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					Mechanical Modeling
The vibration isolation system was modeled using the diagram in Figure 1.
                               Reaction Mass

                         I2                                                  yr


                                                 ks               kd
                           I                           Counter

                                                       Bed Plate

                                   Figure 1. Diagram of System Model.

The mass of the shaker, reaction mass, and the table will be lumped together as, mt.

                                mt g                  Fshaker


                                 k d yt   ksyt

                                   Figure 2. FBD of Lumped Mass, mt.

Summing the forces for mt in the direction indicated in Figure 2 produces:
                      Fy   k d y t  k s y t  Fact  Fsha ker  mt g  mt t .
                                                                             y         (1)

Isolating the constant term produces:
                            mt t  k d y t  k s y t  Fact  Fsha ker  mt g
                               y                                                       (2)

In Eqn.(2), Fact is the force applied by the counterforce actuator to the table and Fshaker is the force
produced by the shaker. The damping and stiffness of the support structure are represented by kd
and ks, respectively.



                               Figure 3. FBD of Shaker Reaction Mass

Summing the forces in Figure 3 for the reaction mass, mr, produces:
                              Fy  Fsha ker  mr g  mr ( r  t )
                                                           y     y                              (3)

Isolating the constant term produces:
                               mr ( r  t )  Fsha ker  mr g
                                    y     y                                                     (4)

Electromagnetic Modeling
To properly model the behavior the system and determine the parameters of the electromagnetic
counterforce actuator, the force in the counterforce actuator must be determined.

Two separate models were developed for the counterforce actuator. Both models assumed an
overall cylindrical shape to maximize the efficiency of the coils. The first model is for an
electromagnet type actuator. It assumes a stationary coil, and a moving magnetic target. The top
and cross section views shown in Figure 4 define the parameters used for the model

                                             Note: Not drawn to scale

        F2     Target                       F2
        2                                   2
                                                    Coil with                   r3 r2
                                                    N turns

                    Iron Stator                                             Target magnet removed
                                                                            to show coil detail.

      Figure 4. Cross Section and Top Views of the Electromagnet, Counterforce Actuator

 The force of on a ferromagnetic material due to a magnetic field gives
                                          F          Ag                                              (5)
                                               2 0
where Bg is the flux density of the air gap, µo is the permeability of free space, and Ag is the cross
sectional area of the corresponding air gap.16 All quantities in Eqn.(5) are constant for a given
geometry except for the flux density. The flux density is given by
                                                Ni
                                          B 0                                                        (6)
where N is the number of turns in the coil, I is the current in the coil, and lg is the size of the air
gap. Substituting Eqn.(6) into Eqn.(5) produces:
                                                N 2I 2
                                        F  0        2
                                                        Ag                                            (7)
                                                 2l g

The cross section above shows the total force on the target split into 2 components, F1 due to the
stator core and F2 which is distributed around the stator rim. Ignoring fringing, the total flux will
be the same throughout the stator core and the air gap. Using Eqn.(7), the forces can be written
                                                N 2I 2
                                       F1   0    2
                                                       Ag1                                       (8a)
                                                2l g1
                                                 N 2I 2
                                        F2   0    2
                                                        Ag 2                                        (8b)
                                                 2l g 2

Assuming the target magnet stays parallel with the top of the stator, the air gap will be the same
for forces F1 and F2. Inserting the cross sectional areas in Figure 4 and summing the two forces
produces the total force from the actuator.
                                             N 2
                                                                  
                                     Fact  0 2 r12  r32  r22  i 2                            (9)
                                             2l g

The air gap, lg, is changing as the target magnet moves with the lumped mass, mt. To account for
this, the term lg is replaced with (lg + yt). The total actuator force becomes

                                   Fact  o
                                                              
                                            N 2 r1 2  r3 2  r2 2 2
                                                2l g  y t 
                                                                     I                     (10)

Knowing the geometry of the stator, the number of turns in the coil, and the position of the target
magnet, the current required to produce the desired force can be determined.

The limitations of this type of actuator are increased modeling complexity due to non-linearity,
introduction of difficult to quantify parameters, and limited effectiveness due to the nature of the
generated force. The force produced is an attractive force, which cannot be reversed. For the
above reasons, a second model was developed for a voice coil type actuator. This type of
actuator can produce a reversible force. Also, with appropriate design considerations its
behavior can be accurately modeled with a simple linear relationship.

Parameters used to model of the voice-coil actuator are defined in
Figure 5.

                   Figure 5. Cross Section of the Voice Coil, Counterforce Actuator
The force on the moving coil in
Figure 5 is proportional to the magnetic flux density, B, the length of wire in the coil, l, and the
current in the coil, I. The actuator force is modeled as:

                                         Fact  B  l  I                                      (11)

Substituting the expressions for the counterforce actuator and shaker forces into the equation of
motion for the lumped table mass, Eqn.(2), produces:
                               mt t  k d yt  k s y t  BlI  FSha ker  mt g
                                  y                                                           (12)

As the coil in counterforce actuator moves up and down, it moves through a magnetic field. This
produces a back electromotive force (emf), which counteracts the supplied voltage. The coil also
has resistance, R, and inductance, L, affecting the flow of current through the supply circuit.
Thus, the dynamics of the circuit supplying the current to the counterforce actuator coil must also
be modeled. The schematic shown in Figure 6 is used to develop the response of the circuit used
to drive the counterforce actuator.

                              Figure 6. Schematic of Voice Coil Circuit

The emf is proportional to the velocity of the coil, the magnetic flux density, and the length of
the coil wire. Summing the voltages around the loop in Figure 6 produces:
                                      IR  L  Bly  Vs                                        (13)

Relating the velocity of the counterforce actuator coil to the movement of system mass produces
the following equations:
                                       IR  L  Blyt  Vs
                                                                                           (14)

Eqn.(14) models the current in the counterforce actuator.

Combined State-space System Model
Finally, the total system then can be modeled using Eqns.(4), (12), and (14). To facilitate
transformation into state space, the equations are solved for the highest order derivatives in each.
Solving for the highest order derivatives in each produces:

                                 k        k         Bl   F          
                          t   d  yt   s  yt    I   Sha ker
                          y      m  m             m   m            g
                                  t        t        t        t      

                                             F          
                                       r   Sha ker
                                       y                  t  g
                                                          y
                                              mr        
                                           R   Bl      V
                                      I    I    y t  s
                                            L  L         L

In state space this requires five variables defined as:

                                                yt  Q0
                                                yt  Q1
                                                y r  Q2                                        (16)
                                                y r  Q3
                                                I  Q4

Using the state variables in Eqn.(16), the state equations became:
                             Q0  Q1

                                  k       k            Bl   F            
                             Q1   d Q1   s
                                   m       m      Q0   Q4   Sha ker
                                                          m      m           g
                                    t       t           t        t        
                             Q2  Q3                                                            (17)

                                 F         
                             Q3   Sha ker   Q1  g
                                   m 
                                      r    
                                    R         Bl  V
                             Q4   Q4   Q1  s
                                     L         L   L

Velocity proportional control was used to simulate the system by replacing the counterforce
actuator voltage supply, Vs in Eqn.(17), with the quantity k*Q1 (where k is the proportionality

The control method used for the shaker allowed the force, FShaker, to be modeled as a simple sine
function with the amplitude and frequency equal to the magnitude and frequency of the desired

Extensive testing was done to determine the voltage needed to produce specified excitation force
over the frequency bandwidth. Using the quartic regression function on a TI-83 graphing
calculator, two 4th order expressions were combined to create a 4th order, piece-wise function,
relating the frequency and the output voltage for a specified force. This relationship was then
entered into a LabView control program.

The definition and determination of the model parameters is shown in the System Calculation


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