Review Finite Element Modeling

Transcription

1 Strctral Dynamic Modeling Techniqes & Modal Analysis Methods [ M ] [ K ] n [ M ] n a [ K ] a [ ω ] [ E a ] Review Finite Element Modeling Peter Avitabile Mechanical Engineering Department University of Massachsetts Lowell.55 Finite Element Review Dr. Peter Avitabile

2 Types of Models for Vibration Analysis Models are developed to assist in the design and nderstanding of system dynamics Analytical models (sch as finite element models) are tilized in the design process Experimental models are also sed for many systems where modeling is not practical or models are too difficlt to develop.55 Finite Element Review Dr. Peter Avitabile

3 Finite Element Model Considerations Finite element models are commonly sed What are we trying to do when generating a model CONTINUOUS SOLUTION DISCRETIZED SOLUTION.55 Finite Element Review 3 Dr. Peter Avitabile

4 Finite Element Model Considerations Modeling Isses continos soltions wor well with strctres that are well behaved and have no geometry that is difficlt to handle most strctres don't fit this simple reqirement (except for frisbees and cymbals) real strctres have significant geometry variations that are difficlt to address for the applicable theory a discretized model is needed in order to approximate the actal geometry the degree of discretization is dependent on the waveform of the deformation in the strctre finite element modeling meets this need.55 Finite Element Review 4 Dr. Peter Avitabile

5 Finite Element Model Considerations Finite element modeling involves the descretization of the strctre into elements or domains that are defined by nodes which describe the elements. A field qantity sch as displacement is approximated sing polynomial interpolation over each of the domains. The best vales of the field qantity at nodes reslts from a minimization of the total energy. Since there are many nodes defining many elements, a set of simltaneos eqations reslts. Typically, this set of eqations is very large and a compter is sed to generate reslts..55 Finite Element Review 5 Dr. Peter Avitabile

6 Finite Element Model Considerations Nodes represent geometric locations in the strctre. Elements bondary are defined by the nodes. The type of displacement field that exists over the domain will determine the type of element sed to characterize the domain. Element characteristics are determined from Theory of Elasticity and Strength of Materials..55 Finite Element Review 6 Dr. Peter Avitabile

7 Analytical Topics for Strctral Dynamic Modeling Strctral element formlations se the same general assmptions abot their respective behavior as their respective strctral theories (sch as trss, beam, plate, or shell) Continm element formlations (sch as D and 3D solid elements) comes from theory of elasticity ν i [ ] ν j EI 6L = 3 L 6L 6L 4L 6L L 6L 6L 6L L 6L 4L F i θ i E, I L θ j F j [ m] ρal = L 54 3L L 4L 3L 3L 54 3L 56 L 3L 3L 4L L.55 Finite Element Review 7 Dr. Peter Avitabile

8 Analytical Topics for Strctral Dynamic Modeling The basis of the finite element method is smmarized below t v s sbdivide the strctre into small finite elements each element is defined by a finite nmber of node points assemble all elements to form the entire strctre within each element, a simple soltion to governing eqations is formlated (the soltion for each element becomes a fnction of nnown nodal vales general soltion for all elements reslts in algebraic set of simltaneos eqations.55 Finite Element Review 8 Dr. Peter Avitabile

9 Finite Element Model Considerations DEGREES OF FREEDOM maximm 6 dof can be described at a point in space finite element se a maximm of 6 dof most elements se less than 6 dof to describe the element TRUSS TORSIONAL ROD STRUCTURAL ELEMENTS 3D BEAM PLATE CONTINUUM ELEMENTS.55 Finite Element Review 9 Dr. Peter Avitabile

10 Finite Element Model Considerations Advantages Models sed for design development No prototypes are necessary Disadvantages Modeling assmptions Joint design difficlt to model Component interactions are difficlt to predict Damping generally ignored.55 Finite Element Review 0 Dr. Peter Avitabile

11 Finite Element Model Considerations A TYPICAL FINITE ELEMENT USER MAY ASK what ind of elements shold be sed? how many elements shold I have? where can the mesh be coarse; where mst it be fine? what simplifying assmptions can I mae? shold all of the physical strctral detail be inclded? can I se the same static model for dynamic analysis? how can I determine if my answers are accrate? how do I now if the software is sed properly?.55 Finite Element Review Dr. Peter Avitabile

12 Finite Element Model Considerations ALL THESE QUESTIONS CAN BE ANSWERED, IF the general strctral behavior is well nderstood the elements available are nderstood the software operation is nderstood (inpt procedres, algorithms,etc.) BASICALLY - we need to now what we are doing!!! IF A ROUGH BACK OF THE ENVELOP ANALYSIS CAN NOT BE FORMULATED, THEN MOST LIKELY THE ANALYST DOES NOT KNOW ENOUGH ABOUT THE PROBLEM AT HAND TO FORMULATE A FINITE ELEMENT MODEL.55 Finite Element Review Dr. Peter Avitabile

13 Finite Element Modeling Using standard finite element modeling techniqes, the following steps are sally followed in the generation of an analytical model node generation element generation coordinate transformations assembly process application of bondary conditions model condensation soltion of eqations recovery process expansion of redced model reslts.55 Finite Element Review 3 Dr. Peter Avitabile

14 Finite Element Modeling Element Definition Each element is approximated by {} δ = [ N]{ x} Shape Fnctions Linear where {δ} [N] {x} - vector of displacements in element - shape fnction for selected element - nodal variable Qadratic Element shape fnctions can range from linear interpolation fnctions to higher order polynomial fnctions. Polynomial.55 Finite Element Review 4 Dr. Peter Avitabile

15 Finite Element Modeling Strain Displacement Relationship The strain displacement relationship is given by where {} ε = [ B]{ x} {ε} [B] {x} - vector of strain within element - strain displacement matrix (proportional to derivatives of [N]) - nodal variable.55 Finite Element Review 5 Dr. Peter Avitabile

16 Finite Element Modeling Mass and Stiffness Formlation The mass and stiffness relationship is given by where [M] [K] [N] {ρ} [B] [C] [ M] = [ N] ρ[ N] V T [ K] = [ B] [ C][ B] V V - element mass matrix - element stiffness matrix - shape fnction for element - density - strain displacement matrix - stress-strain (elasticity) matrix V T.55 Finite Element Review 6 Dr. Peter Avitabile

17 Finite Element Modeling Coordinate Transformation Generally, elements are formed in a local coordinate system which is convenient for generation of the element. Elemental matrices are transformed from the local elemental coordinate system to the global coordinate system sing { x } = [ T ]{ } x LOCAL SYSTEM φ GLOBAL SYSTEM.55 Finite Element Review 7 Dr. Peter Avitabile

18 Finite Element Modeling Assembly Process Elemental matrices are then assembled into the global master matrices sing { x } = [ c ]{ x } where {x} - element degrees of freedom [c] - connectivity matrix {xg} - global degrees of freedom g The global mass and stiffness matrices are assembled and bondary conditions applied for the strctre.55 Finite Element Review 8 Dr. Peter Avitabile

19 Finite Element Modeling Bondary Conditions Bondary Conditions Three Different Methods Elemental matrices are assembled into the global master matrices sing [ K ]{ x } = { F } n n where the eqation for soltion is n where the eqation for the reaction loads is [ Kaa ] [ Kab ] x a Fa = [ K ] [ K ] x F ba bb [ Kaa ]{ xa} + [ Kab ]{ x b} = { Fa } [ K ]{ x } = { F } [ K ]{ x } aa a a ab [ K ]{ x } + [ K ]{ x } = { F } ba a bb b b b b b.55 Finite Element Review 9 Dr. Peter Avitabile

20 Finite Element Modeling Bondary Conditions Bondary Conditions - Method - Decople Eqations Set off-diagonal terms to zero [ K ] aa [ K ] bb x x a b = { Fa } [ Kab]{ xb} [ K ]{ x } bb b.55 Finite Element Review 0 Dr. Peter Avitabile

21 Finite Element Modeling Bondary Conditions Bondary Conditions - Method Stiff Spring Apply stiff spring to bonded dofs (approx zero off-diagonal) [ Kaa ] [ Kab ] [ K ] [ K K ] ba bb x x a = { Fa } [ Kstiff ]{ F } + stiff b b Bondary Conditions - Method 3 Partition Eqations Partition ot bonded DOF [ K ]{ x } = { F } aa a a.55 Finite Element Review Dr. Peter Avitabile

22 Finite Element Modeling Static Soltions typically involve decomposition of a large matrix matrix is sally sparsely poplated majority of terms concentrated abot the diagonal Eigenvale Soltions se either direct or iterative methods direct techniqes sed for small matrices iterative techniqes sed for a few modes from large matrices Propagation Soltions most common soltion ses derivative methods stability of the nmerical process is of concern at a given time step, the eqations are redced to an eqivalent static form for soltion typically many times steps are reqired.55 Finite Element Review Dr. Peter Avitabile

23 Finite Element Modeling - Simple Example Consider the spring system shown below 3 f 3 each spring element is denoted by a box with a nmber each element is defined by nodes denoted by the circle with a nmber assigned to it the springs have a node at each end and have a common node point the displacement of each node is denoted by with a sbscript to identify which node it corresponds to there is an applied force at node 3.55 Finite Element Review 3 Dr. Peter Avitabile

24 Finite Element Modeling - Simple Example The first step is to formlate the spring element in a general sense i j p f ip f jp the element label is p the element is bonded by node i and j assme positive displacement conditions at both nodes define the force at node i and node j for the p element i j Application of simple eqilibrim gives f f ip jp = = p p ( ( i j j i ) = + ) = p p i i + p p j j.55 Finite Element Review 4 Dr. Peter Avitabile

25 5 Dr. Peter Avitabile.55 Finite Element Review This can be written in matrix form to give Now for element # And for element # The eqilibrim reqires that the sm of the internal forces eqals the applied force acting on each node Finite Element Modeling - Simple Example = jp ip j i p p p p f f = f f = 3 3 f f

26 6 Dr. Peter Avitabile.55 Finite Element Review The three eqations can be written as or in matrix form Finite Element Modeling - Simple Example f f f = + = + + = = f f f

27 7 Dr. Peter Avitabile.55 Finite Element Review Finite Element Modeling - Simple Example Now applying a bondary condition of zero displacement at node has the effect of zeroing the first colmn of the K matrix which gives three eqations with nnowns. Solving for the second and third eqation gives = f 0 = f f f

28 8 Dr. Peter Avitabile.55 Finite Element Review Finite Element Modeling - Simple Example Assembly of the stiffness matrix with more elements Notice that the banded natre of the matrix is not preserved when elements are arbitrarily added to the assembly

29 Finite Element Modeling - Beam Elements The force-stiffness eqation for a beam is given by the following eqation F M F M = EI l 3 6l 6l 6l 4l 6l l 6l 6l 6l υ l θ 6l υ 4l θ.55 Finite Element Review 9 Dr. Peter Avitabile

30 Finite Element Modeling - Beam Elements The beam element stiffness matrix is [ K] = EI l 3 6l 6l 6l 4l 6l l 6l 6l 6l l 6l 4l The beam element mass matrix is [ M] = ρal l 54 3l l 4l 3l 3l 54 3l 56 l 3l 3l l 4l.55 Finite Element Review 30 Dr. Peter Avitabile

31 FEM - Beam Element Assembly with Spport A three beam model with spring spport [ K] EI 6l = 3 l 6l 6l 4l 6l l 6l 6l 6l l 6l 4l [ K] EI 6l = 3 l 6l 6l 4l 6l l 6l 6l 6l l 6l 4l [ K] EI 6l = 3 l 6l 6l 4l 6l l 6l 6l 6l l 6l 4l AE [ K] = l [ M] 56 ρal l = l l 4l 3l 3l 54 3l 56 l 3l 3l l 4l AE [ K] = l 56 ρal l l l 54 4l 3l 3l [ M ] = [ M] 3l 3l 3l 56 l l 4l 56 ρal l = l l 4l 3l 3l 54 3l 56 l 3l 3l l 4l.55 Finite Element Review 3 Dr. Peter Avitabile

32 FEM - Beam Element Assembly with Spport The individal stiffness elements assemble as (K + ) 6l 6l 6l 4l 6l l 6l ( + ) ( 6l + 6l) 6l 6l l ( 6l + 6l) (4l + 4l ) 6l 6l ( + ) l ( 6l + 6l) 6l ( 6l (4l 6l l + 6l) + 4l 6l l ) 6l (K + ) 6l 6l l 6l 4l BEAM # BEAM # BEAM #3 SPRING SPRING.55 Finite Element Review 3 Dr. Peter Avitabile

33 MATLAB / MATSAP Script File.55 Finite Element Review 33 Dr. Peter Avitabile

38 Simplistic Model of Mixing Device Natral freqency determination and spport location are reqired for a mixing device MOTOR = 000 LB PROPOSED SUPPORT LOCATIONS.55 Finite Element Review 38 Dr. Peter Avitabile

39 Simplistic Model of Mixing Device The device can be broen down into simplistic pieces of beam elements, mass elements and spport characteristics 3 FT 0 FT 8 FT 8 IN 8 IN OD PIPE IN THICK 6 IN OD PIPE / IN THICK IN 8 IN 0 IN OD PIPE IN THICK MOTOR = 000 LB.55 Finite Element Review 39 Dr. Peter Avitabile

40 Simplistic Model of Mixing Device The characteristics of each individal component are needed in order to mae a rogh model of the system for preliminary evalations. The model is developed to determine the system characteristics to assess whether a more detailed model is needed.55 Finite Element Review 40 Dr. Peter Avitabile

41 MATLAB / MATSAP Script File - Cantilever Beam.55 Finite Element Review 4 Dr. Peter Avitabile