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easier derivation [11]. In any case, the equations of motion can be expressed as where M, C,and K are the inertia, damping, and stiffness matrices, respectively. The vector Q includes the generalized forces and q are the generalized coordinates.
As Fig. 2 shows, in addition to the generalized coordinates from the continuous portion, there are two additional generalized coordinates, one to describe the carriage position uc(t).and another to describe the rotor angular position θm(t). Therefore, the total system order is N = NU +N,with NU =Nu +1 and N.= Nθ.+1.
3 Modal formulation
The solution of the eigenvalue problem [[K].ωj 2[M]]{φj}=[0] relatedtoEq. 9 gives N eigensolutions, each one featuring a natural frequency ωj and a normal mode {φj}. Applying the modal transformation {q}=[φ]{η}, the displacement field presented in Eq. 1 can be expressed as
To compute the axial and angular components of the mode functions, it is necessary to distinguish, in each mode vector {φj}, the elements corresponding to each
generalized coordinate. In this sense, if the generalized coordinates vector q has the following arrangement: each mode vector {φj}can be written as
4 Analysis of the mode functions for different transmission ratios
The mode functions were obtained according to Eqs. 13 and 14 in which the mode vectors correspond to the particular carriage position xc =0.5 L, and a total
moving mass of mc =30 kg. The number of terms in the Ritz series were Nu =Nθ.=4 obtaining a suitable compromise between model complexity and estimation
error [14]. From Eq. 6, it can be seen that the transmission ratio has a direct influence on the degree of coupling between the axial and torsional deformation fields. Therefore, two particular values of transmission ratio (screw lead) are selected for the following study.
Figure 3 shows the axial and angular components of the mode functions for the first four modes for a transmission ratio of 10 mm/rev. Also, in Fig. 3a, the displacement of the carriage was described in each mode as a point value φuc j plotted at x =xc. Similarly, in Fig. 3b, the displacement of the motor rotor was
described by φθm j plotted at xm =.0.2 L. Note that the zero coordinate is located at the rigid bearing; therefore, the middle of the motor is located at negative coordinate values.
Table 2 compares the natural frequencies for the coupled and decoupled solutions for a transmission ratio of 10 mm/rev. The classification, either as torsional or axial, of the modes from the coupled model is done based on the best agreement between the frequencies of both models. The last column shows the error in the natural frequency values from decoupled system solutions; in this sense, a greater error means greater axial–torsional coupling. Similarly, Table 3 shows the results for a screw lead of 32 mm/rev.
5 Mode frequency variation for different operating conditions
The mode shapes and their frequencies are sensitive not only to design parameters, as was shown for different transmission ratios, but also to operation conditions,
like carriage position and total moving mass. The results presented in the previous sections are for a carriage position of xc = 0.5 L and a total mass of mc = 30 kg. Following, the natural frequencies were studied for different carriage positions, different moving masses, and the combination of them, all for the two proposed transmission ratios. In each case, the natural frequency deviation for each mode was computed according to In this way, the rows of Table 4 show the relative deviation of the first natural frequency for the three operation conditions, evaluated at the two transmission
ratios. Similarly, Tables 5 and 6 show the results for the second and third vibration modes, respectively.
According to results in Table 4, the first mode shows to be very sensitive to the carriage position for both transmission ratios. Also it is very sensitive with respect