Time histories of the displacement of the rotating shaft midpoint are presented in Figures 4–7, where only the first mode is excited and . In Figures 4 and 5, only plane has been excited by the initial displacement, i.e. . But, there is vibration in plane (Figure 5). The reason is due to the presence of the gyroscopic effect in Eq. (14). When one plane is excited, the other plane, due to the gyroscopic effect, oscillates too. The data in Figures 6 and 7 is the same as Figure 4, except that ginsenoside rg3 the rotary inertia and, consequently, the gyroscopic term, are neglected . Figure 6 is similar to Figure 4. But, Figure 7 shows that plane does not oscillate. To validate the results of the perturbation solution, the numerical method has been used in Figures 4–7. Both methods agree very well.
In Figure 8, forward and backward amplitudes ( and ) are plotted versus time with earlier data. Because of the damping, amplitudes decay. In addition, both forward and backward amplitudes have close values and curves are coincident, approximately.
Eq. (31) shows that the natural frequencies have two parts: the first, constant parts and , which are linear natural frequencies, the second, the parts that depend on the amplitudes, and consequently on time. Here, nonlinear natural frequencies are defined as:
The nonlinear natural frequencies are functions of and , as well as time. So, to investigate the effects of parameters, and , on the nonlinear natural frequencies, one may set the time on a specified value (e.g. ) and then examine the effects of these parameters. Here, forward and backward nonlinear natural frequencies are defined as: where and are determined from the initial conditions: . FNNF and BNNF, as a function of rotating speed , are shown in Figures 9 and 10 for the first two modes . In both modes, FNNF curves increase and BNNF curves decrease as the rotating speed increases, for all values of . For the same data, the values of FNNF are larger than BNNF. Also, the values of FNNF and BNNF, for the first mode, are smaller than those for the second mode. It is seen from the figures that due to the gyroscopic effects, if rotating speed changes, the change rates of FNNF and BNNF are higher for the larger values of .
In Figures 11 and 12, FNNF and BNNF are plotted versus external damping, “”, for different values of . Only the first two modes are considered. It is observed from the figures that the general characteristics of the curves are similar, and the rate changes of FNNF and BNNF, with respect to external damping, are small. For small “”, FNNF and BNNF have small values and have sharp changes as “” changes. For large values of “”, FNNF and BNNF change slowly.
Comparison between nonlinear natural frequency of a rotating shaft with stretching nonlinearity and a rotating shaft with nonlinearities in curvature and inertia
In this section, we compare the effects of stretching and in-extensionality on the free vibration of a rotating shaft. Hosseini and Khadem  investigated the free vibration of a rotating shaft with nonlinear curvature and inertia. To derive the equations of motion, community succession used the in-extensionality assumption. Since one of the shaft supports was free in an axial direction, this assumption was applicable. In this paper, both supports are fixed in an axial direction and a stretching assumption has been used. We compare the effects of these two types of nonlinearity on the free vibration of a rotating shaft.
From Eq. (42) of Hosseini and Khadem , the analytical solution for the free vibration of an in-extensional rotating shaft is: where: All parameters and variables used in the above equations were defined in Hosseini and Khadem . Here, the analysis is carried out for a forward nonlinear natural frequency; the backward case is similar. To find a simple approximation for the nonlinear natural frequency, the quantity, , is expanded in a Taylor series about up to the first order, as: So, an approximation for the nonlinear natural frequency becomes: where (Hosseini and Khadem ): In addition, we have: which is due to the nonlinear inertia effect and: which is due to the nonlinear curvature effect.