Problem $1$: Folded beam springs $(10$ points total$)$

MEMS devices $($for example, accelerometers$)$ often include moveable structures that are supported

on folded or serpentine beam springs. Examples of folded spring structures are shown in the

figures below; the beam segments labeled $1,2,3,$ and $4$ can bend in order to deflect in the

horizontal direction, while the segments labeled $5$ and $6$ and the central plate that the spring

supports may be considered rigid. In this problem, we will consider a folded spring in which each

deformable beam segment has a length $L,$ width in the plane of a$,$ and thickness $($the dimension

into the page$)$ of $b$ as shown in the diagram.

a$)$ Consider the situation in which a force $F$ is applied to the center of the plate

that is supported by the folded spring, as shown in the diagram at right.

Calculate the horizontal deflection of the plate when the force F is applied.

What is the spring constant of the folded spring structure? You are welcome

to use the beam solutions that are given in the notes. Hint: consider springs

in series and in parallel.

b$)$ Now consider a spring with a larger number of segments. Where

the spring in part a had two segments on each side of the plate, the

new spring has n segments on each side of the plate. $($In the diagram

at right, $n=4,$ but you should consider the general case.$)$ What is the

spring constant of the new structure?