the inertial restrictions of the mechanism must be calculated.Therefore, the CG of each link is used as a way to interpret the inpidual motions of each link, in function of its links and combined restrictions. Each link moves by rotation and translation motions or still with their combination around their CG. The potential energy 𝑉 is evaluated by its contribution from the gravitational potential energy, favoring or not the movement of each position of the links.Therefore,the Lagrangian of the system is calculated for the energies of the three links that have movements in relation to the fixed link of the mechanism, as follows:

The kinetic energy 𝑇cy of the monocylinder possesses an inertia for the translation movement equivalent to that of a cylindrical bar. The kinetic energy 𝑇cr of the crank

possesses an inertia of the rotation movement equivalent to that of a solid cylinder/disk (shaft and crank connection)and thin hoop (wheel and radius). The kinetic energy 𝑇co of the connecting-rod has inertia of the rotation movement equivalent to that of a solid cylinder/disk (shaft and crank connection) and thin hoop (wheel and radius). The gravitational potential energy 𝑉co of the connecting-rod is generated by the vertical movement of its mass 𝑚co on the two dimensional plane 𝑥𝑦. However, the gravitational potential energies 𝑉cy and 𝑉cr do not consider vertical movement of their masses on the two-dimensional plane 𝑥𝑦, then not generating potential energy.

Therefore, one should calculate the kinetic energy of all links of the mechanism in function of their masses and only the potential gravitational energy of the connecting rod,which is the unique link that generates vertical motion.The kinetic energy of the crank just considered the rotational motion, because the crank possesses a rigid coupling with the wheel by means of hoops free to rotate through the bearing coupled to the fixed link of the mechanism.Thus, the calculus of the kinetic energy of the crank is defined as an inertial system equivalent to a solid cylinder and disk for the shaft and the crank combined with the inertia of thin hoops for the wheel and your radius, as follows:

For the calculus of energies it is necessary to define the geometry for the components because the masses generate the moment of inertia equivalent to the components of the compressed air engine. Thus, the mass 𝑚cr related to the moment of inertia is equivalent to a solid cylinder (shaft) and a disk (crank) related to the geometry and radius (𝑟). The mass 𝑚ra related to the moment of inertia is equivalent to a thin hoop with the distribution in function of the wheel middle radius (R/2). The mass 𝑚wh related to the moment of inertia is equivalent to a thin hoop for the hoop of the wheel

with tire which is related to the wheel radius (𝑅). The mass 𝑚co, which is related to the moment of inertia, is equivalent to a slender rod for the connecting-rod and is related to the crank and the cylinder through their angular and linear velocities. Moreover, the mass 𝑚wh related to the moment of inertia is equivalent to a thin hoop for the hoop of the wheel with tire and is related to the wheel radius (𝑅).

However, the kinetic energy of the connecting-rod is calculated considering the momentum of inertia of the masses as to rotational motion and to translational motion,and the connecting-rod possesses two degrees of freedom of motion in 𝑥𝑦 plane.Thus, the calculation of the kinetic energy of the connecting-rod was defined as an equivalent inertia system to a slender rod with axis through end, rotating and translating during the motion in the plane 𝑥𝑦, as given by

Transforming the energy equations in terms of generalized coordinates, trigonometric relations are used to the angles of 𝜃 and 𝛽 (congruent and obtuse) to use in relative

velocity of the connecting-rod in function of the generalized coordinates. As the connecting-rod link is a two degrees of- freedom system, there is two relative velocities related to the crank and its coordinate 𝜃. Therefore, the relative velocity of the connecting-rod considers the translational and rotational motions which is developed by using the cosines’ law, denoting [23, 24]