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Fig. 4. Schematic view of the section with the equivalent heat source. where Q,(t) represents the nth impulse of the equivalent heat input model in Fig. 3, N is the total number of impulses to bs applied to the shaft, Q is the amplitude as introduced before, 6 is an operator for Dirac function as shown in Eq. (5), r, @ and. represent respectively each axis in cylindrical coordinates, zt, indicates the time by which the shaft begins its nth rotatior based on assumption (iii), which means Q,(t) is applied during the nth rotation.

Figure 4 Schematic view of the section with the equivalent heat source. where Q,(t) represents the nth impulse of the equivalent heat input model in Fig. 3, N is the total number of impulses to bs applied to the shaft, Q is the amplitude as introduced before, 6 is an operator for Dirac function as shown in Eq. (5), r, @ and. represent respectively each axis in cylindrical coordinates, zt, indicates the time by which the shaft begins its nth rotatior based on assumption (iii), which means Q,(t) is applied during the nth rotation.

Related Figures (19)

Fig. 1. Sketch for the rotor shaft and the coordinates. In order to avoid the rigid body motion via the use of the inertia frame, the rotating frame named O — éyz is used instead to deduce the governing equations. Before introducing the dynamic equations, the variables of key motions are defined first. The transverse motions in two lateral directions are represented by x(z,t) and y,(z,t) where the expression with brackets means that the motions vary with the axial position and time.
Fig. 1. Sketch for the rotor shaft and the coordinates. In order to avoid the rigid body motion via the use of the inertia frame, the rotating frame named O — éyz is used instead to deduce the governing equations. Before introducing the dynamic equations, the variables of key motions are defined first. The transverse motions in two lateral directions are represented by x(z,t) and y,(z,t) where the expression with brackets means that the motions vary with the axial position and time.
This part of study is the base for obtaining the final temperature distribution in a generic thermal condition. According to assumption ii, the boundary of asymmetric heat flux, shown in Fig. 2, is equivalent to several point heat sources. As mentioned before, in the presence of a broader A,, the temperature solution can be obtained by the method of superposition with the results from several heat sources. Therefore, the heat conduction model with a single heat source is tackled first. Assume that the axial location for A; is around z=Zo and the point with the highest temperature on the according cross-section Sp is Tp, located at (ro,49) as shown in Fig. 4. Therefore, the location for the equivalent heat source is (To,00,Z0). The parameter My in Fig. 4 is the equivalent thermal moment induced by the asymmetric TGs. Thus, the mathematical model of Q(t) is of the following form: Fig. 3. Equivalent impulse heat input.
This part of study is the base for obtaining the final temperature distribution in a generic thermal condition. According to assumption ii, the boundary of asymmetric heat flux, shown in Fig. 2, is equivalent to several point heat sources. As mentioned before, in the presence of a broader A,, the temperature solution can be obtained by the method of superposition with the results from several heat sources. Therefore, the heat conduction model with a single heat source is tackled first. Assume that the axial location for A; is around z=Zo and the point with the highest temperature on the according cross-section Sp is Tp, located at (ro,49) as shown in Fig. 4. Therefore, the location for the equivalent heat source is (To,00,Z0). The parameter My in Fig. 4 is the equivalent thermal moment induced by the asymmetric TGs. Thus, the mathematical model of Q(t) is of the following form: Fig. 3. Equivalent impulse heat input.
Fig. 5. Eigenvalues of #,, with convection coefficient H=(10,190) [W/(m? K)] for the first Bessel function J,(,,r). (a) 2D temperature converted into 1D; (I linearization of nonlinear temperature. The two sub-plots in Fig. 5 give curves of £,, for different convection coefficients respectively when the first Bessel function takes zeroth and fifth order. In this figure, it is illustrated well that a higher order number of m leads to larger eigenvalues of ,, for both orders of the first Bessel function. Besides, it is noticed that the higher the heat convection
Fig. 5. Eigenvalues of #,, with convection coefficient H=(10,190) [W/(m? K)] for the first Bessel function J,(,,r). (a) 2D temperature converted into 1D; (I linearization of nonlinear temperature. The two sub-plots in Fig. 5 give curves of £,, for different convection coefficients respectively when the first Bessel function takes zeroth and fifth order. In this figure, it is illustrated well that a higher order number of m leads to larger eigenvalues of ,, for both orders of the first Bessel function. Besides, it is noticed that the higher the heat convection
Fig. 6. Temperature rise distribution in rotor shaft for H=0. (a) Temperature rise in longitudinal section; (b) in cross section at z=0.50L; (c) in cross sectior at z=0.53L. coefficient H takes, the smaller £,, becomes. However, this influence of H is almost negligible when m is higher than 3 since the biggest relative difference of /,, when m= 4 is less than 1 percent. Hence, it is reasonable to conclude that the major difference of the temperature rise is contributed by the first three orders of £m. As a matter of fact, large values of fm will result in low temperatures with other parameters remained the same in Eq. (11). Hence, it is reasonably concluded that the higher value H takes, the lower the temperature will be. This conclusion matches well with the heat conduction theory. Based on the conclusion above, the distribution of the temperature rise is further calculated for H=O, which indicates the possible biggest temperature rise under the equivalent heat input introduced before. Fig. 6 illustrates how temperature rises longitudinally and in the cross-sections under a certain heat source at the location of (r=R, 9=0, z=L/2) in the cylindrical coordinate system.
Fig. 6. Temperature rise distribution in rotor shaft for H=0. (a) Temperature rise in longitudinal section; (b) in cross section at z=0.50L; (c) in cross sectior at z=0.53L. coefficient H takes, the smaller £,, becomes. However, this influence of H is almost negligible when m is higher than 3 since the biggest relative difference of /,, when m= 4 is less than 1 percent. Hence, it is reasonable to conclude that the major difference of the temperature rise is contributed by the first three orders of £m. As a matter of fact, large values of fm will result in low temperatures with other parameters remained the same in Eq. (11). Hence, it is reasonably concluded that the higher value H takes, the lower the temperature will be. This conclusion matches well with the heat conduction theory. Based on the conclusion above, the distribution of the temperature rise is further calculated for H=O, which indicates the possible biggest temperature rise under the equivalent heat input introduced before. Fig. 6 illustrates how temperature rises longitudinally and in the cross-sections under a certain heat source at the location of (r=R, 9=0, z=L/2) in the cylindrical coordinate system.
Fig. 7. Terms of thermal moment series varying with different values of v and m. Another point can be made by combing Figs. 7 and 5 together with Eq. (15). It has been revealed that m and v can be of quite low values in Fig. 7 and that the eigenvalues ,, are well below 120 [m~'] even for a relatively high order, e.g. 5th order, of the Bessel function. This situation makes ¢m, as shown in Eq. (17), of very small value for generic rotor materials. Therefore, the time constant for thermal process, expressed as 1/e,,, quite big compared to the one for the mechanical process. Therefore the resulted thermal process will vary quite slowly.
Fig. 7. Terms of thermal moment series varying with different values of v and m. Another point can be made by combing Figs. 7 and 5 together with Eq. (15). It has been revealed that m and v can be of quite low values in Fig. 7 and that the eigenvalues ,, are well below 120 [m~'] even for a relatively high order, e.g. 5th order, of the Bessel function. This situation makes ¢m, as shown in Eq. (17), of very small value for generic rotor materials. Therefore, the time constant for thermal process, expressed as 1/e,,, quite big compared to the one for the mechanical process. Therefore the resulted thermal process will vary quite slowly.
In Eq. (28), the inertial terms contributed by the dynamic response only are /°{Xzp) and I,(yzp). They are functions related with eigenfunctions @,(z) for shaft vibration. The orthogonality of &,(z) makes the problem possible to be further simplified. By multiplying the both hands of Eq. (28) with #,(z), where x € (1,2,3,...), and then integrating along the shaft length, a set of differential equations are finally obtained as
In Eq. (28), the inertial terms contributed by the dynamic response only are /°{Xzp) and I,(yzp). They are functions related with eigenfunctions @,(z) for shaft vibration. The orthogonality of &,(z) makes the problem possible to be further simplified. By multiplying the both hands of Eq. (28) with #,(z), where x € (1,2,3,...), and then integrating along the shaft length, a set of differential equations are finally obtained as
In Eq. (37), X:.(0) and Y,,(0) represent the initial displacement while their first derivatives, denoted with superscript of prime (‘), are for the velocities. Seer | we Se aes eS Pee Since the homogeneous solutions have been obtained, a subsequent effort will be put on deriving the particular solutions for Eq. (31). The particular solutions are assumed of the form given in Eq. (39) according to the expressions on the right hands of Eq. (31)
In Eq. (37), X:.(0) and Y,,(0) represent the initial displacement while their first derivatives, denoted with superscript of prime (‘), are for the velocities. Seer | we Se aes eS Pee Since the homogeneous solutions have been obtained, a subsequent effort will be put on deriving the particular solutions for Eq. (31). The particular solutions are assumed of the form given in Eq. (39) according to the expressions on the right hands of Eq. (31)
Parameters setup for cases with different thermal boundaries. Fig. 8. Thermal moment influenced by the convection coefficient H. Table 1
Parameters setup for cases with different thermal boundaries. Fig. 8. Thermal moment influenced by the convection coefficient H. Table 1
Fig. 10. Comparison of the new index and TGs. (a) Different indexes of TGs varying with heat convection coefficient; (b) thermal moment varying with the new index of TGs. Fig. 9. Temperature distributions over section S; with the same heat input. (a) H = 10 [W/(m? k)]; (b) H = 200 [W/(m? k)]
Fig. 10. Comparison of the new index and TGs. (a) Different indexes of TGs varying with heat convection coefficient; (b) thermal moment varying with the new index of TGs. Fig. 9. Temperature distributions over section S; with the same heat input. (a) H = 10 [W/(m? k)]; (b) H = 200 [W/(m? k)]
Fig. 11. Dimensionless vibration, contributed by C,(r,9) only, varying with heating location rz and axial location z. (a) k= 1; (b) ex=2; (c) x=3. The amplitude contribution due to B is ascribed to the shaft geometries and the thermal power of the equivalent heat source, as shown in Eq. (33). The geometries, e.g. the radius R and length L, not only decide the value of B but also have an impact on the «th critical speed which will further influence the values of #, C, and A,. Moreover, the frequency can
Fig. 11. Dimensionless vibration, contributed by C,(r,9) only, varying with heating location rz and axial location z. (a) k= 1; (b) ex=2; (c) x=3. The amplitude contribution due to B is ascribed to the shaft geometries and the thermal power of the equivalent heat source, as shown in Eq. (33). The geometries, e.g. the radius R and length L, not only decide the value of B but also have an impact on the «th critical speed which will further influence the values of #, C, and A,. Moreover, the frequency can
Fig. 13. Thermal vibrations for cases specified in Table 2, with the heating location of r,,=0.5 and vibration point of r,=0.5. Fig. 12. Rotational speeds close to the first two resonance frequencies. (Q; — ith rotational speed, ; — jth resonance speed of the according shaft).
Fig. 13. Thermal vibrations for cases specified in Table 2, with the heating location of r,,=0.5 and vibration point of r,=0.5. Fig. 12. Rotational speeds close to the first two resonance frequencies. (Q; — ith rotational speed, ; — jth resonance speed of the according shaft).
Configurations of geometries and rotational speeds. Table 2
Configurations of geometries and rotational speeds. Table 2
Fig. 14. Thermal vibrations for cases specified in Table 2, with the heating location of r,,=0.25 and vibration point of r,=0.25. It was described in Refs. [29,35,40-—43] that the asymmetric heat input may present in more revolutions. In most cases, the TGs and vibration amplitudes will grow larger due to the close loop with positive feedback. A conservative condition is assumed that heat-input is of the same amount during every revolution. With this assumption, the mathematical model of heat source in Fig. 3 will be adopted, that is the additional heat input is constant for each rotation. The resulted vibrations in inertial frame due to the periodically impulse heat source are illustrated in Fig. 15. The amplitude of the heat input Q is
Fig. 14. Thermal vibrations for cases specified in Table 2, with the heating location of r,,=0.25 and vibration point of r,=0.25. It was described in Refs. [29,35,40-—43] that the asymmetric heat input may present in more revolutions. In most cases, the TGs and vibration amplitudes will grow larger due to the close loop with positive feedback. A conservative condition is assumed that heat-input is of the same amount during every revolution. With this assumption, the mathematical model of heat source in Fig. 3 will be adopted, that is the additional heat input is constant for each rotation. The resulted vibrations in inertial frame due to the periodically impulse heat source are illustrated in Fig. 15. The amplitude of the heat input Q is
Fig. 16. Development of thermal vibrations. (a) Influenced by convection coefficient H [W/(m? K)]; (b) influenced by heat conductivity k [W/(m K)] Fig. 15. Thermal vibrations increase due to periodical heat impulse.
Fig. 16. Development of thermal vibrations. (a) Influenced by convection coefficient H [W/(m? K)]; (b) influenced by heat conductivity k [W/(m K)] Fig. 15. Thermal vibrations increase due to periodical heat impulse.
Fig. 17. Quantitative analysis of thermal vibrations influenced by convection coefficient H and heat conductivity k. (a) By convection coefficient H; (b) by heat conductivity k.
Fig. 17. Quantitative analysis of thermal vibrations influenced by convection coefficient H and heat conductivity k. (a) By convection coefficient H; (b) by heat conductivity k.
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