Advanced Vibrations. Distributed-Parameter Systems: Approximate Methods Lecture 20. By: H. Ahmadian

Transcription

1 Advanced Vibrations Distributed-Parameter Systems: Approximate Methods Lecture 20 By: H. Ahmadian

2 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

3 RAYLEIGH'S PRINCIPLE The lowest eigenvalue is the minimum value that Rayleigh's quotient can take by letting the trial function Y(x) vary at will. The minimum value is achieved when Y(x) coincides with the lowest eigenfunction Y 1 (x).

4 RAYLEIGH'S PRINCIPLE Consider the differential eigenvalue problem for a string in transverse vibration fixed at x=0 and supported by a spring of stiffness k at x=l. Exact solutions are possible only in relatively few cases, Most of them characterized by constant tension and uniform mass density. In seeking an approximate solution, sacrifices must be made, in the sense that something must be violated. Almost always, one forgoes the exact solution of the differential equation, which will be satisfied only approximately, But insists on satisfying both boundary conditions exactly.

5 RAYLEIGH'S PRINCIPLE Rayleigh's principle, suggests a way of approximating the lowest eigenvalue, without solving the differential eigenvalue problem directly. Minimizing Rayleigh's quotient is equivalent to solving the differential equation in a weighted average sense, where the weighting function is Y(x).

6 RAYLEIGH'S PRINCIPLE Boundary conditions do not appear explicitly in the weighted average form of Rayleigh's quotient. To taken into account the characteristics of the system as much as possible, the trial functions used in conjunction with the weighted average form of Rayleigh's quotient must satisfy all the boundary conditions of the problem. Comparison functions: trial functions that are as many times differentiable as the order of the system and satisfy all the boundary conditions.

7 RAYLEIGH'S PRINCIPLE The trial functions must be from the class of comparison functions. The differentiability of the trial functions is seldom an issue. But the satisfaction of all the boundary conditions, particularly the satisfaction of the natural boundary conditions can be. In view of this, we wish to examine the implications of violating the natural boundary conditions.

8 RAYLEIGH'S PRINCIPLE Rayligh s quotient involves V max and T ref, which are defined for trial functions that are half as many times differentiable as the order of the system and need satisfy only the geometric boundary conditions, as the natural boundary conditions are accounted for in some fashion.

9 RAYLEIGH'S PRINCIPLE Trial functions that are half as many times differentiable as the order of the system and satisfy the geometric boundary conditions alone as admissible functions. In using admissible functions in conjunction with the energy form of Rayleigh's quotient, the natural boundary conditions are still violated. But, the deleterious effect of this violation is somewhat mitigated by the fact that the energy form of Rayleigh's quotient, includes contributions to V max from springs at boundaries and to T ref from masses at boundaries. But if comparison functions are available, then their use is preferable over the use of admissible functions, because the results are likely to be more accurate.

10 Example: Lowest natural frequency of the fixed-free tapered rod in axial vibration The 1 st mode of a uniform clamped-free rod as a trial function: A comparison function

11 THE RAYLEIGH-RITZ METHOD The method was developed by Ritz as an extension of Rayleigh's energy method. Although Rayleigh claimed that the method originated with him, the form in which the method is generally used is due to Ritz. The first step in the Rayleigh-Ritz method is to construct the minimizing sequence: undetermined coefficients independent trial functions

12 THE RAYLEIGH-RITZ METHOD The independence of the trial functions implies the independence of the coefficients, which in turn implies the independence of the variations

13 THE RAYLEIGH-RITZ METHOD Solving the equations amounts to determining the coefficients, as well as to determining

14 THE RAYLEIGH-RITZ METHOD To illustrate the Rayleigh-Ritz process, we consider the differential eigenvalue problem for the string in transverse vibration:

15 THE RAYLEIGH-RITZ METHOD

16 Example : Solve the eigenvalue problem for the fixed-free tapered rod in axial vibration The comparison functions

17 Example :

18 Example : n = 2

19 Example : n = 2

20 Example : n = 3

21 Example : n = 3

22 Example : The Ritz eigenvalues for the two approximations are: The improvement in the first two Ritz natural frequencies is very small, indicates the chosen comparison functions resemble very closely the actual natural modes. Convergence to the lowest eigenvalue with six decimal places accuracy is obtained with 11 terms:

23 Truncation Approximation of a system with an infinite number of DOFs by a discrete system with n degrees of freedom implies truncation: Constraints tend to increase the stiffness of a system: The nature of the Ritz eigenvalues requires further elaboration.

24 Truncation A question of particular interest is how the eigenvalues of the (n +1)-DOF approximation relate to the eigenvalues of the n-dof approximation. We observe that the extra term in series does not affect the mass and stiffness coefficients computed on the basis of an n-term series (embedding property):

25 Truncation For matrices with embedding property the eigenvalues satisfy the separation theorem:

26 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

27 Advanced Vibrations Separation Theorem for Natural Systems Lecture 21 By: H. Ahmadian

28 Max-Min Characterization of the Eigenvalues The lowest eigenvalue of a vibrating system is the minimum value Rayleigh's quotient: The question is whether statements similar to those made for 1 st eigenvalue can be made for the intermediate eigenvalues. Modify v by omitting the first eigenvector v 1,

29 Max-Min Characterization of the Eigenvalues Rayleigh's quotient has the minimum value of for all trial vectors v orthogonal to the first eigenvector v 1, where the minimum is reached at v = v 2 : The approach can be extended to higher eigenvalues by constraining the trial vector v to be orthogonal to a suitable number of lower eigenvectors:

30 Max-Min Characterization of the Eigenvalues Consider a given n-vector w and constrain the trial vector v to be from the (n-1)-dimensional Euclidean space of constraint orthogonal to w, (n-1) x (n-1) real symmetric positive definite matrix à corresponding to the (n-1)-dimensional ellipsoid of constraint resulting n-dimensional ellipsoid associated with the real symmetric positive definite matrix A

31 Max-Min Characterization of the Eigenvalues How the eigenvalues of the constrained system à relate to the eigenvalues of the original unconstrained system A? Concentrating first on and introduce the definition: The longest principal axis of the (n-1)-dimensional ellipsoid of constraint associated with à is generally shorter than that corresponding to A, so Iff w coincides with one of the higher eigenvectors, then This is consistent with the fact that constraints tend to increase the system stiffness.

32 Max-Min Characterization of the Eigenvalues The question remains as to the highest value can reach: Consider the trial vector: The choice of V in the given form was motivated by the desire to define the range of as sharply as possible. Indeed, any other choice of V would replace in the right inequality by a higher value.

33 Max-Min Characterization of the Eigenvalues Rayleigh s theorem for systems with one constraint: The 1st eigenvalue of a system with one constraint lies between the 1st and the 2nd eigenvalue of the original unconstrained system, The right side of the inequality, can be reinterpreted as: The 2nd eigenvalue of a real symmetric positive definite matrix A is the maximum value that call he given to min(v T Av/v T v) subject to v T w =0. The preceding theorem can be extended to any number r of constraints, providing a characterization of

34 Max-Min Characterization of the Eigenvalues We consider r independent n-vectors w 1, w 2,..., w r and introduce the definition: In the special case in which the arbitrary constraint vectors w i coincide with the eigenvectors v i of A (i = 1,2,..., r), Assuming: Then

35 Max-Min Characterization of the Eigenvalues The eigenvalue of a real symmetric positive definite matrix A is the maximum value that can be given to min(v T Av/v T v)by the imposition of the r constraints v T w i =0 (i = 1,2,...,r),

36 SEPARATION THEOREM FOR NATURAL SYSTEMS The Courant-Fischer maximin theorem characterizes the eigenvalues of a real symmetric positive definite matrix subjected to given constraints (a reduction in the number of DOFs):

37 SEPARATION THEOREM FOR NATURAL SYSTEMS Next, we assume that the trial vector v is subjected to r -1constraints: Moreover, we assume an additional constraint:

38 SEPARATION THEOREM FOR NATURAL SYSTEMS The constraints are equivalent to:

39 SEPARATION THEOREM FOR NATURAL SYSTEMS To complete the picture, we must have a relation between

40 SEPARATION THEOREM FOR NATURAL SYSTEMS Hence, combining inequalities: Known as the separation theorem,

41 Advanced Vibrations Distributed-Parameter Systems: Approximate Methods Lecture 22 By: H. Ahmadian

42 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

43 Rayleigh-Ritz method (contd.) How to choose suitable comparison functions, or admissible functions: the requirement that all boundary conditions, or merely the geometric boundary conditions be satisfied is too broad to serve as a guideline. There may be several sets of functions that could be used and the rate of convergence tends to vary from set to set. It is imperative that the functions be from a complete set, because otherwise convergence may not be possible: power series, trigonometric functions, Bessel functions, Legendre polynomials, etc.

44 Rayleigh-Ritz method Extreme care must be exercised when the end involves a discrete component, such as a spring or a lumped mass, As an illustration, we consider a rod in axial vibration fixed at x=0 and restrained by a spring of stiffness k at x=l: If we choose as admissible functions the eigenfunctions of a uniform fixed-free rod, then the rate of convergence will be very poor:. The rate of convergence can be vastly improved by using comparison functions:

45 Rayleigh-Ritz method Example : Consider the case in which the end x = L of the rod of previous example is restrained by a spring of stiffness k = EA/L and obtain the solution of the eigenvalue problem derived by the Rayleigh-Ritz method: 1) Using admissible functions 2) Using the comparison functions

46 Example: Using Admissible Functions

47 Example: Using Admissible Functions, Setting n=2

48

49 Example: Using Admissible Functions, Setting n=3

50

51 Example: Using Admissible Functions, The convergence using admissible functions is extremely slow. Using n = 30, none of the natural frequencies has reached convergence with six decimal places accuracy:

52 Example: Using Comparison Function

53

54 Example: Using Comparison Function

55

56 Example: Using Comparison Function Convergence to six decimal places is reached by the three lowest natural frequencies as follows:

57 AN ENHANCED RAYLEIGH-RITZ METHOD Improving accuracy, and hence convergence rate, by combining admissible functions from several families, each family possessing different dynamic characteristics of the system under consideration Free end Fixed end

58 AN ENHANCED RAYLEIGH-RITZ METHOD The linear combination can be made to satisfy the boundary condition for a spring-supported end

59 AN ENHANCED RAYLEIGH-RITZ METHOD Example: Use the given comparison function given in conjunction with Rayleigh's energy method to estimate the lowest natural frequency of the rod of previous example.

60 AN ENHANCED RAYLEIGH-RITZ METHOD

61 AN ENHANCED RAYLEIGH-RITZ METHOD

62 AN ENHANCED RAYLEIGH-RITZ METHOD It is better to regard a1 and a2 as independent undetermined coefficients, and let the Rayleigh- Ritz process determine these coefficients. This motivates us to create a new class of functions referred to as quasi-comparison functions defined as linear combinations of admissible functions capable of satisfying all the boundary conditions of the problem

63 AN ENHANCED RAYLEIGH-RITZ METHOD One word of caution is in order: Each of the two sets of admissible functions is complete As a result, a given function in one set can be expanded in terms of the functions in the other set. The implication is that, as the number of terms n increases, the two sets tend to become dependent. When this happens, the mass and stiffness matrices tend to become singular and the eigensolutions meaningless. But, because convergence to the lower modes tends to be so fast, in general the singularity problem does not have the chance to materialize.

64 AN ENHANCED RAYLEIGH-RITZ METHOD Solve the problem of privious example using the quasi-comparison functions

65 Example: n=2

66

67 Example: n=3

68

69 AN ENHANCED RAYLEIGH-RITZ METHOD

70 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

71 Advanced Vibrations Distributed-Parameter Systems: Approximate Methods Lecture 23 By: H. Ahmadian

72 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

73 The Assumed-Modes Method: System Response known trial functions

74 The Assumed-Modes Method: System Response

75 The Assumed-Modes Method: System Response

76 The Assumed-Modes Method: System Response Example: Use the assumed-modes method in conjunction with a three-term series to obtain the response of the tapered rod of previous Example to the uniformly distributed force

77 The Assumed-Modes Method: System Response

78 The Assumed-Modes Method: System Response

79 The Assumed-Modes Method: System Response

80

81 Damping Effects:

82 Procedure for the Assumed-Modes Method 1. Select a set of N admissible functions. 2. Compute the coefficients k ij of the stiffness matrix. 3. Compute the coefficients m ij of the mass matrix 4. Determine expressions for the generalized forces P i (t) corresponding to the applied force p(x,t). 5. Form the equations of motion.

83 A 2-DOF model for axial vibration of a uniform cantilever bar

84 A 2-DOF model for axial vibration of a uniform cantilever bar

85 Assumed-Modes Method: Bending of Bernoulli-Euler Beams

86 A missile on launch pad

87 A missile on launch pad

88 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

89 Advanced Vibrations Distributed-Parameter Systems: Approximate Methods Lecture 24 By: H. Ahmadian

90 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

91 THE GALERKIN METHOD The approximate solution is assumed in the form known independent comparison functions from a complete set residual Galerkin's method is more general in scope and can be used for both conservative and non-conservative systems.

92 THE GALERKIN METHOD The residual is orthogonal to every trial function. As n increases without bounds, the residual can remain orthogonal to an infinite set of independent functions only if it tends itself to zero, or Demonstrates the convergence of Galerkin's method.

93 THE GALERKIN METHOD Find the natural frequencies of vibration of a fixedfixed beam of length L, bending stiffness EI, and mass per unit length m using the Galerkin method with the following trial (comparison) functions: The governing equation The approximate solution

94 THE GALERKIN METHOD The residual: The Galerkin method gives:

95 THE GALERKIN METHOD

96 THE GALERKIN METHOD

97 THE GALERKIN METHOD

98 THE GALERKIN METHOD Consider a viscously damped beam in transverse vibration.

99 THE GALERKIN METHOD

100 THE COLLOCATION METHOD The main difference between the collocation method and Galerkin's method lies in the weighting functions, the collocation method represent spatial Dirac delta functions.

101 THE COLLOCATION METHOD: A beam in transverse vibration

102 THE COLLOCATION METHOD: The tapered rod Consider the tapered rod fixed at x=0 and spring-supported at x=l. Solve the problem by the collocation method in two different ways: 1)using the locations x i = il/n (i = 1,2,..., n) 2)using the locations x i =(2i-1)L/2n (i=1,2,..., n) Give results for n = 2 and n = 3. List the three lowest natural frequencies for n = 2,3,...,30 and discuss the nature of the convergence for both cases.

103 THE COLLOCATION METHOD: The tapered rod

104 THE COLLOCATION METHOD: The tapered rod

105 THE COLLOCATION METHOD: The tapered rod

106 THE COLLOCATION METHOD: The tapered rod

107 THE COLLOCATION METHOD: The tapered rod For x i = il/n the natural frequencies increase as n increases: The specified locations tend to make the rod longer than it actually is. Because an increased length, while everything else remains the same, tends to reduce the stiffness, The approximate natural frequencies are lower than the actual natural frequencies.

108 THE COLLOCATION METHOD: The tapered rod On the other hand, the locations x i =(2i-1)L/2n tend to make the rod shorter than it actually is, So that the stiffness of the model is larger than the stiffness of the actual system. As a result, the approximate natural frequencies are larger than the actual natural frequencies. This points to the arbitrariness and lack of predictability inherent in the collocation method, with the nature of the results depending on the choice of locations.

109 Distributed-Parameter Systems: Approximate Methods Rayleigh's Principle The Rayleigh-Ritz Method An Enhanced Rayleigh-Ritz Method The Assumed-Modes Method: System Response The Galerkin Method The Collocation Method

110 Advanced Vibrations THE FINITE ELEMENT METHOD Lecture 25 By: H. Ahmadian

111 INTRODUCTION TO THE FINITE ELEMENT METHOD Finite element method is the most important development in the static and dynamic analysis of structures in the second half of the twentieth century. Although the finite element method was developed independently, it was soon recognized as the most important variant of the Rayleigh-Ritz method.

112 INTRODUCTION TO THE FINITE ELEMENT METHOD As with the classical Rayleigh-Ritz method, the finite element method also envisions approximate solutions to problems of vibrating distributed systems in the form of linear combinations of known trial functions. Moreover, the expressions for the stiffness and mass matrices defining the eigenvalue problem are the same as for the classical Rayleigh-Ritz method.

113 INTRODUCTION TO THE FINITE ELEMENT METHOD The basic difference between the two approaches lies in the nature of the trial functions. in the classical Rayleigh-Ritz method the trial functions are global functions, in the finite element method they are local functions extending over small sub-domains of the system, namely, over finite elements.

114 INTRODUCTION TO THE FINITE ELEMENT METHOD In finite element modeling deflection shapes are limited to a portion (finite element) of the structure, with the elements being assembled to for the structural system. The elements are joined together at nodes, or joints, and displacement compatibility is enforced at these joints.

115 ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR Uniform bar element undergoing axial deformation: The shape functions must satisfy the BCs:

116 ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR Considering axial deformation of the uniform element under static loads:

117 ELEMENT STIFFNESS AND MASS MATRICES AND FORCE VECTOR

118 Transverse Motion: Bernoulli-Euler Beam Theory

119 Transverse Motion: Bernoulli-Euler Beam Theory For a beam loaded only at its ends, the equilibrium equation is:

120 Transverse Motion: Bernoulli-Euler Beam Theory

121 Example Determine the generalized load vector for a beam element subjected to a uniform transverse load.

122 Torsion

123 ASSEMBLY OF SYSTEM MATRICES: Scheme for the assembly of global matrices from element matrices for second-order systems using linear interpolation functions

124 ASSEMBLY OF SYSTEM MATRICES: Scheme for the assembly of global matrices from element matrices for fourth-order systems

125 BOUNDARY CONDITIONS The Finite Element formulations inherently satisfy Free boundary conditions. Fixed BC s:

126 BOUNDARY CONDITIONS Supported Spring : Lumped Mass:

127 Example 10.1:The eigenvalue problem for the tapered rod in axial vibration 1. Use the element stiffness and mass matrices with variable cross sections given by: 2. Approximate the stiffness and mass distributions over the finite elements (piece wise constant ) as:

128 Example 10.1:Variable cross section rod element

129 Example 10.1:Variable cross section rod element

130 Example 10.1:Variable cross section rod element

131 Example 10.1:Assembelled Stiffness Matrix

132 Example 10.1:Assembelled Mass Matrix

133 Example 10.1:Exact Parameter Distributions

134 Example 10.1: Convergence Study

135 Advanced Vibrations Superaccurate finite element eigenvalue computation Lecture 26 By: H. Ahmadian

136 Superaccurate finite element eigenvalue computation The consistent finite element formulation: It is theoretically sound and also, provides an assured upper bound on the lowest eigenvalue. Mass lumping producing a diagonal mass matrix An attractive option for the engineer confronted with large complex systems.

137 A Fixed-Free Rod Finite Element Code Consistent Mass Model Lumped Mass Model

138 Convergence Study of the 1 st Mode Consistent Mass Model Lumped Mass Model

139 Superaccurate finite element eigenvalue computation Assembly of the linear finite elements over this mesh using the lumped mass matrix leads to: Provided:

140 Superaccurate finite element eigenvalue computation

141 Superaccurate finite element eigenvalue computation

142 Superaccurate finite element eigenvalue computation

143 Superaccurate finite element eigenvalue computation If: The consistent finite element formulation leads to an overestimation of eigenvalues and The lumped finite element formulation leads to an underestimation of eigenvalue; then it stands to reason that an intermediate formulation should exist that is accurately superior to both formulations.

144 Superaccurate finite element eigenvalue computation Linear combinations of the lumped and the consistent mass matrices give various forms of nonconsistent mass matrices: where the constraint conservation. α + β =1is imposed for mass

145 Optimal element mass distribution Write the general finite difference approximation:

146 Three-nodes string element

147 More details in:

148 PARAMETRIC MODELS AND ERROR ANALYSIS FOR RODS Rod parametric model:

149 PARAMETRIC MODELS AND ERROR ANALYSIS FOR RODS The equation of the ith node in the assembled finite element model th 0 k = EA / dx nd 2 No new req. th 4 θ = 1/12

150 PARAMETRIC MODELS AND ERROR ANALYSIS FOR BEAMS

151 PARAMETRIC MODELS AND ERROR ANALYSIS FOR BEAMS

152 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

153 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

154 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

155 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

156 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

157 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

158 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

159 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES

160 PARAMETRIC MODELS AND ERROR ANALYSIS FOR PLATES