Jamil Ghojel - Fundamentals of Heat Engines

(1.5)

where are the first‐ and second‐order acceleration components in the x direction ( Figure 1.1b,c).

(1.6)

are the first‐and second‐order acceleration components in the y direction.

The first‐order components of the resultant acceleration in the radial direction towards 0 is

Since ω = v / r ,

(1.7a)

Radial acceleration a ris 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 a tis directed perpendicular to OP in Figure 1.1c.

The resultant acceleration is

(1.8)

In the uniform circular motion, , .

Equations 1.4a, 1.7b, and 1.8for 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.

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 1.2Rigid‐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 = a t dm and the moment required to accelerate the same mass is dM = r a t 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 = ∫ r 2 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 I G. 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 and radius of gyration k such that I = mk 2, from which . If I is in units of kg . m 2, k will be in metres. The radius of gyration k can be regarded as the distance from the axis to a point in the plane of motion where the total mass must be concentrated to produce the same moment of inertia as does the actual distributed mass of the body, i.e.