Reduction in number of dofs

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Brianna Daniels
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1 Reduction in number of dofs Reduction in the number of dof to represent a structure reduces the size of matrices and, hence, computational cost. Because a subset of the original dof represent the whole set of dof, some error is introduced. he retained dof are called master dof (masters) and the eliminated ones slave dof (slaves). he number of masters is the size of the reduced system. Masters can be chosen to minimize the loss of accuracy Guyan reduction is a commen method of reduction. 1

2 Guyan Reduction Assumption: inertia forces on the slave dof are negligible compared to elastic forces on them. his implies that the terms in the mass matrix (multiplied by ω ) corresponding to the slaves are negligible compared to the terms in the stiffness matrix. Considering the equilibrium equation of an undamped system under free vibration: [ K ω M] D = Eq. (*) Partitioning the matrices according to masters D m and slaves D s, K K K mm ms ms K ss ω M M mm ms M M ms ss D D m s =

3 Master-Slave ransformation Implementing the assumption in the partitioned equation, = D D M M K K K K s m ms mm ss ms ms mm ω he second row then gives where

3 3 Master-Slave ransformation Implementing the assumption in the partitioned equation, = D D M M K K K K s m ms mm ss ms ms mm ω he second row then gives where over-bars have been omitted since master-slave transformation is not restricted to free vibration. hen, the entire dof set D can be expressed in terms of the masters:

4 Reduced Eigenproblem he j th column of matrix multiplies the j th master dof. If the j th master dof is unity while the other master dof are zero, the j th column of gives the static displacement of the structural dof (namely the j th master dof itself and the slaves). If D is substituted in Eq (*) (two slides back), [ ω ] K = M Dm Pre-multiplying by gives the reduced eigenproblem 4

5 Reduced Forced-Vibration Problem he general equilibrium equation under forcing can also be reduced. hat is, we can transform M D + CD + KD = and obtain the reduced equation R(t) where M D + C D + K D = r m r m Cr = C and Rr = r m Rr (t) R One criterion for choosing masters is that dof having large mass to stiffness ratio are likely candidates. 5

6 Reduction Example he FE model shown has two dof with the eigenproblem equation stated below. We wish to reduce the model to a single dof. We note that m 11 /k 11 =156c/1 while m /k =4c/4=c. Hence the first dof (v ) is chosen as the master. 6

Example (cont.) he submatrices K ss, etc.

system is his gives the only frequency obtainable

7 Example (cont.) he submatrices K ss, etc. are scalars now: K ss = 4EI/L, K ms = - 6EI/L Hence, from the slave is written in terms of the master as he eigenproblem equation of the reduced system is his gives the only frequency obtainable from the reduced system: 1 % higher than with the full system. 7

8 Modal Equations In the Guyan reduction, the full set of displacements (dof) were expressed in terms of a subset of the full set. A system can also be reduced by expressing the displacements as a linear combination of a subset of the system vibration modes. his can be viewed as an example of a Ritz approximation. o do this, it s convenient to scale (normalize) the modes as Consider the eigenproblem written for mode i : [ ] M Di K = ω i 8

9 Premultiplying by D i Eigenvalues [ K ω ] D = i M D = D KD D i i M Di i ω i i i Using the normalization equation ω = D KD for i = 1,..., n ; n = i i i no of dof We can write the above for n modes and gather all of them in a matrix equation: ω 1... ω = D D 1 K D [ D... ] 1 9

Modal Matrix he off-diagonal terms of the product on the right hand side above are zero in light of the orthogonality of the modes, that is, hen we define the

10 Modal Matrix he off-diagonal terms of the product on the right hand side above are zero in light of the orthogonality of the modes, that is, hen we define the modal matrix to be the union of the modes of the system: Rewriting the equation on the previous slide where ω is the diagonal matrix of eigenvalues (spectral matrix). 1

11 Mode Superposition An arbitrary displacement can be written as a linear combination of the vibration modes: where z i is the fraction of mode i that contributes to D. he above is a transformation and we can write it for the three kinematic quantities as z i are called principal or modal coordinates. 11

12 ransformed Equilibrium Equations Substituting the above transformations into M D + CD + KD = R(t) we get the transformed equilibrium equation If proportional damping is used, C φ is diagonal: C = αi βω φ + All of the coefficient matrices on the left side of the transformed equation are then diagonal and the n equations are uncoupled. 1

13 Modal Damping Modal damping is an alternative to proportional damping and works well when damping is small. In modal damping, we replace C φ above by another diagonal ξ i ω i ξi matrix of entries where is the damping ratio for mode i. he n equations are again uncoupled and we can write n scalar equations, which are the modal equations: φ = where is the i th column of φ, the modal matrix. i D i Note that the equilibrium equation of a sdof system (Eq. 9.-) takes the same form if it is divided by m and manipulated. 13

14 Modal runcation z i are solved for from the modal equations, which are scalar equations, and the physical displacements (and other kinematic quantities, if necessary) are obtained from them: Important note: Not all of the n modes need be used in mode superposition. Usually we can use only the lowest several modes. Above we used D=φz which contained all the modes. Instead we can use his approach is called modal truncation. 14

15 Suitability of Mode Superposition With modal truncation, we need only a subset of the modal frequencies and mode shapes. How many modes to use depends on the problem. Few modes may suffice if the excitation frequency is low compared to the modal frequencies of the system when a sinusoidal forcing R(t) is used, the loading varies slowly in time, as in an earthquake loading. Many modes may be needed otherwise, for example, for shock loading. hen mode superposition may not be suitable. 15

FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS

Lecture Notes: STRUCTURAL DYNAMICS / FALL 2011 / Page: 1 FREE VIBRATION RESPONSE OF UNDAMPED SYSTEMS : : 0, 0 As demonstrated previously, the above Equation of Motion (free-vibration equation) has a solution