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Mechanical Modeling The vibration isolation system was modeled using the diagram in Figure 1. Reaction Mass mr I2 yr Shaker yt Table ks kd I Counter -force Actuator 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 mt yt Fact 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) i 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. mr mr yr Fshaker yt 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 development. 2 Note: Not drawn to scale F2 Target F2 Magnet F1 2 2 lg i Core r1 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 2 Bg 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) lg 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 as: N 2I 2 F1 0 2 Ag1 (8a) 2l g1 N 2I 2 F2 0 2 Ag 2 (8b) 2l g 2 3 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 2l g y t 2 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: 4 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: dI IR L Bly Vs (13) dt Relating the velocity of the counterforce actuator coil to the movement of system mass produces the following equations: dI IR L Blyt Vs (14) dt Eqn.(14) models the current in the counterforce actuator. 5 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 (15) 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 constant). 6 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 excitation. 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 section. 7

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posted: | 12/4/2011 |

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