INTRODUCTION
Introduction to roll-to-roll techniques and axially moving system
Roll-to-roll (R2R) is a family of manufacturing techniques involving continuous processing of a flexible substrate as it is transferred between two moving rolls of material (Fig 1) [1] R2R is an important class of substrate-based manufacturing processes in which additive and subtractive processes can be used to build structures in a continuous manner Other methods include sheet to sheet, sheets on shuttle, and roll to sheet; much of the technology potential described in this R2R Technology Assessment conveys to these associated, substrate-based manufacturing methods R2R is a “process” comprising many technologies that, when combined, can produce rolls of finished material in an efficient and cost- effective manner with the benefits of high production rates and in mass quantities [2] Figure 1 illustrates an example of R2R processing of a state-of-the-art nanomaterial used in flexible touchscreen displays
Figure 1 R2R processing of a state-of-the-art nanomaterial used in flexible touchscreen displays
The manufacturing techniques developed based on R2R processing minimize human handling, and, consequently, lead to high quality Moreover, it is well known that the rolled form is convenient for storage and transport Especially, the R2R manufacturing techniques have contributed to the rapid development of flexible electronics
In the field of electronic devices, R2R is the process of creating electronic devices on a roll of flexible plastic or metal foil (Fig 2) In other fields predating this use, it can refer to any process of applying coatings, printing, or performing other processes starting with a roll of a flexible material and re-reeling after the process to create an output roll
Figure 2 R2R applications in printing electronic devices
From the demand for high quality and low-cost production, roll-to-roll (R2R) systems have been admitted to be the most effective system handling flexible materials such as films, textiles, papers, polymers, metal sheets… (Fig 3) [3] Therefore, the R2R systems belong to the class of axially moving system that also represent many mechanisms in civil, aerospace, and automotive engineering such as thread winders, band saws, magnetic tapes, aerial cable tramways, power transmission belts…
Figure 3 Flexible materials used in R2R processing
Axially moving system play an essential role in various engineering systems including continuous material manufacturing lines, roll-to-roll processes, and transport processes (Fig 4) In these systems, the undesirable mechanical vibration of the systems can degrade the associated manufacturing process productivity and even reduce product quality, especially for high speed precision machine systems Motivated by the increasing requirement of production accuracy, the vibration control of axially moving systems has provoked the interests of many researchers for over six decades
Figure 4 Technical textile manufacturing process using R2R techniques
Axially moving systems can be considered as a string model, a beam model, a coupled model, and a plate model depending on the flexibility, the existence of damping, and geometric parameters of the system The moving string/beam/coupled models are a one-dimensional system, whereas the moving plate model is a two-dimensional one Further, a moving string is often utilized to model a continuously moving system without considering the bending stiffness of the material The flexible components whose bending stiffness is significant are generally modeled using a beam model An axially moving string/beam model focuses on the influence of the lateral vibration but ignores the longitudinal vibration, whereas the coupled model accounts for both vibrations [4–6] The coupled model is suitable for modeling materials of significant length The axially moving plate model is appropriate for the analysis of moving materials with considerable width [7] A research in [8] show the distribution of mathematical models of axially moving systems (string, beam, coupled, and plate models), in which the string and beam models are the most commonly used models (Fig 5)
Figure 5 Axially moving string/beam
Literature review
The dynamics of the axially moving system have been studied in many years
It is essentially a typical distributed parameter system with infinite dimensions in the mathematical sense, which makes the control design more difficult in comparison with the rigid mechanical system Although a PDE model can
5 precisely show the dynamic behavior of the system, the analysis and control of the vibrations of axially moving systems by directly using the PDE model is a challenge Therefore, the early investigations related to control were undertaken based on a finite-dimensional set of ordinary differential equations (ODEs), which was established by discretizing the PDE model to a set of ODEs The control design based on an ODE model is convenient for its implementation using the conventional control methods, which are well developed for ODEs
A number of researches has been provided control methods for suppressing mechanical vibrations of the moving materials [4-30] The most common control method of vibration suppression is boundary control The boundary control algorithms constructed by using the measured signals of mechanical vibrations at the left and/or right boundaries can be implemented by laser sensors at the boundary points [19, 21] For example, in [19], a robust adaptive boundary control for an axially moving string that shows nonlinear behavior resulting from spatially varying tension is investigated A hydraulic actuator equipped with a damper is used as the control actuator at the right boundary of the string Followed that a Lyapunov redesign method is employed to derive a robust control algorithm employing adaptation laws that estimate three unknown system parameters and an unknown boundary disturbance Then the uniform asymptotic stability (when the three parameters are all unknown), the exponential stability (when they are known), and the uniform ultimate bounded (with abounded boundary disturbance) of the closed loop system are investigated
Figure 6 Schematic of proposed boundary control of an axially moving string in [19]
In another example, [30] developed an optimal boundary control strategy for the axially moving material system through a mass-damper-spring controller at its right-hand side boundary The partial differential equation describing the axially moving material system is combined with an ordinary differential equation, which describes the MDS The combination provides the opportunity to suppress the flexible vibration by a control force acting on the MDS The optimal boundary control laws are designed using the output feedback method and maximum principle theory through the MDS controller A finite difference scheme is used to simulate and validate theoretical results
Figure 7 Schematic of the axially moving system with MDS controller in [30]
Therefore, in so far as it is convenient to assemble actuators and sensors at the boundaries, the boundary control can be a practical control solution for axially moving material systems However, the disadvantage of the boundary control is that applying the control forces to moving materials might destroy the material surface (for instance, printing process and lithography process in flexible electronics) To avoid strong contact force between actuators and the moving material, distributed control [31] can be possible, where controller provide distributed noncontact forces In some cases the distributed control requires a distributed sensor and actuator networks, which result in a high cost system and therefore, an impractical solution Moreover, the boundary and distributed control are not suitable for portable AMSs, in which installing an actuator is difficult because of narrow space Obtaining a viable control method that not only avoids applying contact forces but also can be implemented by low cost is now requirement, especially in roll-to-roll systems
To overcome this challenge, [8] investigated the transverse vibration control and energy dissipation of axially traveling string system by using a boundary viscous damper By analyzing the nature frequencies of the fixed length and the variable length of the string system, the resonance frequency of the external concentrated force is obtained According to amplitude and the energy reflection ratio, the range and the optimal value of damping coefficient are also obtained The effect of vibration dissipation with a viscous damper and an external concentrated force at the left boundary is investigated respectively by the numerical simulation
Figure 8 Model of axially travelling string system with damping device in [8]
In another study [20, 21], the control method using effects of time-varying transport velocity of the moving material was introduced to suppress transverse vibrations The control technique developed in these papers directly used the system parameter – axial velocity, which could be consider innovative in the literature By observing the state-space equation of the system, the authors identified that linear operator in their system depended on the axial transport velocity As a consequence, a control algorithm based on the regulation of the axial velocity was designed to eliminating lateral oscillation The control algorithm adjusted the axial velocity to track a velocity profile consisting of several slopes instead of the conventional constant-deceleration profile To obtain this profile, an optimal control problem, in which an energy-like function was considered as a cost function, and the axial velocity was used as a control input, was proposed and solved using the conjugate gradient method The effectiveness of this new control algorithm was also examined via numerical analysis
It should be noted that the work [21] was motivated by lithography printing processes, in which the motion type of the moving material is rest to rest Therefore, the control algorithm in the work cannot be applicable for R2R systems where the materials moves continuously
Figure 9 Example of large-area high-throughput roll-to-roll patterning systems used in [20], [21]
In this thesis, as a mean of overcoming the technical challenge mentioned above, a control method that employs the effects of varying tension of the moving material to suppress the transverse vibrations is proposed.
Objectives and Scope
Thus far, the partial differential equations (PDE) have been used to model the AMSs, presenting that AMSs are infinite dimensional systems Although the continuous models describe the AMS dynamics exactly, their infinite number of DOF yields a challenge in analyzing the dynamics as well as deriving control schemes Therefore, instead of working with PDEs, the continuous models can be approximated by spatial discretization methods such as the Galerkin method, in which approximation models employing sets of ODEs enables the use of the control theory for ODEs Therefore, this dissertation investigates an axially moving beam that mimics a R2R system The ODE model of the axially moving beam obtained by using the Galerkin method will be used to derive the proposed control algorithm Main contents of this thesis are shown as below:
First of all, the dynamic model of the axially moving beam using ODEs is derived by using the Galerkin method, where the time-varying transport velocity and time-varying tension are considered Secondly, a nominal model of beam is proposed to formulate the spatially-varying tension Thirdly, a novel control algorithm employing the effects of the time-varying longitudinal beam is derived by using the approximate input-output linearization to suppress the transverse vibration The main advantages of the proposed control method are: (i) To be able to suppress transverse vibration for non-stop R2R processes; (ii) To avoid applying forces to moving material surfaces; (iii) To enable active vibration control for compact AMSs
Moreover, the linear stability theorem are used to find the proposed control parameters The numerical simulation is performed based on the Matlab Simulink and the results is verified based on experiment.
Organization of the dissertation
The dissertation begins with the preliminaries in Chapter 2 that introduce the theoretical techniques used for this thesis such as Hamilton’s principle, Galerkin method, nominal re-model of beam and theory of tracking control via approximate input-output linearization
Chapter 3 introduces the mathematical modeling problems of an axially moving beam with time varying length Using the Hamilton’s principle, the derivations of equations of motion (PDEs) for the AMSs are presented The application of the Galerkin method to reduce the PDEs into sets of ODEs provides the discrete models
Chapter 4 proposes a feedback input-output linearization control In the proposed control method, the nonlinearities of the system are first compensated using nonlinear state feedback and a nonlinear state transformation Then, a linear controller is designed to control the linearized system
Chapter 5 considers the control parameters and illustrates the simulation results to verify the effectiveness of the proposed control algorithm
Finally, chapter 6 summarizes the results and draw conclusion
PRELIMINARIES
Hamilton Principle
In deriving the equations of motion of the axially moving systems, the Hamilton’s principle is a powerful technique Using the Hamilton’s principle, the governing equations of motion and the boundary conditions can be directly obtained, whereas the Euler-Lagrange’s equation only provides the governing equations Moreover, the formulation of the Hamilton’s principle is established from the system kinetic, potential, and work energies, that leads to a great understanding of the system dynamics
In this section, the Hamilton’s principle is applied to derive equations of motion of a beam system Fig 10 shows the axially moving beam travelling between two rollers with time varying ( )v t The left roller is fixed that vertical movement of the beam is restricted Meanwhile, the right roller is allowed to perform small motion along longitudinal direction Therefore, the distance between two rollers is considered as a variable depending on time ( )l t Let t be the time and x be the spatial coordinate along the longitude of motion Denote w( , )x t is transverse displacement The material properties of the beam are the mass per unit length ȡWKHYLVFRXVGDPSLQJc v , the Young modulus E, the cross section A and the moment of inertial of the beam’s cross section about z-axis I
Figure 10 An axially moving beam travelling system
Dynamic model of this system can be represented as partial differential equation (PDE), which is determined based on Hamilton’s principle
Hamilton’s principle is defined as “Of all possible paths between two points along which a dynamical system may move from one point to another within a given time interval from t 0 to t 1, the actual path followed by the system is the one which minimizes the line integral of Lagrangian.”
This means that the motion of a dynamical system from t 0 to t 1 is such that the line integral
G³ Ldt is extremum for actual path This implies that small variation in the actual path followed by the system is zero
The Lagrangian for this system is also obtained based on kinetic energy, potential energy and work done by external forces
Where the first term represents the kinetic energy associated with the longitudinal translational motion and the second term represent the kinetic energy associated with the transverse vibration
The total potential energy can be found as:
Where the first term represents the potential energy associated with the tension force and the second term represent the potential energy associated with bending stiffness
The virtual work done by viscous damping is given as:
In (2.1), (2.2), (2.3), the lettered subscript for w denotes partial differentiation and
Dt t x w w w w (2.4) is the material derivative Thus, the equations of motion for flexible axially moving beam will be derived using Hamilton’s principle:
Finally, PDE model describing the system can be obtained by analyzing equation (2.5).
Galerkin method
In investigating of PDEs, an important issue is to find a solution that satisfies a given PDE and its boundary conditions There are two types of solutions: exact
15 and numerical solutions An exact solution is a function that is obtained by solving PDEs analytically A numerical solution is an approximation to an exact solution Since it is difficult or even impossible to find exact solutions of PDEs, numerical solutions are usually used instead of exact solutions, in which the Galerkin method is very popular for approximating solutions to PDEs by simple functions
(w, , ) x t f t x x w w w w w w (2.6) Based on the Galerkin procedure, the transverse deflection w is approximated by a series of time varying coefficients q t i and linear mode shape function M i z
In which, q t i represents the generalized transverse displacement and weight function M i z represents the effect of the i th eigenvalue of stationary beam The number of functions n in the approximate solution (2.7) affects to the convergence of the sine series as well as the smoothness of the function w(x,t) The method for choosing n can be depend on the practice of system
It should be noted that the set of the basis functions { M i z } i 1 n is orthogonal, ie:
The PDE equation (2.6) is converted to ODEs by projection of equation (2.7) as:
Using the orthogonal property (2.8), (2.9) gives:
It should be noted that the function f can be a nonlinear function, i.e: it can be expressed as: f L N (2.11) where L and N denote the linear and nonlinear parts, respectively If the basis functions are chosen as eigenfunctions of the linear part L, i.e:
Equation (2.10) is then simplified as:
It can be concluded that:
K K n n n ij i j i K j K da t a t a t a t dt O ¦ ¦ J (2.14) where the interaction coefficients J ij are zero when i zj 0 It is shown that the PDE (2.6) has been converted to the ODEs (2.14).
Spatially-varying tension model
To formulate the spatially-varying tension T(x,t), the following nominal model is employed:
Figure 11 Nominal re-model of an axially moving beam travelling system
When forces pull on an object and cause its elongation, we call such stress a tensile stress An object under stress becomes deformed The quantity that describes this deformation is called strain Strain is given as a fractional change in either length or volume or geometry Therefore, strain is a dimensionless number Strain under a tensile stress is called tensile strain When stress is sufficiently low and the deformation it causes in direct proportion to the stress value, the relation between strain and stress can be consider to be linear
In the nominal model, T e (x,t) is the tension resulting from the elastic strain Meanwhile, T v (x,t) represents the tension affected by the transverse vibration of the beam Consider the equilibrium point of the string (in this state, the string does not move)
T x t T x t EA l l l l (2.15) where l 0 is the length of the span roller at t = 0, and l f is the length of the string when the tension of the string is zero
T e (x,t) is the internal tension resulting from the elastic strain Meanwhile,
T v (x,t) represents the tension affected by the transverse vibration of the string When the string moves, we obtain:
T x t l t l l l t (2.17) From the Newton’s second law, we have:
Tracking control via approximate input-output linearization
Tracking control and regulation in applications are common problems attention from control researchers For linear systems, the asymptotic regulation and tracking of signals generated by finite-dimensional linear systems has been studied in a general framework In the nonlinear case, since there exists no generic methods for controller synthesis One major research direction is the use of control Lyapunov functions However, there are no systematic ways of constructing a control Lyapunov function except for systems that are passive or for triangular systems where back-stepping can be applied Another direction of investigation has dealt with feedback linearization Therein, the nonlinearities of the system are first compensated using nonlinear state feedback and a nonlinear state transformation Then, a linear controller is designed to control the linearized system
Briefly, let y d be a desired output trajectory of a nonlinear system The control strategy is as follows:
The idea is to use control of the form u t( ) u t d ( )K x t( d ( )x t( )), where (u d (.),x d (.))is the desired input-state trajectory (found through inversion) satisfying:
With the feedfoward term u d ( ) t , the system is partially linearized Meanwhile the feedback term K x ( d ( ) t x t ( )) is chosen to stabilize the system along the desired state trajectory
It should be note that a nonlinear operator whose application in nonlinear inversion yields a clear connection between unstable dynamics and causal inversion When inversion operator is incorporated into tracking regulators, it could be a powerful tool for control, especially when computation is considered
By choosing the suitable input-output operator, the nonlinearities of the system could be compensated using nonlinear state feedback and a nonlinear state transformation Then, a linear controller is designed to control the linearized system
DYNAMIC MODEL OF SYSTEM
Equations of motion
The equations of motion for flexible axially moving beam will be derived using Hamilton’s principle:
G G G ³ (3.1) where T, P and W are the kinetic energy, potential energy and work done by external forces, respectively
Dt Đ ã ³ ă â á ạ (3.2) where the lettered subscript for w denotes partial differentiation, and:
22 is the material derivative The first term represents the kinetic energy associated with the longitudinal translational motion and the second term represent the kinetic energy associated with the transverse vibration
The total potential energy can be found as
V ³ T EI dx (3.4) where the first term represents the potential energy associated with the tension force and the second term represent the potential energy associated with bending stiffness
The virtual work done by viscous damping is given as:
Hamilton’s principle
Substitute (3.2), (3.4) and (3.5) into extended Hamilton’s principle:
G G G ³ (3.6) and apply the variational operation Use the standard procedure for integration by parts with respect to the temporal variable, one obtains from (3.6)
Dt t x t x x w w w w w w w w w (3.8) Set the coefficient to zero yield the governing equation in the form:
(w tt 2 xt 2 w xx w ) x (w t w ) ( w ) x x x xxxx 0 pA vw v v c v T EIw (3.9)
We formulate T e (x,t) as the internal tension resulting from the elastic strain and T v (x,t) represents the tension affected by the transverse vibration of the string When the string moves, we obtain:
T x t l t l l l t (3.12) From the Newton’s second law, we have:
(3.14) Solving the integration by parts of tension force in (3.9):
Substituting (3.14) into (3.9) and using the following parameters:
AE EI a b c pA pA pA (3.16) Equation (3.9) is rewritten as:
( ) ( ) 2 f f xxxx xx x xt t tt x xx f f l t l l t l a a b v t x l t x v t cv t vw c w l l t l l t
(3.17) With the following boundary conditions:
Galerkin method
Based on the Galerkin procedure, the deflection w(x, t) is approximated by a series of time varying coefficients q t i ( )and linear undamped mode shape function
( , ) ( ) ( ) n i i i w x t ¦ q t M z (3.19) where the linear undamped mode shape function of order i of the axially moving beam is assumed as follows:
The transverse displacement w( , ) x t will be approximated to make q t ( ) the generalized displacement q t i ( )represents the generalized displacement and the orthonormal basis function M i ( ) t represents the effect of the i th eigenvalue of the stationary beam
Substitute (3.19) into our equation (3.16), multiply both side with weight function:
Integrate the obtained equation from 0 to l t ( ), collect all terms of the resultant equations with respect to q t i ( )and its derivate:
S S ¦ (3.25) The governing equation is rewritten as: ij ( ) ( ) C ( ) ( ) i ij i ij ( ) ( ) i 0
( ) 2 i n q q t q q ê ºô ằ ô ằô ằ ô ằơ ẳ Rewrite equation in state space form:
(3.27) Where the mass matrix M t ( ), the damping matrix C t ( ) and the stiffness matrix
DESIGN AN INPUT CONTROL
Overview of model
Consider the dynamic system ij ( ) ( ) C ( ) ( ) i ij i ij ( ) ( ) i 0
With M, C, K is the mass, damping, stiffness matrixes, continuously Note that
M, C, K is the diagonal matrixes and q t ( ) [ ( ) q t 1 q t n ( )] T is the time-dependent vector of generalized coordinates The state space vector can be set as:
For the control design purpose, it is convenient to choose the input control
( ) u t l t l t and rewrite the model of system as:
. nxn nxn ij ij n n n nn n nn
B i q ni i q i n ni q i n ê º ô ằ ô ằ ô ằ ê º ô ằ ô ằ ô Ư ằ ôƯ ằ ô ằ ôơ ằẳ ô ằ ô ằ ôƯ ằ ô ằ ơ ẳ
The control objective is to drive the transverse displacement of beam to a desired one In the control design, the transverse displacement is considered as a control output, which is given as:
Output tracking control design
Consider the dynamic system with desired output transverse y t d ( ) A feedback control input u (t) is designed to guarantee the tracking of the control outputy(t)
That is lim[ ( ) d ( )] 0 t y t y t of The assumption about desired output are made as follows: (i) the desired output and its derivatives is continuous, (ii) y t d ( ) are bounded for all t>0 and are piecewise continuous function
We seek a nominal control input u t d ( ) and desired state s t d ( ) satisfy the following statements:
(i): u t d ( ) and s t d ( ) satisfy the differential equation: d d d ( ) s As Bu t (4.4) (ii): The output tracking is achieved: d d y Hs (4.5)
Take the first and the second derivatives of the control output and using the dynamics equation, the following equations is obtained: y Hs (4.6) y Hs HAs HBu HAs (4.7) y HAs HA s 2 HABu
It should be note that the expression HB=0, follow the step we have:
Set M HA s 2 and J HAB From equation 4.10, we choose the nominal input as:
It can be seen that when the state vector come close to 0 (in example: q=0), the functionJ HAB come to 0 as consequent, which make the nominal input come to infinite Therefore, to avoid the unboundedness of the control input, the proposal control input is used:
1 n n n n i n i i n i n n i n i i ni i ni t sign q q r q q i i n i i n i ni sign r q q r i i n
And r is the set value of the neighborhood of the unbounded point and K P is the control gain:
Remark 1: The control law includes the feedfoward term and feedback term With the feedback term, the system is partially linearized, as proven lately Meanwhile, the feedback term provides a stabilizing effect by adjusting the control gain The block diagram of control method is show below
Figure 13 The block diagram of control method
Remark 2: It should be noted that the number of the basic functions n in (4.13) represents the number of modes of the axially moving beam, which are used in the investigation of the beam dynamics The large number of modes may assure the high accuracy of the results obtained from the investigation In this paper, the following second mode is used to investigate the beam dynamics Therefore, the final proposal control input is shown as:
With control gain K P > K 1 K 2 K 3 K 4 @ and r is the set value of the neighborhood of the unbounded point
The change of the state variables
Introduce the change of state variable:
It can be seen that the matrix N has full rank and consequently, is invertible Therefore, we follow the inverse transformation
( ) 1 ( ) z t N s t (4.16) Using the new variables, the system dynamics becomes:
With the transformation, K s ( , ) F K is the internal dynamic of system H ij , D ij come from (4.3) Define the tracking error e as follows: d d y y e y y ê º ô ơ ằ ẳ (4.21)
Use control law (4.14), the feedback control of the system is now considered as follows:
K I k I k I k I k k ! The Jacobian matrix of the internal dynamics is obtained as:
Where K 0 is the equilibrium point of the internal dynamicK f ( , ) F K The Jacobian approximation is as followed:
K P K K K K is chosen to satisfy J (0, K 0 )to have no eigenvalues on the right side of the imaginary axis and imaginary axis itself In other words, the origin of the system K f ( , ) F K is hyperbolic and asymptotically stable.
The proposed control algorithm
K I D I D and K 4 I 1 D 21 I 2 D 22 is chosen to guarantees the negative of the eigenvalues of matrix J (0, K 0 )
Proof With K 3 I 1 D 11 I 2 D 12 and K 4 I 1 D 21 I 2 D 22 , the J (0, K 0 ) is transform to:
D D c l t l t and D 12 D 21 , the eigenvalues of J (0, K 0 ) is negative
K K is chosen for best control performance and properly input control
For simple K K 1 , 2 is chosen as: 1 1 11 2 12
I I with e is control gain and r is the set value of the neighborhood of the unbounded point S { (.) | ( ) q q t R * ,|| || q f d r }
Then, it is concluded that the final control input is:
(4.25) Remark 3: The hyperbolic system K J (0, K K 0 ) and the Lipchitz condition
| ( f F K a , a ) f ( F K b , b ) | d M F | a F b | M K K | a b | holding for all t guarantee that the approximation of f ( , )F K by J (0, K K 0 ) satisfies the conditions of nonlinear inversion based out tracking [32]
Remark 4: When the moving beam operates in the region
S q q t R q f d r , it is proved that w(x,t) can be push in an arbitrarily small boundedness region by setting a sufficiently small S Therefore, to improve the control performance, the region S should be decreased by choosing small r However, the small r makes the large control input, which is impractical when applying in practice Therefore, the value of r is determined by trial for both practical and effective control solution
SIMULATION RESULTS
Control position and parameter
For mode 2 of the beam, the largest amplitude of vibration is considered to belong to span ( ( ) , ( ) )
Figure 14 The symmetric (1) and anti-symmetric (2) modes of moving beam
In the simulation, the position ( )
4 x l t is chosen for surpassing the transverse displacement Therefore, two position parameters are determined as: 1 1
I 2=1 Let the initial transverse vibration of the axially moving beam (3.17) be
1 0, 3 q (cm), q 2 0, 2(cm), q 1 0,1(cm/s), q 2 0,1(cm/s) The control parameter e,r is selected as follow: e 0, r =0,2 (cm)
In practice, since the exact values of the viscous damping, Young modulus, 2 nd moment of inertia… of the beam may not be obtainable, the estimates values based on theoretical calculation is used in order to simulate the propose control algorithm (4.25)
The axially moving system is considered with the system parameters listed in Table 1 These parameters are used in numerical simulations
Mass per unit ȡ 1800 kg m / 3 The viscous damping c v 0.001 Nm s /
With these parameters, numerical simulations is used to determine the transverse vibration of the beam with and without applying the control algorithm (4.25) The simulations is worked on Matlab R2016a software The model system (4.2) is solved using function ode45 of Matlab The transverse displacement according to time is drawn from (4.3).
Simulation results
To verify the effectiveness of the control algorithm, numerical simulation is performed by Matlab2016 There are two profiles for regulation of the axial transport velocity In the first profile, the axial transport velocity increased from the value of 0 m/s with acceleration of 0.2 m s / 2 in 5.5 seconds then decreased with the same deceleration in next 5.5 seconds (Fig 15) In the second profile, the velocity increases from 0 to 2 m/s within a second Afterwards, it is kept as constant in a period of 10 seconds Finally, this axially velocity decreases rapidly from 2 to 0 m/s within a second (Fig 18)
In the case of no vibration control, the transverse displacement according to time of position ( )
4 x l t are observed (Fig 16), (Fig.19) When applying control algorithm (4.25), the transverse displacement tend to decreases rapidly, which can be easily noticeable (Fig 17), (Fig 20) The detail results of each transport velocity profile are shown below
5.2.1 The axial transport velocity profile 1
Figure 15 The time-depended axial velocity v(t) profile 1
The transport velocity profile 1 (Fig 15) is a triangular motion profile and is characterized by equal acceleration and deceleration times (and distances), with no time spent at a constant velocity In other words, a triangular motion profile divides the time allowed for the move into two halves, an acceleration period and a deceleration period This profile is commonly used for applications that don’t require a period of constant velocity since it provides the fastest movement between two points
Figure 16 demonstrates the variation of transverse displacement according to time of transport velocity profile 1 in the case when the control algorithm is not applied It can be seen that the transverse vibration depends only on the value of
40 the viscous damping coefficient of the beam material Therefore, vibration suppression that relies only on the viscous damping force usually requires plenty of time It took nearly 6 seconds for vibration suppression
Figure 16 The transverse displacement according to time of velocity profile 1 when not apply the control algorithm Figure 17 demonstrate the transverse displacement varies with respect to time when applying the control algorithm The value of w(x,t) decreases rapidly After 0.9 seconds, value of w(x,t) approaches 0 It is obviously with control algorithm, the time to eliminate the vibration is significant improvement
Figure 17 The transverse displacement according to time of velocity profile 1 when apply the control algorithm
5.2.2 The axial transport velocity profile 2
In the transport velocity profile 2, the time-depended axial velocity v(t) profile (as shown as Fig.18), which is widely used in practice, is employed The difference between this profile and the first profile is the appearance of constant velocity, which is called a trapezoidal profile In detail, the axially velocity increases from 0 to 2 m/s within a second Afterwards, it is kept as constant in a period of 10 seconds Finally, this axially velocity decreases rapidly from 2 to 0 m/s within a second
Figure 18 The time-depended axial velocity v(t) profile 2
Figure 19 The transverse displacement of velocity profile 2 according to time when not apply the control algorithm
Figure 19 demonstrates the variation of transverse displacement according to time of transport velocity profile 2 in the case when the control algorithm is not applied It can be concluded that the transverse vibration depends on the value of the viscous damping coefficient of the beam material From Fig 19, it is clear that without control algorithm, the transverse vibration occurs quite long, and it took approximately 5.1 seconds for vibration suppression
Figure 20 The transverse displacement according to time of velocity profile 2 when apply the control algorithm
To suppressing the transverse displacement quickly, control algorithm is applied Figure 20 illustrates the transverse displacement varies with respect to time After 0.4 seconds, value of w(x,t) approaches 0 It is obviously with control algorithm, the time to eliminate the vibration is significant improvement
This control algorithm is expected to impart two improvements to performance of the system First, the transverse displacement is expected to decay and
44 suppressed completely Second, the profile for the control input, which is the velocity of the movable roller, needs to be practical
Furthermore, the control input is also illustrated in Fig 20, which shows that the maximum velocity needed for vibration suppression is 1.7 meter per second Therefore, this velocity is easy to be generated by mechanism such as hydraulic motor or servo motor
Figure 21 The velocity profile of pushing roller Overall, figure 15 to figure 21 shows the transverse displacement at position
4 x l t x when and when not applying the control algorithm in two cases: a trapezoidal axial transport velocity input and a triangular axial transport velocity input The results were collected from the numerical simulations illustrating the effective of the proposed control algorithm As show in the figures, the time needed for suppressing the transverse displacement decreased noticeable from 6
45 second to 0.9 second that is nearly 85 percent respect to time in profile 1 and from 5.1 second to 0.4 second that is nearly 92 percent respect to time in profile 2 It is obviously with control algorithm, the time to eliminate the vibration is significant improvement However, the velocity profile of moving roller needed for vibration suppression is achievable, but the chattering problem caused by fast dynamics response can somewhat yields challenges in practical approach
CONCLUSIONS 46 REFERENCES
In this thesis, a control algorithm to suppress transverse vibrations of an axially moving beam is presented The equations of motion of the axially moving beam are derived by using Hamilton’s principle With regard to the dynamics of the axially moving beam the Galerkin method was applied to reduce the PDEs into sets of ODEs, which were rewritten into state-space equations The proposed control algorithm is designed based on the linearization input-output approximate method
The advantage of the proposed control law is to regulate the transverse displacement of the moving beam without applying external force to the material surface directly and therefore, to prevent damage of the material surface.Through a comparison of the simulation results (time of vibration suppression) for control and no control situation, the considerable improvement effected by applying the control algorithm to the beam system for vibration suppression were verified.
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I=8.5*10^-11;%m4 lf=0.96; %m o1=1/sqrt(2); %x=lt/4 o2=1; %x=lt/4 a=E/p;b=E*I/(p*A);c=cv/(p*A);
%input velocity profile 2 rn;dt=0.001;n=round(rn/dt); for i = 1:1000 t(i)=i*dt; dv(i)=2; v(i)=dv(i)*t(i); end for i = 1001:10000 t(i)=i*dt; dv(i)=0; v(i)=2; end
52 for i = 10001:11000 t(i)=i*dt; dv(i)=-2; v(i)=2+dv(i)*(t(i)-10); end
%input initia q1(1)=0.003;dq1(1)=0.001; q2(1)=0.002;dq2(1)=-0.001; l(1)=0.965;dl(1)=0;ddl(1)=0; y(1)=q1(1)*o1+q2(1)*o2; yc= y(1)*0.1;
%control output and system values e 0; r=0.0002; for i = 2:n
Denominator(i)=4/3/l(i-1)*((sign(abs(q1(i-1)*o2-q2(i-1)*o1)-r)+1)*(q1(i- 1)*o2-q2(i-1)*o1)+((sign(r-abs(q1(i-1)*o2-q2(i-1)*o1))+1)*r));
Proposedcontrol(i)=Numerator(i)/Denominator(i); ddl(i)=Proposedcontrol(i) ; if y(i-1)>= yc dl(i)=dl(i-1)+dt*ddl(i); else if l(i-1)>=l(1) dl(i)=-l(i-1)*c;
53 else dl(i)=0; end end l(i)=l(i-1)+dt*dl(i);
K11(i)=b*pi^4/(2*l(i)^3)-pi^2*(v(i)^2+dv(i)*l(i))/(2*l(i))+(dv(i)+a*(l(i)- lf)/(l(i)*lf))*pi^2/4+pi^2*v(i)*dl(i)/(2*l(i))-pi^2*dl(i)^2/(6*l(i));
K21(i)=-20/9*(a*(l(i)-lf)/(l(i)*lf)+2*v(i)*dl(i)/l(i)-2*dl(i)^2/l(i))+4/3*((c*dl(i)- c*v(i))-32/9*dv(i))+8/3*ddl(i)/l(i);
K12(i)=-20/9*(a*(l(i)-lf)/(l(i)*lf)+2*v(i)*dl(i)/l(i)-2*dl(i)^2/l(i))-4/3*((c*dl(i)- c*v(i))-8/9*dv(i))-8/3*ddl(i)/l(i);
K22(i)*b*pi^4/(2*l(i)^3)-4*pi^2*(v(i)^2+dv(i)*l(i))/(l(i)*2)+(dv(i)+a*(l(i)- lf)/(l(i)*lf))*pi^2+4*pi^2*v(i)*dl(i)/(l(i)*2)-4*pi^2*dl(i)^2/(6*l(i));
Cx(i)=q1(i-1)*(M(i)/dt^2+C11(i)/dt)+dq1(i-1)*M(i)/dt+q2(i-1)*C21(i)/dt; Ex(i)=M(i)/dt^2+C11(i)/dt+K22(i);
Fx(i)=q2(i-1)*(M(i)/dt^2+C11(i)/dt)+dq2(i-1)*M(i)/dt+q1(i-1)*C12(i)/dt; q1(i)=(Fx(i)*Bx(i)-Ex(i)*Cx(i))/(Dx(i)*Bx(i)-Ex(i)*Ax(i)); q2(i)=(Fx(i)*Ax(i)-Dx(i)*Cx(i))/(Ex(i)*Ax(i)-Bx(i)*Dx(i)); dq1(i)=(q1(i)-q1(i-1))/dt;
%system output y(i)=o1*q1(i)+o2*q2(i); end figure; plot(t,y); xlabel('time [s]'); ylabel(' w[l/4]’); title('Profile2’);
Full name: GIANG HOANG NGUYEN
Emails: 1770539@hcmut.edu.vn giangnguyenhoang.tl@gmail.com Cell phone: 098 819 6759
Degree Field Institution Date Conferred
Hanoi University of Science and Technology
Ho Chi Minh City of Technology