Two nonlinear springs, \(S_{1}\) and \(S_{2}\), are connected in two different ways as indicated in Fig. 1.88. The force, \(F_{i}\), in spring \(S_{i}\) is related to its deflection \(\left(x_{i}\right)\) as

\[F_{i}=a_{i} x_{i}+b_{i} x_{i}^{3}, \quad i=1,2\]

where \(a_{i}\) and \(b_{i}\) are constants. If an equivalent linear spring constant, \(k_{\mathrm{eq}}\), is defined by \(W=k_{\text {eq }} x\), where \(x\) is the total deflection of the system, find an expression for \(k_{\text {eq }}\) in each case.

Transcribed Image Text:

S 000 W (a) S W (b) $2 FIGURE 1.88 Two nonlinear springs connected in series and parallel.