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Slovak University of Technology Faculty of Material Science and Technology in Trnava MECHATRONICS Lecture 12 – PowerPoint PPT presentation

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Slovak University of Technology Faculty of
Material Science and Technology in Trnava
  • Lecture 12

  • An electric drive is a kind of electromechanical
    system consisting of three components
  • electrical motor,
  • mechanical part (gearing etc.), 
  • control system providing for an optimal control
    of technological process.
  • Electric motor becomes the most important part of
    the machine aggregate. Working quality of the
    drive, accuracy of performed technological
    operations too, even the dynamical load of all
    the mechanical parts in the machine aggregate
    depend on its static and dynamic characteristics.
    In the contribution, influence of separate
    excited DC motor parameters on machine aggregate
    dynamics is observed.

Dynamic model of a machine aggregate with
a direct current motor
A dynamic model of an aggregate with a DC drive
using separate excited direct current electric
motor and its torque characteristics are on
Dynamic model of machine aggregate with
a separate excited DC electric motor and its
torque characteristics
  • The motion equation of the machine aggregate
    model with stiff bindings (k ? 8) is described by
    a differential equation

IS - reduced inertia moment of the machine
aggregate including DC motor, w - mechanical
angular speed of the machine aggregate (k ? 8),
Md - driving electromagnetic torque of the DC
motor, Mz - loading torque on the DC motor
shaft, k - stiffness of elastic coupling.
The following text describes an exact and
simplified way of solving for a simple system
with constant inertia (I const.) and harmonic
load torque
where M0 is the amplitude of variable loading
torque component and O is the loading torque
angular frequency.
  • The dynamic torque characteristics of a separate
    excited DC motor have a form

This equation is of the 2nd order and allows for
analysing new qualities, say electro-mechanic
resonance. Its solution with respect to ? and
with notation
c kF - DC motor constant,
- electromechanical time constant,
- stiffness of the static characteristics,
- Laplace operator,
Operator form looks as follows
  • The motion of the aggregate driven by the DC
    drive is described by a equation set (supposing
    stiff couplings)

or by an equivalent blocs schematics
Block scheme of the machine aggregate with the
separately excited DC motor
Using the expression for the static
characteristics of the DC motor we can write the
equations in the next form
Amplitude-frequency characteristics of the
machine aggregate
  • The second component of operators equation refers
    to dynamic difference of angular speed from the
    steady state, in this case from no load speed.
    The transfer function with respect to loading
    torque is

For the static characteristics the transfer
function has the form
The natural circular frequency is
After writing p iO the corresponding
amplitude-frequency characteristics is

Figure represents amplitude-frequency
characteristics for a DC motor of 200 W and
Id 2.5.10-5 kgm2
td 0.05 s,
tm 0.02 s.
Amplitude-frequency characteristics for a DC
  • On following Figures are represented the
    amplitude-frequency characteristics for IS 2Id
    and IS  4Id respectively. It is obvious that
    amplitude-frequency characteristics and hence
    running unsteadiness solved with static and with
    dynamic characteristics are significantly
    different, the second giving an abrupt increase
    of unsteadiness of the angular velocity in an
    area of aggregate parameters, yielding a quite
    new dynamic effect - electromagnetic resonance.

Electromechanical resonance
The nature of transients are determined by the
roots of characteristic equation in the transfer
For m lt 4 the equation has complex roots and the
electric drive can be supposed to be an
oscillating system with the damping equal to
ratio of the both time constants. Smaller
armature circuit resistance makes tm smaller and
td bigger. During dynamic modes of operation, due
to periodical component of loading torque or
controlling signal, in the area around W0 the
amplitude of oscillation increases sharply
(unsteadiness of mechanical angular speed and
driving torque of the machine aggregate) due to
the resonance in spite of absolutely stiff
couplings in mechanical part of the aggregate.
The phenomenon can be observed in the Fig.
showing the amplitude-frequency characteristics
for various values of m.
The resonance is visible for m 2. Within the
interval 0.64 m 2 the damping is evident.
Further decrease of m causes an increase of
resonance peak
  • A machine aggregate is a dynamic system,
    consisting as a rule from a driving machine,
    gearing mechanism with its binding, controlling
    and commanding accessories and a driven plant.
    The system characteristics, as well as
    characteristics of the individual subsystems are
    a result of their mutual accouplement and
    interference during operational activity. The
    system characteristics depend not only on the
    subsystems' initial characteristics, on their
    depreciation and overall status within the
    relevant time, but mainly on external phenomenon
    valid for the operating time of the system. The
    gearing, a most common reducing mechanism of a
    machine aggregate, is often an element
    determining the dynamic attributes of the whole

Dynamic model of geared motor
Dynamic model of the geared motor with spur
  • Equations of motion of this geared motor as a
    machine aggregate using a linear dynamic
    characteristics of an electric motor are

where I1, I2 - reduced inertia moments of the
rotor and the loading mechanism, j1, j2 - angular
displacement of the shafts, w1, w2 - mechanical
angular speeds of the electric motor and driver
shafts, w0 - synchronous angular velocity of the
induction motor or angular no load speed of
the DC motor, Md - total reduced driving
(electromagnetic) torque of the motor, Mz - total
reduced loading torque, Mz1 - resistance
torque, k - rigidity of elastic binding, Mk -
torque of elastic binding Mk k(j1 - j2)
kDj, Dj - total reduce clearance and kinematic
bindings, p d/dt - differential operator, Te
- electromagnetic time constant of the electric
motor, b - rigidity of the static torque
characteristics of the motor.
Dynamic loading in the geared motor
Influence of the kinematic deviations
  • Now the change of loading nature can be shown,
    taking constant of aggregate parameters into
    account. Here Md const, Mz1 const, Mz
    const while k is the finite rigidity. Equations
    of motion of the model have then the form

The total reduced clearance in kinematic bindings
can be expressed by
Equations in an autooscilation form allows for
analysing the influence of kinematic deviations
to the movement of the drive. The kinematic
deviations are a source of internal periodical
exciting forces with amplitude proportional to
the maximal angular deviation Djmax growing with
increased rigidity of the elastic binding k. The
frequency of the excitement is proportional to
the motor's rotor angular velocity In most cases
Djmax is small and the corresponding change of
DMmax is small too, it could be even neglected,
but the increase of dynamic load caused by
kinematic inaccuracies turns to be dominant in
the case of resonance.
If the frequency of excitement at maximal
operating speed of the motor wmax gt w0, (w0 is
the angular eigenfrequency) the case can occur
that w0 w, i.e. the resonance occurs. So an
increased dynamic load from gearing inaccuracies
results in decrease of lifetime and in failures
of drives. Also, positioning of machine aggregate
is less exact and production process
worsened. The system of equations due to
existence of dj2r/dj2 is non-linear. In praxis,
pulsating value of the gearing ratio is small
(2-5 ) and so for purposes of analysis of
machine dynamics it is allowed to assume that
dj2r/dj2 1.
Then, we have
The there-in-before assumption allows transiting
from the system of differential equations with
variable coefficients to a system of linear
differential equations with constant
Then the system is given an adjusted form
The system of equations is linear with a constant
excitation (amplitude) DM  kDjmax affecting both
discs in counterphase with a frequency w -
proportional to angular velocity of the rotor. It
is to be emphasises that the excitement is a
constituent of elastic binding torque.
From the there-in-before debate on kinetic
conditions of geared machine aggregates results,
that kinetic equations are structurally analogy
to machine aggregate kinetic equations with
variable gear ratio. For cylindrical gears,
existence of - inevitable minimal clearances
due to production, - possible changes in tooth
dimensions due to heating is typical. Parameter
values grow at work, hence an adverse influence
to working conditions of the drive. Due to
clearances generated strokes causes additional
dynamic load for a part of the drive. Correct
determination of dynamic loading components is of
major importance for increasing the gearing
reliability and the life-time. Fundamentally the
most important step is determination of elastic
binding torque-of transferred torque-among
individual elements of machine aggregate.
Influence of the gearing clearence
The clearances in kinematic bindings and gearing
cause transients within the aggregate. During
meshless run (no contact of teeth) no mechanical
binding exists between bodies I1, I2. In the
simplest case of machine aggregate parameters
being constant, the body I1 runs in uniformly
accerelated rotary movement with an angular
velocity of
where eF is angular acceleration of the body I1.
To overcome a clearance the body I1 needs time
t, during which the angular speed w1 is changed
to w1z according
During the time t is the body I2 in standstill or
in an uniform movement. Hence, after the meshless
run an elastic stroke occurs and the kinetic ener
accumulated is changed to elastic deformation of
e.g. teeth and to heat, hence the dynamic load is
increased. If the instant of leaving the meshless
run is the initial time t 0, than the
transients describes the start of the system.
Dynamic model of geared motor with gearing
clearance consideration
Considering parameters of the aggregate to be
constant, i.e. Md const., Mz const. and
finite rigidity k the system can be modified to a
form for elastic binding torque
is a natural angular frequency,
is an average value of the angular acceleration.
Solving for initial conditions
and for a rigid binding k is
- average value of loading,
- amplitude of periodic value of elastic binding
From the solutions results that Mk elastic
binding torque is a periodic function. It means
that in a machine aggregate additional mechanical
oscillation, hence strokes come to existence.
This is why the value of maximal torque compared
with the average value MzF - (a static load) is
The fact is expressed by the so called dynamic
For the process watched and a maximal value of
elastic binding torque according to Kd can be
Resulting from this equation the dynamic
coefficient grows with increased value eF and
inertia I2. It can be more times bigger than the
static load in the case of mechanisms of big
Considering all the clearances in kinematic
bindings and gearing, the elastic binding torque
is a periodical function, as confirmed by
existence of additional mechanical oscillation
and stroke load of the mechanical
aggregate. Solving for initial conditions
and for rigid k is solution in following for
After overrunning the clearance an elastic stroke
comes into action and the accumulated kinetic
energy is changed to elastic deformation of e. g.
teeth and to heat, hence an increased dynamic
loading. The change of the loading is
characterised by the dynamic coefficient, defined
as a ratio of maximal elastic binding torque and
the average value MzF
This equation yields the fact that with a given
inertia I1 and reduced clearance Dj the gearing
dynamic load depends - on the acceleration in
the instant when the clearance is overrun and -
on the ratio of inertia of the drive side and
load side. The real load than may be more times
bigger than the static one. The problems treated
in the article has been oriented to the analysis
of impacts caused by kinematic and dynamic
deviations in gearing to dynamic attributes of
machine aggregates. As shown, a mechanical
gear-box in a machine aggregate is a source of
excitations with a large scale of frequencies. A
mechanical part of the drive with respect to an
electrical one is a controlled subject. The
control shall provide for (ensure) an optimal
moving modes of the system withs respect to a
given technology process