Non contact control of an axially moving beam by varying tension force

Keywords: Axially moving systems; Roll to roll systems; Transverse vibration control; Velocity tracking; Spatially varying tension; Galerkin method; linearization input-output approxima

HO CHI MINH CITY OF SCIENCE AND TECHNOLOGY

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MASTER THESIS

NON-CONTACT CONTROL OF AN AXIALLY MOVING BEAM BY

VARYING TENSION FORCE

ĐIỀU KHIỂN DAO ĐỘNG HỆ THỐNG VẬN CHUYỂN VẬT LIỆU MỀM

MECHATRONICS CODE: 8520114

HO CHI MINH CITY, 6 December 2019

ACKNOWLEDGEMENT

After more than a year of work, thanks to my efforts and the support of many people, my MS thesis “Non-contact control of an axially moving beam by varying tension force” was completed

First of all, I would like to express my sincere gratitude to my advisor Prof Nguyen Quoc Chi of the faculty of mechanical engineering at Ho Chi Minh city

of Technology for his passionate advice for his patience, motivation, and immense knowledge Besides, he gave me the access to the laboratory and provided me the best research conditions He also steered me in the right the direction whenever he thought I needed it

I am also grateful to the members of Control and Automation Laboratory of the faculty of mechanical engineering at Ho Chi Minh city of Technology for helping me to design and perform the research Without their precious support I

would not be able to conduct this research

Last but not the least, I must express my very profound gratitude to my parents for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis

I hereby declare that the thesis has been done without any plagiarism violations and does not conflict with any issue in ethics

Ho Chi Minh, 17 December, 2019

Nguyen Hoang Giang

ABSTRACT

The main topic of the thesis is presenting a control algorithm to suppress transverse vibrations of an axially moving beam In this thesis, the equations of motion of the axially moving beam are derived by using Hamilton’s principle The Galerkin method is applied in order to reduce the PDEs describing the dynamics of the axially moving beam and into sets of ODEs For control design purposes, these ODEs are 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 Finally, the effectiveness of the proposed control algorithm is verified by numerical simulations and experiments

Keywords: Axially moving systems; Roll to roll systems; Transverse

vibration control; Velocity tracking; Spatially varying tension; Galerkin method; linearization input-output approximate method

CONTENTS ACKNOWLEDGEMENT

ABSTRACT

LIST OF FIGURES

LIST OF TABLES

CHAPTER 1 INTRODUCTION 1

1.1 Introduction to roll-to-roll techniques and axially moving system 1

1.2 Literature review 4

1.3 Objectives and Scope 9

1.4 Organization of the dissertation 10

CHAPTER 2 PRELIMINARIES 12

2.1 Hamilton Principle 12

2.2 Galerkin method 14

2.3 Spatially-varying tension model 16

2.4 Tracking control via approximate input-output linearization 18

CHAPTER 3 DYNAMIC MODEL OF SYSTEM 20

3.1 Equations of motion 21

3.1.1 Total kinetic energy 21

3.1.2 Total potential energy 22

3.1.3 Work done 22

3.2 Hamilton’s principle 22

3.3 Galerkin method 24

CHAPTER 4 DESIGN AN INPUT CONTROL 28

4.1 Overview of model 28

4.2 Output tracking control design 30

4.3 The change of the state variables 32

4.4 The proposed control algorithm 34

CHAPTER 5 SIMULATION RESULTS 36

5.1 Control position and parameter 36

5.2 Simulation results 38

5.2.1 The axial transport velocity profile 1 39

5.2.2 The axial transport velocity profile 2 41

CHAPTER 6 CONCLUSIONS 46 REFERENCES

LIST OF FIGURES

Figure 1 R2R processing of a state-of-the-art nanomaterial used in flexible

touchscreen displays 1

Figure 2 R2R applications in printing electronic devices 2

Figure 3 Flexible materials used in R2R processing 3

Figure 4 Technical textile manufacturing process using R2R techniques 3

Figure 5 Axially moving string/beam 4

Figure 6 Schematic of proposed boundary control of an axially moving string in [19] 6

Figure 7 Schematic of the axially moving system with MDS controller in [30] 6 Figure 8 Model of axially travelling string system with damping device in [8] 8 Figure 9 Example of large-area high-throughput roll-to-roll patterning systems used in [20], [21] 9

Figure 10 An axially moving beam travelling system 13

Figure 11 Nominal re-model of an axially moving beam travelling system 17

Figure 12 Axially moving beam with one movable roller 21

Figure 13 The block diagram of control method 32

Figure 14 The symmetric (1) and anti-symmetric (2) modes of moving beam 36

Figure 15 The time-depended axial velocity v(t) profile 1 39

Figure 16 The transverse displacement according to time of velocity profile 1 when not apply the control algorithm 40

Figure 17 The transverse displacement according to time of velocity profile 1 when apply the control algorithm 41

Figure 18 The time-depended axial velocity v(t) profile 2 42 Figure 19 The transverse displacement of velocity profile 2 according to time

when not apply the control algorithm 42

Figure 20 The transverse displacement according to time of velocity profile 2

when apply the control algorithm 43

Figure 21 The velocity profile of pushing roller 44

LIST OF TABLES

Table 1 Control parameters 37 Table 2 System parameters 38

CHAPTER 1 INTRODUCTION 1.1 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

1.2 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

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

1.3 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

1.4 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

CHAPTER 2 PRELIMINARIES 2.1 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 ρ, the viscous damping c 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 t0 to t1, 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 t0 to t1 is such that the

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:

0

1 ( (w ) (w ) ) 2

l

V = ∫ T +EI dx (2.2) Where the first term represents the potential energy associated with the tension

force and the second term represent the potential energy associated with bending

is the material derivative Thus, the equations of motion for flexible axially moving

beam will be derived using Hamilton’s principle:

0

f t

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

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

Consider the PDE:

2 2

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 functionϕi( )z

i 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 { ϕi( )z }i=1 n is orthogonal, ie:

2 2

2 2

n

n x

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:

2 2

2.3 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 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:

2 1 ( , ) ( , )

2.4 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:

( ) ( ( )) ( ( )) ( )

x t = f x t +g x t u t (2.19)

( ) ( ( ))

y t =h x t (2.20) 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:

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

CHAPTER 3 DYNAMIC MODEL OF SYSTEM

Travelling string or beam are the simplest models to describe axially moving material systems The strings are suitable to model slender flexible materials, where their bending stiffness is relatively small When the bending stiffness of a slender moving material is sufficiently large, it should be considered as a moving beam Plates are used to model flexible materials that have considerable width such as substrates in flexible electronics, textiles, papers, and thin metals Unlike string and beam models, the plate is a three-dimensional continuum Mathematically, the axially moving plate models are more complex than the axially moving beam or string models

For a plate model, since the value of the length l is much larger than the values

of the width b and the thickness h, the in-plane stiffness is much higher than the

out-plane stiffness Therefore, it can be assumed that the in-plane displacements are very small in comparison with the out-plane displacement Based on this assumption, in this chapter, only suppression of the transverse displacement of

axially moving beam is focused Equations of motion of the axially moving beam described in chapter 2 are derived by using Hamilton’s principle Moreover, the dynamic analysis of system is performed based on the reduced order model obtained by the Galerkin method

Figure 12 Axially moving beam with one movable roller

Let l(t) be the length of the beam, A the cross-sectional area, ρ the mass per

unit length, I the moment inertia of the beam, and E the Young’s modulus

t

T W P dt

∫ (3.1)

where T, P and W are the kinetic energy, potential energy and work done by

external forces, respectively

3.1.1 Total kinetic energy

The total kinetic is:

2 2

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

3.1.2 Total potential energy

The total potential energy can be found as

0

1 ( (w ) (w ) ) 2

l

V = ∫ 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

Substituting (3.14) into (3.9) and using the following parameters:

(0, ) 0, ( , ) 0, xx(0, ) 0, xx( , ) 0

3.3 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

w x t q t ϕ z

=

=∑ (3.19)

where the linear undamped mode shape function of order i of the axially moving

beam is assumed as follows:

1

i x( , ) ( ) sin

( )

n i i

l t

πφ

=

=∑ (3.20) 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 ϕi( )t represents the effect of the i theigenvalue of the stationary beam

Substitute (3.19) into our equation (3.16), multiply both side with weight function: