MECHATRONICS

Lecture 10

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

STATIC & DYNAMIC CHARACTERISTICS OF DRIVES AND LOADS

The external attributes of both main aggregate components are defined by mechanical characteristics, giving the relation between main quantities. The main quantities are mostly: load torque Mz, drive/driving torque Md, position φ and its 1st

and 2nd derivations - angular speed ω, angular acceleration ε .

The operating regime of the aggregate is given by the nutual relation of mechanical characteristics of drive and load.

Practical equation to solve the dynamics of an aggregate is

Mdyn = Md(φd, ωd) – Mz(φ, ω)

where Mdyn is the reduced dynamical torque of the system - inercity of the system

and its motional state given as Idω/dt or more exactly I(dω/dt) + ½(dI(φ)/dφ)ω2.

Mechanical characteristics of the plant, load, representing a sum of torques Mz due

to all the forces affecting the aggregate. Mz in general depends on the (angular)

velocity ω and periodically on the (angular) position φ of the output element

Mz(φ, ω) = Mz(φ+2, ω)

Mechanical characteristics of the drive(s), defined as functional dependance of the driving torque Md from (angular) speed ω of the output element of the drive.

Md = Md(ωd)

Mechanical characteristics of the plant load Loading torques and their specification

The general expression for the loading torque is

Mz(φ, ω) = MC(φ) ± MR(ω) = MC(φ) - sgnω MR(ω)

Given that the loading torque Mz(φ,ω) can have 2 separated members,

• one, dependent on the angular position φ of the main/reference element only, MC(φ)

• the other, dependant on the angular speed ω of the main/reference element only, MR(ω).

The first member MC(φ) includes torques being in context with the change of

potential energy, in context with conservative forces of the aggregate.

The second member MR(ω) includes torques originated in the technology process

(e.g. plastic deformation as forming, shaping) and friction torques. They change their sign after the speed direction is changed, they effect always against the movement of the main/refference aggregate element.

Hence,the loading torques can be devided into two groups:

- active loading torques

- passive (resistance loading torques).

Resistance/passive loading torquesClasification of plants from the mechanical characteristics point of view

In general, for the single direction of revolution, funcion of load vs. (angular) speed can be expressed as

MT0 - is friction torque at zero speed

MzN - is rated loading torque (at rated (angular) speed ωN)

x - is an exponent indicating a specific mechanical characteristics form

Often MT0 = 0 , then the equation is

MR(ω) = c ωx

where c = MzN / ωNx is a constant and

depends on the type of the plant.

The equation allows for better classification of mechanical characteristics of plants, that are as a rule symmetrical nonlinear functions

Characteristics No 1 ... x = 0

The characteristics is independant on the speed (MR = const.). The characteristics is valid for elevating equipments, traction units of cranes, distributors, feeders etc. In general: equipments moving at low speed along horizontal track. Also the dry (Coulomb) friction by single direction move can be of constant load with MzN = MT0 gets the form

MT(ω) = MT0 sgn ω .

This characteristics appears e.g. when displacing a mechanical part and by some kinds of shaping machines, where MT0 may come from a large interval of values.

Characteristics No 2 ... x = 1

Load is linear and proportional to the speed. This is the case of viscous-friction forces and torques.

Characteristics No 3 ... x = 2

Load is non-linear, increasing with speed, valid for fans and centrifugal pumps. Characteristics are also valid for movement/traction in gases or liquids, expressing there the resistance of medium.

Characteristics with ... x = 3

For high speed airplanes, hydroturbines etc. (Mach No, cavitation).

Characteristics No 4 ... x = -1

The load falls hyperbolically with the speed. This characteristics is valid for a group of winding, reeling equipments in paper, textile, steel industry.

The above quoted characteristics describe but the typical and basic loads from industry. Beside them there is a large number of other characteristics, that are a combination of these above basic characteristics. The total load of such ones is a sum of partial loads for individual x-es, as appropriate for the given case of plant:

Mz,total = k(-1)ω-1 + sgn(ω).k0.ω0+k1ω1+k2.ω2 +...

As an example: the basic components for a reeler are those for winding

(x = -1) and for friction in bearings (x = 0), hence

Mreel = k(-1)ω-1 + k0ω0 = k(-1)ω-1 + k0

Active loading torques

Quite frequent in aggregates are loads with periodically variable component. This

kind of load depends as a rule on the angular position of the reference/main

element of the aggregate and is a result of- periodical change of external load - non-linear kinematical coupling - crank-, cam- or slotted-link mechanisms.

In the case given we have supposed that active loading torques depend

periodically on angular position φ of the reference element, so that

MC(φ) = MC(φ+2)

This component of loading torques, expressed in the form of Fourier sequence

For a number of machinery aggregates the load torque from plant resistances is periodical and depends on the time t.

The analysis of permanently effective harmonic load is very important for reliable dynamics of machine aggregates

MC(t) = MC0 sin ωt for t 0

Typical technological loads of machine aggregates

The periodic functions represent external (technological) loads of machine aggregates and can be also expressed in the form of Fourier sequence

Torque characteristics of the drivesAn aggregate is driven by a drive. Whatever kind of drive it is, for solving the dynamics of the aggregate we need to know but those attributes of the drive, which are relevant for mutual cooperation of aggregate and drive. Now our choice is an electric drive. An up-to-date electric drive is a system consisting from subsystems: electric motor producing the driving torque, a converter supplying the motor, a control unit incl. sensors for feedbacks. The drive subsystem representing the electromechanical energy conversion is an electric motor. Thus we have to know the relevant characteristics of the electric motor. Again, instead of a real motor we work with a model of a motor. If the attributes of a motor are given by couplings between input and output parameters of the motor, we work with so called external model of a motor). Main input and output quantities are:

input quantity u(t) energising the processes of electromechanical energy conversion,

law for changing the value of output quantity qd(t) ,

and general driving force Qd(t) or the torque

Md .Schematics of an electric motor

Ideal characteristics of motorsThe angular or linear speed of the output element be in any instant dependant

only on the input quantity u. Then the ideal kinematic characteristic of the motor is dq

)(ufqd

This characteristics describes approximately the motor attributes and is useful for the first stage of designing the motor. Can be used for drives where the load has abbandonable influence on the motor speed.

Supposing the general driving force Qd is not dependant on the speed of the

output element and is a function of input quantity u, the ideal force characteristics of the motor is

dq

)(uQQ dd

b) a)

0dM

dM

0q

dq

u0 u0 u u

a)ideal kinematic characteristic,

b) ideal moment characteristic

If qd is the position coordinate of the output element, the general force is the driving torque and characteristics by the form

Static motor characteristics express the relation between the constant speed of the output element for the constant value of quantity u. They can be used for load changes or speed changes ale slow.

),( dds quM ),(3 dMuf

b) a)

0dsM

0dsM

dsM

0q

dq

Md

0q

dq

u0

u u

u0

a) static ideal characteristic, b) static moment characteristic

Dynamic characteristics of motorsDue to inertial electro-magneto-mechanical processes of energy change inside the motor, the speed value of the output element in the given time instant depends not merely on the actual load value, but also on the law of energy change in time. This fact can be taken into consideration by inserting the derivation of general driving force by time into the characteristics of the motor

dq

),d

d,(3 q

t

QTQufq d

dd

or

),,(d

dqquQQ

t

QT dsd

d

),,(d

dqquMM

t

MT dsd

d

Parameter T is the motor time constant and equations are the linear dynamic characteristics. Without the parameter q in the static characteristics the dynamic characteristics are simpler:

),(d

dquQQ

t

QT dsd

d

Linearization of motor characteristics

Linearity of characteristics is an ideal case, unfortunately static characteristics of (electro)motors are rarely linear. Some are in reality or at least theoretically (in some approximation, within some limits, under some conditions etc.) linear or linearizable.

If the characteristics are not linear, they can be linearized ina vicinity of some (working) points

Coefficients β are co called rigidity of the static motor characterstics in the given point of the characteristics. In fact it is the direction of the tangent in the given point, as a rule in a working point of the characteristics

Linear dynamic motor characteristics: qurMt

MT dd

d d

d

Torque characteristics of induction electromotor

The induction electromotor, due to its simple construction is a well known motor for machine aggregates.

This is why i tis worth of analyzing its torque characteristics. A dynamic model of an aggregate with induction electromotor is a two-disk elastic system To specify the driving torque Md(ωd) we need besides the moving equations also equations of the induction electromotor torque characteristics.

Ideal characteristics AC motors

p

f

pq f

ss

2

f - is frequency of supplying network voltage,

p - is number of pole pairs (number of poles = 2p),

ωs - is synchronous angular speed of the motor.

Dynamic model of a machine aggregate with induction electromotor (with AC drive)

Static characteristics of AC motors

Static torque characteristic of induction electromotor

Rather than speed, with induction motors the slip i sused, defined as

s

d

s

dss

1

The static (torgue) characteristics in the form of full form Kloss characteristics is

KK

K

KKds

sas

s

s

ssaM

M2

)1(2

s  slip sK is critical slip corresponding to MK

MK maximal/critical toque, a ratio of real resistances of stator and rotor windings

The linear(ised) torque characteristics can now be written in the form

)(d

ddsdd

d Mt

MT

Kse sp

T

1

is electromagnetic time constant of the motor,

Ks

Kd s

M

2is rigidity coeficient of the static characteristics.

Dynamic models and static torque characteristics of DC motors

The linear(ised) torque characteristics can now be written in the form

)(d

ddsdd

d Mt

MT

For the DC motor with external excitation

For the DC series and compound motors the linearization in some point, say vicinity of static equilibrum