First, we performed simulations to compare Rayleigh and Timoshenko models based on different fundamental frequency (FF). In this investigation, the structure of roller is made from one steel layer. The initial value of thickness, inner diameter and length are sequential 5mm, 250 mm and 1800 mm. The parameters of the DMPC are presented in Table 1. The simulations results are shown in Figure 5.
Table 1 The parameters of the DMPC The control horizon (N ) c 5 The prediction horizon (Np) 20 The control parameter (r ) T 5x10-6
In Figure 5(a), it can be observed that the different FF of two models was round 8%
when thickness of steel layer was changed from 5 mm to 50 mm. Next, we consider variation of length to diameter ratio in Figure 5(b). The FF is inversely proportion to the length to radius ratio. Two simple simulations proved important effectiveness of tranverse shear force. The model expressed by Timoshenko beam theory is more accuracy and reliability.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Figure 5 Compare different fundamental frequencies of Rayleigh’s and Timoshenko roller model.
0 5 10 15 20 25 30 35 40 45 50 10
15 20 25 30 35 40 45
Fundamental frequency (rad/s)
Thick of nylon layer (mm)
Figure 6 Express the variation of fundamental frequency when increasing the thickness of nylon layer.
The next purposed simulation was carried out in order to demonstrate the effective of nylon and steel materials. These materials are frequently provided to manufacture roller layers. In this illumination, the length, inner and outer of roller are fixed. First, it is assumed that the roller only made from steel material, and then the thickness of nylon is increased with direction from outside to inside surface of the roller. Figure 6 has shown the variation of the FF of roller when reducing the thickness of steel layer and simultaneously increasing the thickness of nylon layer. It is easy to see that the roller made from only nylon requests bending force entirely lower than the steel roller is. Besides, the nylon is softer and has smaller mass density. These characteristics give more advantages for processing fabric, designing, manufacturing and maintaining roller, and saving energy in operation. However, the nylon roller is easily damaged, torn or destroyed when receiving
great external forces. Therefore the combined roller made by inner steel and outer nylon layer with suitable thickness of these materials is a good choice.
0 100 200 300 400 500 600 700 800 900 1000
26 26.5 27 27.5 28 28.5 29 29.5 30 30.5 31
Fundamental frequency (rad/s)
Velocity of the roller (rpm)
Figure 7 Relationship between the roller velocity and fundamental frequency.
Figure 7 shows the simulation results in order to demonstrate the effect of roller velocity with FF. In this figure, the FF has a large variation in the high velocity;
nevertheless, when velocity of roller varies from 0 to 300 rpm, the variation of FF is rather small.
0 0.5 1 1.5 2 2.5 3 3.5 4
(b) The unconstrained force applied by each cylinder
0 0.5 1 1.5 2 2.5 3 3.5 4
(c) The first time derivative of unconstrained force
Figure 8 Express the variation of unconstrained force when changing the number of cylinders in the multi-cylinder system.
Table 2 Physical parameters of the combined roller
Roller length (L) 1800 mm
Inner diameter of steel layer (d ) 0 270 mm Outer diameter of steel layer (d ) 1 290 mm Outer diameter of nylon layer (d ) 2 350 mm
Young’s modulus of steel (E ) 1 1.962x105 N/mm2 Young’s modulus of nylon (E ) 2 1.96x103 N/mm2
Mass density of steel (1) 7.85x10-6 kg/mm3 Mass density of nylon (2) 1.14x10-6 kg/mm3
Roll velocity () 80 rpm
Poission’s ratio of steel (1) 0.3 Poission’s ratio of nylon (2) 0.4
The parameters of roller for totally next illuminations are presented in the Table 2.
The effective of the purposed approaches were verified by next simulations. In this simulation, a number of cylinders applied for bending roller were varied. Two different simulations are presented to express important role and effectiveness of the number cylinders to the maximum magnitude of designed force for bending roller. It was assumed that the forces applied by these cylinders were simultaneous time and the same magnitude force. Firstly, the first time derivative and the magnitude of control force were not limited.
In the Figure 8, the designed force applied by only one cylinder was approximately 1.25x105 N for bending 1 mm at the middle point of the roller length. When the
multi-cylinder system had sequential 3, 7 multi-cylinders, the forces provided by each multi-cylinder were
(b) The constrained force applied by each cylinder
0 0.5 1 1.5
(c) The first time derivative of constrained force
Figure 9 Investigate the variation of constrained force when changing the number of cylinders in the multi-cylinder system.
Secondly, Figure 9 shows simulation results with the first time derivative and the magnitude of control force constraints. The number cylinders varied sequential 3, 7 and 15 cylinders, the maximum bending force of each cylinder was 3.5x104 N while the maximum first time derivative of force was 2.2x105 N/s and the minimum was -1.5x105 N/s. It is easy to see that when multi-cylinder system had 3 cylinders, the total force was not enough to gain set point displacement 1 mm. This case gave poor dynamic response performance because the undesired oscillation occurred at the output of system. When using 7 or 15 cylinders, the total forces were large enough to derive the desired displacement, give high dynamic response performance of system and the multi-cylinder system with 15 cylinders had better dynamic response performance when comparing with the multi-cylinder system had 7 cylinders.
Dynamic response performances of two different DMPC were compared in next illustrations. The first controller is unconstrained DMPC which obtained optimal value of incremental control trajectory by taken derivative first order objective function to zero without inequality constraints. The name of the second controller is constrained DMPC. In this controller, the optimal value of incremental control trajectory were gained by solving objective function which has the same form standard quadratic programming subject to inequality constraints of inputs. Set point displacement of line 1 and 2 were 1.5 mm and line 3 and 4 were 3 mm at the middle point of roller length. In Figure 10, line 1 and 3 had presented the obtained dynamic response performances of unconstrained DMPC, while line 2 and 4 has shown simulations results of DMPC with inequality constraints. It can be clearly seen that when the desired displacement in the range operation that means force requirement for bending was in the range power actuation system, the results of dynamic
response performances of both DMPC are similar at the same time, however, when the force requirement is over the limited ability of actuation system, the simulation results show that the unconstrained DMPC gave poor performance with occurring oscillation in the steady state of system, whereas constrained DMPC was able to damp the oscillation.
This is a very one of great properties of the constrained DMPC. In some circumstances, the system can become unstable when receiving some excitedly undesired disturbance that violate input saturation. The designed controller needs enough ability to control system for returning stable state as soon as possible. This is the reason why constrained DMPC is proposed in this study.
Conclusions
This project sets out to show that the new model of a combined roller based on Timoshenko beam theory is more accuracy and reliability. The second useful major of this study proved that using a multi-hydraulic cylinder system and putting it inside roller can overcome limited power of bending hydraulic actuation system, increase the magnitude and the first time derivative of bending force. Therefore, not only putting up the range of desired displacement, but even decreasing so fast the transient time of dynamic response.
Finally, constrained DMPC with great properties is suitable effectiveness not only to control and improve the dynamic response performance but also open wide the operation range of the BRCS.
To suppress the undesired roller vibration quickly and derive more effective deflection of roller, the control force supplied by the BRCS has to obtain more powerful magnitude and fast response than the external force does. The model predictive controller
is applied to the BRCS because it allows the control system to derive a high dynamic response performance and use more effective power of actuation system. The controller bases on optimal algorithms and quadratic cost functions to obtain control trajectory action at each sample time by solving online optimization problem in systematic way. Further, the constrained model predictive controller can allow the control system to return stable state quickly and operate closed to constrained operating range. Therefore, this controller can open larger operating range of the bending roller control system.
0 0.5 1 1.5
(b) The force applied by each cylinder
0 0.5 1 1.5
(c) The first time derivative of force
Figure 10 Compare dynamic response performances of the BRCS with the unconstrained and constrained DMPC.
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