2.5 A Jeffcott Rotor Model with an Offset Disc

Figure 2.20(a) show a more general case of the Jeffcott rotor when the rigid disc is placed with some offset from the mid-span. With a and b locate the position of the disc in a shaft of length l. The spin speed of the shaft is considered as constant. For such rotors apart from two transverse displacements of the center of disc, i.e., x and y, the tilting of disc about the x and y-axis, i.e., φ_x andφ_y, also occurs; and it makes the rotor system as a four DOFs. For the present analysis, the rotary inertia of the disc is considered, however, the effect of the gyroscopic moment has been neglected. In Fig. 2.20(b) points C and G represent the geometrical center and the center of gravity of disc, respectively. The angle, Φ, represent the phase between the force and the response.

Jeffcott rotor with an offset disc

From Figure 2.20(b), we can have the following relations for the eccentricity

(2.66)

where e_x and e_y are components of the eccentricity, e, in the x and y -directions, respectively (in fact these components of eccentricity are in the plane of disc that is inclined).

From Figure 2.20(c) equations of motion of the disc in the y- and φ_x directions can be written as

(2.67)

and

(2.68)

where m is the disc mass, I_d is the diametral mass moment of inertia about the x-axis, f_y is the reaction force and M_yz is the reaction moment. It should be noted that the moment is taken about the point G. From above equations it can be observed that equations are non-linearly coupled with the angular (titling) component of displacement, φ_x.

(2.69)

and

(2.70)

where I_d is the diametral mass moment of inertia about the y-axis, f_x is the reaction force and M_zx is the reaction moment. Equations (2.69)  and  (2.70) are also non-linearly coupled with the angular component of displacement, φ_y. However, two transverse planes (i.e. y-z and z-x) motions are not coupled and that will allow two-plane motion to analyze independent of each other, i.e., set of equations (2.67) and (2.68) and equations (2.69) and (2.70) can be solved independent of each other.