑行者站立在脚踏上的情况。
- Target safety factor against yield = 2.0, ensuring robust operation without permanent deformation. | 目标屈服安全系数 = 2.0，确保运行可靠且无永久变形。
- Minimum service life: 10⁶ load cycles, demanding adequate fatigue resistance. | 最低使用寿命：10⁶ 次载荷循环，要求具备足够的抗疲劳性能。
- Lightweight design to improve bicycle handling and efficiency. | 轻量化设计，以改善自行车操控性和效率。
- Cost-effective manufacture for mid-range bicycle market. | 面向中端自行车市场，制造成本合算。
2. Load and Force Analysis | 载荷与受力分析
The crank arm can be modelled as a cantilever beam fixed at the bottom bracket end and loaded at the pedal end. When the pedal is at the top-dead-centre horizontal position, the full downward force produces a pure bending moment about the fixed end. Any horizontal force component is neglected for this static case.
曲柄可以简化为一个在中轴端固定的悬臂梁，在脚踏端受载。当脚踏处于上止点水平位置时，向下的力对固定端产生纯弯矩。在此静态工况下忽略水平分力。
Bending moment M = F × L = 1500 N × 0.17 m = 255 N·m
This bending moment is the primary design load. In reality, a radial force also causes torsion when the crank is not perfectly horizontal, but the cantilever bending condition usually governs the maximum stress.
该弯矩是主要设计载荷。实际上，当曲柄不完全水平时径向力还会引起扭转，但悬臂弯曲工况通常决定最大应力。
3. Material Selection | 材料选择
Two candidate materials are considered for the crank arm: aluminium alloy 7075-T6 and chromium-molybdenum steel AISI 4130. Their relevant properties are compared in the table below.
考虑两种曲柄候选材料：铝合金 7075-T6 和铬钼钢 AISI 4130。下表对比了它们的相关性能。
| Property | 7075-T6 Aluminium | 4130 Steel |
|---|---|---|
| Density ρ (kg/m³) | 2810 | 7850 |
| Yield strength σy (MPa) | 503 | 460 (normalised) |
| Ultimate tensile strength (MPa) | 572 | 560 |
| Fatigue strength at 10⁶ cycles (MPa) | ~160 (smooth) | ~280 |
| Approx. cost ratio per kg | 3.5 | 1.0 |
7075-T6 aluminium is selected for its superior strength-to-weight ratio, which is critical for a lightweight component. Although steel offers better fatigue strength and lower raw material cost, the weight penalty and higher inertial load during pedalling make aluminium the preferred choice for performance bicycles. The lower fatigue limit of aluminium must be carefully managed through design.
选择 7075-T6 铝合金是因为其卓越的比强度，这对轻量化部件至关重要。尽管钢材具有更好的疲劳强度和较低的原材料成本，但重量劣势和踩踏时更大的惯性载荷使铝合金成为性能自行车的首选。铝材较低的疲劳极限必须通过设计谨慎应对。
4. Stress Analysis and Cross-Section Design | 应力分析与截面设计
For initial sizing, the crank arm cross-section is approximated as a solid rectangle of width b = 22 mm and height h = 30 mm, tapering slightly towards the ends. The section modulus Z for a rectangular beam about the axis of bending is:
初步定寸时，将曲柄截面近似为宽度 b = 22 mm、高度 h = 30 mm 的实心矩形，两端略呈锥度。矩形梁绕弯曲轴的截面模量 Z 为：
Z = b h² / 6 = (0.022 m) × (0.030 m)² / 6 = 3.3 × 10⁻⁶ m³
Maximum bending stress σmax at the fixed end is calculated as:
固定端最大弯曲应力 σmax 计算如下：
σmax = M / Z = 255 N·m / 3.3 × 10⁻⁶ m³ ≈ 77.3 MPa
This stress is well below the yield strength of 503 MPa. The static safety factor is 503 / 77.3 ≈ 6.5, which comfortably exceeds the target of 2.0. However, fatigue performance will dictate the final design. Stress concentration at the pedal thread and the taper transition must be analysed, with a fatigue stress concentration factor Kf typically around 2.2 for threaded regions. The local peak stress can then be estimated as σpeak = Kf × σnom.
该应力远低于 503 MPa 的屈服强度。静态安全系数为 503 / 77.3 ≈ 6.5，轻松满足目标值 2.0。但疲劳性能将决定最终设计。必须分析脚踏螺纹处和锥度过渡区的应力集中，螺纹区域的疲劳应力集中系数 Kf 通常约为 2.2。此时局部峰值应力可估算为 σpeak = Kf × σnom。
5. Factor of Safety and Fatigue Life | 安全系数与疲劳寿命
For aluminium alloys, there is no well-defined endurance limit; the fatigue strength continues to decrease beyond 10⁶ cycles. The design must therefore target a finite life. Using the S-N curve for 7075-T6, the allowable alternating stress for 10⁶ cycles is approximately 160 MPa for polished specimens. Applying a surface finish factor ka = 0.8, size factor kb = 0.85, and a reliability factor kc = 0.814, the corrected fatigue strength Se becomes:
铝合金没有明确的耐久极限，疲劳强度在超过 10⁶ 次循环后仍会持续下降，因此设计须针对有限寿命。根据 7075-T6 的 S-N 曲线，抛光试样的 10⁶ 次许用交变应力约为 160 MPa。计入表面加工系数 ka = 0.8、尺寸系数 kb = 0.85 和可靠度系数 kc = 0.814，修正后的疲劳强度 Se 为：
Se = 160 × 0.8 × 0.85 × 0.814 ≈ 88.5 MPa
The nominal stress of 77.3 MPa is below the corrected fatigue strength, implying infinite life is possible. However, incorporating Kf = 2.2 raises the peak stress to 77.3 × 2.2 = 170 MPa, which exceeds Se. The effective safety factor in fatigue becomes Se / σnom ≈ 88.5 / 77.3 ≈ 1.14, which is marginal. To achieve the desired reliability, the cross-section can be gently increased, or shot-peening can be introduced to induce compressive residual stresses.
名义应力 77.3 MPa 低于修正疲劳强度，意味着可能实现无限寿命。但计入 Kf = 2.2 后峰值应力升至 170 MPa，超过 Se。实际疲劳安全系数变为 Se / σnom ≈ 88.5 / 77.3 ≈ 1.14，非常紧张。为达到预期可靠性，可略微加大截面，或引入喷丸处理产生残余压应力。
6. Manufacturing Process Selection | 制造工艺选择
Common processes for crank arms include casting, forging, and CNC machining from billet. The following table summarises their merits.
曲柄的常见制造工艺包括铸造、锻造和由坯料 CNC 加工。下表汇总了其优缺点。
| Process | Advantages | Disadvantages |
|---|---|---|
| Casting | Low unit cost, complex shapes possible | Porosity risk, lower fatigue strength, thicker sections needed |
| Hot forging | Excellent grain flow, high strength, consistent quality | High tooling cost, limited to simpler shapes, secondary machining required |
| CNC billet machining | Very high precision, flexible design, no tooling investment | High material waste, slower production rate, higher unit cost for volume |
Hot forging of 7075 aluminium is chosen for medium to high production volumes. The forged blank is then heat-treated to T6 condition and finish-machined on CNC mills to achieve the threaded pedal hole and precise spindle interface. Forging aligns the grain structure along the crank arm’s length, significantly enhancing fatigue resistance.
对于中高产量，选择 7075 铝热锻。锻造毛坯随后进行 T6 热处理，并在 CNC 铣床上精加工出螺纹脚踏孔和精密的中轴接口。锻造使晶粒沿曲柄长度方向排列，显著增强抗疲劳性能。
7. Quality Control and Testing | 质量控制与测试
To ensure every crank arm meets the design specification, the following quality control measures are implemented:
为确保每件曲柄符合设计规范，实施以下质量控制措施：
- Dimensional inspection using a coordinate measuring machine (CMM) for critical interfaces. | 使用三坐标测量机对关键接口进行尺寸检测。
- Visual and dye-penetrant inspection to detect surface cracks. | 目视和渗透探伤以检测表面裂纹。
- Brinell hardness testing to verify T6 heat treatment (hardness ≥ 150 HB). | 布氏硬度测试以验证 T6 热处理（硬度 ≥ 150 HB）。
- Proof load testing on a sample basis: apply 1.5 × F (2250 N) and check for no permanent deformation. | 抽样验证载荷试验：施加 1.5 倍 F（2250 N），检查无永久变形。
- Fatigue testing of randomly selected samples with a target of passing 10⁶ cycles at design load. | 抽样疲劳试验，目标是在设计载荷下通过 10⁶ 次循环。
Statistical process control (SPC) is employed to monitor key process parameters such as forging temperature and machining tolerances, ensuring consistent quality across the batch.
采用统计过程控制（SPC）监控关键工艺参数，如锻造温度和加工公差，确保整批质量一致。
8. Environmental and Sustainability Considerations | 环境与可持续性考量
Aluminium is highly recyclable, with only about 5% of the energy required for primary production needed for remelting. The design encourages end-of-life recycling by avoiding mixed-material inserts where possible. The lightweight crank arm (estimated mass 0.22 kg) reduces the overall bicycle mass, contributing to lower energy consumption during use. Forging generates less machining waste than billet machining, improving the material utilisation rate. The environmental impact can be assessed using life cycle analysis (LCA), which shows that the use phase dominates energy consumption, justifying the effort to minimise mass.
铝材可高度回收，重熔所需能源仅为初级生产约 5%。设计尽可能避免混合材料嵌件，鼓励终端回收。轻质曲柄（估计质量 0.22 kg）减轻了自行车总重，有助于降低使用阶段能耗。锻造比坯料加工产生的切屑少，提高了材料利用率。环境影响可用生命周期评价（LCA）评估，结果显示使用阶段主导能源消耗，因此努力减轻质量是合理的。
9. Cost Analysis and Economic Viability | 成本分析与经济可行性
A simplified cost model for a production batch of 10,000 units is outlined:
以下概述 10,000 件生产批次的简化成本模型：
| Cost element | Cost per unit (£) |
|---|---|
| Raw material (0.35 kg forging blank × £5/kg) | 1.75 |
| Forging and heat treatment | 2.50 |
| CNC finishing and threading | 3.20 |
| Surface treatment (anodising) | 0.80 |
| Quality inspection (amortised) | 0.50 |
| Total manufacturing cost | 8.75 |
With an estimated selling price of £15–20, the gross margin is attractive. The tooling investment for forging dies can be amortised over large volumes, making the approach economically viable. For lower volumes, CNC machining from extruded bar might be more economical.
估算售价为 15–20 英镑，毛利颇具吸引力。锻模的投资可在大量生产中摊销，使方案经济可行。对于较小批量，用挤压棒材直接 CNC 加工可能更经济。
10. Conclusion and Recommendations | 结论与建议
The analysis confirms that a hot-forged 7075-T6 aluminium crank arm with a 22 mm × 30 mm rectangular section can meet the strength and fatigue requirements while remaining lightweight. The static safety factor of 6.5 and the corrected fatigue safety factor of 1.14 indicate that local stress relief features, such as generous fillet radii and possibly shot peening, should be adopted to improve fatigue reliability. It is recommended that finite element analysis (FEA) be used to refine the shape and identify high-stress regions before prototyping. Further weight reduction may be possible through topology optimisation, and future designs could explore carbon-fibre-reinforced polymer composites for even higher strength-to-weight ratios.
分析证实，采用 22 mm × 30 mm 矩形截面的 7075-T6 铝合金热锻曲柄能够满足强度和疲劳要求，同时保持轻量。6.5 的静态安全系数和 1.14 的修正疲劳安全系数表明，应通过大圆角和可能的喷丸处理等局部应力消除措施来提高疲劳可靠性。建议在原型制作前使用有限元分析（FEA）优化形状并识别高应力区域。通过拓扑优化可进一步减重，未来设计可探索碳纤维增强聚合物复合材料以获得更高的比强度。
Published by TutorHao | Engineering Revision Series | aleveler.com
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