This mechanism has restrictions to move in the vertical direction because of the cylinder but can translate free in 𝑥 direction. This vertical displacement restriction enables the system to move in relation to its length(𝑠CGcy), which generates the angular movement because of its connection to the connecting-rod.

These mechanisms presented in Figure 1 are a single degree-of-freedom system with the displacement of the pneumatic cylinder varying the Top Dead Center (TDC) up to the Bottom Dead Center (BDC), thus rotating the output crank in 2𝜋 rad. The linear position of the cylinder in relation Figure 1: Schematic draw of a compressed air engine composed of pneumatic cylinder and a crank-connecting-rod.

to its CG in the plane 𝑥𝑦 (𝑥CGcy) through the length of CG(𝑠CGcy) can be described as presented in 𝑥CGcy = 𝑟cos𝜃 + 𝑙cos𝜙 − 𝑠CGcy.(1)

In the way to determine the linear position of (𝑥CGcy) in relation to the angular position of the crank 𝜃 and the linear speed of the cylinder, the following geometrical relations are made out as presented in (2) to (4) [20].

In kinematics, (5) presents the (𝑥CGcy) position in relation to 𝜃 and (6) shows the (VCGcy) linear velocity, which is further used to calculate the kinetic energy.

2.1. The Modelling of the Governing Equations of Motion. In this subsection, the mathematical equations that emulate the motion of the nonlinear connecting-rod-crank system with horizontal geometry and mono cylinder motor are developed using the energy method of Lagrange, where its function is represented by  

The Lagrangian is expressed in terms of the generalized coordinate 𝜃, the masses concentrated in the CG of each link,the geometry, and the rigid materials.Therefore, the equation of Euler-Lagrange is described as (8), where 𝑄𝜃 represents thenonconservative forces.

In this way, the resulting force 𝐹R composed of the nonconservative forces in function of the virtual displacement can be obtained by the virtual work of

The nonconservative force 𝐹cae is considered as a damping force, considering the whole damping forces of the system,which are 𝐹cy; 𝐹cyco; 𝐹cocr, and 𝐹cr (Figure 1). Moreover, the compressed air engine is considered as a viscous-damping,that is, a relation proportional to the angular velocity of the wheel of the system multiplied by a constant, as given by

The 𝐹a force is another dissipative force of the system. It is provided due to two dissipative forces of the system, one is From the own motion of the mechanism, and the other is from the transmission of the wheel motion to the bicycle, which dissipate energy of the system. The connecting-rod crank system is excited by the air force of the pneumatic pressure via a control signal to a pressure regulator valve. Therefore,the air force 𝐹a is proportional to the area of the cylinder and air pressure 𝑃a determined by the control SDRE, given by

where 𝑃a is the nonconservative pressure which generates the force 𝐹a of the air that excites the movement of the system with 𝑑 being the cylinder diameter of the compressed air engine.

The friction force 𝐹f depends on the pressure, tire section,and wheel length, as much as the type of the contact surface.The considered value to this resistance force of the bearing is approximately 9Nto big section tires and 2.3Nto racing tires of thin section, low friction, and high performance [21]. 𝐹f is proportional to the normal force (𝐹N), whose component is Generated by the angle of in clination of the ground (𝛽) applied to themass of the bike (𝑚b) and the mass of the cyclist (𝑚c) as function of gravity (𝑔). Therefore, (13) depends on the rolling coefficient (𝑐r) in function of the masses which varies with the following types of parameters: bearings, ground, tire, and hoop, as well as tire pressure [22].

In this way, all forces relation of (8) of Euler-Lagrange to the calculus of resulting work of nonconservative forces was defined. However, to determine the full equations of motion, it is needed to estimate the conservative forces and energies that balance the mechanism movement. Thus, for the analysis of the equilibrium of the conservative forces, the kinetic energy 𝑇 for translational movements and rotation of