9.3 Modeling Aspects for Dynamic Analysis

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Section 9.1 and Section 9.2 provide guidelines for problem and analysis classification. Once a decision

has been made about the type of problem and the method of analysis to be used, the designer still has to

finalize many details to create a working model for the structure. This section provides a brief discussion

of the main aspects required to do this. This includes choice of model size and master DoF, lumped and

consistent mass modeling, and use of symmetry. Another important aspect of modeling, damping, will

be dealt with in Section 9.4.

9.3.1 Model Size and Choice of Master Degrees of Freedom

Using the same model for both static and dynamic analysis will make the FE solution to the dynamic

problem more complicated and therefore much more memory and CPU resources will be required. Also,

the results may not be necessarily more useful or accurate. Therefore, FE models for dynamic analysis

should, in general, be kept simple.

Depending on the type of dynamic analysis, it may be possible to construct a much simpler model for

the dynamic problem, possibly using spring, beam, and mass elements. If we consider modal analysis, for

example, the objective is to find the mode shapes of the structure and the corresponding natural

frequencies. In practice, we seldom need more than the very few lower natural frequencies and mode

REMARKS

* Structural loading may be classified into four categories: steady (constant with time), cyclic

(harmonic with period T ¼ 2p and frequency f ¼ 1=T or general periodic that may be

decomposed into harmonics by Fourier analysis), transient with a duration T; and random.

* Random forcing functions are commonly specified as the magnitude of the input

acceleration squared vs. frequency (PSD). Typical PSD curves have been compiled for

various random events and are normally available in commercial FE programs.

* The natural frequencies of a system are those frequencies that the system tends to vibrate

under conditions of free vibration. The mode shapes give the relative displacements of each

point in the structure when it vibrates in one of the natural frequencies. Symmetric

structures will generally have symmetric and antisymmetric mode shapes.

* If the loading is harmonic with a frequency of less than approximately one third of the first

natural frequency of the structure, static analysis for the peak load levels suffices. In

transient loading, if the longest natural period of a system is less than about one third of the

rise time, loading is quasi-static and static analysis suffices.

* In transient loading, if the longest natural period corresponding to the first natural

frequency of the structure is more than about twice the rise time of the load, it is necessary

to classify the loading as shock or impact loading and transient dynamic analysis would be

required.

* If the longest natural period falls between the quasi-static and shock conditions, a transient

dynamic analysis would be required.

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© 2005 by Taylor & Francis Group, LLC

shapes for a structure. Using a static model that may have thousands of DoF in a modal analysis is very

time consuming and does not serve the purpose of the mode shape analysis. It may be better and more

efficient to construct a mass spring/beam model for the modal analysis. Figure 9.23 shows a static analysis

model for a shaft and a corresponding much simpler mode shape analysis model. In the mass/beam

model for modal analysis, the beam properties may change to accommodate specific variations in the

shaft geometry. It is also possible to extract accurate values for the spring and beam stiffnesses in 3-D

using a detailed static model.

Most programs provide means for condensing or reducing the DoF in a dynamic analysis. A

common condensation method used is the “Guyan reduction” (see e.g., Freed and Flanigan, 1991;

Bouhaddi and Fillod, 1992; Deiters and Smith, 2000). In these condensation methods, the analyst

specifies certain DoF to be masters and to be retained by the program, while all other DoF would be

considered slaves and would be eliminated or condensed. The dynamic analysis is then carried out

using the master DoF. Information pertaining to slave DoF is still available and may be extracted in

subsequent runs that may involve stress or other detailed analyses or postprocessing. Reduction has the

same effect as modal superposition in terms of reducing the problem size and the required CPU time.

Fundamental differences between the two methods may be realized due to the nature of reducing the

system of equations. In the modal superposition technique, reduction is accomplished by assuming that

the superposition of the structural response due to the first few lower mode shapes adequately represents

the response of the structure. In other reduction methods, such as the Guyan reduction technique, we

assume that the response of the system is accomplished from the response of certain predetermined

master DoF. In the modal superposition technique, a modal analysis must be performed first and the final

equations of motion are decoupled, whereas in the other reduction methods no modal analysis is

necessary and the final equations of motion are generally full and coupled. Reduction of the total DoF to

a set of master DoF has the effect of imposing displacement constraints on the system that increase the

stiffness and subsequently overestimate the natural frequencies. The least affected modes are the

lower ones. Therefore, reduction and modal superposition techniques should not be used, in general,

for shock and impact loading (the modal superposition technique entirely ignores the response from

higher modes).

The choice of master DoF may be performed automatically by the program. There is no assurance,

however, that this will always produce the best choice of masters. The analyst may like to use the program

for a first choice of master DoF and then augment the choice manually to improve the accuracy. Simple

rules may be used to identify the master DoF (see ANSYS User’s Manual, 2003). The general rule is to

choose DoF with large mass and small stiffness or large mass to stiffness ratios as masters. Master DoF

should be chosen in the direction of the expected response of the structure, e.g., in a beam or shell model,

lateral DoF would be more appropriate than axial or membrane DoF, assuming that the load produces

lateral deflections. Rotational DoF are seldom used as masters. It is also important to spread out and not

cluster the master DoF so that they will be capable of producing the structural response. Finally, DoF with

concentrated mass, a specified input force, or displacement should be retained as masters.

FIGURE 9.23 Static and dynamic models for a shaft.

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© 2005 by Taylor & Francis Group, LLC

The question of how many master DoF are adequate is not an easy one to answer. The rule of thumb is

to have having at least double the number of master DoF compared with the number of modes of

interest. The other guideline is to use 10 to 50% of the total DoF as masters. In practice, the only sure way

to test the adequacy of the number of master DoF is to run the analysis with two different choices and

compare the results to make sure that the modes of interest are not substantially different.

9.3.2 Lumped and Consistent Mass Modeling

The mass characteristic of the structure may be modeled as a lumped or consistent mass matrix. Lumped

mass simply means dividing the total mass of the element by the number of nodes. This produces a

diagonal mass matrix and may significantly reduce the numerical effort required to solve the dynamic

equations. Considering rotational inertia terms in lumped mass assumptions is arbitrary and is normally

ignored. If the lumped mass matrix has zero diagonal terms, the matrix will be positive semidefinite and

the natural periods corresponding to the zero diagonal terms will be zero. This will have serious

implications on the choice of the time step in explicit direct integration solution methods and will be

discussed in Section 9.4. A lumped mass assumption implies a discontinuous acceleration field in the

element and it usually gives good accuracy for lower modes and frequencies (Chan et al., 1993; Jensen,

1996). A consistent mass matrix is derived by introducing inertia forces in the virtual work formulation

of a dynamic FE problem. This would normally lead to a full mass matrix making contributions to both

translational and rotational DoF as well as coupling terms. Consistent mass representation usually

provides better accuracy for higher modes and frequencies (Kim, 1993).

It may be concluded that, if no reduced integration is used to compute element stiffness, the element is

compatible, and a consistent mass matrix is used, then the calculated natural frequencies will be an upper

bound of the model frequencies (Cook et al., 1989). In some applications such as plates, shells, and 3-D

solid elements, it has been found that the convergence rate of the natural frequency calculations for the

lumped mass discretization is the same as that for a consistent mass formulation (Chan et al., 1993;

Jensen, 1996).

Some FE programs provide a modified lumped mass modeling in which the diagonal translational

terms are proportional to the diagonal terms of the corresponding consistent mass matrix. This yields

more accurate results, in general, than the traditional lumped mass assumption. Several studies have been

published on the merits of combining consistent and lumped mass formulations into some kind of

modified lumped mass matrix (Chan et al., 1993; Kim, 1993; Jensen, 1996). The modified lumped mass

matrix is generally a linear combination of the lumped and the consistent mass matrices. For an

improved accuracy, the consistent mass should be weighted more than the lumped mass.

REMARKS

* In general, dynamic analysis models should be simpler and much smaller in terms of the

number of DoF (than static analysis ones). In modal analysis, it is better and more efficient

to construct a mass spring/beam model for the structure.

* “Guyan reduction” may be used to reduce the model size by selecting master DoF for the

analysis and condensing out slave DoF.

* Master DoF may initially be chosen by the program. The analyst should, however, ensure

that the choice includes nodes with a large mass to stiffness ratio and with DoF in the

direction of the expected response of the structure. Also, it is important to spread out and

not cluster the master DoF.

* As a rule of thumb, the number of master DoF should be at least double the number of

modes of interest or about 10 to 50% of the total DoF.

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© 2005 by Taylor & Francis Group, LLC

In building a lumped mass model for the structure, the analyst should realize that one of the objectives

is to capture the few lower natural frequencies and mode shapes of the structure. To achieve this, the

model should minimize the strain energy and maximize the kinetic energy of the structure. Minimizing

the strain energy of the structure may be realized by accurately modeling the soft links in the load path.

Structural supports are particularly difficult in this regard. Changing the support model from completely

rigid to slightly flexible may shift the first natural frequency by as much as 40% (although it will not have

much impact on the mode shape). Maximizing the kinetic energy is simply realized by accurate modeling

of the system mass, especially in areas where the mass is large and the stiffness is small.

Care should be exercised when modeling masses attached to the system, e.g., a large “nonstructural”

mass on top of a building (e.g., a mass of a machine, reservoir, etc.) or the mass of an engine in a car

model. If the stiffness characteristic of the nonstructural mass is not important, it may be simply modeled

as a rigid lumped mass with an offset mass center from the nearest node. Most programs have a rigid link

capability to connect this mass to the structure. This kind of model is transformed internally to a

nondiagonal mass matrix and may only be used with a consistent mass formulation for the structural

mass. If the structural mass is modeled as lumped, the nonstructural mass will simply be added to the

nearest node, which may not be an accurate representation of the physical model.

9.3.3 Use of Symmetry

Use of symmetry to reduce the analysis cost and computer time is rather tricky in modal analysis. As

shown in Table 9.2, symmetric structures have both symmetric and nonsymmetrical mode shapes. If

we only use a symmetric model with symmetric boundary conditions, all nonsymmetrical modes will

be undetected. This will change the interpretation of the results and will have an impact on

subsequent analyses using the modal data. If the objective is to perform only a modal analysis, it may

be advisable to have a less detailed full model rather than a detailed half or quarter model. Also, if the

REMARKS

* A lumped mass matrix is a diagonal mass matrix that may be obtained by simply dividing

the total mass by the number of DoF or alternatively by certain combinations of lumped

and consistent mass terms. A consistent mass matrix is derived from FE virtual work

equations and is normally a full mass matrix having translational, rotational, and coupling

terms.

* Lumped mass may significantly reduce the numerical effort required to solve the dynamic

equations without impacting the accuracy.

* Zero diagonal terms in the mass matrix should be avoided as much as possible.

* A lumped mass assumption usually gives good accuracy for lower modes and frequencies,

whereas consistent mass representation usually gives better accuracy for higher modes and

frequencies.

TABLE 9.2 Symmetric and Antisymmetric Boundary Conditions

Plane of symmetry Symmetric Boundary Conditions Antisymmetric Boundary Conditions

ux uy uz ux uy uz ux uy uz ux uy uz

XY — — 0 0 0 — 0 0 — — — 0

XZ — 0 — 0 — 0 0 — 0 — 0 —

YZ 0 — — — 0 0 — 0 0 0 — —

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© 2005 by Taylor & Francis Group, LLC

objective is to perform frequency response or transient analysis using the modal data, it may be

advisable to rely on a full model. Experienced analysts may use symmetric models and run the modal

analysis with symmetric and antisymmetric boundary conditions to capture the response of all

modes. Table 9.2 summarizes symmetric and antisymmetric boundary conditions for translational

and rotational DoF.

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