Let us examine circular motion of two equal forces from 0 to π (Pic. 1). One of the forces is always tangent to the rotation circle; the other one is directed upright or downright. Some intermediate transformations are omitted due to their simplicity.
Aa – work of force acting tangentially to the circle:
Aa = ωMa
Ma = FaR
Aa = πRFa (1.1)
Where: ω — angle of rotation from 0 to π
Ma — torque from the tangential force
R — circle radius
Fa — tangential force
Ar — work of force acting vertically :
Mr = Frsinφ
Ar = 2RFr (1.2)
Where:Mr — torque from the vertical force
Fr — vertical force
φ — rotation angle
It follows from this transformation that if the forces are equal, torques created by these forces are not equal.
I consider these equations as this is the way forces act in the rotor piston planetary transmission mechanism. Fluid pressure through transformation in the planetary gear is always tangent to the rotation circle, and the buoyancy force or gravity in fluid is always directed vertically. From now on, all the transformations will be performed regarding the buoyancy force acting in fluid. The difference between gravity action and buoyancy force is only in the opposite vectors direction of these forces.
Let us use a well-known statement to determine forces: gravity action depends on height of motion and does not depend on the motion path. Let us examine uniform motion of a cylinder with a piston in a fluid with height equal to the diameter of the transmission mechanism piston rotor (Pic. 2). In this case, fluid pressure will be applied to the piston, and the buoyancy force caused by the air block in the cylinder will be applied on the cylinder. At the same time it is supposed that the reverse end of the cylinder is not filled with fluid and air can escape the cylinder at normal pressure. Motion is vertical and downward, so in this case pressure will increase uniformly, and action of the buoyancy force will decrease uniformly as the piston moves. In this case, piston functioning (Aa) is described by the following equations:
Aa = PсрL
Pср = RSpg
Aa = πR²Spg (2.1)
Where: Pср — total pressure at the piston
L= πR – piston length
S — piston area
p — fluid density
g — ускорение свободного падения
The buoyancy force (Fr) and work of the buoyancy force (Ar) are described by the following equations:
Fr = LSpg
Ar = πR²Spg (2.2)
Therefore, work done by the piston inside the cylinder and work of the buoyancy force acting on the cylinder along the height of the transmission mechanism are equal.
The values of the force acting on the piston (Fa) and the buoyancy force acting on the cylinder (Fr) are found by the formula:
Fa = Fr = A/h = πR²Spg/2R = 1/2πRSpg (3.1)
Where: A = Aa = Ar
h = 2R — motion height
By inserting the obtained values into formulas 1.1 and 1.2, respectively, we obtain the following results for the work of the fluid pressure acting on the piston circumferentially:
Aa = πRFa = 1/2π²R²Spg (4.1)
For the work of the buoyancy force acting on the piston circumferentially:
Ar = 2RFr = πR²Spg (4.2)
The difference between actions of these forces is calculated by the formula:
dA = Aa-Ar = 0,57πR²Spg (4.3)
This is the formula for motion of the rotary piston transmission mechanism. This inequality can be the basis for designing engines that transform the pressure difference in fluid acting along the height of the transmission mechanism piston rotor.
In addition to this, I calculated one more method to create a difference between actions of the fluid pressure and the buoyancy force. It involves changing the radius of the buoyancy force action vector by means of changing geometry parameters of transmission mechanism piston rotor. But this transformation will be given later.