Marcel Thomas

A quick note on sensitivity analysis: it is often convenient to perform a linear multivariate Taylor series expansion about an operating point. That might sound complicated, but it can allow you to perform quick estimations in your head. For example, if you make the flexure 10% thicker, you should expect ~30% increase in stiffness, ~10% drop in motion range, and ~15% increase in natural frequency.

A simple starting example is capacitance. Capacitance ( $C$ ) for a parallel-plate capacitor depends on the dielectric constant ( $\epsilon$ ), the capacitive area ( $A$ ), and the gap distance ( $g$ ): $C = \frac{\epsilon A}{g}.$ A linear (i.e. first-order) multivariate Taylor series expansion shows how small variations in geometry ( $\delta A$ is a variation in area, and $\delta g$ is a variation in gap distance) cause a small variation in capacitance ( $\delta C$ ): $\delta C \approx \frac{\partial C}{\partial \epsilon} \delta \epsilon+ \frac{\partial C}{\partial A}\delta A + \frac{\partial C}{\partial g} \delta g$ $\delta C \approx \frac{A}{g} \delta \epsilon + \frac{\epsilon}{g} \delta A - \frac{\epsilon A}{g^2}\delta g.$ What's really cool is that if you divide by capacitance, then you get a simpler relationship: $\frac{\delta C}{C} \approx \frac{\delta \epsilon}{\epsilon}+ \frac{\delta A}{A} - \frac{\delta g}{g}.$ This allows you to see a percent change in capacitance as a function of percent changes of input geometry (e.g. a 10% increase in separation distance causes ~10% drop in capacitance.)

More generally, if you have a quantity $Q$ that is a function of several other variables ( $q_i$ ), i.e. $Q = f(\{ q_i \}) = f(q_1,q_2,q_3,...),$ then the total multivariate Taylor series gives the variation in the quantity as a function of the variation of the input variables: $\delta Q = \sum_i \left( \sum_{n=1}^\infty \frac{1}{n!} \frac{\partial^n Q}{\partial q_i^n} (\delta q_i)^n \right).$ The first order ( $n=1$ ) Taylor series is often good enough for quick estimations: $\delta Q \approx \sum_i \frac{\partial Q}{\partial q_i} \delta q_i.$ If the quantity happens to be in a convenient form, namely $Q = \alpha \prod_{i}q_i^{N_i},$ where $\alpha$ is a constant, and $N_i$ is an exponent associated with $q_i$ , then the percent variation in $Q$ turns out to be $\frac{\delta Q}{Q} \approx \sum_i N_i \frac{\delta q_i}{q_i}.$

Now for a practical example involving cantilever beam stiffness: the bending stiffness ( $k$ ) of a cantilever beam is given by $k = \frac{3EI}{L^3} = \frac{Ebh^3}{4L^3},$ which implies that $\frac{\delta k}{k} \approx \frac{\delta E}{E} + \frac{\delta b}{b} + 3\frac{\delta h}{h} - 3 \frac{\delta L}{L}.$ Now we can vary the parameters and see the change in stiffness and compare it to the theoretical prediction. A 5% increase in $b$ causes a 5% increase in stiffness for both the theoretical and approximate analyses (a linear approximation works perfectly well for a linear function.) A 5% increase in $h$ causes a 15.8% increase in stiffness, and the approximation predicted a 15% increase. A 5% increase in $L$ causes a 13.6% drop in stiffness, and the approximation predicted a 15% drop. The error caused by the approximation should increase with the order of the quantity (e.g. $q^4$ diverges from linear faster than $q^3$ ), and the error grows with the input variation (i.e. the linear approximation works better at predicting an output variation if the input variation is smaller.)

Sensitivity Analysisï¿¼