Energy optimization in jerk decoupled translatory feed axes

Muhammad Ammad Ejaz

Under Supervision of Dr. Chen Silu

What is jerk decoupling

“Decreasing Vibrations with passive filter i.e a spring and damper element” which is normally mounted between movable part of linear motor and frame structure

Mechanical Model

Primary part

Secondary Part

Friction forces

Coulomb (Non velocity

dependent)

Reynolds (Velocity dependent)

Stribeck (Dropping of friction with

higher velocities in low velocity range)

The slide of primary part Mp is guided with longitudinal ball rail system and forces acting on it can be given by following equation

Primary Part

The secondary path comprises two bodies . Jerk decoupled slide and equivalent mass. Following equation can be Set up for equivalent frame

While forces acting on second body of this path is given by

following equation

Secondary Part

Transfer function describes complete mechanical model comprising primary part , jerk decoupling slide , equivalent frame and linear motor .Following is the transfer function of primary equation

In the same manner TF of equivalent and jerk decoupled frame given

&

Transfer Function

Linear motor is being placed between primary path and jerk decoupling slide of secondary path in order to get TF of linear motor

Note : TF of linear motor is very important for the designing of the velocity control circuit

Transfer Function of linear motor

Linear motors in machine are based on synchronous principle for this reason a model of rotating field main machine is applied .Three different ways of controls are commonly used

Trapezoidal commutation

Sinusoidal commutation

Field oriented control

System Design and Simulation

Stator coordinate system

Rotor coordinate system

Motor Model

Rotor coordinate system

Field oriented synchronous linear motor

To get rotor voltages

For getting direct and quadratic voltages

Conversion from stator voltage to direct &quadratic voltages

Model of Linear Motor

If using one pair of poles , equation for motor model is given as

Drive force can be given as

Note: After calculating these currents and drive force another transformation from rotor to orthogonal coordinate system needed

-

-

=

Equations for linear motor model

Visualizing a field oriented control

Energy Observer

Primary & secondary

Part ‘s

Potential Energy

Kinetic Energy

Friction Energy

Potential differential work and potential work are given as

Kinematic differential work and kinematic work given as

And same for friction energy

Energy equations for Primary part

The potential and kinematics energies of EF is similar to primary one but in this case instead of friction there is material damping ,work given as

The work stored in the spring can be given as

Energy equations for Equivalent Frame

Potential and kinematic works are similar to first two parts but in this case elastic work

work caused by damping

work caused by friction

Energy equations for Jerk–decoupling slide

SO this is how it works

Mechanical Model

Synchronous linear direct drive

Energy Observer

Friction is the tangential reaction force between two surfaces in contact

The friction force as function of velocity for constant velocity motion is called Stribeck

The force required to overcome the static friction and start motion is called Break away force

Friction Phenomena

Static friction models

The main idea is that friction opposes motion and it’s magnitude is independent of velocity and contact area and described as

Fc is proportional to normal load i and it is being described as coulomb friction

Viscous friction is used for force component and which is

normally described as

Static Model Examples

Static Model Examples

Static Model Examples

Static Model Examples

It was developed to overcome the zero velocity detection

problem and to avoid switching between different state equations for sticking and sliding .It was developed to overcome different state equations for sticking and sliding

For velocities within internal state may be non zero but output block is maintained at zero

Drawback: This model strongly coupled with rest of system

The Karnopp Model

When a object is subject to stress the friction force increases gradually until rupture occurs

Dynamic Models: The Dahl Model

The friction force is only position force is only displacement dependent

For a time domain Dahl model

For

Dynamic Models: The Dahl Model

Bilman and sorine stress rate independence .The magnitude of friction depends only on sign V and given as

The models are expressed as linear systems in space variable

The first order model is given by

First order model

Model by Bliman and Sorine

The first order model does’t give stiction nor does it give a friction peak this can be achieved by second order

The model is parallel connection of fast and slow Dahl Model

Model by Bliman and Sorine

Neither Dahl or LuGre model are capable of realizing the nondrifting property in presliding.There is model named extension of LuGre model is capable of capturing the property but it does’t solve the problem of lack of nonlocal memory

The Leuven model elaborated the LuGre model further by including hysteresis but hysteresis function Fh while maintaining the LuGre formulation was not with implementing difficulties

Relating the existing friction models to Model structure

The LuGre Model

Rate depende

nt friction

Phenomena of surfaces are pushed

aprt by lubricant

Stribeck effect

Model can be given as

Z denotes the average bristle deflection .The model behaves like a spring for small displacement .Linearization of around velocity and zero state gives

Sigma is the stiffness of bristles and sigma 1 the damping

The LuGre Model

This model overcomes the problem we faced earlier

`

Pros: friction characteristics, stribeck effect in sliding

frictional lag , hysteresis behavior with nonlinear memory which is not shown by LuGre

Maxwell slip model