**2.1 Dynamic modeling**

The Modeling of FPLA and numerical simulation of its performance have already been done by a lot of researchers. Usually the model consists of a dynamic calculation of the piston motion and a thermodynamic calculation of the engine's thermodynamic events. The piston motion is governed by the interaction of forces that act on the piston simultaneously, which are in-cylinder pressure force from each cylinder, electromagnetic force, friction force and inertia force, as is shown in Fig.2.

Fig. 2. Free body diagram for FPLA

The Newton's second law is applied:

$$m\frac{d^2\mathbf{x}}{dt^2} = \left(p\_L - p\_R\right)A - F\_f - F\_e \tag{1}$$

Since the FPLA doesn't have a crankshaft, it is free of the dynamic friction caused by the crankshaft mechanism and the friction force is caused mainly by the interactions between cylinder and piston rings, cylinder and piston skirt, which is small compared with the load of the linear alternator. Therefore, the friction force is considered to be constant during the whole cycle in the calculation [13].
