MECHATRONICS Lecture 09 Slovak University of Technology Faculty of Material Science and Technology in Trnava

MACHINE AGGREGATE DYNAMICS Machine or a machine aggregate is a system, performing some kind(s) of mechanic movement needed for given working process. Machine Aggregate Description Schematics of a rotary machine aggregate

Actual trends and the main designers objectives in machine and machine aggregates construction are: Higher power can be reached by reaching as extreme parameters (in absolute values) as possible and reaching the extreme parameters possible per the smallest volume/mass. The above mentioned parameters are technological and economical, such as: power, speed, efficiency, lifetime, price, reliability, potential market etc.. Due to higher speed the excitig frequences go up too. Due to bigger volume or due to mass-spare the own angular frequences ω go down. There is a danger of resonances. higher speed/revolutions (P = Fv or P = Mω) of the aggregate, bigger volume/mass (P = k 1 V or P = k 2 m) of the aggregate.

Basic subsystems (single-DOF ) are: drive (driving unit), gearbox or other drive-load coupling element (force/torque and speed transmitting unit), working machine (driven unit taken as a working machine or as a working process). A machine aggregate, as a system is composed of a few basic subsystems, in some reasonable combination

A drive consists of a motor (combustion-, hydraulic-, electric- etc.) and its appliances. E.g. an electric drive consists of a converter incl. (transformer) + electric motor/machine + controller(s) + apliances. Note: A machine can be either motor or load, depending on the overall situation and control! This is valid for electrical machines mainly. See another text for more. A gearbox represents a large family of gearing mechanisms. It is to say that motors are supposed to be generators of rotary motion only, even if there is a number of motors generating a translatory/linear motion (linear motors, linear actuators). In general then it is rational to speak of rot-rot, rot-transl, transl-rot, transl-transl gearboxes or any moving conditioning units. A working machine (plant) comes from a very large and versatile family of mechanisms, performing the desired job in production, science, services (e.g. traffic) etc., see another chapter for more.

BASIC TERMS AND CHARACTERISTICS OF MACHINE AGGREGATES Operation of the aggregate, is controlled by an input quantity u, the output parameter is a common ordinate of some output aggregate part: position q d or the speed which performs rotating or translating moving under the common driving force Q d coming from the drive/motor. The output force of the motor (a part of the drive) is characterized a functional dependancy of Q d on parameters u, q d and i.e. the mechanic characteristics If q is an angular quantity, than the common force is the driving torque, so that: for given constant moment of inertia I d.

Working mechanism, load, is more or less complex group of subsystems that change and transfer the moving of the input shaft and torque comming from the gearbox to the rotary or translatory movings of terminal working units needed to perform the production or other processes. The number of input parameters as a rule is equal to number of drives/motors and tells what is the aggregate´s number of „DOF“. The output parameters of the working machine are position coordinates of the individual terminal working units x 1, x 2,..., x n. The terminal working units operates as transformers of the parameters, so that x j = f j (q 1, q 2,..., i n ) j = 1,2,..., n Supposing that we discusse a machine aggregate with single DOF x = f(q).

Working (technological, productional) process is a source of loading forces for the machine aggregate force F z. For the dynamical analysis we shall know the functional dependency of the load on the output parameters and their derivatives, i.e. mechanic characteristics of the load If x, are angular deviation and angular speed, than the common force is the loading torque M z so that the mechanic characteristics of the load (of the working mechanism) is M z = M z (t, φ, ω) The working mechanism has a moment of inertia of its own. It may be  constant, I m = const., the aggregate has the constant parameters,  variable, periodically variable the parameter/position coordinate of the output unit, say φ, then I m (φ) = I m (φ + 2  ), and the aggregate has the variable parameters.

DYNAMIC MODELS OF AGGREGGATES The basic method of creating a dynamic model of mechanic system is „reduction of mass and force relevant parameters“ of the mechanic system. It means, the dynamic model is created supposing that inertial bodies of the system transfer all their masses and moments of inertia onto the driving motor shaft and become „ideal“ i.e. massless and moment of inertia-less. Ideal bodies are then bound together by ideal massless bindings, as the real bindings transfered their ekastic and damping properties onto the driving motor shaft, too. The equivalency law demands the equivalency of kinetic energies, works or powers in real system and its model. In the case of conservative forces there is a demand of equivalency of potential energies. Let us analyze now a system with n output shafts having torques M 1, M 2,....M n having moment of inertia I 1, I 2,....I n and having angular speeds ω 1, ω 2....ω n. The driving motor shaft, i-th in the number and having the angular speed ω i be the main component of the system. Reduction of mass and force relevant parameters of a mechanic systems

Reference torque and reference moment of inertia Reference torque The reference torque is a proportional sum of the of motor´s own loading torque plus a sum of all driven subsystems´ loading torques, whatever is their displacement within the machine aggregate and whatever is their kind of motion, taking the speed ratio into account. Let us use indexes for values of torque or moment of inertia as follows: motor-refreference value on the motor shaft load-motordriving motor´s own value on the motor shaft load-rotvalue for a rotary movement of the loading equipment shaft load-translvalue for a translatory movement of the loading equipment rod motor value of the motor shaft For a suposed ideal efficiency (equal to 100 percent):

Let us use quantity unites as follows: P... [W] M... [Nm] Ω... [rad.s -1 ] or [s -1 ] v... [m.s -1 ] P motor-ref = P load-motor +  P load rot +  P load-transl M motor-ref ω motor =  i (M load-rot-i ω load-rot-i ) +  i (F load-transl-j v load-transl-j ) Hence where i ωi = ω motor / ω load-rot-i is a speed ratio,  vj = ω motor / v load-transl-i.

The reference moment of inertia is a proportional sum of the of motor´s own moment of inertia plus a sum of all driven subsystems´ moments of inertia, whatever is their displacement within the machine aggregate and whatever is their kind of motion, taking the speed ratio into account. E motor-ref = E load-motor +  E load rot +  E load-transl ½ I motor-ref ω 2 motor =  i (½ I load-rot-i ω 2 load-rot-i ) +  i (½ m load-transl-j v 2 load-transl-j ) Hence Remember: From the above equations it is evident that there is no need to analyze the kinematic scheme along the power transfer path of the machine aggregate to investigate the individual ratios i ωi,  vj. All what is needed is to find out just all the i-th values of I i, ω i, and all the j-th values of m j, v j. Reference moment of inertia

Reference of kinematical parameters angular displacement angular speeds i ωi = ω motor /ω load-rot-i is a speed ratio. k j is parameter of stiffness, b j is parameter of damping. Reference of parameters of stiffness and damping angular acceleration

Equations of motion of machine aggregates Basic types of models The common designers practice knows three basic rotational aggregate types. Their models (the models are theoretically correct but admissible simplified descriptions of the real objects, as a rule visually unsimillar with the real objects) are in graphical schematics represented in Fig. elastic system consisting of three bodies elastic system consisting of two bodies single body rigid mechanical element. Graphical representation of mathematical dynamical models of rotational machine aggregates

Motion equations of basic aggregate types The Lagrange equations of the 2 nd kind are very efective in writing the motion equqtions. If the aggregate representing equation has n degrees of freedom and the equation only has holonomous couplings, the equations are of the form general speed E k kinetic energy of the system (function of general coordinates of speed and time) general force, given by the sum of elementery works δA j, of all forces active in the system to make an elementary displacement δq j q j general coordinate where

For a conservative system wit potential energy E P = E P (q 1, q 2,..., q n ). Then and the Lagrange equations can be modified into the form where L = E K – E P is the Lagrange function, also called the kinetical potential. In the case of oscillating systems it is convenient to use the Lagrange functions of the 2 nd kind where D is the disipative (Raileigh) function representing the damping energy of the system Q j * represents the general forces identical to exciting forces in the system (the others are involved in E P and D ).

In the 3-body elastic system with mass the general coordinates are angular deflections φ 1, φ 2, φ 3 and their angular speeds ω 1, ω 2, ω 3. The kinetic energy of the system is Now the elementary work of all the forces and torques for angular shift δφ 1, shall be found for the general force Q 1 δA 1 = (M d – M z1 – M 1 ) δ  1 where M 1 = k 12 (  2   1 ) is the torque of the elastic coupling among I 1 and I 2. Then the general force Q 1 = M d – M z1 – M 1 In the same way general forces Q 2 and Q 3 can be found Q 2 = M 1 – M z2 – M 2 Q 3 = M 2 – M z3 where M 2 = k 23 (  3 -  2 ) is the torque of the elastic force beween   and I 3.

The motion equations of 3-body model of the system. In the case of the 2-body system the system of motion equations can be found from supposing k 1 = k, M 2 = 0, M z3 = 0, I 3 =0 where M 1 = k(  2 -  1 ).