The Rayleigh Ritz Method Computer Science Essay

The Rayleigh Ritz rule Computer Science EssayThe precondition subsidization is done with the soul purpose of developing an intense knowledge and apprehensiveness of vibrational behaviour and dynamic response of structures.The assignment aims to apply up to date rooms of structural dynamics in aerospace and aerospace system engineering. Here we using up Rayleigh-Ritz manner and delimited piece manner to regain the natural relative frequence and panache experimental condition of the given basintilever diffuse.1. Rayleigh-Ritz MethodRayleigh-Ritz method is an extension service of the Rayleigh method which was developed by the Swiss mathematician and physicist Walter Ritz. Its one of the widely apply method to calculate more accurate measure of fundamental frequency, win it also gives nearnesss to the higher frequencies and mode plaster casts.In the Ritz method the single shape break eat up is replaced by a series of shape break downs multiplied by constant c oefficients, that is the single execute of deflection choose in Rayleigh method is fake to be a sum of several crops multiplied by constant coefficients. The coefficients value ar modified by reducing the frequency with respect to each of the coefficients, which result in n algebraic equations in. The antecedent of these equations pull up stakes give the value of natural frequency and mode shapes of the system. It should be taken into account that the success of the method is solitary(prenominal) possible so extensive as the shape function taken satisfies the geometric landmark conditions of the difficulty. The method should also be differentiable to the found of the derivatives of the equations. Here the function bug outhouse ignore discontinuities like shear due to concentrated great deal that involve third derivatives in communicate.The Rayleigh-Ritz method is done by anticipate the deflection curve of the transfer byThe function argon the simulated chemise fu nctions that satisfy geometrical boundary conditions.For a sesstilever burn the boundary conditions argonThey argon selected such that it is possible to obtain a good nearness to each of the required natural modes by superposition.The quantities are generalized coordinates representing contributions of each fictitious functions.For a beam divided into n span wise stations the total differential equation send word be take inulate using Lagrange equation asPutting as a solution , where the amplitude of the displacement is, is the frequency and is the phase angle.This educate of characteristics equations can be solved for n discrete values of . This equation can easily be put into a hyaloplasm form for numerical calculation asFor a beam divided into n span wise station the wad and stiffness terms can be formulated into matrices asWhere = hyaloplasm of sham modes= mass matrix= matrix of weighting coefficients= rigidity matrixHence we write asThe above equation is considered t o be convenient for computation, but has limitations in the modal verbity of expressing the tug life force.Given DataLength L=1.5Modulus of Elasticity E=74 GPaPoissons Ratio P=0.33Material densityThe sense of the beam tapers uniformly from 0.3 at the decided end to 0.1 at the drop by the wayside end.The breadth of the beam tapers uniformly from 0.02 at the unbending end to 0.005 at the free end.The fictive modes are given by the polynomial functionMATLAB Operation L=1.5L =1.5000x=0,0.15,0.3,0.45,0.6,0.75,0.9,1.05,1.2,1.35,1.5x =0 0.1500 0.3000 0.4500 0.6000 0.7500 0.9000 1.0500 1.2000 1.3500 1.5000 s=x/Ls =0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000 V1= 2*s.2-(4/3)*s.3+(1/3)*s.4V1 =0 0.0187 0.0699 0.1467 0.2432 0.3542 0.4752 0.6027 0.7339 0.8667 1.0000 V2=(10/3)*s.3-(10/3)*s.4+s.5V2 =0 0.0030 0.0217 0.0654 0.1382 0.2396 0.3658 0.5111 0.6690 0.8335 1.0000 V=V1V2V =0 0.0187 0.0699 0.1467 0.2432 0.3542 0.4752 0.6027 0.7339 0.8667 1.00000 0.0030 0.02 17 0.0654 0.1382 0.2396 0.3658 0.5111 0.6690 0.8335 1.0000 dV1=(1/(L.2))*(4-8*s+4*(s.2))dV1 =1.7778 1.4400 1.1378 0.8711 0.6400 0.4444 0.2844 0.1600 0.0711 0.0178 0 dV2= (1/(L.2))*(20*s-40*(s.2)+20*(s.3))dV2 =0 0.7200 1.1378 1.3067 1.2800 1.1111 0.8533 0.5600 0.2844 0.0800 0 dV=dV1dV2dV =1.7778 1.4400 1.1378 0.8711 0.6400 0.4444 0.2844 0.1600 0.0711 0.0178 00 0 .7200 1.1378 1.3067 1.2800 1.1111 0.8533 0.5600 0.2844 0.0800 0Weighting matrix can be formulated using Trapezoidal rule, Simpsons rule and Lagranges Interpolation formula.By Lagranges insertion formula if the beam is divided into 10 equal parts with spacing d then weighting matrix is computed asMATLAB Operation d=0.15d =0.1500 W1=(d/3.7266)*1,6.616,-3.020,16.954,-16.216,26.599,-16.216,16.954, -3.020, 6.616,1W1 =0.0403 0.2663 -0.1216 0.6824 -0.6527 1.0706 -0.6527 0.6824 -0.1216 0.2663 0.0403 W=diag(W1)W =0.0403 0 0 0 0 0 0 0 0 0 00 0.2663 0 0 0 0 0 0 0 0 00 0 -0.1216 0 0 0 0 0 0 0 00 0 0 0.6824 0 0 0 0 0 0 00 0 0 0 -0.6527 0 0 0 0 0 00 0 0 0 0 1.0706 0 0 0 0 00 0 0 0 0 0 -0.6527 0 0 0 00 0 0 0 0 0 0 0.6824 0 0 00 0 0 0 0 0 0 0 -0.1216 0 00 0 0 0 0 0 0 0 0 0.2663 00 0 0 0 0 0 0 0 0 0 0.0403Mass matrix is a diagonal matrix representing the mass per strong length at the eleven span wise stations.The matrix can be calculated byMaterial density = 2700The depth of the beam at a station with a distance x from the fixed end is given byDepthSimilarly the breadth of the beam at a station with a distance x from the fixed end is given byBreadthMATLAB Operation h=0.3-(s*0.2)h =0.3000 0.2800 0.2600 0.2400 0.2200 0.2000 0.1800 0.1600 0.1400 0.1200 0.1000 b=0.02-(s*0.015)b =0.0200 0.0185 0.0170 0.0155 0.0140 0.0125 0.0110 0.0095 0.0080 0.0065 0.0050 m=2700*diag(b)*diag(h)m =16.2000 0 0 0 0 0 0 0 0 0 00 13.9860 0 0 0 0 0 0 0 0 00 0 11.9340 0 0 0 0 0 0 0 00 0 0 10.0440 0 0 0 0 0 0 00 0 0 0 8.3160 0 0 0 0 0 00 0 0 0 0 6.7500 0 0 0 0 00 0 0 0 0 0 5.3460 0 0 0 00 0 0 0 0 0 0 4.1040 0 0 00 0 0 0 0 0 0 0 3.0240 0 00 0 0 0 0 0 0 0 0 2.1060 00 0 0 0 0 0 0 0 0 0 1.3500The atomic add 16 moment of area of the beam is given byMATLAB Operation I=diag(h)*(diag(b).3)/12I =1.0e-006 *0.2000 0 0 0 0 0 0 0 0 0 00 0.1477 0 0 0 0 0 0 0 0 00 0 0.1064 0 0 0 0 0 0 0 00 0 0 0.0745 0 0 0 0 0 0 00 0 0 0 0.0503 0 0 0 0 0 00 0 0 0 0 0.0326 0 0 0 0 00 0 0 0 0 0 0.0200 0 0 0 00 0 0 0 0 0 0 0.0114 0 0 00 0 0 0 0 0 0 0 0.0060 0 00 0 0 0 0 0 0 0 0 0.0027 00 0 0 0 0 0 0 0 0 0 0.0010Rigidity matrix is the diagonal matrix that gives the product of modulus of elasticity and the second moment of area of the beam almost the neutral axis.EI=74000000000*IEI =1.0e+004 *1.4800 0 0 0 0 0 0 0 0 0 00 1.0933 0 0 0 0 0 0 0 0 00 0 0.7877 0 0 0 0 0 0 0 00 0 0 0.5511 0 0 0 0 0 0 00 0 0 0 0.3723 0 0 0 0 0 00 0 0 0 0 0.2409 0 0 0 0 00 0 0 0 0 0 0.1477 0 0 0 00 0 0 0 0 0 0 0.0846 0 0 00 0 0 0 0 0 0 0 0.0442 0 00 0 0 0 0 0 0 0 0 0.0203 00 0 0 0 0 0 0 0 0 0 0.0077 replace in Rayleigh-Ritz equationThis givesSimplifyingThe above equation is a qu adratic equation in , which can be solved=Result The approximate values of the first and second natural frequencies of the given beam under flexural vibrations, by the use of Rayleigh- Ritz method, was found to be2. manner shapesConsider the equationSubstituting the values of in the above equation and simplifyingThe column matrix that represents the mode shape at the eleven stations is obtained by putting,=0.0578Substituting the value of in the above equation and simplifyingThe column matrix that represents the mode shape at the eleven stations is obtained by putting,= 0.06933. Finite Element MethodFinite Element Method (FEM) is considered to be one of the profound developments in the static and dynamics analysis of continuous systems. It provides a discrete approximation to vibration of continuous systems. The finite part method can be developed as a special case of the Rayleigh -Ritz method. The method was originally developed for the static- stress analysis of convoluted dist ributed parameter structures. Now a days FEM is widely utilise to disciplines of heat transfer, electro magnetics, fluid flow and vibrations.In finite portion method the structure is divided into a wide-ranging follow of small but finite parts called elements which are matching at points called nodes. For each element a displacement function is assumed which satisfies the geometric boundary condition so that continuity is achieved between the elements. The variations in displacement of each element( which can be linear, quadratic etc.), are assumed oer the length of the element. This method allows the displacement of whatsoever point in the element to be expressed in terms of the displacement at the end of the element. These displacements by finite element terminology are called nodal variables. contrasted Rayleigh-Ritz in finite element method the global coordinate is replaced by a local coordinate where is the length of the element. The kinetic and strain zip of the element is obtained by integrating along the elements length, in terms of the nodal variables. By superposing the energies contributed by the individual elements into which the structure is divided, we can obtain the kinetic and strain button of the structure or system in terms of the nodal variables of the whole structure. The finite element method is mainly based on variational principles.The method is considered very much versatile and can be use to somatogenetic problems with arbitrary shapes, oodles and support conditions. The finite element model has a close resemblance to the actual structure.Many general finite element code packages have been written over the years with user warm windows and menus (GUI) which allow for easy geometry setup, boundary condition manipulation and evaluation/ bet on processing of common structural problems. Some of the most popular codes in the industry are ANSYS, MSC Nastran and MARC. ANSYS will be the code utilize for this assignment.ANSYS Operati onDe o.k. Material clapperclaw 1 Set preferencesPreferences are set in order to filter quantities that pertain to this discipline. mistreat 2 Define constant material properties.Modulus of elasticity, Poissons ratio and Density are defined. mensuration 3- ModelingCreate the beam with required geometry.Generating troth gradation 4 Define element typeTwo element types are defined a 2-D element and a three-D element. The beam cross-sectional area is betrothaled with 2-D elements, and then the area is to be extruded to create a 3-D volume. The occupy will be extruded along with the geometry so 3-D elements will automatically be created in the volume.Step 5 Mesh the areaMesh control are specified in order to obtain a particular absorb density.Element edge length is set at 0.01Note Mesh density is very important. If the mesh is too coarse your result can contain serious errors. If the mesh is too fine, would cause waste of computer resources, experience excessively long run time, th e model may be too large to be run on the computer system. Unfortunately it can non be definitively specified how fine the mesh density should be. But one way to find out is to perform the analysis with what seems to be a reasonable mesh. Then reanalyse the problem with twice as many elements in the critical region and comparability the results. If the two mesh give the same result then the mesh probably be adequate. If there is substantial difference between the two results then further refinement of the mesh is required.Step 6- extrude the meshed area into a meshed volume.The 3-D volume is generated by first changing the element type to SOLID 45, which is defined as element type 2, and then extruding the area into a volume.The subjugate of element divisions is set as 10Offsets of extrusion are set as 0, 0, 1.5Tapering ratio is given as 0.33333, 0.25, 0Apply loadsStep 7- Unselect 2-D elementsBefore applying constraints to the fixed end of the wing, unselect all the PLANE42 elem ents employ in the 2-D area mesh since they will not be used for the analysis.Step 8- Apply constraints to the modelConstraints will be applied to all nodes located where the wing is fixed to the body. Select all nodes at z = 0, then apply the displacement constraints.Obtain SolutionStep 9- Specify analysis type and optionsSpecify a modal analysis type. topic of modes to be extracted is given as 5Number of modes to be expanded is given as 5Step 10- SolveObserve resultsStep 11-List natural frequencies***** INDEX OF info SETS ON RESULTS FILE *****SET TIME/FREQ LOAD STEP SUBSTEP accumulative1 11.964 1 1 12 40.840 1 2 23 100.05 1 3 34 144.08 1 4 45 182.70 1 5 5Step 12- Animate the two mode shapes.Set the results for the first mode to be animated. Observe the first mode shape. Animate the next mode shape. Observe the second mode shape. Repeat the same procedure to obtain the other three mode shapes.First Mode ShapeSecond Mode ShapeThird Mode ShapeFourth Mode ShapeFifth Mode Shape4. re lation of Rayleigh-Ritz Method and Finite Element MethodRayleigh-Ritz methodRayleigh-Ritz method uses the principle of conservation of energy to formulate the matrix equation.one major advantage of this method is that it allows us to pretermit the non-applied forces like forces at a point of rolling contact, forces at frictionless guides etc.Considering the method to be an extension of the Rayleigh method it has an improved the rightful(a) by assuming the deflection curve of the beam to be all the same the assumed function should satisfy all the boundary conditions and should be linearly independent. This can be easily achieved by using polynomial normal to derive the deflection function.Disadvantages and limitationsThe selection of assumed deflection function requires a good knowledge and expertise of the methodGood approximation of the true natural modes are only possible as the assumed function are limited in numbers and natureAll n modal solutions will not give a good appro ximation to the true mode, so it is necessary to discard some higher frequency modes.The approximations are only good for lower modes and it becomes worse for higher modes.The major limitation of this method is in the manner in which the strain energy can be expressed.Finite Element MethodThe finite element method (its practical application is often as Finite Element Analysis)is a powerful technique developed in the analysis of complex structural mechanics. In this method the structure is divided into large number of finite parts or elements which are interconnected at points called nodes. The elements will have properties like thickness, Youngs modulus, Poissons ratio etc. An assumption is made over the variation over the length of the element.This allows to find the displacement at any point in the given structure by introducingDisadvantages and limitationsThe method is not considered convenient for simple structures.Its a time consuming operation.Its accuracy depends on the numbe r of elements the structure is divided.It does not provide a closed-form solution, denying analytical resume of the effects of changing parameters.It needs a reliable program for support.Creating a good model requires experience.A good amount of data are required and voluminous output must be sorted and studied.Comparison of resultsObtained by Rayleigh-Ritz MethodObtained by Finite Element Method% divagation0.77%26.82%There is difference of 26.82% for the second frequency of the system. The value obtained by the Rayleigh-Ritz method can be brought closer to accuracy by increasing the number of assumed functions and by improving their nature. Normally the approximation becomes worse as we move to higher modes .With use of only two assumed functions, the solutions obtained are considered to be satisfactory.Errors in Rayleigh-Ritz methodTo use the method with ease the assumed functions are kept as simple as possible by using simple polynomial functions and at fewer times only the fun ctions of sine and cosine are used. There is no exact settlement as to which function the good approximate value can be obtained.There are always some terms omitted in the function which results to an ineffective solution.This method is considered to be inflexible as the actual displacement of the structure is restricted to only the shape generated by superposing the finite number of functions selected by the analyst. recommendation to improve Rayleigh-Ritz methodThe iterative process can be carried out with each time adding the term in the assumed function until it gives the exact value.Errors in Finite Element method ill-shapen mesh can result in flawed stiffness and mass termsErrors are always presented at joints and constrained boundaries due to uncertainty.Recommendation to improve Finite Element methodIterative method has to be applied to see the number of elements actually required to break down the structure so as to obtain more accurate value. enchant methods should be emp loyed for remeshing like HYPERMESH, Mesh++ based on a posterior error.CONCLUSIONThe first and second natural frequencies of the given beam are found out by using both Rayleigh-Ritz method and Finite element method, and mode shapes for these frequencies are drawn.