Consider-a-system-consisting-ofw-to-masses-m-1-and-m-2-where-the-pendulum-rod-has-an-nolinear-elastic-coefficient-k-k-0-1-2-with-being-the-angle-between-the-rodand-the-vertical-directi

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Question Number 211677 by MrGaster last updated on 16/Sep/24

$$\mathrm{Consider}\mathrm{a}\mathrm{system}\mathrm{consisting}\mathrm{ofw}$$$$\mathrm{to}\mathrm{masses}.{\mathit{m}}_1\mathbf{and}{\mathit{m}}_2,\mathrm{where}\mathrm{the}\mathrm{pendulum}\mathrm{rod}\mathrm{has}\mathrm{an}$$$$\mathrm{nolinear}\mathrm{elastic}\mathrm{coefficient}\mathit{k}\left(\mathit{\theta }\right)={\mathit{k}}_0(1+\mathit{\alpha }{\mathit{\theta }}^2),\mathrm{with}\mathit{\theta }\mathrm{being}\mathrm{the}\mathrm{angle}\mathrm{between}\mathrm{the}$$$$\mathrm{rodand}\mathrm{the}\mathrm{vertical}\mathrm{direction}.$$$$\mathrm{Suppose}\mathrm{the}\mathrm{angle}\mathrm{is}\mathrm{smallh}$$$$\mathrm{enoug}\mathrm{that}\mathit{\theta }\mathrm{The}\mathrm{square}\mathrm{of}\mathrm{can}\mathrm{be}\mathrm{ignored}$$$$\mathrm{andthe}\mathrm{motion}\mathrm{equation}\mathrm{of}\mathrm{thes}$$$$\mathrm{sytem}\mathrm{can}\mathrm{be}\mathrm{approximated}\mathrm{as}\mathrm{as}$$$$\mathrm{imple}\mathrm{swing}\mathrm{with}\mathrm{lineary}$$$$\mathrm{elasticit}.\mathrm{However}\mathrm{for}\mathrm{largere}$$$$\mathrm{angls}\mathrm{nonlinear}\mathrm{factors}\mathrm{becomei}$$$$\mathrm{sgnificant}\mathrm{and}\mathrm{must}\mathrm{bed}$$$$\mathrm{considere}\mathrm{in}\mathrm{the}\mathrm{equation}\mathrm{ofo}$$$$\mathrm{motin}.\mathrm{The}\mathrm{motion}\mathrm{equation}\mathrm{ofh}$$$$\mathrm{tis}\mathrm{system}\mathrm{under}\mathrm{the}\mathrm{action}\mathrm{ofv}$$$$\mathrm{graity}\mathrm{and}\mathrm{including}\mathrm{nonlineara}$$$$\mathrm{elstic}\mathrm{coefficient}\mathrm{is}\mathrm{derived}.$$

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