MECHATRONICS

1. Slovak University of Technology Faculty of Material Science and Technology in Trnava MECHATRONICS Lecture 11

2. MECHATRONICS IN TECHNOLOGY EQUIPMENTS AND SYSTEMS The machine aggregate as a model of technology equipment The machine aggregate representit dynamic system to drive a plant (mechanical load, mechanical working equipment) and to control the technological process.. There are 3 subsystems there: • (electric) drive, i.e. the (electro)motor and the gear • driven mechanical equipment that represents not only some kind of equipment („HW“) for electromechanical energy conversion, but also both by it realized technological process and the product of the process (production, manipulation, shaping, traffic etc.) - from physical pouint of view represents all the torques and/or forces generated by the technological process), • the control system performing an optimal control of the machine aggregate from both the technology process/product and the statics/dynamics of the aggregate (as a whole) point of wiew. There are an older and simpler analogous control type and a modern digital type with a (hierarchical set of) control processors .

3. Block diagram of the machine aggregate A machine aggregate and its purposive control respecting the mutual energetic interaction of its subsystems is a mechatronic system consisting of • the supply - electric AC or DC network as a primary electric energy source +  power semiconductor converter of some kind as a econdary electric enery supply, • electric AC or DC drive with proper kind of electromotor(s) • a plant subsystem • control electronics (analog or digital,digital - programmable microcontroller system).

4. The machine aggregate as a mechatronical system

5. The control subsystem is an information subsystem. Hence, from another point of view, a mechatronic system is an integration of a power electromechanic system of machine aggregate and plant generating the torques and forces needed by the process, under prescribed speed, position etc. b energy supplying power electronic system modifying the electrical energy constant parameters of the primary source (supplying network) to by process postulated elecrical energy variable parameters of the secondary energy source (the converter). c information control electronics. The power electromechanic system with the energy supplying power electronic system perform an electromechanical energy conversion under the purposive management of the programmable control electronics. The goal is an optimal control with respect to technological process or the aggregate dynamics as a whole.

6. Machine aggregates with controlled drives often need a more-level, hierarchic control. In the basic level of, say a speed system, the (angular) speed ω of the motor/drive is controlled by a speed controller, perhaps with the aid of a subsidiary current system (current loop), by a current controller. The control of both curent and speed loop controllers can be designed starting with the curent controller in a number of ways. The classical control design procedures are based on „standard“ forms of the time response of the (curent, speed) controlled system on the input step of their input control signal or input disturbance signal. To design the position control systems, the above designed speed control system (speed feedback loop) becames subsidiary to a position control loop. Cascade, parallel and feedback groupings of controllers are available, just to refer some of the design procedures.In the higher control levels of technology control, operational quantities/parameters are managed, with thw goal to keeo the conditions of the process optimal. Conditions are variable, depending on the orientation and type of the controlled process and the machine aggregate. The system approach finds our mechatronic aggregat to be merely a subsystem, but an internally controlled part of the system, control algorithms for the current acceleration, speed, position or any technological parameter control are known for analog and digital control branche. Very up-to-date algorithms and principles are worth to be mentioned.

7. Organizing the information links Important item of the technology is the organization of information links within the controlled drive system From the links among individual drive subsystems and control subsystems point of viewthe structures as follows are available for the transfer of information: • star structure(a) for centralized control, • multiplex channel(b) - common bus • loop structure (c,d) - allows for higher transfer reliability of sophiticated drive systems • hierarchy structure (e) applies authonomy muliplex channels, elastic control systems and central control to build up authonomy of individual control (sub)systems Model of the logical-control couplings

8. Modelling the machine aggregates • Formulate and describe mathematically all the individual construction parts of the machine aggregate and interactions among them • .Create the main program based on the previous point and the help programs for the main program. • Perform identification measurements on the model, i.e. perform the simulation experiments and postprocesse them. Make statement, how truly the model substitutes the reality object/process. • Make an expert opinion on the model. The goal is to get the best possible working model of the reality - real object or process, anyway the simplest one. The quality of the model depends on the quality definition, as an example the quality be a compromise between the best possible stability of the model, its minimal computational time and accuracy of results. Another criteria may be: 1. the model accepts input data in form how data are available (no need to transform them), 2. the model gives results in the needed form (say both graphical pictures and numerical tables), the model is maintainable (program documents allow modifications if possible or if needed) etc. The whole sequence to create a machine aggregate mathematical model has 3 parts: 1. Modelling of energetic interactions among the subsystems

9. 2. Reducing the model to formulate the control laws Create a linearized model of control, say, for the machine aggregate working point shifted within a small displacement zone. Create a linearized model in the working part of the torque-speed characteristics. Create a (simple structure) nonlinear model by the „constants assigning method for the nonlinear model“ using the regres principle. 3. Creating the control law. Its verification within the validity area of the reduced model, based on the simulation experiments. Our solution method and the postprocessing of results come out of the above text. First, the structure of the machine aggregate (mechatronic system is analyzed - its interdisciplinary character, and - interactions between the state of 1. mechanical and control parts of the drive 2. parameters of the pant (technology process). Second, time responses (dynamic characteristics) are solved, using the numerical or analytical simulation, on the models of individual subsystems. The influence of the control subsystem on the aggregate dynamics is analyzed.

10. NUMERICAL OR ANALYTICAL SIMULATION Numerical or analytical simulation is optional, depends on the CAMS program (program system, package) in use. Say the MATLAB can work analytically but perhaps its main domain is the discrete simulation. The MATHEMATICA works analytically. As for the dynamic modelling and simulation, the kernal view is whether the integration of the mechatronical system describing set of differential equations is solved by algebraical or numerical integration. Third, criteria for the choice of e.g. dynamic errors of the drives, prescribe the coincidence of some number of characteristic frequency components etc. The dynamic errors in drives are caused mainly by final compliance of individual elements of transmission mechanism, backslash in kinemathical twins and excitings. Determination of these errors or at least determination of their assessment is one of the basic tasks of the dynamical analysis of the drives. The final problem solving goal is the synthesis of control laws for the machine aggregate, ensuring the optimal cooperation of the drive and plant/load/technology from the process or the product point of view.

11. Mechatronic tasks in the machine aggregate dynamics The basic equation to be solved is so-called „motion equation“ (or „torque equation“) saying thet the sum of torques present in the system is equal to zero: where Mdyn is the main,reference, reduced torqueof the machine aggregate. It represents the inercity of the system and its moving state. It is the power of the reference inercity moment I(φ) and dω/dt. For rigid couplings and in aggregate with constant gearing it is equal to or for the variable gearing it is equal to: where MD is driving torque (from electric drive/motor) reduced on the main element of the machine aggregate.

12. MLФ is the average value of the loading torque, MLa is the amplitude of the variable loading torque component, ω is angular frequency of the variable loading omponent In the case of electromotor can be axpressed within a small wicinity of working point Ф by a linear dynamic equation, ML is the loading torque originated by the working process reduced to the main element. MLcan be expressed Solving the above motion equation for the ω we find that it is equal to Equations are parametric equations of so called dynamic characteristics of a machine aggregate. The working point ф(ωΦ, MdynΦ) defined as a cross-section of torque characteristics and load characteristic moves along a closed trajectory with the same angular frequency as the speed/revolutions. Steady-state of the motion - dynamical characteristic of a machine aggregate

13. where in general: M is matrix of torques u  is the matrix of voltages The mechatronical tasks in machine dynamics can be described by two basic physical equations in a matrix form: The above equations, a frapant organic linking of torques (mechanics) and voltages (electrotechnics) are a mathematical shorthand writing the term „mechatronic“. They represent two sets of equations linked mutually. For decades the both matrix sets represented the mathematical description of the task in Laplace transform, because of easier and faster solving the task. Lately the PC programs for analytical way of solution (say Mathematics) pass the transformation by. Anyway, Laplace transform is a strong custom for a generations of engineers and Laplace (LC,LW,LH...) are taken as a natural way of thinking. The above equations are de facto a system of ODEs with a set of algebraic equations substituted into the ODEs. The dynamical phenomen can of the equations be still stressed by a time invariancy and by interlinks of structures and parameters

14. In this case the mechatronical tasks in the dynamics of machines can be split to two distinct groups: 1/ investigation of dynamic quality of electric drive (control+converter+motor+mecha-nical gear) - investigation of internal dynamics -has two possible subgroups: a - investigation of dynamic characteristics of gears for given external loads and given motor torque-speed characteristic. The goal is to find a controlling procedure for the internal dynamic of the drive bychanging the converter or drive characteristic but respecting the real attributes of the mechanical gear that allows for optimal cooperation between the drive and load We can see a pair of mechatronical interpretations - Let us suppose a real gear, let us change the attributes of torque-speed characteristic of the motor by the control subsystem, let us suppose the given parameters of the plant (working mechanizm). - Let us control the parameters of technology process, while characteristic(s) of the drive are given. b - investigating the drive parameters change due to internal dynamic of the drive. The mechatronical interpretation supposes a real motor, to find is a gear control(rigidity, damping, impedance)for programmed technological regimes. The goal is to find a control algorithm using variable gear for controlling the drive internal dynamic that allows for optimal mutual interaction anf the plant/load/working mechanizm

15. 2/ investigation of drive influence onto machine aggregate dynamic -investigation of so called external dynamics tasks. Mechatronical interpretation of this kind of tasks: respecting the real plant (technology machine) to control torque-speed characteristics of both the motor and mechatronical gear simultaneously. The goal of this type of tasks is: by controlling the drive to find possible optimal cooperation (interaction) of the drive and plant (working machine) from the technological process point of view.

16. Mathematical models of Rotary Electro-Mechanical Drives /REMD/ with absolutely stiff mechanical components, frequently under use in design and construction, not always allow to determine exactly enough the real character of dynamic activities in electric drives under various operational regimes. It can be documented by a number of research works that motor characteristics for some operational regimes have a significant influence on elastic vibration of the mechanic subsystem. On the other side, elastic vibration of the mechanic subsystem has an influence on electric subsystems of both controlled and open-loop drives. It means that the mathematical model of the REMD and its equivalent, simulation model should consist of: TRANSFER FUNCTION • mathematical model of the mechanic subsystem, i.e. a system of motion equations, describing the behaviour of mechanic subsystem of the drive, • mathematical model of the electric subsystem. i.e. a system of equations describing the behaviour of electric supply (battery or network, converter, auxiliary power circuitry) and control circuitry (controllers, sensors), • system of equations describing external and internal load and parasitic influences on the REMD.

17. As the REMD models are created as objective-oriented and structured dynamic systems (DS), the model description may differ in nonessential aspects. The linear description of the REMD in a matrix form has the next basic form: x(t) is the vector of space variables u(t) is the input vector, vector of inputs y(t) is the output vector, vector of outputs A is the system matrix B is the input matrix C is the output matrix The linear REMD are described by an algebro-differential system of equations, consisting of a system of the ordinary differential equations (ODEs) with constant coefficients and a system of algebraic equations. From the users point of view the resultant mathematical model may have be simply a package of programs, where the main program describes the above equations in a proper programming language. Visually it is a text form. Very useful form of resultant mathematical model, however, is a graphical one, visually a block schematics - let us refer to Matlab and Simulink respectively.

18. Models of the mechanic subsystem The kinematic scheme of REMD mechanic subsystem may be very sophisticated. There is a number of rotating/translating inertial masses, interconnected by shafts/rods, gearboxes and clutches. Let us discuss here but the rotating subsystems. Elasticity and inertia are specific for all the mechanic parts in the drive. They should be treated as objects with distributed parameters and described by a system of PDEs. To simplify the solving process, allowing so to use a system of ODEs with constant coefficients we supose that: • deformation of the mechanic parts are elastic and obey the Hook´s law, • inertial masses of drive parts are „gathered“/„lumped“ into fly-wheels, that are interconnected by elastic and mass-less shafts, • external torques are allowed to drive the fly-wheels, but not the shafts. So, the real systems are reduced onto systems with lumped parameters.and an error is so introduced into the calculation. Anyway, the modeling and analysis become simpler and the results are still well interpretable. The mathematical error may be minimized by the use of convenient methode, say FEM. As a rule, inertity values Ji are „moved,reduced“ from its position and gathered on the motor shaft, so called „main“ part. The interconnections of Ji are mass-less, elastic, characterized by stiffness kij , by internal damping bij and by back-slashes ψij. If needed, external damping due to the dry friction, air-resistance etc. is recognized too

19. Transient Properties of the Rotational ElectroMechanical Drive Transfer Function The overall REMD atributes are described by a system of partial nonlinear differential equations. As a rule, corect and admittable simplifications and abandonings allow to describe many REMD by a system of ordinary differential equations with constant coefficients (nonlinearities, time variancy of parameters, ambient influence etc. can be abandoned). A convenient method allows to implement nonlinearities later, i.e. after creating a running linear model the nonlinearities and additional modifications can be realised. The linearisation allows for principle of superposition saying that the total reaction (behaviour) of the system on the individual excitations (external stimuli) is equal to the sum of reactions on the individual excitations. This aspect is very valuable to analyze systems.

20. Dynamic properties of the REMD can be expressed by: • transfer functions: answer the type of dynamics, displacement of poles and zeros, • transients, time response characteristic: answer the systems response on the input stimulus step, • impulse characteristic: answers the system response on the input impulse • etc. All these options for expressing or visualizing the dynamic properties are described by a differential equation, describing the relation between output and input quantity: y is the output quantity u  is the input quantity ai , bi are constants n is the order of the differential equation, for real systems m ≤ n. Transforming it into tle Laplace transformation we are given a transfer function, i.e. the ratio of the output signal Laplace transform and input signal Laplace transform (ratio of output and input signal pictures) under zero initial conditions. The system defined by has the transfer function as follows: where s is the Laplace operator.

21. The denominator of is the characteristic polynom. It can be written in the form pi are poles of the transfer function. The left side coefficients ai are real, hence the system poles on the right side can be real or complex conjugate twins. Another writing is possible, too: nj are roots of polynom or zeros of the transfer function. They also can be real or complex conjugate. The transfer function can be written evidently also as where G is the gain (amplification) of the transfer function. Negative inverse values of poles and zeros are called time constants Ti. Zeros, poles, gain, time constants tell very much about the system atributes.

22. Transient characteristic- graphical image, output quantity vs time of the solved DE describing the time response of the system on the input quantity unit jump under zero initial conditions. Analytically it is the result of back-Laplace-transformed of output quantity: Pulse characteristic-graphical image, output quantity vs. time, of the solved differential equation describing the time response of the system on the input quantity unit pulse under zero initial conditions. Analytically it is the result of back-Laplace-transformed transfer function The relation between the transfer and pulse characteristics is defined by the the pulse characteristic is the first derivation of the transfer characteristic by time. Also: the pulse is the first derivation of the step function, and the input characteristic is the first derivation of the transfer characteristic. Convolution: Given a signal described by a pulse characteristic. The time response on arbitrary signal u can be found by the convolution. The picture of the output signal is:

23. Then the original of the output signal is: Frequency transfer function-the ratio of Fourier output picture and Fourier output picture for imaginary variable jω , j = √(-1), ω = 2πf, under zero initial conditions. Comparing the frequency transfer function can be found by simple changing in the transfer function the operator s for the jω. Frequency characteristic-graphical interpretation of the frequency transfer function modified by logarithming the expression The complex lin/lin plane is used with ω as independent variable. Often log/lin or log/log planes are more convenient for displaying the next expressions: • resulting into two curves: • amplitude-frequency characteristic F[db] = 20.log|F(jω)| • phase-frequency characteristic φ[ ° ] = arg F(jω) .

24. The frequency axis is as a rule logarithmic, therefore the above two characteristics are called logarithmic-frequency characteristics. The dynamic character of the system is described by the both ones together. However, the phase characteristics of standard transfer functions can be memorized, then the amplitude characteristic only shall be found. Even if in the computers era the time space computation is easy, computation in Laplace (LW=LC=LH) operator space is still “in” thanks to evident advantage. This is why the next text keeps working with transfer functions. Simple REMD examples may help to understand the praxis of Laplace transform.