Active Vibration Control

Problem Statement of Project

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To build a model in MATLAB© based on FEM analysis in ANSYS© for a simply supported beam and identifying the significant modes for the design of controller.

Scope of Project

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A given structure can vibrate with many modes. The design of controller for all the modes is very difficult. However, all the modes do not contribute significantly to the overall disturbance. Hence, we filter out the modes which cause the maximum disturbance. Hence a controller can be designed to control only these modes.

Conclusion of Project

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MATLAB© model was extracted from ANSYS© model. Displacement v/s Time plot for all modes included model and reduced model shows no visible difference for settling time and nature of curve. Hence, by controlling only the significant modes, controlling objective can be achieved. An attempt is made to control the first mode and the second mode of vibration in ANSYS©. Better results can be obtained by varying patch size and location.

Future Scope of Project

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Designing controller based on the MATLAB© results for reduced model and controlling the significant modes of vibration. An experimental setup can be fabricated based on this analysis.

Concept of Smart Structures

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The dictionary definition of the word “Smart” (brisk, spirited, mentally alert, bright, clever) is not quiet adequate for an engineer.

For an engineer “A Smart Structure” means a structure that can sense an external disturbance and respond to that with active control in real time to maintain the mission requirements. A Smart Structure typically consists of a host structure incorporated with sensors and actuators coordinated by a controller.

The integrated structured system is called Smart Structure because it has the ability to perform self diagnosis and adapt to environmental change. One promising application of such smart structure is the control and suppression of unwanted structural vibrations.

Smart Structure

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Smart Structure comprise of host structure along with Sensors, Actuators & controller networks. Consider a simply supported beam as a smart structure.

Basic Elements of a Smart Structure

The major components are:

Sensor Patch
It is bonded to the host structure (Beam). It is generally made up of piezoelectric crystals (one of the smartest materials). It senses the disturbance of the beam and generates a charge which is directly proportional to its strain. Direct piezoelectric effect is used here.

Controller
The charge developed by the sensor is given to the controller. The controller lines the charge according to suitable control gain and then the charge is fed to the actuator. Controller also forms the feed back transfer function for this system.

Actuator Patch
The lined up charge from the controller is fed to the actuator. An actuator is a piezoelectric patch bonded to the host. Due to the input voltage, actuator causes pinching action (or generates shear force along the surface of the host which acts as the damping force and helps in the attenuating vibration motion of the beam. Converse piezoelectric effect is used here.

Active Vibration Control & Smart Structures

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Active vibration control is defined as a technique in which the vibration of a structure is reduced by applying counter force to the structure that is appropriately out of phase but equal in force and amplitude to the original vibration. As a result two opposing forces cancel each other, and structure essentially stops vibrating.

Techniques like use of springs, pads, dampers, etc have been used previously in order to control vibrations. These techniques are known as ‘Passive Vibration Control Techniques’. They have limitations of versatility and can control the frequencies only within a particular range of bandwidth. Hence there is a requirement for ‘Active Vibration Control’.

‘Active Vibration Control’ makes use of ‘Smart Structures’. This system requires sensors, actuators, a source of power and a compensator that performs well when vibration occurs. Smart Structures are used in bridges, trusses, buildings, mechanical systems, space vehicles, telescopes, and so on. The analysis of a basic structure can help improve the performance of the structures under poor working conditions involving vibrations.

Active Vibration Control & Controller Objective

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According to the way in which controllers are designed, the control techniques have been classified such as Independent Modal Space Control (IMSC) and Coupled Control (CC).

Independent Modal Space Control (IMSC)
Space structures are characterized by a large number of degrees of freedom, so control of such systems requires a new approach. In IMSC method the control laws are designed in the modal space for each mode independently. The problem reduces to a set of independent second order systems so that control is possible.

IMSC requires an appreciable less amount of energy than the coupled control. It gives larger choice of control techniques including non linear control. In this method the Performance Index ‘J’ is independent of the actuator location.

IMSC requires that the number of actuators must be equal to the number of modes of vibration. With this constraint the possibility of actuator failure becomes critical and there is a requirement of a spare actuator. IMSC cannot be used with available actuators. Actuators incompatible with IMSC cannot provide the relative force distribution IMSC demands.

Modified Independent Modal Space Control (MIMSC)
One of the predominant difficulties in active control of flexible structures is that such structures have a number of vibration modes within or beyond the bandwidth of the controller. In active control of flexible structures, spill over can occur because only a few vibratory modes are dealt with by the controller. Although modal space based optimal control is known to avoid spill over, it requires a large number of sensors and actuators. MIMSC algorithm proposed by Baz-et-al is used to minimize the number of actuators. With it an artificial neural network is also used to identify the system characteristics and reduce the number of sensors. MIMSC control has an excellent closed loop structural damping.

Coupled Controllers
Coupled controllers are used when simultaneous control of multimodes is required. Two feed back laws are used for Coupled control design: state feedback and output feedback.

Output feedback allows us to design plant controllers of any desired structure. In output feedback method, the feedback gain k is chosen to minimize the quadratic cost or the performance index. Performance index PI is the functional relationship involving some system characteristics in such a manner that the optimum conditions as per the requirements may be determined from it.

Theoritical Model Analysis of a Simply Supported Beam

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The theoretical analysis of a simply supported beam is necessary to find out the exact values of the natural frequencies. The exact frequencies would be used further to validate the results obtained by the analysis software ANSYS©.

As cross sectional dimensions of our beam are small as compared to its length, it can be treated as Euler Bernoulli Beam [10]. The reference figure for the ANSYS© is shown below.

Continuous beam Analysis

Let
A = Cross-sectional area of the beam
E = Modulus of elasticity of the beam material
ρ = Density

Consider an element dx of the beam subjected to shear force Q and bending moment M.

Assumptions
1. No axial forces are acting on the beam.
2. Effects of shear deflection are neglected.
3. The deformation of the beam is assumed due to moment and shear force.

Model Analysis and Convergence Study in ANSYS

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A host structure such as a cantilever or simply supported beam and its surface bonded with piezoelectric patches acting as sensor and actuator is called as an ‘Integrated Structure’. The first step in understanding the working of smart structures is to analyze the host structure using finite element analysis. The smart structure can then be controlled accordingly.

Finite Element Modal is created in ANSYS© using the various tools present. Size of the beam selected is 500x25.4x0.8 mm.

Meshed Model of beam

The ends of the beam are constrained in the UZ direction. Modal Analysis is performed in ANSYS© using the Block Lanczos Method. The number of modes of vibration to be extracted and the frequency range is specified. To determine the mesh size of the beam it is important to perform a convergence study. The first three natural frequencies of the vibration of the beam are calculated for the various mesh sizes by performing modal analysis and are compared with the theoretical values as given below:

In order to compromise between accuracy and computational time the mesh size of (60*8*1) is selected whose natural frequency is found to be 7.155 Hz which is very close to 7.151 Hz.

Modelling of Integrated Structure using ANSYS

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After finding the appropriate mesh size and the natural frequency of the host structure, the modeling of the piezoelectric patch is carried out. The piezoelectric patch is modeled with the dimensions 72x25.4x0.71 mm. After modeling the beam and the piezoelectric patch, the key points are merged to fix the patch with the beam. The material properties are defined for both, the patch and the beam. The element type selected for the beam is SOLID 45 and that for the piezoelectric patch is SOLID 5.

Meshed Model of Beam with Piezoelectric Patch

After modeling, the electrical degrees of freedom of the patch are coupled at the outer surfaces of the patch and the intermediate common surfaces between the patch and the beam. After modeling, the first natural frequency of the integrated structure is calculated by performing the modal analysis. The natural frequency is found out to be 5.6514 Hz.

The mode shapes at the natural frequencies are of extreme importance, as the deformation of the beam is maximum for these mode shapes. So in any vibration control methodology, the main aim should be to avoid the natural frequencies at any circumstances.

Simply Supported Beam Vibrating in First Mode, 5.65 Hz

Simply Supported Beam Vibrating in Second Mode, 27.03 Hz

Simply Supported Beam Vibrating in Third Mode, 66.72 Hz

Analysis using MATLAB

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Performing matrix computation in MATLAB© is not difficult. However, the direct modeling of a physical structure in MATLAB© is very complicated. ANSYS© model directly corresponds to the geometry and material properties of the physical structure. Its FEM analysis is accurate and reliable. But performing complex matrix computation directly in ANSYS© is not possible. So it is necessary to extract model in MATLAB© from the FEM analysis in ANSYS©.

Obtaining Nodal Solution using MATLAB

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We obtain the first six natural frequencies of the integrated structure by modal analysis as they are more relevant. Nodes that are along the central axis in the X direction and on the surface of the beam are selected. As the number of element divisions along the x axis is 60, the number of nodes selected would be 61. As consideration of all the nodes would increase computation effort and time, eleven equidistant nodes are selected. The displacement of each node from the equilibrium position, for each of the six frequencies is calculated and stored as an Eigen vector file. The values of all the six frequencies are stored as a frequency file. As expected the displacement of the end nodes is zero for all frequencies.

Obtaining D.C. Gain using MATLAB

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All the Eigen vector values for the six frequencies are uploaded in the MATLAB© Workspace. All the frequencies obtained might not contribute to the systems disturbance significantly. Hence, only the frequencies that dominate are selected. This selection is made on the basis of the D C Gain value for each mode. This D C Gain value is now calculated using MATLAB©. It is assumed that the force is applied at the node, which for a particular mode, is peak. As the case is of simply supported, again the same node is selected for the output. Now, the D C Gain is given as follows[8]:

D C Gain is calculated for each mode shape. D C Gain values obtained for each mode are as follows:

Graph of D.C. Value v/s Frequency is plotted.

MATLAB© plot for DC value vs. Mode number

From the graph, it is observed that as the mode number increases there is a significant decrease in DC Gain value. High frequency modes have less nodes displacement. For a given structure, modes with negligible D C Gain value do not contribute much to the disturbance. Hence, these high frequency modes can be ignored for controlling purpose.

Filtering of Significant Modes using MATLAB

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We rank the relative importance of the contribution of the each of the individual mode. The elimination of low D C value modes is performed iteratively. It is necessary that after elimination of high frequency modes there should be no significant in response of system. It is observed that last three modes have negligible D C Gain value, hence these modes are eliminated. Transient response result for all modes included and first 3 modes included are obtained. For transient analysis, node3 is selected for application of the force and same node is considered for output.

Simply Supported Node 3 Displacement v/s Frequency for 6 Modes included

Graph shows the magnitude of displacement for node 3 with respect to frequency. Five peaks in the graph signify five modes of vibration.

Simply Supported Node 3 Displacement v/s Time for 6 Modes included

MATLAB© result for node 3 displacement with respect to time is plotted. The disturbance vanishes at 2.5 sec. Same analysis is repeated again, but including only first 3 modes.

Simply Supported Node 3 Displacement v/s Frequency for 3 Modes included

Graph shows the magnitude of displacement for node 3 with respect to frequency. Three peaks in the graph signify three modes of vibration.

Simply Supported Node 3 Displacement v/s Time

MATLAB© result for node 3 displacement with respect to time is plotted for all modes included model and first three modes included model on same graph. It is observe that there is no significant difference in response.

Hence, for given structure, first three modes are of importance for controlling purpose.

Obtaining Reduced Model using MATLAB

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Although there is no significant difference observed for response of three modes included system, still an error is introduced as D C gain contribution of eliminated modes are not included in overall D C Gain. In order to eliminate this error, the MATLAB© function “modred” (MODel order REDuction) is introduced. In “modred” function assumptions are made are made about some modes being more important than other. This allows reducing size of the problem to that of the “ important modes”, while adjusting the overall D C Gain to account for the D C Gains of eliminated modes. The “mdc” or “Matched DC” gain option for the function “modred” reduces defined states by setting the derivatives of the state to be eliminated to zero, then solving for remaining states[8].

The other option for “modred” is the “del” option, which simply eliminates the defined states, typically associated with high frequency modes.

Node 3 Displacement with 3 Modes included with Modred "DEL" Option

Graph shows the node 3 displacement for three modes included model, using modred “del” option, where three high frequency modes are eliminated. It is observed that at high frequency the reduced curve attenuates with frequency similar to “all modes” curve.

Node 3 Displacement for 3 Modes included with Modred "MDC" Option

Graph shows the node 3 displacement for three modes included model, using modred “mdc” option. A rise in the high frequency portion of the magnitude curve as a result of the reduction is observed.

Simply Supported Node 3 Displacement v/s Time for all Modes, Reduced Modes with "MDC" & "DEL" Option

Response of node 3 for all modes included model, three modes included model with “mdc” option and three modes included model with “del” option are plotted on same graph. No visible difference in transient analysis is observed.

Transient Analysis using ANSYS

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FEM analysis software ANSYS© provides many features related to the analysis of the vibrations in a structure. One such feature is transient analysis. Transient analysis in ANSYS© is carried out for any application to study the system properties with respect to time. Transient vibration analysis gives useful information about system damping and other effects of the controlling forces on vibration with a function of time. ANSYS© provides excellent coding features by which the controller logic can be built in itself and controller simulation can be carried out. This feature of ANSYS© is used for active vibration control in the present case.

Controlling analysis is carried out in this section by iterative selection of controller gain. Controller gain is not designed based on the earlier results obtained from MATLAB©. In this section an attempt is made to control the first and second mode of vibration.

Simulation Study to Control First Mode using ANSYS

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The closed loop controller is introduced in ANSYS© by means of a macro. The sensor output is taken by taking the difference between the X direction displacement (u1 and u2) of two nodes, in the same plane on either side of the centre line. This is divided by the distance between them to get strain. It is then appropriately amplified by multiplying it with sensor gain ks. The controller is then defined by the error function and the voltage va to be applied is calculated by multiplying the error function with appropriate actuator gain (ka).

The generic macro used to simulate the controller action is given below:

*set,dt,0.0088
*set,ts2
*set,nv,559
*set,nr1,1117
*set,nr2,455
*set,dx,18*(8.33e-3)
*set,ks,1000
*set,kv,1000
*set,kc,1
*set,va,0
*do,t,3*dt,ts,dt
*get,u1,node,nr1,u,x
*get,u2,node,nr2,u,x
err=0-ks*(u2-u1)/dx
va=kc*kv*err
d,nv,volt,va
time,t
solve
*enddo

Where,

ts : Time for which analysis is carried out
nv : Node at which voltage is applied
nr1 : Higher node considered for strain calculation
nr2 : Lower node considered for strain calculation
dx : Distance between nodes nr1 and nr2
ks : Sensor gain
kv : Actuator gain
kc : Proportional (controller) gain
va : Applied voltage
err : Error function

The value of kc is varied to obtain desired controller effect. But constraint is placed on the maximum value of kc due to voltage. The values of Rayleigh damping are taken as 0.002.

Displacement v/s Time Plot for different kc values

Voltage v/s Time Plot for different kc values

Though in above case, the vibration settling time is decreased. But the rate of decrease is quite low. A steeper decrease of vibration settling time can be obtained by increasing the patch size.

Simulation Study for Controlling Second Mode using ANSYS

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Second mode of vibration can be controlled by placing the patch at a distance of 0.25 times the length of the beam, from any one end of the simply supported beam. The force for the transient analysis is applied at the same point to excite the second mode of vibration.

Meshed Integrated Structure for Control of Second Mode

All parameters like ks, kv and damping values are kept constant. The value of kc is varied for finding the controller effectiveness. The graphs of displacement v/s time and voltage v/s time are plotted.

Displacement v/s Time Plot for different kc values

Voltage v/s Time Plot for different kc values

Though in above case, the vibration settling time is decreased. But the rate of decrease is very low. A steeper decrease of vibration settling time can be obtained by increasing the patch size or using more than one patch.

References

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Amod Deshpande, Anil Salve, Ganesh Kasar, Chandresh Dubey, Active Vibration Control of Simply Supported Beam, project report for B.E. 2004.
A. V. Srinivasan, D. M. McFarland, Smart Structures: Analysis & Design, University Press, Cambridge, 2001.
Devdas shetty, Richard kolk, Mechatronics system design, Thomson learning, New Delhi, 2007.
F. Lewis, V. L. Syrmos, Optimal Control, Wiley, New York, 1990.
Iyer Subramaniam, Abhishek Bagwe, Nitin Gurav, Mangesh Ghandat, Rishikesh Bhere, Active Vibration Control, project report for B.E. 2005.
M. R. Hatch, Vibration Simulation using MATLAB & ANSYS, CRC Press, LLC, Boca Raton, FL, 2000.