Inclusion of Missing Mass Correction - CAESAR II - Help

CAESAR II Users Guide

Language
English
Product
CAESAR II
Search by Category
Help
CAESAR II Version
12

The response of a system under a dynamic load is often determined by superposition of modal results, with CAESAR II specifically providing the Spectral Analysis method for use. One of the advantages of modal analysis is that usually only a limited number of modes are excited and need be included in the analysis. The drawback to this method is that although displacements may be obtained with good accuracy using only a few of the lowest frequency modes, the force, reaction, and stress results may require extraction of far more modes, possibly far into the rigid range, before acceptable accuracy is attained. The Missing Mass option offers the ability to include a correction which represents the quasi-static contribution of the higher order modes not explicitly extracted for the modal/dynamic response, thus providing greater accuracy with reduced calculation time.

The dynamic response of a linear multi-degree-of-freedom system is described by the following equation:

Ma(t) + Cv(t) + Kx(t) = F(t)

Where:

M = n x n mass matrix of system

C = n x n damping matrix of system

K = n x n stiffness matrix of system

a(t) = n x 1, time-dependent acceleration vector

v(t) = n x 1, time-dependent velocity vector

x(t) = n x 1, time-dependent displacement vector

F(t) = n x 1, time-dependent applied force vector

Assuming harmonic motion and neglecting damping, the free vibration eigenvalue problem for this system is

KF - MF w2 = 0

Where:

F = n x n mode shape matrix

w2 = n x n matrix where each diagonal entry is the angular frequency squared of the corresponding mode

The modal matrix F can be normalized such that FT M F = I (where I is the n x n identity matrix) and FT K F = w2.

Partition the modal matrix F into two sub-matrices:

F = [ Fe Fr ]

Where:

Fe = mode shapes extracted for dynamic analysis (that is., lowest frequency modes)

Fr = residual (non-extracted) mode shapes (corresponding to rigid response, or the "missing mass" contribution)

The extracted mode shapes are orthogonal to the residual mode shapes, or:

FeT x Fr = 0

The displacement components can be expressed as linear combinations of the mode shapes:

x = FY = Fe Ye + Fr Yr = xe + xr

Where:

x = Total System Displacements

xe = System Displacements Due to Extracted Modes

xr = System Displacements Due to Residual Modes

Y = Generalized Modal Coordinates

Ye = partition of Y Matrix Corresponding to Extracted Modes

Yr = Partition of Y Matrix Corresponding to Residual Modes

The dynamic load vector can be expressed in similar terms:

F = K F Y = K Fe Ye + K Fr Yr = Fe + Fr

Where:

F = Total System Load Vector

Fe = Load Vector Due to Extracted Modes

Fr = Load Vector Due to Residual Modes

Y = Generalized Modal Coordinates

Ye = Partition of Y Matrix Corresponding to Extracted Modes

Yr = Partition of Y Matrix Corresponding to Residual Modes

Normally, modal superposition analyses completely neglect the rigid response the displacement Xr caused by the load Fr. This response, of the non-extracted modes, can be obtained from the system displacement under a static loading Fr. Based upon the relation\-ships stated above, you can estimate Fr as follows:

F = K Fe Ye + K Fr Yr

Multiplying both sides by FeT and considering that FeT Fr = 0:

FeT F = FeT K Fe Ye + FeT K Fr Yr = FeT K Fe Ye

Substituting we2 for FeT K Fe and solving for Ye:

FeT F = we2 Ye

Ye = FeT we-2 F

The residual force can now be stated as

Fr = F - K Fe Ye = F - FeT K Fe w e-2 F

As seen earlier

FT M F w2 = I w2 = FT K F

Substituting FeT MFe we2 for FeT K Fe:

Fr = F - FeT M Fe we2 we-2 F = F - FeT M Fe F

Therefore, CAESAR II calculates the residual response (and includes it as the missing mass contribution) according to the following procedure:

  1. The missing mass load is calculated for each individual shock load as:

    Fr = F - FeT M Fe F

    The load vector F represents the product of:

    • the force set vector and the rigid DLF for force spectrum loading;

    • the product of the mass matrix, ZPA, and directional vector for non-ISM seismic loads;

    • and the product of the mass matrix, ZPA, and displacement matrix (under unit ISM support displacement) for seismic anchor movement loads.

    Tthe missing mass load varies depending upon the number of modes extracted by the user and the cutoff frequency selected (or more specifically, the DLF or acceleration corresponding to the cutoff frequency). "Rigid," for the purposes of determining the rigid DLF, or the ZPA, may be designated by the user, through a setup parameter, to be either the DLF/acceleration associated with the frequency of the last extracted mode, or the true spectral DLF/ ZPA that corresponding to the largest entered frequency of the input spectrum.

  2. The missing mass load is applied to the structure as a static load. The static structural response is then combined (according to the user-specified combination method) with the dynamically amplified modal responses as if it was a modal response. This static response is the algebraic sum of the responses of all non-extracted modes—representing in-phase response, as would be expected from rigid modes.

  3. The Missing Mass Data report is compiled for all shock cases, whether missing mass is to be included or not. The percent of mass active is calculated according to:

    % Active Mass = 1 - (å Fr[i] / å F [i])

    summed over i = 1 to n, where n is the number of modes included

The maximum possible percent that is theoretically possible for this value is of course 100%; however numerical inaccuracies may occasionally cause the value to be slightly higher. If the missing mass correction factor is included, the percent of mass included in the correction is shown in the report as well.

Because the CAESAR II procedure assumes that the missing mass correction represents the contribution of rigid modes, and that the ZPA is based upon the spectral ordinate value at the frequency of the last extracted mode, we recommend that you extract modes up to, but not far beyond, a recognized "rigid" frequency. Choosing a cutoff frequency below the spectrum’s resonant peak [point (1) below] provides a non-conservative result, because resonant responses may be missed. Using a cutoff frequency higher than the peak (2), but still in the resonant range, will yield conservative results, because the ZPA/rigid DLF will be overestimated. Extracting a large number of rigid modes for calculation of the dynamic response may be conservative (4), because all available modal combination methods (SRSS, GROUP, ABS, and so forth) give conservative results versus the algebraic combination method which gives a more realistic representation of the net response of the rigid modes. Based upon the response spectrum shown below, an appropriate cutoff point for the modal extraction would be about 33 Hz (3).