1. MECHATRONICS Lecture 12 Slovak University of Technology Faculty of Material Science and Technology in Trnava

2. ELECTROMECHANICAL PROPERTIES OF SEPARATELY EXCITED DIRECT-CURRENT MOTOR 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.

3. 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 Figure. Dynamic model of machine aggregate with a separate excited DC electric motor and its torque characteristics

4. The motion equation of the machine aggregate model with stiff bindings (k → ∞) is described by a differential equation I  -reduced inertia moment of the machine aggregate including DC motor,  -mechanical angular speed of the machine aggregate (k → ∞), M d -driving electromagnetic torque of the DC motor, M z -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 M 0 is the amplitude of variable loading torque component and Ω is the loading torque angular frequency.

5. The dynamic torque characteristics of a separate excited DC motor have a form - DC motor armature circuit inductance (the sum of inductances), - electromagnetic time constant of the electric drive, - static torque of the DC motor - no load angular speed, - DC motor armature circuit resistance (the sum of resistances), U- DC motor armature source voltage, k- DC motor designing constant, Φ- exciting magnetic torque of the DC motor. Then This equation is of the 2 nd order and allows for analysing new qualities, say electro- mechanic resonance. Its solution with respect to ω and with notation c = kΦ-DC motor constant,, - Laplace operator, - electromechanical time constant, - stiffness of the static characteristics,

6. 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

7. 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

8. After writing p = iΩ the corresponding amplitude-frequency characteristics is where, hence also Figure represents amplitude-frequency characteristics for a DC motor of 200 W and parameters I d = 2.5.10 -5 [kgm 2 ]  d = 0.05 [s],  m = 0.02 [s]. Amplitude-frequency characteristics for a DC motor

9. On following Figures are represented the amplitude-frequency characteristics for I Σ =2I d and I Σ = 4I d 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. Amplitude-frequency characteristics for DC motor a) ; b)

10. Electromechanical resonance The nature of transients are determined by the roots of characteristic equation in the transfer function Hence For m < 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  m smaller and  d bigger. During dynamic modes of operation, due to periodical component of loading torque or controlling signal, in the area around  0 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

11. DYNAMIC PROPERTIES OF A GEARED MOTOR 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 system. Dynamic model of geared motor Dynamic model of the geared motor with spur gearing

12. Equations of motion of this geared motor as a machine aggregate using a linear dynamic characteristics of an electric motor are where I 1, I 2 - reduced inertia moments of the rotor and the loading mechanism,  1,  2 - angular displacement of the shafts,  1,  2 - mechanical angular speeds of the electric motor and driver shafts,  0 - synchronous angular velocity of the induction motor or angular no load speed of the DC motor, M d - total reduced driving (electromagnetic) torque of the motor, M z - total reduced loading torque, M z1 - resistance torque, k- rigidity of elastic binding, M k - torque of elastic binding M k = k(  1 -  2 ) = k ,  - total reduce clearance and kinematic bindings, p = d/dt- differential operator, T e - electromagnetic time constant of the electric motor,  - rigidity of the static torque characteristics of the motor.

13. where M z is the total reduced loss of torque in the motor, is the maximal angular deviation of the gearing. Dynamic loading in the geared motor Now the change of loading nature can be shown, taking constant of aggregate parameters into account. Here M d = const, M z1 = const, M z = const while k is the finite rigidity. Equations of motion of the model have then the form: Influence of the kinematic deviations The total reduced clearance in kinematic bindings can be expressed by where  is total reduced clearance,  j - real clearances in gearing expressed by a total deflection of the j-th element, - gearing ratio of the main and j-th element.

14. Value of the clearance  j depends on the gearing module. By reduction to the motor shaft and reduction down (i.e. multiplying by the gearing ratio ) it has a dominant impact to the total reduced clearance. 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  max 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  max is small and the corresponding change of  Mmax 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  max >  0, (  0 is the angular eigenfrequency) the case can occur that  0 = , 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 d  2r /d  2 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 d  2r /d  2 »1.

15. 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 coefficients. Then the system is given an adjusted form The system of equations is linear with a constant excitation (amplitude)  M = k  max affecting both discs in counterphase with a frequency  - proportional to angular velocity of the rotor. It is to be emphasises that the excitement is a constituent of elastic binding torque.

16. 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 I 1, I 2. In the simplest case of machine aggregate parameters being constant, the body I 1 runs in uniformly accerelated rotary movement with an angular velocity of where   is angular acceleration of the body I 1. To overcome a clearance the body I 1 needs time t, during which the angular speed w1 is changed to  1z according

17. During the time t is the body I 2 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. M d = const., M z = 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.

18. Solving for initial conditions and for a rigid binding k  is where - amplitude of periodic value of elastic binding torque. - average value of loading, From the solutions results that M k 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 M z  - (a static load) is bigger.

19. The fact is expressed by the so called dynamic coefficient For the process watched and a maximal value of elastic binding torque according to K d can be evaluated Resulting from this equation the dynamic coefficient grows with increased value   and inertia I 2. It can be more times bigger than the static load in the case of mechanisms of big inertia. 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

20. 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 M z  This equation yields the fact that with a given inertia I 1 and reduced clearance  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