Dynamics analysis and experiment of banana-shaped vibrating-dewatering screen

4.1. Dynamic analysis based on MATLAB/Simulink

In order to further analyze the dynamic characteristics of the BVDS, the simulation model was established based on the MATLAB/Simulink simulation module, and the specific parameters are shown in Table 1. The simulation models of $X$ and $Y$ directions of the BVDS are shown in Fig. 4. Use the Sine Wave in the Sources module library as the sine excitation signal. The Scope oscilloscope in receiver module library is used to output displacement, velocity and acceleration signal curves.

Fig. 4Simulation model of the BVDS based on MATLAB/Simulink

According to Table 1, the corresponding sinusoidal excitation was applied, the exciting angular velocity was set to 102.05 rad∙s^-1 ($n=$975 rpm), and the simulation time was set to 10 s. The simulation result of displacement time domain response of the BVDS is shown in Fig. 5, and the Lissajous diagram of displacement during the stable period is shown in Fig. 6.

As can be seen from the simulation results in Fig. 5, when the BVDS is running at a set operating frequency, the screen body enters a state of disturbed vibration in the first 0-7 s, and the displacement amplitudes of the screen body in the $X$ and $Y$ directions are relatively large. The maximum displacement amplitude in the $X$ direction is 5.36 mm, and the maximum displacement amplitude in the $Y$ direction is 5.12 mm. After 7 s, the screen body entered the stable operation stage. In the stable stage, the displacement amplitude of the screen body in the $X$ and $Y$ directions is about 3.45 mm. In addition, it can be found from the displacement curve that the phase difference of the screen body in the $X$ and $Y$ directions is approximately equal to 0°. As can be seen from the Lissajous diagram of displacement during the stable stage in Fig. 6, the displacement trajectory of the screen body is a diagonal line of 45° along the $X$ direction.

Fig. 5Time-responses of displacement by MATLAB/Simulink simulation

Fig. 6Lissajous diagram of displacement by MATLAB/Simulink simulation

Fig. 7Time-responses of velocity by MATLAB/Simulink simulation

Fig. 8Time-responses of acceleration by MATLAB/Simulink simulation

The time domain response simulation results of velocity and acceleration for the BVDS along the $X$ and $Y$ directions are shown in Fig. 7 and 8. Compared with the displacement curve, it takes less time for the velocity and acceleration waveforms in $X$ and $Y$ directions to reach the stable value. The velocity and acceleration waveforms in the stable stage are standard sine waves. It can also be seen from Fig. 7 that the velocity amplitudes of the screen body in the $X$ and $Y$ directions are stable at 350.72 and 352.50 mm∙s^-1, respectively. It can also be seen from Fig. 8 that the acceleration amplitudes of the screen body in the $X$ and $Y$ directions are 35.79 and 35.97 m∙s^-2, respectively. The phase difference of the velocity and acceleration curves of the screen body in the $X$ and $Y$ directions are approximately equal to 0°. The above MATLAB/Simulink simulation analysis results provide a solid theoretical basis for the development of the BVDS.

4.2. Vibration analysis based on the dynamic experiment

Fig. 9 shows the time-domain response curve of the screen body displacement of the BVDS in different time periods. The corresponding angular rotation velocity $\omega$ is 102.05 rad∙s^-1 ($n=$ 975 rpm). The time-domain response process of displacement can be divided into three different stages: starting stage (0-20 s), stable operation stage (40-60 s) and shutdown stage (85-105s), and the displacement signals of each stage are significantly different. It can be seen that the main vibration direction of the vibration system is $X$ and $Y$ axis. Relative to the $X$ and $Y$ axes, the amplitude in the $Z$ axis is small and negligible. According to the time-domain response results, the time-space characteristic curve of the displacement of the screen body position at different periods and the displacement Lissajous diagrams can be obtained. The results are shown in Fig. 10 and 11.

The displacement time domain response of the BVDS at starting stage (0-20 s) is shown in Fig. 9(a), with the angular velocity $\omega$ gradually increases from 0 to 102.05 rad∙s^-1. In this process, when the vibration passes near the natural frequency of the screen body, the amplitude of the screen body increases significantly, making the vibrating screen appear resonance. The maximum displacement amplitude of the screen body in the $X$ direction reaches 4.74 mm, and the maximum displacement amplitude in the $Y$ direction reaches 8.45 mm, which is significantly larger than the working amplitude. The BVDS has a large vibration and swing, and the displacement time-space characteristic curve and Lissajous diagrams are more complicated, as shown in Fig. 10(a) and Fig. 11(a).

After starting 20 s, the BVDS enters the steady stage. As can be seen from Fig. 9(b) and 9(c), the displacement waveform in $X$ and $Y$ directions are sine wave, and the vibration amplitude is stable as working amplitude. The average displacement amplitude of the screen body in the $X$ direction is 3.27 mm, and the average displacement amplitude in the $Y$ direction is 3.65 mm. The space-time curve is regular, as shown in Fig. 10(b), The displacement trajectory is a regular oblique line, as shown in Fig. 11(b).

Fig. 9Time domain response curves of displacement over different time periods

Fig. 10Time-space characteristic curve of displacement over different time periods

Fig. 11Lissajous diagrams of displacement over different time periods

Fig. 12Time-domain characteristic curves of velocity signals

The time-domain response of displacement in the shutdown stage is shown in Fig. 9(d). When the vibrating screen is in the shutdown stage for 85-105 s, the motor velocity and angular velocity gradually decrease. After passing through the resonance region again, the displacement in the $X$ direction increases slowly first and then decreases sharply. The displacement curve in the $Y$ direction changes several times, and the vibration in the $Y$ direction is more intense than that in the $X$ direction. The maximum displacement amplitude in $X$ direction and $Y$ direction reaches 4.28 mm and 13.05 mm respectively. Similar to the starting stage, the displacement time-space characteristic curve and the Lissajous diagrams in the shutdown stage are also relatively complex, as shown in Fig. 10(c) and 11(c).

In addition, the velocity and acceleration of the screen body are also important index of the vibration intensity. Fig. 12 shows the time-domain velocity variation curve of the screen body along the $X$ and $Y$ directions. The velocity of the screen body along the $Z$ axis is smaller than that in the $X$ and $Y$ directions. Similar to the time-domain response curve of displacement, the velocity curve of the screen body also has a phase difference of nearly 0°. The velocity of the screen body varies greatly during the starting and stopping stages, as shown in Fig. 12(a). The velocity amplitude of the screen body along the $Y$ direction is close to 454.3 mm∙s^-1 in the starting stage, and close to 720.4 mm∙s^-1 in the stopping stage, which is much higher than the velocity amplitude in the stable operation stage. As shown in Fig. 12(b), the average velocity amplitude of the screen body in the $X$ direction is 336.21 mm∙s^-1, and the average velocity amplitude along the $Y$ direction is 365.87 m∙s^-1. The velocity amplitude of the screen body along the $Y$ direction is larger than that along the $X$ direction.

Fig. 13 shows the time-domain acceleration curve of the screen body along the $X$ and $Y$ directions. Different from the time-domain response curves of displacement and velocity, the acceleration curve at the start and stop stages is relatively stable without drastic changes, as shown in Fig. 13(a). The maximum acceleration in the start-up and shutdown phases is close to that in the stable operation phase. As shown in Fig. 13(b), the average acceleration amplitude of the screen body in the $X$ direction is 33.58 m∙s^-2, and the average acceleration amplitude in the $Y$ direction is 37.56 m∙s^-2. The amplitude of acceleration in the $Y$ direction is slightly larger.

Fig. 13Time-domain characteristic curves of acceleration signals

The specific values of the experimental test results and theoretical results are compared, as shown in Table 2. It can be seen from the table that the experimental results of the motion characteristics of the vibrating screen have a strong agreement with the theoretical results. Under stable operation, the relative error between experimental results and theoretical results is less than 7 %, and the maximum error is 6.58 %. The experimental results show that the dynamic theoretical model can perfectly describe the motion of the BVDS.

4.3. Vibration characteristics with different rotational velocity

In this section, the influence of rotational velocity on the dynamic characteristics of the screen body of the BVDS is theoretically discussed. Other parameters are shown in Table 1. Based on Eq. (3), the curve of screen body amplitude under different rotational velocity is shown in Fig. 14.

As can be seen from Fig. 14(a), before the angular velocity $\omega$ reaches the resonant angular frequency, the amplitude $A_x$ of the screen body along the $X$ direction and the amplitude $A_y$ in the $Y$ direction slowly increase with the increase of $\omega$. When the angular velocity $\omega$ reaches the resonance region, the amplitude $A_x$ and $A_y$ increase sharply. When $\omega$ exceeds resonance region, the amplitude $A_x$ and $A_y$ decrease rapidly. Then the screen body enters the transition region and working region. In the transition region and working region, $A_x$ and $A_y$ decrease slowly with the increase of $\omega$.

Fig. 14Vibration amplitude of the screen body under different ω

When the vibration is in the resonance region, the vibration amplitude of the screen body is too large, which leads to unstable running of the vibrating screen. When the vibration is in the transition region, the angular velocity $\omega$ is relatively slow, and the velocity and acceleration of the screen body are low, resulting in insufficient vibration strength and low screening performance. As shown in Fig. 14(b), when the vibration is in the working region, the amplitude of the screen body gradually becomes smaller, the vibration is relatively stable, and high screening performance can be obtained in practice. Moreover, when the angular velocity $\omega$ is too high, the alternating load imposed on the spring and the machine noise will increase, while reducing the service life of the machine.

In order to further investigate the effect of the angular velocities $\omega$ in the working region on the dynamic properties of the screen body, the angular velocities $\omega$ of 94.2, 102.05, and 109.9 rad∙s^-1 (i.e. $n$ of 900, 975, and 1050 rpm, respectively) were taken for discussion. The time-domain changes of displacement, velocity, acceleration ($x$, $\dot{x}$, $\ddot{x}$) and ($y$, $\dot{y}$, $\ddot{y}$) of the screen body along the $X$ and $Y$ directions under different angular velocities are shown in Fig. 15. In general, with the increase of angular velocity, the displacement amplitude of the screen body decreases obviously, the velocity amplitude decreases slightly, and the acceleration amplitude is basically unchanged.

Fig. 15Vibration characteristics of the screen body at different ω

As can be seen from Fig. 15(a), in the vibration process of one period, the time-domain change curve of screen body displacement along the $X$ direction presents a sinusoidal change rule. When $\omega$ is 94.2 rad∙s^-1, in one period (0-0.067 s), the displacement gradually increases from 0 to 4.04 mm, then gradually decreases to –4.04 mm, and then gradually increases to 0 (positive and negative signs indicate the displacement direction), and the displacement amplitude $A_x$ is 4.04 mm. When the rotational velocity $\omega$ increase to 102.05 rad∙s^-1, the displacement amplitude $A_x$ reduces to 3.44 mm. When $\omega$ is further increased to 109.9 rad∙s^-1, the displacement amplitude $A_x$ continues to decrease to 2.96 mm.

Fig. 15(b) shows the time-domain variation curve of screen body displacement $y$ along the $Y$ direction. The phase Angle between the displacement curve $X$ and $Y$ is 0°, and the corresponding displacement curve in the $Y$ direction of the screen body also presents a sinusoidal change law. When the angular velocity $\omega$ increases from 94.2 to 102.05 rad∙s^-1, the displacement amplitude $A_y$ of the screen body along the $Y$ direction decreases from 4.07 mm to 3.45 mm. Similarly, when $\omega$ is further increased to 109.9 rad∙s^-1, the displacement amplitude $A_y$ decreases to 2.97 mm.

Fig. 15(c) and Fig. 15(d) are the time-domain response curves of the screen body velocity along the $X$ and $Y$ directions, which are derived from the first derivative of the displacement curves shown in Fig. 15(a) and Fig. 15(b) respectively. In the process of screen body vibration, the velocity of screen body shows harmonic fluctuation. As the angular velocity $\omega$ increases from 94.2 to 102.05 rad∙s^-1, the velocity amplitude of the screen body along the $X$ direction $\left|\dot{x}\right|$ decreases from 380.73 to 350.72 mm∙s^-1, and as the rotational velocity $\omega$ continuously increases to 109.9 rad∙s^-1, the velocity amplitude $\left|\dot{x}\right|$ continuously decreases to 325.13 mm∙s^-1. Similarly, when $\omega$ is increased from 94.2 to 109.9 rad∙s^-1, the velocity amplitude of the screen body along the $Y$ direction $\left|\dot{y}\right|$ decreases from 383.02 to 326.56 mm∙s^-1. Fig. 15(e) and 16(f) are the time-domain response curves $\ddot{x}$, $\ddot{y}$ of the screen body acceleration along the $X$ and $Y$ directions drawn according to the second derivative of the displacement curve. As $\omega$ increases from 94.2 to 109.9 rad∙s^-1, the acceleration amplitude of the screen body in the $X$ direction $\left|\ddot{x}\right|$ slowly decreases from 35.86 to 35.72 m∙s^-2, and the acceleration amplitude of the screen body in the $Y$ direction$\left|\ddot{y}\right|$ slowly decreases from 36.08 to 35.89 m∙s^-2.

It should be noted that when $\omega$ increases from 94.2 to 109.9 rad∙s^-1, the displacement amplitude of the screen body along the $X$ direction $A_x$ decreases by 26.73 %, and the velocity and acceleration amplitude$\left|\dot{x}\right|$, $\left|\ddot{x}\right|$decrease by 14.60 % and 0.39 %, respectively. Similarly, the displacement amplitude of the screen body along the $Y$ direction $A_y$ decreases by 27.03 %. the velocity amplitude and acceleration amplitude $\left|\dot{y}\right|$, $\left|\ddot{y}\right|$ decrease by 14.74 % and 0.53 %, respectively. Therefore, when the angular velocity $\omega$ exceeds a certain range, the further increase of $\omega$ will reduce the vibration amplitude of the screen body, and the angular velocity $\omega$ should be controlled within a certain range.