ACTIVE VIBRATION CONTROL OF A FLEXIBLE PLATE SYSTEM. A Thesis Presented. Charles Anthony Sidoti

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1 ACTIVE VIBRATION CONTROL OF A FLEXIBLE PLATE SYSTEM A Thesis Presented by Charles Anthony Sidoti to The Department of Mechanical and Industrial Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering in the field of Mechanics Northeastern University Boston, Massachusetts August

2 Abstract Vibration control and/or attenuation can significantly improve the performance and operation of systems and machines in various industries. As technology advances, the methods of vibration control also become more involved and therefore allow for control of more complex structures. This thesis focuses on vibration control of a flexible plate system. The basic modeling of the physical system is developed as a precursor to the controller design. Linear controllers are explored along with advanced controllers. These controllers require an accurate model of the physical system of interest. For this reason, an experimental setup is designed and developed for implementation. A major part of this thesis is devoted to the experimental design, setup, and implementation of different controllers to achieve broadband attenuation within a frequency range from 400 to 2000 Hz. An experimental setup is designed to demonstrate attenuation of flexible modes of the plate system. Actuators, accelerometers, and force sensors are specified to apply appropriate disturbance forces of varying complexity and to measure and record data necessary to update the theoretical model and design an effective controller. The experimental setup is initially made up of three relatively large stacked aluminum plates. After proof-of-concept results are obtained, the setup is scaled down to represent a more applicable system. The miniaturized setup is made up of a large base plate to distribute the input forces and a small flexible plate to be controlled. The control scheme involves non-collocation control of a location to which the accelerometer is mounted. With a desired range of attenuation, the controller is designed and developed. The initial stages involve experimental development of a PID controller using time delays and filters to attenuate multiple resonant frequencies. The controller showed 58% and 81% attenuations of the first two resonant peaks at 618 Hz and 1001 Hz, respectively. These results represent a proof-of-concept using the measured force input as feedback to the controller. In parallel, a model reference adaptive controller is explored, and data is recorded for later use in updating the model for increased accuracy. Significantly greater 2

3 attenuation is envisioned with the completion of the development of this advanced controller. 3

4 Acknowledgements I would like to acknowledge my advisor Dr. Nader Jalili for his help, teaching, and support throughout this thesis. I would like to acknowledge Troy Lundstrom for his work in conjunction with myself for a majority of the experimental results obtained in this thesis. His work in modeling the system as well as analyzing results was paramount to the project. I would like to thank my thesis advisory committee members, Dr. Rifat Sipahi and Dr. Bahram Shafai, for their help in preparation for my thesis defense. I would like to acknowledge Matt Jamula for his work previously completed in relation to this project. 4

5 Table of Contents Abstract... 2 Acknowledgements... 4 Table of Contents... 5 Table of Figures... 7 List of Symbols Introduction Theory Dynamic System Modeling Distributed-Parameters System Modeling Lumped-Parameters System Modeling Control Theory Linear Controllers Advanced, Robust, and Adaptive Controllers Experimentation Setup, Design, and Component Selection General Setup Components Selection Suspension Design System Characterization Control Schemes Initial Results Inertial Actuator Patch Actuator Miniaturized Setup

6 3.5.1 Miniaturized Experimental Setup Miniature System Characterization Results Discussion of Results Future Work Appendix A: The Modal Shop 2007E Actuator Specifications Appendix B: PCB 208C01 Force Sensor Specifications Appendix C: PCB 352C22 Accelerometer Specifications Appendix D: ANSYS Flexible Excitation Results Appendix E: Data Acquisition Block Diagrams Appendix F: PCB 712A01 Piezoelectric Actuator Specifications Appendix G: PCB 712A02 Piezoelectric Actuator Specifications Appendix H: Calibration Constants Appendix I: ControlDesk Data Acquisition Window Appendix J: Two Stage Controller Block Diagram References

7 Figure Captions Figure 2.1: Discretized SDOF Beam System [1] Figure 2.2: Block Diagram for a PID Controller Figure 2.3: Single Mass System Functional Block Diagram, PID Control Figure 2.4: Functional Block Diagram for 2 Mass System, PID Control Figure 2.5: SimMechanics Block Diagram for 2 Mass System, PID Control Figure 2.6: Model Reference Adaptive Control Scheme Figure 3.1: General Stacked Plate Experimental Setup Figure 3.2: Flexible Mode Excitation of Middle Plate Figure 3.3: Wire Rope Isolator Figure 3.4: Single and Four Actuator Configuration Setups Figure 3.5: Underside of Suspension System Figure 3.6: LDV Data Compared to Accelerometer Data Figure 3.7: System Characterization Frequency Response Figure 3.8: Passive Implementation of the Inertial Actuators Located Near the Center of the Center Plate Figure 3.9: New Inertial Actuator Mounting Location Figure 3.10: Passive Results with Inertial Actuator at New Mounting Location. 31 Figure 3.11: Active Attenuation Using the PCB 712A02 Inertial Actuator Figure 3.12: Experimentally Determined Mode Shapes for 515 Hz, 754 Hz, and 919Hz, Respectively Figure 3.13: Piezoelectric Patch Mounted to the Central Plate Figure 3.14: Time Domain Manual Attenuation Using Patch Actuator Figure 3.15: Broadband Attenuation of the First Four Resonant Peaks Figure 3.16: Miniaturized Flexible Plate Figure 3.17: Miniturized Setup with Thicker Baseplate Figure 3.18: Miniature System Characterization Figure 3.19: Broadband Attenuation Feedback Loop Figure 3.20: Broadband Attenuation Results

8 Figure A.1: Actuator Specifications Sheet (top) Perspective View of Actuator (bottom)...45 Figure B.1: Force Sensor Specifications Sheet (top) Picture of Force Sensor (bottom)...46 Figure C.1: Accelerometer Specifications Sheet (top) Picture of Accelerometer (bottom)...47 Figure D.1: ANSYS Results Isometric View (top) Side View (bottom). Displacements are Represented by Arrows and Show Clear Flexible Excitation...48 Figure D.2: ANSYS Results Plotted to Show Flexible Motion at 400 Hz (top) and 1200 Hz (bottom...49 Figure E.1: Block Diagrams Showing Data Acquisition Schemes For a Characterization Run (top) and a Control Run (bottom)...50 Figure F.1: Engineering Drawing for PCB 712A01 Actuator...51 Figure F.2: Specifications for PCB 712A01 Actuator...52 Figure G.1: Engineering Drawing Page 1 for PCB 712A02 Actuator...53 Figure G.2: Engineering Drawing Page 2 for PCB 712A02 Actuator...54 Figure G.3: Specifications for PCB 712A02 Actuator...55 Figure I.1: ControlDesk Data Acquicition Setup...57 Figure J.1: Block diagram for Multi-Stage PI Controller to Demonstrate Broadband Attenuation

9 List of Symbols ω ωn φn xd Kp Ki Kd frequency natural frequency mode shapes desired trajectory proportional gain integral gain derivative gain 9

10 1 Introduction As control technology advances, dynamic systems present more complex problems with less obvious solutions. One common application for these advanced controllers is vibration attenuation. In high precision systems, vibrations can decrease accuracies and efficiencies, and cause potentially disastrous damage. The ability to control these vibrations and minimize amplitude response to disturbances is valuable to many different industries across the board. This thesis focusses on non-collocation vibration control of a flexible plate system. Advanced controllers make up a broad category within the controls field. These controllers are capable of achieving non-collocation control as well as being robust and adaptive to variances in system parameters. Advanced controllers can be made effective when the system is modeled accurately. With the goal of more effective controllers, the dynamic system model pays an important role in the controller design. Pole placement is a simple way to design a stable system. A state space representation of the system is used to create a full state feedback controller that will perform based on the desired pole placement. This controller is not designed to be robust or adaptive. Observers and estimators can be designed to help improve both robustness and adaptive properties of the controller. A model reference adaptive controller can also be designed to be robust and adaptive. This method involves using a well-designed system model as a reference for control. The control parameters are updated with an algorithm that aims to match the reference model s response with the measured system response. Once this controller is developed, the robustness and adaptive properties can be optimized. One of the more complicated aspects of the controller of focus in this thesis is non-collocation control. This means that the location to be controlled is not collocated with the controller input to the system. This presents additional challenges since simple feedback control may not work and time delays will play 10

11 a significant role. Many techniques exist to help deal with this issue including the use of time delays with simple feedback controllers. A more ideal approach involves a complete and accurate reference model that essentially contains frequency response functions between all inputs and the output to be controlled. With this method, if the disturbance input can be measured at the source, the noncollocated output can be estimated using the reference model and the output can be controlled appropriately with a control input. Before adaptive control can be implemented, simpler control schemes are explored and experimentally implemented. This involves a carefully designed experimental setup that allows for maximum adaptability as control methods evolve. The main methods that are explored in this thesis involve flexible control with a patch actuator and control with an inertial actuator. To implement each of these, separate experimental setups must be designed, built, characterized, and tested. The focus is on the systems response within a defined frequency range. A system that can attenuate vibrations within a defined range is highly desirable for many different industries involving sensitive equipment. By reducing or eliminating vibrations, a system can be made more accurate and also more durable. This allows for higher precision and longer lasting devices. At a higher level, beyond vibration control, non-collocation control can be applied to many control problems. While this thesis focusses on vibration control, similar control methods and theory can be implemented in position control, flow control, and various other control fields making the technology exceedingly valuable. 11

12 2 Theory The theory behind all control begins with a strong foundation in dynamics. Dynamic systems are modeled and manipulated in various ways by adding control systems with the ultimate goal of achieving stability. Models can be continuous or discrete and can be utilized for a wide range of accuracies based on applications and control specifications. While control may be the ultimate goal, a controller is only as good as the model it was developed from. 2.1 Dynamic System Modeling Any physically realizable system can be modeled to some limited degree of accuracy to simulate an expected output based on a given input. Systems can be characterized as linear and nonlinear. Most systems that physically exist are nonlinear [1]. Nonlinear systems are more difficult to model and allow for limited mathematical representations. With the use of functional block diagrams, a complex nonlinear system model can be simulated. This method still has limitations and can get very complicated as the system becomes increasingly complex. To limit the complexity of a dynamic system model, nonlinear systems are usually represented by a series of simplified linear models. This reduces the accuracy of the model but is significantly easier to model and manipulate. With the system simplified to be linear, the equations of motion can be converted into state space representation which allows for easier controller design based on many different methods from full state feedback to model reference control Distributed Parameters System Modeling The equations of motion that truly represent a dynamic system with the highest resolution are derived as a function of position. These equations that represent the system at any position are referred to as distributed parameters or continuous system models. This type of modeling is often more complex and may not be 12

13 necessary depending on the application. A common way to derive continuous system equations is using energy method and/or Hamilton s Principle. D Alembert s Principle of virtual work leads to the generalized equations of motion as seen in Equation 2.1. Q j = d ( T ) d ( T ) (2.1) dt q j dt q j Where Qj are all external forces and T is the kinetic energy of the system. The qj terms are the generalized time-dependent coordinates. Most systems of equations of motion are developed using multiple q terms, hence the subscript j. The result of Lagrange s equation is a system of ordinary differential equations that represent a distributed parameters dynamic system in order to model its behavior. For this thesis, due to the complexity of the plate structures and difficulty with the boundary conditions, a distributed parameters model is not developed Lumped Parameters System Modeling A lumped parameters, or discrete model is a simpler and more efficient way than continuous modeling. A system is broken down into the desired degrees of freedom (DOF) and therefore the model can be made more computationally efficient. The model therefore is an approximation of the dynamic system and will not exactly match the continuous equations. To derive a discrete model, the desired DOF must first be determined. Depending on which rotational and translational DOF are of interest, the element type and a mesh are decided. The continuous system is essentially replaced with a series of mass, springs, and dampers. A discretized single DOF system is depicted in Figure

14 Figure 2.1: Discretized SDOF Beam System [1] A SDOF system is governed by the differential equation given in Equation 2.2. mx (t) + cx (t) + kx(t) = f(t) (2.2) where m is the mass, c is the damping coefficient, k is the stiffness, and f is the force acting on the system. For multi DOF systems, a similar approach can be used, however some manipulation is necessary. For each DOF, there will be a differential equation similar to the one shown above to represent its motion, so the system of equations can be written as: [M]x (t) + [C]x (t) + [K]x (t) = f (t) (2.3) Where [M], [C], and [K] represent the mass, damping, and stiffness matrices, respectively. In this equation, the x variable is expressed as a vector and represents all DOF included in the discretization. f is also expressed as a vector and represents the forces acting on the corresponding DOF. To find the natural frequencies of the system, an eigen solution (ignoring damping) produces the following equation: det[k ω 2 M] = 0 (2.4) ω represents the natural frequencies of the system, and K and M represent the same mass and stiffness matrices mentioned previously. With the eigenvalues representing the natural frequencies, the mode shapes can be calculated as well. By solving the following equation for φn the eigenvectors which represent the mode shapes can be derived. 14

15 [K ω 2 n M]φ n = 0 (2.5) Each calculated natural frequency will have a corresponding mode shape. These natural frequencies and mode shapes are very important to controlling the system. As previously mentioned, higher resolution results in more accurate natural frequencies and mode shapes, and eventually a more effective controller. A system of differential equations representing a discretized physical system technically makes a model of the system. However, to make the equations more easily used for control theory, the system is usually converted into state space. This can be done using the mass, damping, and stiffness matrices and either calculated manually or using MatLab to convert to state space form, which is represented in the equations below. x (t) = A(t)x(t) + B(t)u(t) (2.6) y(t) = C(t)x(t) + D(t)u(t) (2.7) Where x(t) is a vector representation of the states, and y(t) is the output that is monitored. A(t), B(t), C(t), and D(t) represent coefficient matrices and are derived from the equations of motion, and u(t) represents the control input to the system, typically a forcing function. In the state space form, many techniques exist to manipulate the coefficient matrices for controller design or the system could be used as a reference model. 2.2 Control Theory All controllers are designed based on equations of motion that represent a dynamic system, therefore their performance is limited by the accuracy and resolution of the model. For the structure of interest in this thesis, both linear and more advanced, robust, adaptive controllers are explored Linear Controllers Linear controllers make up the most basic division of control and are typically a good fit for position control. For dynamic control, linear controllers tend to be 15

16 less effective but may be combined with other blocks to increase effectiveness and prevent instabilities. One common type of linear controller is the Proportional Integral Derivative (PID) controller. This controller takes an input signal and applies proportional, integral, and derivative gains to the signal, its integral, and its derivative, respectively. By manipulating these gains, the controller can be designed to maintain stability and attain the desired settling time. The PID controller can be applied theoretically to a state space model, or a transfer function model. The block diagram representation of a PID controller is shown in Figure 2.2. Figure 2.2: Block Diagram for a PID Controller The transfer function representation of a PID controller is shown in Equation 2.8. K d s 2 +K p s+k i s (2.8) With a working model, theoretical results can be simulated using programs such as MatLab to determine the efficiency and performance of the controller. A PID controller can be designed using Routh Hurwitz stability analysis, Bode plot analysis, Nyquist stability analysis, or root locus method. For the dynamic model in state space form, MatLab can be used to convert the state space matrices into a transfer function representation. With this transfer 16

17 function, the final value theorem, Equation 2.9, can be used to determine steady state error of the system. lim t (f(t)) = lim(g(s)) (2.9) s 0 For development of the PID controller, since the dynamic model was being developed in parallel with experimental development, the majority of the tuning was done on the test setup. For experimental tuning of the PID controller, the Ziegler-Nichols method [2] was primarily used. A PID controller can also be theoretically tuned using functional block diagrams. Due to an incomplete experimental setup, the goal of the following exercise was merely to see if active control of the system was plausible using a basic setup and easily implementable components. Initially a simple one mass system was tested using a disturbance signal made up of white noise and different sine wave signals within a frequency range from Hz. Figure 2.3 shows the Simscape block diagram representing a simplified single mass system to assess the general PID controllers performance. This diagram also compared the results from a previously developed Simulink diagram to the Simscape representation. Results showed that the PID controller was effective and could potentially be further tuned to show significant attenuation of vibrations. This also matched the results generated in Simulink. 17

18 Figure 2.3: Single Mass System Functional Block Diagram, PID Control A more complex system was then evaluated under the same disturbance conditions. The Simscape block diagram for this can be seen in Figure 2.4. This again showed theoretical results that matched the Simulink models and showed attenuation. 18

19 Figure 2.4: Functional Block Diagram for 2 Mass System, PID Control Finally another model to still more realistically represent a physical system was developed using SimMechanics. This interface implements realistic physical components with varying size and mounting capabilities to more closely represent the dynamic system of interest. This diagram is shown in Figure 2.5. With this model, the results did not match as closely as the Simscape models had shown, however correlation was still seen. Again the PID controller showed attenuation of the vibrations within the frequency range of interest. 19

20 Figure 2.5: SimMechanics Block Diagram for 2 Mass System, PID Control It is clear from the basic models, that PID control is a viable path for attenuation of vibrations in the system. This being said, alternate routes are explored including advanced controllers to improve robustness and adaptive characteristics Advanced, Robust, and Adaptive Controllers Linear controllers can often perform well with position control, but may not prove as effective with active control such as vibrations. For vibration control, more advanced controllers can provide higher levels of attenuation and account for varying system parameters over time. The ability of a controller to perform despite changes in system parameters is known as adaptive control, while its ability to reject unknown disturbances is known as robustness. To achieve desired levels of vibration attenuation, a controller that is both robust and adaptive is desired. A common form of adaptive control is known as Model Reference Adaptive Control (MRAC). To implement MRAC, an accurate theoretical model representing the physical system is necessary. This system output is compared with that of the reference model, and based on the error, the control parameters 20

21 are adjusted. This is done to force the physical model to match the reference model output. A diagram representation of a typical MRAC system is shown in Figure 2.6. Figure 2.6: Model Reference Adaptive Control Scheme The limitations seen in MRAC systems are based on the accuracy and resolution of the reference model. With a model that closely represents the physical system, the control scheme can be tuned to perform under adapting conditions. Another key to the controller is the adjustment algorithm. MRAC control is a wide category of control and can be implemented in many different ways. This results in many different algorithms and many different ways of updating the system. Depending on the model, it is usually important to update mass, stiffness, and damping [3] [4] [5]. 21

22 3 Experimentation One of the main objectives of this project was to implement vibration control schemes on a stacked plate system. To implement control schemes, a properly designed experimental setup is necessary. This section covers the development of a reconfigurable experimental setup as well as implementation of control schemes and results. 3.1 Setup, Design, and Component Selection The system of interest is made up of multiple plates stacked on top of each other and separated by spacers. The controller has an attenuation target range from 400 Hz to 2000 Hz. The system is required to show control of flexible modes of the system and therefore must be capable of demonstrating flexible motion. The entire setup must be capable of recording accurate acceleration data as well as force input data. The following sections detail the experimental setup design General Setup For ease of machining, the stacked plates were designed to be constructed from aluminum sheet. Three plates of different sizes and thicknesses are mounted to each other via the four corners, and then separated using spacers. This is the general stacked plate setup that is used for all experiments conducted. The desired system will be disturbed through the bottom plate and the top plate is controlled via some form of force input to the middle plate. The general setup is shown in Figure 3.1. The dimensions pictured are approximate sizes in inches. 22

23 Figure 3.1: General Stacked Plate Experimental Setup To isolate the stacked plate dynamic system, a suspension is necessary. Section outlines the design of such a suspension Components Selection To be capable of inducing vibrations and measuring them at various points, equipment was needed. Actuators were needed to apply the disturbance signal to the system while force sensors or an impedance head were used measure the input. A micro-accelerometer was used to measure the signal at the position of interest. Various actuators were also used for control, and are described further in Section 3.3. The actuators, accelerometer, and force sensors needed to be specified based on the frequency range of interest and the desire to excite flexible modes of the system. ANSYS models were created to simulate the system response to various disturbances. The purpose of these models was to be certain that the specified actuators were capable of exciting the flexible modes of the system. After researching prices, sizes, and output specifications, the best candidate for the disturbance actuator(s) was The Modal Shop s 2007E electromagnetic shaker. The specifications for this piece of equipment are shown in Appendix A. To be The ANSYS simulations revealed that the flexible modes of the system could be excited using the specified actuator. Results from the ANSYS simulation are 23

24 shown in Figure 3.2 and additional results for other frequencies are presented in Appendix D along with the ANSYS elemental systems. 1.00E Hz Deforma t=0.01s 5.00E E E E-07 Deflec on (in) -1.50E E E-07 R² = E E E E-07 X - Posi on (in) Figure 3.2: Flexible Mode Excitation of Middle Plate The accelerometer was specified based on the power of the disturbance actuator. Again, the ANSYS model was used to simulate disturbances to the system and the max accelerations during an applied frequency sweep force input were observed. Results led to the selection of the PCB 352C22 ICP Accelerometer. Detailed specifications for this sensor are shown in Appendix C. To correctly map the frequency response function (FRF) for any system, both force input and acceleration output (or any desired output) must be measured and recorded. Force sensors are placed between each of the disturbance actuators and the system to measure the input forces. These sensors were specified based on the output capabilities of the disturbance actuators. PCB s 208C01 Force Sensor was determined to be adequate; detailed specifications are shown in Appendix B. In summary: actuators were verified through simulations to be capable of exciting flexible modes of the system, accelerometers were specified to maximize resolution while paying attention to limits, and force sensors were selected based on the input limits to the system. 24

25 3.1.3 Suspension Design In an effort to prevent dynamic interactions within the control frequency range, a stiff mounting rig isolating the test article from the rest of the experimental setup is necessary. The suspension system must also be capable of mounting the actuators and accommodate the Laser Doppler Vibrometer (LDV). This suspension system must also serve as a mount for the 2007E shaker(s). A key component of the suspension system is the isolator. A wire rope isolator was determined to be the best candidate for isolating the test article. With high damping and relatively low stiffness the isolators, shown in Figure 3.3, prevent the additional mode shapes that appear when the support system interacts dynamically with the test article. Four isolators are mounted to the sides of the base plate to suspend the test article from the support structure. Figure 3.3: Wire Rope Isolator The support structure is made up of two 12 x 12 square plates. The bottom plate has holes necessary for mounting the 2007E actuators and aluminum channel posts. There are also holes for mounting the entire structure to the isolation table. The top plate has matching holes for mounting to the aluminum channel and a large cutout to be capable of taking laser measurements on the test article. Pictures of the test article suspended in the Figure 3.4 and Figure

26 Figure 3.4: Single and Four Actuator Configuration Setups Figure 3.5: Underside of Suspension System 3.2 System Characterization To determine the frequency response of the system and to verify all sensors and associated data acquisition equipment, the experimental setup had to be characterized. First, a single actuator configuration with the accelerometer mounted to the plate where vibrations are to be attenuated was wired to the dspace for data acquisition and disturbance signal output. 26

27 Acceleration (m/s^2) To verify the measurements recorded from the accelerometer, LDV data was taken at the same location and compared. The results can be seen in Figure 3.6 showing that that the collocated data matched within the accuracies of the equipment Acceleration comparison for accelerometer and LDV for linear sine chirp from 400 Hz-800 Hz PCB 352C22 accel. LDV Time (sec) Figure 3.6: LDV Data Compared to Accelerometer Data The block diagrams used for acquiring the data are given in Appendix E. With verified accelerometer measurements, the system dynamics could be recorded. To determine the natural frequencies of the system, a frequency sweep input force from 0 to 2000 Hz was applied. The acceleration output at the top flexible plate was recorded and a Fourier transform was performed to see the response in the frequency domain. The initial data taken from the LDV and the accelerometer is shown in Figure 3.7. The first three most significant resonant frequencies are obtained to be 515 Hz, 752 Hz, and 927 Hz. These frequency values are used in designing the controller. The data recorded is also used in evaluating the controller s performance. 27

28 Figure 3.7: System Characterization Frequency Response Figure 3.7 clearly shows narrow peaks representing each of the resonant frequencies. These peaks extend far above the noise floor and will be the initial targets for attenuation. 3.3 Control Schemes To achieve attenuation of parts of the system, multiple control schemes were explored. The first idea was based on the use of an inertial actuator. Inertial actuators can be added to a structure to add force as opposed to many actuators which apply inter-component forces. The inertial actuator can also be tuned to match the resonant frequency of the structure by changing the inertial mass. When the actuator is tuned to the correct frequency it allows for greater output forces and is typically more effective. The inertial actuator available is fairly large due to the high cost of custom piezoelectric actuators. The large mass does cause dynamic effects to the system. The added mass appears similar to added damping to a system, in the experimental data. A second control scheme involved the use of a piezoelectric patch actuator. The actuator is bonded to a flexible region of the system and expansion and contraction of the patch apply forces in the direction perpendicular to the plane of 28

29 the patch. The patch is small and causes little change in the dynamics of the system. A third method explored was cantilever beam actuators. Technology developed at Northeastern University at the Piezoactive Systems Laboratory ( shows that piezoelectric cantilever beams can be used to generate forces as well as sense acceleration simultaneously. This type of control actuator would be ideal and could prove very valuable. The technology, however, is complicated and far off from implementation. The advantages are significant and would allow for much easier scaling for larger or smaller application as well as control of more DOF. The cantilever beam actuator applies forces to both rotational and translational DOF which allows for more advanced control and potentially higher performing controllers. 3.4 Initial Results Two main types of controllers were implemented experimentally. The first experimental implementation was with the inertial actuator. This was used both passively and then actively to achieve higher levels of attenuation and to portray the advantages of active control. The second type of controller that was implemented was the patch actuator. This control scheme has little passive effects and therefore was used actively to demonstrate attenuations Inertial Actuator The Peizoactive Systems Laboratory owns two PCB inertial actuators. Each of these actuators has a stiffness associated with them that can be used along with the inertial mass to calculate the resonant frequency. By matching the resonant frequency of the actuator with one of the resonant frequencies of the system, the actuator will act like a tuned absorber. When the inertial actuator is then actively powered, it will further attenuate vibrations by absorbing more energy from the system. 29

30 Due to the high flexibility and significant motion of the center plate, the mounting point for the inertial actuator was chosen to be near the center of the center plate. This area was thought to allow for the most significant attenuations of vibrations in the smaller top plate. The inertial mass was calculated and tuned to match the natural frequency of the inertial actuator to the first peak at about 515 Hz. Using the PCB 712A01 and 712A02 inertial actuators, (specifications shown in Appendix F and G) the inertial masses to match the first resonant peak were calculated to be 159g and 22g, respectively. Initially data was taken with the inertial actuator acting passively H(f) No absorber (2 g nut at CB59 PCB 712A02 w. 22 g at CB59 PCB 712A01 w. 159 g at CB g at CB Frequency (Hz) Figure 3.8: Passive Implementation of the Inertial Actuators Located Near the Center of the Center Plate Figure 3.8 clearly shows that the passive results do not look like a typical tuned absorber should. The tuned absorber should have the effect of splitting the peak it is tuned to. It was determined that this was due to the mounting location. As can be seen in Figure 3.8, both passive tuned absorbers have the same effect as mounting a large mass at the same location. The absorbers are not acting like absorbers but simply appearing as added damping due to the large mass. This location is simply too flexible, therefore, the actuator mounting location was moved closer to the standoffs in an area of higher local stiffness, shown in Figure

31 Figure 3.9: New Inertial Actuator Mounting Location In this new location, the test was repeated, however, the 712A01 actuator was not used due to the extremely large relative mass. The passive results using the 712A01 actuator with a 22g mass and 20g mass are shown in Figure Now it is clear that the inertial actuator is splitting the peak into two peaks of lower magnitude. The passive results show that the inertial actuator is in fact tuned properly and mounted in an acceptable location H(f) No absorber (2 g plastic nut at CB37 PCB 712A02 w. 22 g at CB37 PCB 712A02 w. 20 g at CB Frequency (Hz) Figure 3.10: Passive Results with Inertial Actuator at New Mounting Location With the inertial actuator tuned to the correct resonant frequency, active attenuation could be explored. Initially, a manually tuned sine wave was 31

32 implemented with a time delay. For these tests, the disturbance signal was input as a sine wave at the resonant frequency. The acceleration was measured at the non-collocated area being controlled. The time delay made up for phasal differences between the disturbance input to output, and the control input to output. As expected, the tuning was successful and able to attenuate vibrations about 70% more than the passive attenuation results. These results, shown in Figure 3.11, served as a proof of concept for the inertial actuator control scheme, however they are not true control. Since the input to the controller is manually set, it has not knowledge of the output at the control area and therefore does not adjust automatically. An attempt to use a PID controller proved unstable. Using Ziegler Nichols experimental tuning methods [2] the PID controller was tuned to see some minimal attenuation, however, the system would quickly go unstable under normal conditions. Figure 3.11: Active Attenuation Using the PCB 712A02 Inertial Actuator The inertial actuator requires a better theoretical model and a more advanced controller. The instabilities seen during testing are due to the poor dynamic 32

33 performance of the linear PID controller. This development is explained further in Section Patch Actuator The large relative mass of the inertial actuator proved to be an issue with experimental implementation. The piezoelectric patch actuator does not present this problem. To implement the patch actuator, the mounting location is critical. Experimental velocity data was taken over a grid across the center plate. This data was then used to experimentally determine the mode shapes at the first three resonant frequencies, shown in Figure By visual inspection of the mode shapes, an area of high strain was selected for mounting the patch. The patch is shown mounted to the center plate in Figure Figure 3.12: Experimentally Determined Mode Shapes for 515 Hz, 754 Hz, and 919Hz, Respectively Please note that the blank areas in the experimentally determined mode shapes are points under the top plate where the LDV was unable to obtain data. Since this area was not a candidate for mounting the patch, it did not affect the determination of the mounting location. 33

34 Figure 3.13: Piezoelectric Patch Mounted to the Central Plate Similarly to the implementation of the inertial actuator, the system was tuned manually first. Contrary to the inertial actuator, however, the patch actuator is not tuned to a resonant frequency. There is also no significant passive attenuation seen with the patch actuator. Therefore, active attenuation was explored initially. Using the same manual control methods described in Section 3.4.1, the area of interest was significantly attenuated. The time domain results are shown in Figure

35 Figure 3.14: Time Domain Manual Attenuation Using Patch Actuator The time domain results served as a proof of concept for attenuation capabilities using the patch actuator actively. The time domain results did not, however, show broadband attenuation. The desired attenuation range from Hz contains each of the first three resonant frequencies. To show attenuation throughout this range, a PID controller with a time delay was explored. Again, experimental tuning methods were used to dial in the gains. It was experimentally determined that the derivative gain caused the system to go unstable at higher frequencies and therefore a PI controller was used. The gains and time delay were set, and a linear frequency sweep was applied as the disturbance signal. Figure 3.15 shows the broadband results. The zoomed in views show the attenuation levels. The first, third, and fourth peaks showed attenuations of approximately 76%, 30%, and 46%, respectively, while the second peak showed a magnification of about 22%. 35

36 H11(f) PI controller on PI controller off Frequency (Hz) H11(f) H11(f) H11(f) H11(f) Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz) Figure 3.15: Broadband Attenuation of the First Four Resonant Peaks 3.5 Miniaturized Setup As devices are continually becoming smaller, the natural progression was to miniaturize the system. This demonstrates scalability and the implementation of control in a more easily applicable form. The first step is to miniaturize the physical system Miniaturized Experimental Setup The baseplate, which serves to distribute forces, is thickened to prevent its flexible motion. This allows for the modeling assumption that the force distribution plate is rigid and also prevents the appearance of extra modes within the control frequency range. The flexible plate, where control is to take place is, miniaturized to a 1/32 thick plate approximately x The Accelerometer is used to record data at the point of interest, while simultaneously acting as a small mass that could represent a sensitive object mounted to the 36

37 flexible plate. The miniaturized experimental setup is shown in Figure 3.16 and Figure Figure 3.16: Miniaturized Flexible Plate Figure 3.17: Miniturized Setup with Thicker Baseplate A miniature patch actuator is used for the control input to the system. The patch acts similarly to the larger one mentioned in Section

38 3.5.2 Miniature System Characterization With the miniaturized system, the model is no longer representative of the physical system. To update the model and to determine the new resonant frequencies, data is recorded throughout the frequency range of interest. The data reveals the first two resonant frequencies at 614 Hz and 1001 Hz. The results of the frequency sweep system response are shown in Figure X: Y: X: 1001 Y: 77.7 G Frequency (Hz) Figure 3.18: Miniature System Characterization To show broadband attenuation, these first two peaks are singled out as targets. To be certain that the patch actuator is capable of attenuating these peaks, a manual test is done. A sine wave signal is sent at each resonant frequency as the disturbance input. That same signal is delayed and amplified and used as the control input. The delay is adjusted to create destructive interference and the amplitude is adjusted to attempt to attenuate the vibrations. With some tuning, it becomes clear that the patch is capable of delivering ample force to the system to attenuate the vibrations caused by the disturbance signal. 38

39 3.6 Results To attenuate each of the first two resonant frequencies, two separate PID controllers are used. A force sensor, measuring the input force to the system is used as feedback. First, Ziegler-Nichol s methods for experimentally tuning a PID controller were explored. Due to the size of the baseplate, self-resonance is not achievable and therefore the tuning must be done live. A sine wave with a frequency of 614 Hz is input as the disturbance signal. The proportional gain is increased to a safe value and the time delay is adjusted to attempt to achieve higher levels of attenuation. The proportional gain is then adjusted as necessary, and this back and forth tuning of the time delay and the proportional gain is repeated until maximum attenuation is achieved. The same is done for the integral and derivative gains. It is found that the derivative gain induces instability and does not help to attenuate vibrations in this case. This process is repeated separately for the second resonant frequency of 1001 Hz. To attenuate both peaks, each PID controller must only be active when its respective peak is excited. A simple Butterworth bandpass filter is used for this. The pass frequencies are set at the respective resonant frequencies and the passbands are set to 40 Hz. A control loop is established with the input force measurement as feedback. The respective time delays, PI gains, and filters are combined and the structure shown in Figure 3.19 is generated. A complete Block diagram is given in Appendix J. Figure 3.19: Broadband Attenuation Feedback Loop 39

40 Dspace Output Channel Figure 3.19: Broadband Attenuation Feedback Loop The controller is tested with a frequency sweep to show the effectiveness in attenuating each of the first two resonant peaks. The disturbance signal is applied and data is taken with each controller on separately and then together. The results are shown in Figure The controllers clearly split each of their respective peaks into two peaks of lower amplitude. The attenuation at the resonant frequencies exceed 90%, however, the split peaks have attenuations of 58% and 81% for the first and second resonant peaks, respectively X: Y: X: Y: X: Y: X: Y: X: 1001 Y: 77.7 X: 1022 Y: G No controller Peak 1 attenation Peak 2 attenation Both peaks attenuated Frequency (Hz) Figure 3.20: Broadband Attenuation Results 40

41 4 Discussion of Results The results described in Section 3.6 show that active non-collocated attenuation of a flexible plate system is not only possible but scalable as well. The attenuation levels of 58% and 81% are promising as more complex controllers are developed. These results are to be improved upon to be applicable throughout industry, however, they come with a setup that can be used to develop and test advanced control schemes in an easily adaptable way. As technology advances and systems become more complex and more precise, the need for vibration attenuation systems increases. The experimental setup designed as part of this thesis presents a system where control is easily implementable. The ability to mount up to four actuators and apply up to four separate disturbance signals allows for maximum potential adaptability. The setup exhibits a mounting scheme that allows for many different control actuators to be used and all the necessary measurements to be taken with ease. The stiffened suspension rig prevents dynamic interaction with the test article allowing for clean accurate test data to be taken. This setup is paramount to ongoing work, explained in more depth in Section 0. While the results obtained are not significant enough to be used for applications, they do show potential. The experimental setup has all aspects necessary for continued development of control schemes. Using this setup and building upon the results already obtained, higher levels of attenuation should be attainable. 41

42 5 Future Work Work will continue with the experimental setup to develop a more advanced robust, adaptive controller. Initially the setup will be modified slightly to represent a more realistic situation where inter-component forces are measured as opposed to the force input from the disturbance actuators. Data will be taken to produce frequency response functions and to fully characterize the system. Data will also be taken to update the theoretical model for improved accuracy. The new model will be used in the more advanced controllers. A model reference adaptive controller will be explored. The model accuracy is critical as it will dictate the effectiveness of the controller. A controller will be designed based on the model with the ability to be updated based on certain parameters. As the controller performs, the real time results will be compared with the expected results from the model and the error will be used to update the model. This updated model will then update the controller to improve performance. Once tuned this controller should be capable of rejecting unknown disturbances and adapting to changes in system parameters over time. The development of the MRAC will greatly increase the potential for applications. Once correctly designed and appropriately packaged, the control system could be a simple add-on to any system requiring vibration attenuation. 42

43 References [1] N. Jalili, Piezoelectric-Based Vibration Control, New York: Springer Science+Business Media, [2] R. C. Dorf and R. H. Bishop, Modern Control Systems, Upper Saddle River: Prentice Hall, [3] A. M. Annaswamy, "Model Reference Adaptie Control," Control Systems, Robotics, and Automation, vol. 10, [4] J.-k. Lee, B.-h. Suh and K.-i. Abe, "Model Reference Adaptive Control of Nonlinear System using Feedback Linearization," in Society of Instrument and Control Engineers, Sapporo, [5] R. Prakash and R. Anita, "Robust Model Reference Adaptive PI Controller," Journal of THeoretical and Applied Information Technology, pp , [6] A. Morse, "Overcoming the Obstacle of High Relative Degree," Yale University, New Haven, [7] J. H. Williams, Fundamentals of Applied Dynamics, Wiley, [8] N. T. Nguyen and K. Krishnakumar, "An Optimal Control Modification to Model-Reference Adaptive Control for Fast Adaptation," in AAIA Guidance, Navigation and Control Conference and Exhibit, Honolulu,

44 Appendix A: The Modal Shop 2007E Actuator Specifications Figure A.1: Actuator Specifications Sheet (top) Perspective View of Actuator (bottom) 44

45 Appendix B: PCB 208C01 Force Sensor Specifications Figure B.1: Force Sensor Specifications Sheet (top) Picture of Force Sensor (bottom) 45

46 Appendix C: PCB 352C22 Accelerometer Specifications Figure C.1: Accelerometer Specifications Sheet (top) Picture of Accelerometer (bottom) 46

47 Appendix D: ANSYS Flexible Excitation Results Figure D.1: ANSYS Results Isometric View (top) Side View (bottom). Displacements are Represented by Arrows and Show Clear Flexible Excitation. 47

48 400Hz Deforma t=0.01s 1.00E E E E E-07 Deflec on (in) -1.50E E E-07 R² = E E E E-07 X - Posi on (in) 1200Hz Deforma t=0.027s 1.34E E E-05 Deflec on (in) 1.33E E E-05 R² = E E E X-Posi on (in) Figure D.2: ANSYS Results Plotted to Show Flexible Motion at 400 Hz (top) and 1200 Hz (bottom). 48

49 Appendix E: Data Acquisition Block Diagrams Figure E.1: Block Diagrams Showing Data Acquisition Schemes for a Characterization Run (top) and a Control Run (bottom). 49

50 Appendix F: PCB 712A01 Piezoelectric Actuator Specifications Figure F.1: Engineering Drawing for PCB 712A01 Actuator 50

51 Figure F.2: Specifications for PCB 712A01 Actuator 51

52 Appendix G: PCB 712A02 Piezoelectric Actuator Specifications Figure G.1: Engineering Drawing Page 1 for PCB 712A02 Actuator 52

53 Figure G.2: Engineering Drawing Page 2 for PCB 712A02 Actuator 53

54 Figure G.3: Specifications for PCB 712A02 Actuator 54

55 Appendix H: Calibration Constants % Northeastern University % Piezoactive Systems Laboratory % 04/22/2013 % MEMS Based Active Vibration Isolation % This mfile list all transducer calibration constants for devices % associated with Raytheon MEMS project. This file also lists serial % numbers for future reference. % Force transducer constants (mv/n) % S/N LW35899 kf1 = 110.5; % S/N LW35901 kf2 = 108.3; % S/N LW35970 kf3 = 115.9; % S/N LW35971 kf4 = 115.8; % Impedance head constants % Mod. #: 200D01, S/N 1772 % Force constant (mv/n) kif=22.4; % Acceleration constant (mv/m/s^2) kia=10.2; % Accelerometer constants (mv/m/s^2) % S/N LW ka1 = 1.070; % Mod. #: Y352C23, S/N ka2 = 0.527; 55

56 Appendix I: ControlDesk Data Acquisition Window Figure I.1: ControlDesk Data Acquicition Setup 56

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