故事 · 「全员同时作弊」的概率有多小
Story · How rare is "everyone cheats at once"?
想象一根由五段杆件首尾相接拼成的长轴,每段都有 ±0.1mm 的加工容差。总长误差最大能有多少?
最坏情况法回答:万一五段全偏长 0.1mm,总长就偏 0.5mm —— 按这个定容差,绝不会装不上。
可工艺工程师会反问:五段同时、同方向偏到极限,这概率有多小?
如果每段误差是独立随机的,它们更可能互相抵消 —— 有的偏长、有的偏短,加起来接近于零。
统计学给出答案:当各误差独立时,总误差的标准差是各标准差的平方根和,而非直接相加。
于是 RSS 用 √(0.1² × 5) ≈ 0.224mm 取代了 0.5mm —— 同样的装配良率,容差却宽松一倍多,子件加工成本大降。
这就是为什么镜头组、电机轴系、堆叠 PCB 这类多件叠加场景,行业默认用 RSS 而不是最坏情况。
Picture a long shaft built from five rods stacked end-to-end, each with a ±0.1 mm machining tolerance. What's the worst total length error?
Worst-case logic says: if all five rods drift long by 0.1 mm, the assembly drifts by 0.5 mm — size the spec for that and nothing ever fails to fit.
But the manufacturing engineer pushes back: how likely is it that all five drift to the same extreme in the same direction at the same time?
If the errors are independent random draws, they're much more likely to cancel each other out — some long, some short, the sum close to zero.
Statistics confirms it: for independent errors, the standard deviation of the sum is the square root of the sum of squares, not the linear sum.
So RSS replaces 0.5 mm with √(0.1² × 5) ≈ 0.224 mm — same assembly yield, but more than twice as much elbow room, and dramatically lower per-part machining cost.
That's why lens stacks, motor shaft trains, board-on-board PCBs — anywhere parts pile up — default to RSS instead of worst-case.
1 容差堆叠:T_wc 线性相加 vs T_rss 平方根和 Tolerance Stack: linear T_wc vs root-sum-of-squares T_rss
拖子件容差drag a slider2 实际装配误差分布(为什么 RSS 够用) Real assembly error distribution (why RSS is enough)
独立子件误差相加 → 总误差近似正态分布,绝大多数装配体挤在中间窄窄一带, RSS 边界(约 3σ)已覆盖 99.7% 的装配。 最坏情况边界远在外侧,那里几乎没有真实装配体 —— 为极小概率付全额保险,太贵。 Sum independent part errors → the total is approximately normal, with nearly every assembly clustered in a narrow band near the center. The RSS limits (about 3σ) already cover 99.7% of assemblies. The worst-case limits sit far outside, in a region that holds almost no real assemblies — paying full insurance against an event that essentially never happens is expensive.
3 现实里的容差堆叠 Tolerance stacks in the real world
容差分配:把装配总公差合理拆给各子件,关键件收紧、非关键件放宽。
Tolerance allocation: divide the assembly total across parts thoughtfully — tighten the critical ones, loosen the rest.
RSS vs 最坏情况:最坏情况绝对安全但贵,RSS 用统计抵消换更经济的容差。
RSS vs worst-case: worst-case is bulletproof but pricey; RSS trades on statistical cancellation for a cheaper spec.
装配公差:电机轴系、镜头组、堆叠 PCB 等多件叠加场景,RSS 是行业常用法。
Assembly tolerance: motor shaft trains, lens stacks, board-on-board PCBs — anywhere parts pile up, RSS is the default.
成本-良率:容差越紧加工越贵,RSS 在可接受良率下把容差放到最经济。
Cost vs yield: tighter tolerances cost more to machine — RSS pushes tolerance to the most economical point that still meets yield.
一句话
In one line
两个公式的差别,本质是对「同时性」的两种假设。最坏情况法 T_wc = Σ|tᵢ| 默认
所有子件同时偏到同一侧极限 —— 这是一个近乎不可能的联合事件,却要为它买全额保险。
RSS T_rss = √(Σtᵢ²) 承认统计现实:独立随机误差会互相抵消,不会集体作弊,
总误差因此服从正态、集中在中心,3σ 就够覆盖 99.7%。
关键洞察是 —— 子件越多,平方根和相对线性和省得越多:5 个等容差子件,RSS 只有 WC 的约 45%。
但 RSS 有前提:各容差近似独立、近似正态、且分布对称居中。
若子件强相关、偏度大,或工艺中心严重偏移,RSS 会低估总公差 —— 这时得回到更保守的最坏情况法或做修正。
The two formulas differ on one assumption: simultaneity. Worst-case T_wc = Σ|tᵢ| assumes every part drifts to the same extreme at the same time — a joint event that's nearly impossible, yet you pay full insurance against it.
RSS T_rss = √(Σtᵢ²) embraces statistical reality: independent random errors tend to cancel — they don't collude.
The total error is approximately normal, clustered around the center, and 3σ already covers 99.7%.
The key insight: the more parts you stack, the bigger the RSS-vs-linear savings. Five equal-tolerance parts: RSS is roughly 45% of worst-case.
RSS has preconditions, though: tolerances must be approximately independent, approximately normal, and centered.
If parts are strongly correlated, heavily skewed, or the process mean is biased, RSS will underestimate the total — fall back to worst-case or apply a shift correction.
常见误用
Common mistakes
不管前提一律用 RSS。RSS 要求各容差独立、近似正态、居中对称,否则会低估总公差。
Reaching for RSS without checking the preconditions. RSS demands independent, near-normal, centered tolerances — otherwise it underestimates the total.
对安全关键尺寸也用 RSS 省料。失效后果严重的关键链宁可保守,用最坏情况法兜底。
Using RSS to shave tolerance on safety-critical features. When failure is severe, stay conservative and use worst-case as a floor.
忽略工艺中心偏移,只算容差。中心偏移会吃掉余量,需用偏移修正或动态 RSS 评估。
Computing tolerance alone while ignoring process mean shift. Mean shift eats your slack — apply a shift correction or dynamic RSS evaluation.