In a paper by Camescasse et al.: "Bistable buckled beam: Elastica modeling and analysis of static actuation", in equation 46, the authors state a non-linear ode system for a bistable buckled beam. In the following section, they discuss the numerical approach and analysis. For that, they define boundary conditions (48) and (49).
For ease of discussion, let's start with some assumptions in my implementation to simplify/explain the model a bit. In my case, $\hat{\delta}^\pm = 0.5$, which erases all distinctions into $\pm$. Logically speaking, this means that the Force $F$ applies in the center of the beam. $F$ is given. $k$ , defined in (38) detemines the slenderness ratio of the beam and is also given. $\frac{d\vec{Q}}{ds}$, defined in (3), is the position vector the point at section $s$ on the neutral axis on the beam, so effectively $\frac{dx}{ds}$ and $\frac{dy}{ds}$. $\vec{\tau}$ is given in (6). All values are normalized, so e.g. $x(1)$ is the value of $x$ at the right end of the beam.
$\tag{46}\left\{\begin{array}{} \frac{d^2\theta}{ds^2} + \hat{\delta} F \cos(\theta) - k \delta P F \cos(2\theta) - \frac{1}{2}*k[P^2-\delta^2F^2]\sin(2\theta) = 0 \\ \frac{d\vec{Q}}{ds}=(1+kN)\vec{\tau} \end{array}\right.$
with
$\tag{47} N = - P \cos(\theta) + \delta F \sin(\theta) $,
$\tag{3+37a} \vec{Q} = X \vec e_1 + Y \vec e_2$
and
$\tag{6} \vec \tau = \cos \theta(s) \vec e_1 + \sin \theta(s) \vec e_2 $
I want to implement the following scenario: The beam is precompressed, so the joint $B$ is at the position $x_B$. Then, a centered actuation Force $F$ pulls the beam downwards. My goal is to analyze the buckling/snap-through behaviour, but i have difficulties implementing all boundary conditions.
$\tag{48}\left\{\begin{array}{} X(0) = 0 \\ Y(0) = 0 \\ \theta'(0) = 0 \\ \theta(0) = \theta_A \end{array}\right.$ $\tag{49}\left\{\begin{array}{} X(1) = x_B \\ Y(1) = 0 \\ \theta'(1) = 0 \\ \theta(1) = \theta_B \end{array}\right.$
First of all, i don't understand the last conditions in (48) and (49). Solving $\theta(s)$ automatically gives $\theta_a$ and $\theta_b$, so this seems superfluous and is IMO not a boundary condition.
Second, i don't have an extensive numerical background, so there might very well be a wrong assumption in the following problem: the model effectively has four integration constants/degrees of freedom (from integrating to $\theta(s)$, $\frac{d\theta}{ds}$, $x(s)$ and $y(s)$). With the parameter $P$ unknown, this should leave us with a total of 5 boundary conditions to solve. But without counting $\theta_a$ and $\theta_b$ we need six. In my understanding this makes it overdetermined and i have no idea to approach this problem. I already successfully implemented a version in matlab that simply ignores one boundary condition, e.g. $y(1) = 0$, but this obviously causes the buckling beam to drift downwards when actuated with $F$ and not buckling.