Rayleigh Ritz Method

6 Key excerpts on "Rayleigh Ritz Method"

• 

  eBook - ePub

  Mechanical Vibrations

  Theory and Application to Structural Dynamics

  • Michel Geradin, Daniel J. Rixen(Authors)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  The Rayleigh method (Lord Rayleigh 1894) which consists in approximating the fundamental eigenmode of a continuous system by one kinematically admissible function (i.e. satisfying the kinematical conditions of the problem) and to substitute it in a variational expression to get an approximation to the corresponding eigenvalue has been a giant step in the development of engineering tools. With Lord Rayleigh's method, the error characterizing the approximation to the fundamental eigenvalue is of an order of magnitude less than the error on the eigenshape, and the numerical value obtained is always an upper bound to the exact solution.

  An equally important step was made by Ritz (1909) when he proposed to generalize Rayleigh's method by building the approximation to the continuous solution from a finite set of selected kinematically admissible functions. The coefficients of the series expansion are then the discrete unknowns of the problem, which can be interpreted as generalized displacements. Applying the variational principle when the solution is expressed as a finite sum of functions leads to a linear system of equations similar to the form governing lumped models. The mass and stiffness matrices so obtained1 are those associated to the unknown amplitudes of the chosen representative functions. The behaviour so computed corresponds to the best approximation to the solution of the continuous problem in the subspace spanned by the interpolation functions. With the Rayleigh–Ritz method, discretization methods were born.

  The works of Lord Rayleigh and W. Ritz on variational methods together with B.G. Galerkin's weighted-residual approach (Galerkin 1915) form a consistent theoretical framework for the reduction of a continuous problem with an infinite number of degrees of freedom to a discrete one. However, in their original form they were all still lacking the flexibility needed to describe continuous systems with complex geometry and subject to arbitrary boundary conditions and loading distribution.

  The Rayleigh–Ritz method has been largely used until 1960 for the development of approximation methods in structural dynamics. Aeroelasticity is probably the engineering discipline in which the method has found its widest field of application, as demonstrated by the classical works by R. Bisplinghoff and his co-workers (Bisplinghoff et al.

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  eBook - ePub

  Approximation Techniques for Engineers

  Second Edition

  • Louis Komzsik(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  10 ].

  Let us now restrict our focus to the linear symmetric eigenvalue problem of order n

  A

  x ¯

  j

  =

  λ j

  x ¯

  j

  .

  Here j = 1, 2, …, n , although in practice the number of eigenvalues computed is usually much less. Let us assume that we have a matrix Q with m orthonormal columns

  Q T Q = I .

  The Rayleigh–Ritz procedure computes the matrix Rayleigh quotient of

  T m = Q T AQ .

  This matrix has m rows and columns and is tridiagonal if A is symmetric and Q spans a Krylov subspace. The eigenvalues of this matrix are called the Ritz values and are computed from the Ritz problem of

  T m

  u ¯

  j

  =

  θ j

  u ¯

  j

  .

  The approximate eigenvectors (also called Ritz vectors) of the original eigenvalue problem are recovered as

  x ¯

  j

  = Q

  u ¯

  j

  .

  This procedure, in essence, provides an approximate solution to the eigenvalue problem by projecting the matrix into a subspace spanned by the columns of Q . The residual error of the approximate eigenvector is computed as

  r ¯

  j

  = r (

  x ¯

  j

  ) = A

  x ¯

  j

  -

  θ j

  x ¯

  j

  = A Q

  u ¯

  j

  -

  θ j

  x ¯

  j

  .

  This error bounds the eigenvalue as

  (

  θ j

  - | |

  r j

  | | ) ≤

  λ j

  ≤ (

  θ j

  + | |

  r j

  | | ) .

  The quality of the Rayleigh–Ritz procedure depends on the selection of the subspace spanned by Q . The performance of the method depends on the construction of the subspace. The aforementioned Krylov subspace is a good selection; however, computing it efficiently is where the Lanczos method exceptionally shines.

  10.3   The Lanczos method

  The Lanczos method [9 ] is an excellent way of generating the Krylov subspace of A

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  eBook - PDF

  Vibration of Laminated Shells and Plates

  • Mohamad Subhi Qatu(Author)
  • 2004(Publication Date)
  • Academic Press

    (Publisher)

  If a complete set of functions is used, the Ritz method has excellent con-vergence characteristics and is relatively easy to program (Qatu 1989). The Ritz method, sometimes referred to as Rayleigh–Ritz, has been used successfully to analyze thin cylindrical shells (Ip et al . 1996; Sharma et al . 1996; Kumar and Singh 1996; Heyliger and Jilani 1993). It has also been used to study thin shallow shells (Chun and Lam 1995; Qatu and Leissa 1991a,b; Qatu 1993a, 1994b,c, 1995b,c; Messina and Soldatos 1999a,b, Sheinman and Reichman 1992). The Ritz method has also been used to study thick shallow shells by Singh and Kumar (1996) and spherical shells by Chao and Chern (1988), and other shells by Lee (1988). Raouf and Palazotto (1992) used the Ritz method with harmonic balance to analyze non-linear vibrations of shells. One has to start with a displacement field that satisfies at least the geometric boundary conditions when using the Ritz method. Furthermore, it is somewhat difficult to obtain a displacement field for a relatively complex shell structure or set of such structures. For example, a problem of a plate attached to a shell at a certain angle is challenging if the Ritz method is to be used, as found by Young and Dickinson (1997). Special treatments were introduced to overcome the difficulty of using the Ritz method with general boundary conditions (Kumar and Singh 1996; Li and Mirza 1997). Vibration of Laminated Shells and Plates 64 3.3. THE GALERKIN METHOD The Ritz method is used to obtain approximate solution to problems having an energy (or variational) expression. There are problems in mechanics which do not allow for such an expression. The weighted residual methods are basically a generalization of the Ritz method to handle problems that do not allow an energy expression. The Galerkin method is a special case of the general weighted residual methods. In these methods, the governing differential equations and corresponding boundary conditions are needed.

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  Energy Principles and Variational Methods in Applied Mechanics

  • J. N. Reddy(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  The one-parameter Ritz solution in this case is U1(x, y) = 5f0a 2 16T  1 - x 2 a 2   1 - y 2 a 2  , (19) which is more accurate than the one-parameter solution in Eq. (8). 6.3. THE RITZ METHOD 323 (b) For a choice of trigonometric functions, we can use φ1(x, y) = cos πx 2a cos πy 2a , φ2(x, y) = cos 3πx 2a cos πy 2a , φ3(x, y) = cos πx 2a cos 3πy 2a , . . . . (20) In closing this example, we note that Eq. (15) arises, with a different interpretation of the variable u and the data (T, f0), in a number of other fields. For example, it arises in the study of the torsion of a square cross-section prismatic bar where T = 1 and f0 = 2Gθ, where G denotes the shear modulus and θ is the angle of twist per unit length. Variable u is the Prandtl stress function. It also arises in the study of conduction heat transfer in a square isotropic medium with conductivity k (i.e., T = k) and internal heat generation of f0 per unit area, and u denotes the temperature. 6.3.6 The Ritz Method for General Boundary-Value Problems 6.3.6.1 Preliminary Comments In the previous sections, the Ritz method was introduced as an approximate method that utilizes a variational principle, such as the principle of minimum total potential energy. Since the variational principle is equivalent to a set of governing equations, the Ritz method provides approximate solution of the un- derlying equations. That is why the Ritz method is termed a direct variational method. For problems outside the field of solid and structural mechanics, the construction of an analog of the minimum total potential energy principle or its equivalent is needed to use the Ritz method. In this section a procedure for constructing an integral form that is equivalent to a differential equation, called weak form, is presented, so that the Ritz method can be used for prob- lems outside of solid and structural mechanics.

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  eBook - PDF

  Advanced Vibration Analysis

  • S. Graham Kelly(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  Since the Rayleigh–Ritz method uses an energy inner product, it can be applied when the operator is self-adjoint and positive definite with respect to a valid inner product. The Rayleigh–Ritz method provides the best approximation to the solution from the vector space spanned by the chosen basis functions. The method does not speak to how to choose the basis elements. The actual value of the minimum of D E ( w ) depends upon the choice of the basis for W. The question whether the accuracy of the approximation improves as the number of basis elements is increased can be addressed in the following manner. The elements of any basis are, by definition, linearly independent. Because of this and the positive definiteness of L , it can be shown that the matrix represented by Equation 8.19 is nonsingular. Thus the Rayleigh–Ritz approximation exists and is unique. This implies that any choice of a basis for W leads to the same Rayleigh–Ritz approximation. For a chosen basis, the Gram–Schmidt process can be used to determine an orthonormal basis spanning W. Thus in assessing the accuracy of a Rayleigh–Ritz approximation it may be assumed that the basis functions are orthonormal with respect to the energy inner product, ð w i ; w j Þ E Z d i ; j (8.21) Let W k be a k dimensional subspace of S, the domain of L , with k orthonormal basis vectors satisfying Equation 8.21. Let ~ w k be the Rayleigh– Ritz approximation to the solution of Lu Z f from W k . From Equation 8.11, ~ w k Z X k i Z 1 ð f ; w i Þ w i (8.22) Let W k C 1 be a ( k C 1)th dimensional vector space whose basis elements are those of W k and w k C 1 , which is orthonormal with the other basis elements. The Rayleigh–Ritz approximation from W k C 1 is ~ w k C 1 Z ~ w k C ð f ; w k C 1 Þ w k C 1 (8.23) Rayleigh–Ritz and Finite-Element Methods 529

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  eBook - ePub

  Finite Elements and Approximation

  • O. C. Zienkiewicz, K. Morgan, K. Morgan(Authors)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Example 2.6 , we construct an approximation

  thus automatically satisfying the essential condition at x = 0, but not the natural condition at x = 1. The constants are now determined by the Rayleigh-Ritz method of minimizing Π with respect to variations in a 1 ,a 2 ,...,a M .

  For example, using a two-term approximation means that and this attains its minimum value when and Performing the integrations produces the equation set

  As would be expected from previous observations, these are just the equations produced when the Galerkin method of solution is applied to this problem, and therefore the solution a 1 = 11.7579, a 2 = 3.4582 of Example 2.6 is reproduced, and with the one and two-term approximations again illustrating convergence to the natural boundary condition on

  d

  /dx at x = 1.

  EXERCISES

  6.13.   Obtain, by the Rayleigh-Ritz method, an approximate solution to a problem of steady one-dimensional heat conduction with a distributed heat source governed by the equation

  and the boundary conditions = 0 at x = 0,

  d

  /dx = 1 at x = 1. Use piecewise linear finite element trial functions with a distance of between successive nodes.

  6.14.   Return to the problem of Example 3.4 of a bar under the action of axial body forces b per unit volume, where the governing equation was written

  subject to = 0 at x = 0,

  d

  /dx = 0 at x = 1, and where A is the cross section of the bar, E is Young’s modulus for the bar, and is the displacement at any point. Show that the variational form of this problem is equivalent to minimizing the total potential energy of the system. When A, E, b

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