An ordinary incremental encoder that generates a square pulse per pre-determined rotation angle of the shaft is used in this study. As explained above, the incremental encoder generates a specific number (pulses per revolution: PPR) of pulses per revolution. In case of no rotational vibration or disturbance, the output of the encoder would be in the shape of that in Fig. 2.

Fig. 2 
Example of the output signal from an incremental encoder

If the numbers of pulses from the encoder is N, the rotation angle (θ) of the shaft can be calculated with Eq. (1) by dividing 2πN with PPR.

The absolute encoder generates the voltage output proportional to the rotational angle of the shaft from a specific reference point. An example voltage output from the absolute encoder is explained in Fig. 3 where the output voltage increases linearly from V_min to V_max according to the rotation angle in the range from 0 to θ_max. In this plot the disturbances and rotational vibration of the shaft are assumed to be negligible. And the rotation angle of the shaft at a specific time slot can be obtained by measuring the output voltage at the given instant. Also, the number of rotations of the shaft can be calculated with the number of triangular pulses in the given time interval.

Fig. 3 
Example of the output signal from an absolute encoder.

In this section, the process for the rotational vibration measurement of the incremental encoder is provided. In case of the shaft rotating with a constant angular velocity, the incremental encoder will generate pulses in a constant time interval (period) as shown in Fig. 2. But in the case of variable angular velocity due to the rotational vibration, the encoder will generate a series of pulses with variable periods. So, it is possible to find out the variation of angular velocity of the shaft by checking the periods of the individual pulses.

The angular velocity of the shaft corresponding to the i^th pulse in Fig. 4 can be calculated as Eq. (2), following equation.

Fig. 4 
Pulse output of incremental encoder installed at shaft with and without torsional vibration

Accordingly, the angular velocity variation of the shaft can be obtained by calculating angular velocity corresponding to each pulse in Fig. 4 using Eq. (2).

In this section, the principle of the measurement of rotational vibration of the shaft using an ordinary absolute encoder will be explained. In the first step of the process, the analogue signal of is sampled with a specific sampling rate to obtain a series of digital signal as shown in Fig. 5.

Fig. 5 
Example of the digital sampling of an absolute encoder output signal

In the case of the constant angular velocity, the slopes of individual digital data will be constant since the slopes are proportional to the angular velocity at the time slot. On the other hand, the slopes for the individual points will have variation with time when the shaft have rotational or torsional vibration. So, the rotational vibration of the shaft can be calculated based on the slopes of individual data points in Fig. 6.

Fig. 6 
Pulse output of absolute encoder installed at shaft with and without torsional vibration

Figure 6 explains arbitrary 3 points on the sampled voltage data. Rotation angle (θ_i) and angular displacement (∆θ_i) for the i^th time slot can be obtained as Eq. (3a) and Eq. (3b) following equations.

Angular velocity of the i^th time slot can be obtained by dividing ∆θ_i with time interval (∆t_i=t_i+1-t_i). In addition, since the voltage output of the encoder is sampled at a constant time internal, ∆t_i is constant for all the data points as ∆t. Consequently, angular velocities for each point can be calculated as Eq. (4).

By repeating this process for a specific time interval, the rotational velocity variations or rotational vibration of the shaft for the time interval can be obtained.

In this section, rotational vibration measurement results are simulated using computer simulations. The rotational speed of the shaft in these simulations is assumed to be as Eq. (5) with the first and second order rotational vibration along with nominal speed of rotation.

In this equation Ω is the nominal speed of shaft and Ω₁ and Ω₂ are the magnitudes of the first and second order rotational vibration of the shaft. The rotation angle of the shaft can be obtained as Eq. (6) by integrating the speed of shaft with respect to time.

In this simulation, the imaginary shaft is assumed to be rotating with the angular velocity given in Eq. (5) along with the parameters given in Table 1.

As explained above, the incremental encoder generates a square pulse as θ increases in pre-determined amount 2π/PPR and the absolute encoder generates a triangular pulse as θ increases by 2π rad. Also, the output signals from the encoders are sampled by a data recorder for the analyses in the computers. The simulation parameters for the output signals and data recorders are given in Table 2.

In addition, 0.5 % random error is added to the output signals from the encoders to consider the imperfectness of the encoders.

The simulation results for the two encoders under the conditions described in Table 1 and Table 2 are compared with each other and with rotational vibration which is introduced intentionally. The time data for the variations of angular velocities are compared in Fig. 7(a) and Fig. 7(b).

Fig. 7 
Results of simulation to verify the measurement process (key: ◦ ◦ ◦ Theoretical, - - Simulation)

As one can see in the figures, ∆ω_i from the incremental and absolute encoder and those for the intentionally introduced are quite well match each other. But, the effect of the random error is more dominant in the result of the absolute encoder. In addition, Fourier transform results of above data are explained in Fig. 8(a) and Fig. 8(b).

Fig. 8 
Results of frequency analysis on the result of the simulation

As shown in the figure, the first (2.5 Hz) and second order (5.0 Hz) rotational vibrations which are intentionally introduced in the simulations are clearly identified in the simulation results for the two processes. Also, the magnitudes of the vibration of individual components obtained from two processes are listed in Table 3 along with theoretical values.

As shown in the table, the magnitudes of the first and second order vibrations are almost same as those of intentional rotational vibrations. Consequently, based on the results in Fig. 7 and Table 3, it can be concluded that the proposed processes have enough accuracies in the measurement of the rotational vibration of the shaft.