Stack up dimensional tolerances of parts to evaluate assembly gap using worst-case, RSS, and Monte Carlo simulation. Visualize each part's contribution via tornado chart.
| Part name | Nominal [mm] |
+ tolerance | - tolerance | Sensitivity a |
|---|
The fundamental goal is to find the total variation, $U$, in a critical assembly dimension (like a clearance). This dimension is a linear combination of $n$ individual part dimensions $x_i$, each with a nominal value and a tolerance $t_i$. The sensitivity $a_i$ is usually +1 or -1, depending on if the part adds or subtracts from the total.
$$U = \sum_{i=1}^{n}a_i x_i$$Since each $x_i$ varies, $U$ will vary. The different stackup methods calculate the likely range of $U$.
1. Worst-Case Tolerance: The simplest, most conservative method. It assumes all part dimensions simultaneously reach their maximum or minimum limits.
$$T_{\text{WC}}= \sum_{i=1}^{n}|a_i \cdot t_i|$$2. Root-Sum-Square (RSS) Tolerance: A statistical method. It assumes part variations are independent, random, and normally distributed within their tolerance ranges, which are treated as ±$k\sigma$ limits (e.g., ±3σ).
$$T_{\text{RSS}}= k \sqrt{\sum_{i=1}^{n} (a_i t_i)^2}$$Here, $k$ is the "Sigma Level" from the simulator. If tolerances are set as ±3σ ($k=3$), then $T_{\text{RSS}}$ estimates the ±3σ range for the assembly.
Automotive Engine Assembly: Pistons, rings, and cylinders all have tightly controlled tolerances. Using Worst-Case analysis might suggest impossible-to-machine parts, while RSS or Monte Carlo shows that statistically, 99.73% (for 3σ) of assemblies will have proper clearance, enabling feasible manufacturing.
Consumer Electronics (Phone/Camera): The gap between a screen and its bezel, or the alignment of lens elements inside a camera module, is critical for aesthetics and function. Stackup analysis ensures the gap looks consistent across millions of units and that lenses stay in focus despite part variations.
Aerospace Structures: In wing or fuselage assembly, thousands of parts with individual tolerances must align for rivet holes to match. Monte Carlo simulation helps predict the probability of fit-up issues, guiding where to allocate tighter (and more expensive) tolerances.
Medical Device Manufacturing: For a syringe plunger, the clearance between the plunger and barrel affects both smooth operation and fluid leakage. Setting a "Min. Clearance Target" in the simulator directly relates to ensuring the device functions reliably and safely for every patient.
First, are you under the impression that "the RSS method always yields a better (tighter tolerance range) result than the worst-case method"? In reality, when the number of parts is extremely low (e.g., 2-3 parts), the RSS result can be nearly the same as or even lead to misjudging the safety factor compared to the worst-case method. For example, if just two parts have a tolerance of ±0.1mm, the worst-case stack is ±0.2mm, while the RSS method gives about ±0.14mm. If you design based solely on blind faith in the RSS result here, you risk a higher defect rate than anticipated. Remember that the statistical averaging effect becomes more powerful as the number of parts increases, which is where the true value of the RSS method shines.
Next, the assumption that "the sensitivity coefficient a_i is always either +1 or -1, right?". While this holds for simple linear stacks, it's a different story when geometric relationships or angles are involved. For instance, when calculating the play in a linkage mechanism, the length tolerances of parts affect the result through sines or cosines of angles. In such cases, the sensitivity coefficient takes on values completely different from 1. If your tool allows you to adjust the "sensitivity coefficient," I encourage you to try values other than 1 (like 0.5 or -0.707) and see how the results change.
Finally, the trap of "distribution settings" in Monte Carlo simulation. Tools often assume a normal distribution for simplicity, but in practice, this is frequently not the case. For example, in machining, tool wear can cause dimensions to skew to one side, or parts after selective assembly may have a truncated distribution. Ignoring these real-world distributions and calculating with the default normal distribution can lead to significant misjudgment of the actual defect rate. The greatest strength of using Monte Carlo simulation lies in this "freedom of distribution," so whenever possible, develop the habit of estimating the distribution from actual measurement data.