Fundamentals of Heat Engines. Jamil Ghojel
the force is variable and moving along a straight line,
Newton' Second Law for a particle can be written as
For an incremental change in distance, ds = vdt; hence
Finally,
(1.3)
The work done by a force is equal to the change in kinetic energy. This equation is the simplest form of the conservation of energy equation.
1.1.4 Circular Motion
Rotary motion is the most convenient means for transferring mechanical power in almost all driving and driven machinery. This is particularly so in heat engine practice where thermal energy is converted to mechanical work, which is then transferred via rotating shaft to a driven machinery (electrical generator, propeller, wheels of a vehicle, pump, etc.). Consider the non‐uniform circular motion shown in Figure 1.1, in which particle P at angular position θ has linear tangential velocity v and angular velocity ω.
Figure 1.1 Non‐uniform circular motion in Cartesian coordinates: (a) initial position and velocity; (b) first‐order components of resultant acceleration; (c) second‐order components of resultant acceleration.
The components
(1.4a)
(1.4b)
The accelerations in the same directions are
(1.5)
where
(1.6)
The first‐order components of the resultant acceleration in the radial direction towards 0 is
Since ω = v/r,
(1.7a)
Radial acceleration ar is directed opposite to OP in Figure 1.1b
The second‐order components of the resultant acceleration in the tangential direction is
Since the angular acceleration
(1.7b)
Tangential acceleration at is directed perpendicular to OP in Figure 1.1c.
The resultant acceleration is
(1.8)
1.1.4.1 Uniform Circular Motion of a Particle
In the uniform circular motion,
Equations 1.4a, 1.7b, and 1.8 for velocity and acceleration become:
(1.9)
(1.10)
These equations apply to any point on the outer surface of a machinery shaft rotating at constant angular velocity, such as reciprocating and gas turbines engines.
1.1.5 Rotating Rigid‐Body Kinetics
The motion of a particle can be fully described by its location at any instant. For a rigid body, on the other hand, knowledge of both the location and orientation of the body at any instant is required for full description of its motion.
The motion of the body about a fixed axis can be determined from the motion of a line in a plane of motion that is perpendicular to the axis of rotation (Figure 1.2). The angular position, displacement, velocity, and acceleration are, respectively, θ, dθ,
Figure 1.2 Rigid‐body rotational motion.
The tangential and radial components of the acceleration at P and the resultant acceleration are, respectively,