# The Dynamic Finite Element Engineering Essay Essay

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- LITERRATURE REVIEW
- LQG ( Linear Quadratic Gaussian )
- .
- Aim
- Formulation
- FINITE ELEMNET FORMULATION
- ELEMENT GEOMETRY AND DISPLACEMENT FIELD
- =
- STRAIN DISPLACEMENT REALTION
- DIRECT AND CONVERSE PIEZOELECTRIC RELATION
- ELECTRIC POTENTIAL IN PIEZOELECTRIC PATCH
- DYNAMIC FINITE ELEMENT EQUATION
- =
- STATE-SPACE REPERSENTATION
- LQR OPTIMAL FEEDBACK
- DETERMINATION OF WEIGHTING MATRIX
- =
- LQG OPTIMAL FEEDBACK
- .

Composite stuffs are made up of two or more stuffs holding different chemical and physical belongingss which are separate and distinguishable in a macroscopic degree within the finished construction. Composite stuffs are extensively used in the industry of aerospace constructions and attempt is being put for the development of smart and intelligent constructions. A batch of research is traveling on this proficient development particularly in the field of wellness monitoring, quiver and control of flexible constructions utilizing detectors and actuators arrangement in the host construction. Composite stuffs being light weight construction has low internal damping and higher flexibleness and are susceptible to big quiver holding big decay clip. These construction require suited integrating of active control for better public presentation under operation.Piezoelectric stuffs which has flexible construction can move as detectors and actuators and it can supply self-monitoring and self-controlling capablenesss to these constructions. In many practical conditions, these construction experiences mechanical burden which are needed to be controlled for mechanical responses. Detectors, actuators and a accountant are required for this type of quiver control. The design procedure of this sort of system has three chief stages of structural design, optimum arrangement of detectors and actuators and design of accountants. A complete tool for the electro-mechanical analysis of such construction is necessary to work out such job and control strategy is required for control.

Functionally Graded Materials ( FGM ) , is characterized by changing the composing and construction in volume in such a manner that the local stuff belongingss of the stuff is achieved. By rating the stuff belongingss the consequence of emphasiss on the complexs due to sudden alterations in the stuff belongingss can be minimized. The FGM has many utilizations in these industrial universe. Aircraft and aerospace industry and computing machine circuit industry are much interested in this stuffs as it can defy high thermic emphasiss. This is done by utilizing a ceramic bed with a metallic bed. The ceramics material provide heat and caustic opposition and the metal provides strength and stamina.

Accountants are required to happen the optimum addition to minimise the public presentation index. Open-loop and closed-loop are used for the arrangement of actuators and detectors. Design of accountant avoids the work of happening out the control addition randomly to work out the aims and get the better of the jobs of impregnation. Linear Quadratic Control ( LQR ) has been used to happen the optimum addition by cut downing the public presentation index in quiver control utilizing the weighting matrices [ Q ] and [ R ] . These burdening matrices affects the end product public presentation and the input cost, therefore, the choice of burdening matrices is of much importance. Linear Quadratic Gaussian ( LQG ) is another method for finding of optimum addition

A figure of plants have been done for the active quiver control of smart constructions and besides for mechanical burden of these constructions. Till now LQR technique has been found effectual for the active quiver control with burdening matrices, which gives optimum control addition by minimising the public presentation index. In this present work LQG control strategy has been proposed for commanding the dynamic oscillation due to mechanical burden gradient and to command all types of quiver of different types of burden of functionally graded composite stuffs.

## LITERRATURE REVIEW

Composite stuffs offers superior belongingss of high strength and high stiffness to the metallic stuffs which has resulted in the big usage of composite stuffs is the aircraft and aerospace industry. In order to develop such stuffs, Koizumi [ 1 ] gave the construct of FGM ( functionally graded composite stuffs ) . Bailey and Hubbard [ 2 ] used the angular speed at the tip of cantilever beam with changeless addition and changeless amplitude and by experimentation achieved the control addition. Bhattacharya et Al. [ 3 ] used LQR method for quiver suppression of spherical shells made up of laminated complexs by test and mistake method of choice of burdening matrices [ Q ] and [ R ] . Ang et al [ 4 ] proposed a method of choosing burdening matrices which is entire energy method. Narayan and Balmurugun [ 5 ] presented a finite component patterning with distributed actuators and detectors and used LQR method to command the supplanting by test and mistake choice of matrices [ Q ] and [ R ] . Christensen and Santos [ 6 ] proposed an active control system to command blade and rotor quiver in a twosome rotor blade system utilizing tip mass actuators and detectors. Roy and Chakraborty [ 7 ] developed a GA based LQR control strategy to command the quiver of smart FRP constructions with piezoelectric surface at surface minimising the maximal supplanting. Abdullah et al [ 8 ] used GA to at the same time put actuators and detectors in the multi floor bulding and utilizing the end product feedback as the control jurisprudence in footings of leaden energy of the system and concluded that the determination variable is enormously dependent on the choice of burdening matrices [ Q ] an [ R ] . Robandi et al [ 9 ] proposed a usage of GA for optimum feedback control in multi machine power system. Yang et al [ 10 ] presented a coincident optimisation by puting the detectors and actuators and the size of the detector and actuator and feedback control addition for the quiver suppression of merely supported beam by minimising the entire mechanical energy of the system. They did non see the input energy and so the actuator electromotive force is was shown. Wang [ 11 ] presented the optimisation of detectors and actuators pairs for torsional quiver control of a laminated composite cantilever home base utilizing end product feedback control. Reddy and Cheng [ 12 ] presented three dimensional solutions for smart functionally graded home bases. He and Liew K M [ 13 ] presented active control of FGM home bases with incorporate piezoelectric detectors and actuators. Huang and Shen [ 14 ] presented the kineticss of functionally graded ( FG ) plate fall ining two piezoelectric beds at its top and bottom surfaces undergoing nonlinear quivers.

## LQG ( Linear Quadratic Gaussian )

LQG ( Linear Quadratic Gaussian ) control trades with unsure additive system which is distributed by linear white Gaussian noise, LQG accountant is a combination of LQE ( Linear Quadratic Estimator ) and LQR ( Linear Quadratic Regulator ) . LQG control can be applied to additive clip variant system and additive clip invariant system. Linear clip changing system is used for the designing of additive feedback accountant for non-linear system. The solution of LQG job is the most appropriate consequence in the control and system field.

The quadratic loss map is

= E [ x + ]

Where N & A ; gt ;

, and = semi definite symmetric matrices

Discrete clip additive equations

Where, i = distinct clip index

, = distinct clip Gaussian

Minimized quadratic cost map

J = E (

F, 0,

Discrete clip LQG accountant

= E (

The Kalman addition

Where is determined Riccati difference equation which runs frontward in clip,

The feedback addition matrix

## .

Where is determined Riccati difference equation which runs backward in clip,

## Aim

To happen the addition utilizing the Linear Quadratic Gaussian ( LQG ) control and Linear Quadratic Regulator ( LQR ) control.

To happen the responses of the quiver utilizing unit impulse and unit measure maps

To happen the feedback utilizing the addition and response and eventually commanding the quiver.

## Formulation

## FINITE ELEMNET FORMULATION

Here the Reissner – Mindlin premise has been considered to depict the kinematics utilizing the first order shear distortion theory. The basic premises are

The consecutive line normal to the mid surface may non stay consecutive during distortion

The consecutive line matching to the emphasis constituent which is extraneous to the mid surface is disregarded.

Fig1 smart bed shell component

Fig 2 shell component with assorted co-ordinate system

The general smart shell component with composite and piezoelectric beds is shown in figure 2. The piezoelectric spots are bonded to the surface of the constructions and the bonding beds are thin. The planetary co-ordinate system ( X-Y-Z ) represents the displacement constituents of the mid-point of the normal, the nodal co-ordinates, planetary stiffness matrix and applied force vectors. , , are three reciprocally perpendicular vectors at each nodal point. Vector is perpendicular to and parallel to x-z plane and is assumed to be parallel to x-axis. is obtained from the cross merchandise of and. , , are the unit vectors in the way of, , . ? – ? – ? is a natural co-ordinate system, where ? and ? are curvilineal co-ordinates and ? is the additive co-ordinate with ?= -1 and ?= 1 in the top and bottom surfaces.

## ELEMENT GEOMETRY AND DISPLACEMENT FIELD

The co-ordinate of a point within a component in isoparametric preparation is obtained as

( 1 )

Where

## =

is the shell at the K node.

The supplanting field has five grade of freedom, three supplanting of its mid-point and two rotary motions ( . The supplanting of a point calculated from the two rotary motions by

( 2 )

Where, , are the supplanting of node K along the mid surface of the planetary co-ordinate system, is the form map at node K.

## STRAIN DISPLACEMENT REALTION

Five strain constituents in the local co-ordinate system is

= ( 3 )

The strain-displacement matrix can be formed utilizing the supplanting derived functions in the planetary co-ordinate system. The relation between strain constituents in planetary co-ordinate system and the nodal variable can be expressed as

( 4 )

## DIRECT AND CONVERSE PIEZOELECTRIC RELATION

The additive piezoelectric constitutive equations which is coupled with elastic and electric field is expressed as

( 5 )

( 6 )

Where = electric supplanting vector, emphasis vector, = strain vector and = electric field vector. = piezoelectric yoke changeless and = insulator changeless matrix.

## ELECTRIC POTENTIAL IN PIEZOELECTRIC PATCH

The component is assumed to hold one grade of freedom at the top of the piezoelectric actuator and detector spots, and severally. The electric potency over an component is assumed to be changeless all over and it varies linearly through the thickness of the piezoelectric spot. As the electric field is dominant in the thickness way the electric field can be accurately approximated with a non-zero constituent merely. The electric field potency is expressed as with this estimate

( 7 )

( 8 )

Subscript a and s represents the actuators and detector spot, and are the electric field gradient matrices of the actuators and detectors severally.

## DYNAMIC FINITE ELEMENT EQUATION

The coupled finite element equation derived for one component theoretical account after the application of variational rule and finite component discretization becomes

( 9 )

Structural mass: =

Structural stiffness:

Dielectric conduction: –

The overall dynamic finite component equation is

## =

Where [ ] = planetary mass matrix, [ ] = planetary elastic stiffness matrix, [ ] and [ ] = planetary piezoelectric matching matrices of actuator and detector spots severally. [ ] and [ ] = planetary insulator stiffness matrices of actuator and detector spots severally.

## STATE-SPACE REPERSENTATION

The manners of quiver which are low have lower energy and are most excitable 1s. These are more important to the planetary response of the system. A average matrix ? s used as transmutation matrix between the generalised co-ordinates d ( T ) and the average co-ordinate matrix ? ( T ) . The displacement vector can be approximated as

Where = [ …….. ] is the abbreviated modal matrix.

The decoupled dynamic equation when modal damping is considered is

Where is the muffling ratio.

In the province infinite signifier it can be written as

[ ] = [ A ] [ X ] + [ B ] { } +

Is the system matrix

Is the perturbation matrix, = perturbation input vector, = control input

[ ] = and [ X ] =

The detector end product equation is

{ Y } = [ ] [ X ]

Where [ ] depends on the modal matrix [ ? ] and the detector matching matrix [ ]

## LQR OPTIMAL FEEDBACK

Linear quadratic regulator ( LQR ) is used to happen the control additions. Here, the feedback control is used to minimise the cost map which is relative to the system responses. The cost map is

J = dt

Where, [ Q ] = positive definite burdening matrices on the end product

[ R ] = positive definite burdening matrices on the control input

The Ricatti equation in steady matrix is

[ K ] + [ K ] [ A ] – [ K ] { B ] [ K ] + [ Q ] [ C ] = 0

The optimum addition after work outing the Ricatti equation is

[ ] = [ ] [ K ]

The end product electromotive force can be calculated utilizing the end product feedback by

## DETERMINATION OF WEIGHTING MATRIX

In LQR optimisation procedure burdening matrices [ Q ] and [ R ] are of import constituents as it influences the system public presentation. [ Q ] and [ R ] are assumed to be semi-positive definite matrix and positive definite matrix severally by Lewis. As per Ang et Al. [ Q ] and [ R ] can be determined by

[ Q ] = and [ R ] = ?

The leaden energy of the system is

## =

Where = dielectric matching matrix of the actuator

and = coefficients associated with entire kinetic energy, strive energy and input energy severally.

## LQG OPTIMAL FEEDBACK

LQG is besides used to happen the control additions like the LQR control. It is used to minimise the quadratic loss map which is dependent on the system responses

The quadratic loss map is

= E [ x + ]

Where N & A ; gt ;

, and = semi definite symmetric matrices

Minimized quadratic cost map

J = E ( , F, 0,

Discrete clip LQG accountant

= E ( ,

The Kalman addition

Where is determined Riccati difference equation which runs frontward in clip,

The feedback addition matrix

## .

Where is determined Riccati difference equation which runs backward in clip,