MECHATRONICS

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MECHATRONICS
Slovak University of Technology Faculty of
Material Science and Technology in Trnava

• Lecture 11

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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 .

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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).

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The machine aggregate as a mechatronical system
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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.

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• Machine aggregates with controlled drives often
  need a more-level, hierarchic control. In the
  basic level of, say a speed system, the (angular)
  speed ? 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.

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• 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 view the
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
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Modelling the machine aggregates

• The whole sequence to create a machine aggregate
  mathematical model has 3 parts
• 1. Modelling of energetic interactions among the
  subsystems

• 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.

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• 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.
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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.

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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(f) and d?/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.
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• In the case of electromotor can be axpressed
  within a small wicinity of working point ? by
  a linear dynamic equation, ML is the loading
  torque originated by the working process reduced
  to the main element. MLcan be expressed

ML? is the average value of the loading
torque, MLa is the amplitude of the variable
loading torque component, ? is angular
frequency of the variable loading omponent

Solving the above motion equation for the ? we
find that it is equal to
Equations are parametric equations of so called
dynamic characteristics of a machine aggregate.
The working point ?(?F, MdynF) 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
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• The mechatronical tasks in machine dynamics can
  be described by two basic physical equations in a
  matrix form

where in general M is matrix of torques u
 is the matrix of voltages
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

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• 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 (controlconvertermotormecha-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 by changing 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

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• 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.

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• TRANSFER FUNCTION

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

• 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.

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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.
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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 Hooks 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 ?ij. If
needed, external damping due to the dry friction,
air-resistance etc. is recognized too
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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.
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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.
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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.
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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
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Then the original of the output signal is
Frequency transfer function - the ratio of
Fourier output picture and Fourier output picture
for imaginary variable j? , j v(-1), ? 2pf,
under zero initial conditions.
Comparing the frequency transfer function can be
found by simple changing in the transfer function
the operator s for the j?. Frequency
characteristic - graphical interpretation of the
frequency transfer function modified by
logarithming the expression
The complex lin/lin plane is used with ? 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 Fdb
  20.logF(j?)
• phase-frequency characteristic f arg
  F(j?) .

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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 (LWLCLH) 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.