Application Area 9: Appliance Design

A mechanical accelerometer is used to measure the acceleration of a rocket test sled. It contains a linear spring-damper system. See the image below. By tuning the stiffness and damping, the accelerometer can be designed to estimate the acceleration it feels within an acceptable time.

Objective: We can compute the deviation of any system response $x$ from its target value $x^{*}$ as follows:; (1)
$S T E P (t_{t a r g e t}) = {\begin{matrix} 0 i f t \leq t_{t a r g e t} \\ 1 i f t > t_{t a r g e t} \end{matrix}$
(2)
$f (x) = {\int_{t_{s t a r t}}^{t_{e n d}}}_{}^{} S T E P (t_{t a r g e t}) * | x - x^{*} | d t$; The problem is formulated as:; $m i n ({x - x^{*})^{2} |}_{s t e a d y s t a t e}$; $s u c h t h a t f (x) < 1 e - 3$
Design Variables: The stiffness of the spring, $k$ , and its damping, $c$ , are the design variables.
Results: By properly selecting the spring stiffness and damping, the accelerometer can be designed to respond in the desired time frame.; Figure 2.