I am trying to solve the following problem for $p$ (the pressure) between a piston skirt and a cylinder lining. This depends on the secondary motion of top and bottom of the piston skirt $e_t$ and $e_b$ respectively.

These values are given by $\begin{pmatrix} m_{pin}(1-a/l)+m_{pis}(1-b/l) && m_{pin}a/l+m_{pis}b/l\\I_{pis}/l+m_{pis}(a-b)(1-b/l) && m_{pis}(a-b)(b/l)-I_{pis}/l \end{pmatrix}\begin{pmatrix}\ddot{e_t}\\\ddot{e_b}\end{pmatrix}=\begin{pmatrix}F+F_s+F_ftan(\phi)\\M+M_s+M_f\end{pmatrix}$

However, $F$,$F_f$, $M$ and $M_f$ depend on the pressure, which depends on $e_t$ and $e_b$. For example $F$ is given by the integral of the pressure over the surface of the skirt that is in contact with the oil ($A$= {$0\leq y\leq L, \quad2\pi-\theta_1/2\leq\theta\leq\theta_1/2$ }:

$$F=R\int\int_A p(\theta,y)\cos(\theta)d\theta dy$$

where the pressure is calculated from the reynolds lubrication equation:

$$\frac{\partial}{\partial x}\left(\phi_x(h)h^3\frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial y}\left(\phi_y(h)h^3\frac{\partial p}{\partial y}\right)= 6\mu U\left(\phi_c(h)\frac{\partial h}{\partial x}+\sigma\frac{\partial \phi_s(h)}{\partial x}\right)+12\mu\phi_c(h)\frac{\partial h}{\partial t}$$

With $x=R\theta$ and BC:$$p=0 \quad\theta_1\leq\theta\leq\theta_2\\\frac{\partial p}{\partial x}|_{\theta=0}=\frac{\partial p}{\partial x}|_{\theta=\pi}=0\\p(\theta,0)=p(\theta,L)=0$$

The coefficients $\phi_i(h)$ all depend on $h$ (complicated and long expression which we can call $f(h)$ for now. The coupling is that $$h(\theta,y,t)=C+e_t(t)\cos(\theta)+[e_b(t)-e_t(t)]\frac{y}{L}\cos(\theta)$$ and $$\frac{\partial h}{\partial t}=\cos(\theta)[\dot{e_t}(1-y/L)+\dot{e_b}y/L]$$

Is this possible to solve in Mathematica? And if so what is the best way to go about it? Dsolve or Nsolve? Apologies for the rather crude questions but I am very new to Mathematica, i only started using it yesterday as i had spent an awful lot of time trying to solve this problem in Matlab via finite difference method and an iterative scheme but with no success. I played around with Mathematica trying to set up the problem but i can't seem to figure out how to do so. Any help or tips would be greatly appreciated.

Thanks in advance.

  • 3
    $\begingroup$ It is quite possible that the system can be solved. But in the form you have given it, it does not look closed. $\endgroup$ Commented Jul 31, 2023 at 11:41
  • 1
    $\begingroup$ Several coupled ODE-PDE systems have been discussed on this site. See mathematica.stackexchange.com/a/188406/1063 for a partial list. $\endgroup$
    – bbgodfrey
    Commented Jul 31, 2023 at 21:59


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.