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(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,
Referring to Figure 1.2, the force required to accelerate mass dm at P is dF = atdm and the moment required to accelerate the same mass is dM = r at dm.
The resultant moment needed to accelerate the total mass of the rotating rigid body is
For a constant angular acceleration,
(1.11)
where I = ∫ r2 dm is the moment of inertia of the whole mass of the rigid body rotating about an axis passing through 0. Equation (1.11) indicates that if the body has rotational motion and is being acted upon by moment M, its moment of inertia I is a measure of the resistance of the body to angular acceleration α. In linear motion, the mass m is a measure of the resistance of the body to linear acceleration a when acted upon by force F.
In planar kinetics, the axis chosen for analysis passes through the centre of mass G of the body and is always perpendicular to the plane of motion. The moment of inertia about this axis is IG. The moment of inertia about an axis that is parallel to the axis passing through the centre of mass is determined using the parallel axis theorem
(1.12)
where d is the perpendicular distance between the parallel axes.
For a rigid body of complex shape, the moment of inertia can be defined in terms of the mass m
