9.5 Various Dynamic Analyses

Back

9.5.1 Modal Analysis

This is normally the first, and in certain cases the only, type of analysis required for a structure. It is

used to obtain the dynamic characteristics of a system in terms of its natural frequencies or eigenvalues

TABLE 9.4 Typical Damping Ratios for Various Systems and Materials

System/Material Damping Ratio

Various metals in elastic range , 0.01

Small diameter piping systems 0.01 – 0.02

Large building during earthquake 0.01 – 0.05

Large diameter piping systems 0.02 – 0.03

Welded joints and rigid metal structures 0.02 – 0.04

Prestressed concrete structures 0.02 – 0.05

Metal structures with joints 0.03 – 0.07

Transmission lines (aluminum or steel) 0.04

Reinforced concrete structures 0.04 – 0.07

Rubber 0.05

Bolted joints 0.07

Shock absorbers 0.30

REMARKS

* Damping is a source of energy dissipation in the structure and is generally difficult to

quantify. Damping sources in a structure include internal, Coulomb, and viscous friction.

* The most common forms of damping provided by FE programs are discrete, proportional,

and modal viscous damping.

* Discrete damping is modeled by special spring and damping elements in FE applications.

The damping matrix resulting from these elements cannot, in general, be decoupled as in

Equation 9.12. Hence, these elements may be used only with direct integration solution

methods.

* Proportional viscous or Rayleigh damping (Equation 9.17) leads to a decoupled damping

matrix and may be used in both direct integration as well as modal superposition

approaches.

* A commonly used method to determine the coefficient of proportionality is through

defining two modal damping ratios (Equation 9.19).

* In modal viscous damping, the damping ratios are directly specified for the participating

modes and the damping matrix is assumed to be diagonal.

Finite Element Applications in Dynamics 9-33

© 2005 by Taylor & Francis Group, LLC

and associated undamped free vibration natural modes or eigenvectors. These eigenmodes may then be

used to decouple the general equations of motion in other types of dynamic analyses. Modal analysis is

the first step required before performing mode superposition harmonic or transient analysis as well as

spectrum analysis. The natural frequencies of the system may be obtained by solving the eigenvalue

problem:

ðK 2v2MÞw􀀊 ¼ 0 ð9:20Þ

For a nontrivial solution, the coefficient matrix ðK 2 v2MÞ is singular and its determinant is equal to

zero, which yields the values of the natural frequencies. For each eigenvalue, vni , a corresponding mode

shape, w􀀊i, can be obtained from Equation 9.20. If the mass matrix has some zero diagonal values, the

number of eigenpairs will be less than the total number of DoF by the number of zero entries on the

diagonal of the mass matrix. For properly constrained structures, K is positive definite and all natural

frequencies are positive. If the structure is unconstrained or partially constrained, however, K is positive

semidefinite and the eigenvalues will contain zero frequencies representing the rigid-body modes. The

calculated mode shapes from Equation 9.20 are normalized and they satisfy the orthogonality

properties: wT

i Mwj ¼ 0 and wT

i Kwj ¼ 0 for i – j and wT

i Mwi ¼ 1:0 and wT

i Kwi ¼ v2i

; where wi ¼

w􀀊i=

ffiffiffiffi

mi p and mi ¼ w􀀊T

i Mw􀀊i:

Several eigenvalue extraction methods are available. Among the most commonly used are the block

Lanczos, Subspace Iteration, Power Dynamics, and Reduced Householder methods (Mirovitch, 1980;

Belytschko and Hughes, 1983; Bathe, 1996). The Lanczos method is recommended for large models with

many modes to be extracted (about 50 or more). It is also better suited for models containing different

types of elements such as solid, shell, and beam elements with the possibility that some elements are

distorted. The method is fast and efficient but requires more memory than the Subspace Iteration

method. In contrast, the Subspace Iteration method is well suited for extracting lower numbers of modes

(less than 50) in models with well-shaped elements and it requires less memory than the Lanczos method.

The Power Dynamics method is used for very large models, with more than 100,000 DoF, and is especially

useful for obtaining a solution for the first several modes of the model. The Reduced Householder

method is recommended for finding all the modes in small to medium models of less than 10,000 DoF

(ANSYS Users Manual, 2003).

The information obtained from an eigenvalue or modal analysis is vital in the analysis of dynamic

systems. The designer’s goal is to ensure that the loading frequency is not close to any natural

frequency of the system and that the mode shapes do not produce excessive deformation in weak

sections of the structure. With respect to the mode shapes, it is desirable to avoid mode shapes that

are similar to deformation patterns obtained from the static loading on the structure. As stated

previously, the comfortable range for an operating frequency is three times higher than the nearest

natural frequency from the lower side and three times smaller than the nearest natural frequency

from the higher side. The natural frequencies may be altered by changing the mass distribution or

adjusting the geometry of the system. If the loading frequency is higher than one or more of the first

natural frequencies then large vibrations or “shudder” may occur during startup and in certain cases

special startup procedures may have to be devised. If the loading frequency is close to one of the

natural frequencies, a large factor of safety should be used. For example, an operating frequency 20%

away from a natural frequency may call for doubling the nominal loads applied on a shaft. In some

very special applications, it may be desirable actually to operate the system at one of its natural

frequencies and utilize the amplification in the response, e.g., ultrasonic welding equipment and

certain nanoscale measuring instruments.

Modal analysis requires no special boundary conditions and the structure may be completely free or

partially constrained. Special attention should be given to the application of constraints, since

overconstraining a system will result in overprediction of the first modes, which are generally

nonconservative in a modal analysis.

9-34 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

9.5.2 Transient Dynamic Analysis

9.5.2.1 Direct Integration Method

As indicated above, with the direct integration method, the general equation of motion is integrated

directly without transformation. The input loads are specified as a function of time and the response is

calculated in the time domain. Transient analysis could be viewed as solving an equivalent static analysis

problem for each time step in the domain. The response quantities may include the time history of

displacements, velocities, and accelerations at each point in the structure. Most programs will also

provide a stress history output that is calculated in a postprocessing phase. The output results are

straightforward and easy to interpret. As in static analysis, structures undergoing transient analysis

should be fully constrained and all rigid-body motion should be eliminated. Initial conditions of

displacement and velocity are also required to perform the analysis.

As discussed in Section 9.4.2, direct integration may be performed through explicit or implicit

schemes. It should be noted that while implicit schemes do not require a critical time step, the size of the

time step will affect the accuracy of the solution. Most programs are capable of automatic time stepping,

which should be seriously considered. If automatic time stepping is not available in the program, the rule

of thumb is to use a time step that is at least one tenth of the load period. Smaller time steps may be

required to capture peak response and sharp changes in the input load.

The equations of motion (Equation 9.2) may first be reduced in size before the start of the integration

process by choosing certain master DoF as discussed above. The reduced equations are then solved by one

of the direct integration schemes and the results may then be expanded to the full set of DoF.

The following steps may be used as general guidelines in performing direct integration transient

dynamic analysis:

* Build the FE model: This will include identifying the problem type and the type(s) of elements to be

used in the model. The mesh should be fine enough to capture the highest mode required for the

analysis. If stresses are required, the intensity of the mesh should be adequate for stress calculations.

* Apply the loads and boundary conditions: All rigid-body modes should be constrained and the model

should be fully constrained. Time-dependent loads are normally specified in a table format.

* Specify initial conditions: Initial conditions of displacement and velocity are required for

transient analysis. Most programs assume zero initial conditions for acceleration. If initial

accelerations are not zero, the analyst may apply appropriate acceleration loads over a small time

interval. It should be noted that initial conditions are required for all unconstrained DoF.

REMARKS

* Modal dynamic analysis is used to obtain the dynamic characteristics of a system in terms of

its natural frequencies (eigenvalues) and natural modes (eigenvectors). It is normally the

first (and may be the only) dynamic analysis to be performed and its results are required to

decouple the equations of motion for other analyses.

* Common eigenvalue extraction methods include: Lanczos (recommended for large models

with many modes to be extracted; about 50 or more); Subspace Iteration (recommended for

extracting lower number of modes, less than 50); Power Dynamics (recommended for very

large models of more than 100,000 DoF); and Householder (recommended to find all

modes in small to medium sized models of less than 10,000 DoF).

* Modal analysis requires no special boundary conditions and the structure may be

completely free.

Finite Element Applications in Dynamics 9-35

© 2005 by Taylor & Francis Group, LLC

* Set solution control parameters, start solution and review results: This includes mass and damping

formulation, damping parameters, time integration, and time stepping parameters as well as

parameters to control output and postprocessing options.

9.5.2.2 Mode Superposition Method

In this method, the uncoupled modal equation 9.13, repeated here for convenience,

q€ þ diagð2jrvrÞq_ þ diagðv2r

Þq ¼ {fr ðtÞ} ¼ FTpðtÞ ð9:21Þ

consist of m independent second-order differential equations with constant coefficients, where m is the

number of the participating modes used in the mode superposition process.

At time t ¼ 0; each modal equation is subject to the initial conditions qr ð0Þ ¼ wT

r Mu0 and q_ rð0Þ ¼

wT

r Mu_ 0: For the underdamped case ðjr , 1Þ; the solution to a typical modal equation (excluding rigidbody

modes) is

qr ðtÞ ¼ e2jrvr t ½ar sin v􀀊r t þbr cos v􀀊r t􀀉 þ

ðt

0

fr ðtÞhr ðt 2 tÞdt ð9:22Þ

where hr ðt 2 tÞ is the unit-impulse response function, defined by

hr ðt 2 tÞ ¼

1

v􀀊r

e2jrvr ðt2tÞ sin v􀀊r ðt 2 tÞ ð9:23Þ

The first term in Equation 9.23 represents the free vibration response (homogenous solution) and the

second term represents the particular solution (Duhamel integral). ar and br are constants evaluated

from the initial conditions and v􀀊r is the damped natural frequency given by

v􀀊r ¼ vr

ffiffiffiffiffiffiffiffiffi

1 2 jr

2

q

ð9:24Þ

The integral in Equation 9.22 may be evaluated in closed form if the applied loading pðtÞ; and

consequently fr ðtÞ; is a step input, ramp input, piecewise linear function of time, or harmonic input;

otherwise, numerical integration is utilized to obtain the response.

The general steps of a typical modal transient analysis include building the model, extracting the

required modes for the analysis, obtaining the modal transient analysis and, finally, expanding the modal

superposition solution. It should be noted that most programs do not allow change of displacement

constraints after the mode extraction step is performed; i.e., all displacement constraints should be

specified before performing the modal analysis to extract the required modes. Some programs may

require the full loading conditions to be specified in the modal extraction step. Initial conditions should

be specified before performing the modal transient solution. This may include initial displacement

and velocities.

REMARKS

* In transient analysis, the input loads are specified as a function of time and the response

(displacement, velocity, acceleration, and stress) is calculated in the time domain. The

structure should be fully constrained and initial conditions are required.

* Direct integration transient analysis may be performed with explicit (conditionally stable)

or implicit (unconditionally stable) schemes.

* Prior to performing integration, the size of the equations may be reduced by choosing

master DoF and condensing out slave DoF.

* Transient analysis with mode superposition requires an initial modal analysis run followed

by the superposition of modal responses (Equation 9.22) obtained from the solution of the

decoupled equations. The integral in Equation 9.22 may be evaluated in closed form for step,

ramp, piecewise linear, and harmonic inputs; otherwise, numerical integration is utilized.

9-36 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

9.5.3 Harmonic Response Analysis

9.5.3.1 Direct Integration Method

Harmonic response analysis involves computing the steady-state response of a structure to a set of

harmonic concentrated loads, pressure loads, and harmonic ground motion. The harmonic loads may be

defined in terms of different amplitude and phase spectra. In general, the load vector may be represented

in the form:

piðtÞ ¼ piðVÞsin½Vt þ CiðVÞ􀀉 ð9:25Þ

where piðtÞ is a component of the forcing function having a magnitude, piðVÞ; phase shift, ciðVÞ, and

forcing frequency, V:

Substituting Equation 9.25 into Equation 9.2, we obtain:

Mu€ þ Cu_ þ Ku ¼ pðtÞ ¼ p􀀊ðVÞsin½Vt þ C􀀊 ðVÞ􀀉 ð9:26Þ

Using direct integration schemes to solve Equation 9.26 is theoretically viable but not recommended

practically for a few reasons. A large number of time steps would be normally required to obtain a

meaningful response to a harmonic excitation and the equations have to be resolved for each forcing

frequency. Also, if the damping is zero and a particular forcing frequency coincides with one of the

natural frequencies, the matrix becomes singular and the solution process fails. If the forcing frequency is

close to one of the natural frequencies, the matrix becomes ill conditioned and poses solution problems

even if the damping is not zero. For these reasons, it is recommended to perform harmonic response

analyses using the mode superposition method.

9.5.3.2 Mode Superposition Method

In the mode superposition method, Equation 9.26 is transformed into generalized modal coordinates

and decoupled. For the rth mode, the equation takes the form:

q€ r þ 2jrvrq_ r þv2r

qr ¼ f􀀊r eiVt ð9:27Þ

where f􀀊r is the amplitude of modal load given by f􀀊r ¼ wT

r p􀀊 : The steady-state modal response is given by

qrðtÞ ¼ q􀀊reiVt ð9:28Þ

which when substituted into Equation 9.27 gives

q􀀊rðVÞ ¼ HrðVÞ·f􀀊rðVÞ ð9:29Þ

and

Hr ðVÞ ¼

1

v2r

1 2

V2

v2r

􀁻 !

þ i 2jr

V

vr

" 􀀏 􀀐# ð9:30Þ

Equation 9.29 in the frequency domain is equivalent to Equation 9.22 in the time domain and Hr ðVÞ is

the Fourier transform of the unit-impulse response function hr ðtÞ:

The physical response is recovered from the generalized modal response through the transformation

u􀀊 ¼ Fq􀀊 : The velocity and acceleration components are obtained from the corresponding displacement

component through multiplication by V and V2; respectively. Finally, the stress components are

obtained by the use of modal stresses (Equation 9.16). Most programs provide the option that the user

can obtain the responses in either an amplitude – phase format or real – imaginary format and the actual

value of all responses can also be obtained for a given value of Vt:

Finite Element Applications in Dynamics 9-37

© 2005 by Taylor & Francis Group, LLC

The input to harmonic analysis constitutes the forcing frequency and the magnitude of the load vector

at various points of load applications. For each loading case, all forces should have the same frequency. A

change in the forcing frequency gives rise to a new load case. In harmonic analysis, as is the case in

transient and static analyses, the structure should be fully constrained and all rigid-body motions should

be eliminated. If the structure has rigid-body modes in one of the DoF, the analyst may use soft springs to

ground the structure or utilize symmetry to provide constraints.

The general steps of a typical modal harmonic analysis are quite similar to those for the modal

transient analysis, i.e., they include building the model, extracting the required modes for the analysis,

obtaining the modal harmonic analysis and, finally, expanding the modal superposition solution. It

should also be noted that most programs do not allow changes in displacement constraints after the

mode extraction step, i.e., all displacement constraints should be specified before performing the modal

analysis to extract the required modes. Some programs may require the full loading conditions to be

specified in the modal extraction step.

9.5.4 Response Spectrum Analysis

This analysis is an efficient alternative to transient dynamic analysis for estimating the maximum

response under support excitations without regard to the time at which the maximum occurs. If the

input loading is classified as a shock or impulse, this analysis is termed “shock spectrum analysis.” In

theory, one may use a direct integration scheme to find the response as a function of time and then use

the maximum value of the response. This may be prohibitive because of the very large number of time

steps that will be required to capture the peak response. As a more efficient alternative, a modal analysis

should be performed first and then a sufficient number of modes should be retained for the response

spectrum analysis. Then the solution of the individual decoupled modal equation (Equation 9.13) for a

viscous underdamped system under ground motion wðtÞ is

qr ðtÞ ¼

1

v􀀊r

ðt

0

w€ ðtÞe2jrvr ðt2tÞsin½v􀀊r ðt 2 tÞ􀀉dt ð9:31Þ

The response function in Equation 9.31 is scanned over time to find the maximum scalar value ðqr Þmax:

Then the physical value of the response of the ith DoF due to ðqr Þmax may be given by

u􀀊 ir ¼ FirðqrÞmax ð9:32Þ

in which Fir is the component of the modal vector, r; in direction i: Equation 9.32 is different from

Equation 9.11 in which all the components of the modal vector, Fi, and the generalized response, qi, are

used and, therefore, actual and accurate physical displacements are obtained.

REMARKS

* Harmonic response analysis involves computing the steady-state response of a structure to a

set of harmonic loads and ground motion (Equation 9.26).

* Using direct integration schemes in harmonic analysis is theoretically viable but not

recommended or used practically.

* Harmonic analysis with mode superposition requires an initial modal analysis run followed

by the superposition of modal responses (Equation 9.28) obtained from the solution of the

decoupled equations.

* The input to harmonic analysis constitutes the frequency and the magnitude of the load

vector at all points of load applications. For each load case, all forces should have the same

frequency. The structure should be fully constrained.

9-38 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

It can be readily seen that the quantities ðqrÞmax; and hence u􀀊 ir ; are only functions of the natural

frequency and damping (in addition to the ground excitation). A shock (or response) spectrum curve for

a certain value of damping may be defined as the maximum responses of all such single DoF systems with

a given damping and plotted as a function of natural frequency. Similarly, the maximum acceleration and

maximum velocity may also be determined and are termed as spectral acceleration and spectral velocity,

respectively.

The mass matrix in response spectrum analysis may be lumped or modified lumped based on the

consistent mass matrix. Response spectrum analysis using consistent mass generally lowers the resulting

internal forces and moments in the structure and therefore produces less stresses and may be considered

generally nonconservative (Gregory, 1990).

Unlike transient dynamic analysis, the contributions of the physical responses for each of the modes

cannot be directly summed to obtain the total response. This is because the maxima for each mode occur

at different times and the information on the time of maxima is not available in the shock spectra.

Reasonable, but arbitrary, estimates of the maxima may be obtained by using one of the following

commonly used modal combination methods (Wilson et al., 1981; NISA User’s Manual, 1992; Roussel,

1994; Joshi and Gupta, 1998):

1. The sum of absolute magnitudes (ABS or PEAK): The absolute sum of the modal responses is

given by

Rtot ¼

Xm

r¼1

lRr l ð9:33Þ

in which Rr is the physical response (displacement, velocity, or acceleration) due to mode r and m is

the number of modes considered. This method is conservative and is used if the natural frequencies

are closely spaced (within 10% of each other) and/or when damping is large. It is also shown that the

method may lead to unrealistically high calculated responses, e.g., in the coupled analysis of light

secondary systems attached to heavy primary structures or in the decoupled analysis of systems when

the centers of mass and stiffness do not coincide (Mertens, 1994).

2. The square root of the sum of the squares (SRSS or RMS): The square root of the sum of the squares of

the modal response is given by

Rtot ¼

ffiffiffiffiffiffiffi

Xm

r¼1

R2r

vuut

ð9:34Þ

This method is applicable if the modes are statistically independent, which is the case when the natural

frequencies are far apart and/or when damping is small. The SRSS method is commonly used in a wide

range of applications. Some studies indicate, however, that the method may underestimate the response

of structures with high-frequency modes, e.g., long-span bridges exposed to high wind velocities (Joshi

and Gupta, 1998).

3. The complete quadratic combination (CQC): The complete quadratic combination is given by the

following formula (Wilson et al., 1981; Der Kiureghian and Nakamura, 1993):

Rtot ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Xm

r¼1

Xm

s¼1

Rr ·RsPrs

vuut

ð9:35Þ

in which

Prs ¼

8

ffiffiffiffiffi

jrjs

p

ðjr þ gjsÞg 3=2

ð1 2 g2Þ þ 4jrjsð1 þ g 2Þ þ 4g 2ðjr

2 þ js

2Þ ð9:36Þ

where jr and js are modal damping ratios and g ¼ vs=vr : This method encompasses the SRSS and ABS

procedures for jr ¼ js: If g ¼ 0, the CQC method reduces to the former, and if g ¼ 1, it reduces to the

latter. Certain modifications of the CQC method exist, aimed at improving response estimates for

structures with high frequency modes (Der Kiureghian and Nakamura, 1993).

Finite Element Applications in Dynamics 9-39

© 2005 by Taylor & Francis Group, LLC

4. Combination of ABS and SRSS: The absolute maxima of the modal responses is added to the square

root of the sum of the squares of the remaining modal responses as follows (NISA User’s Manual, 1992):

Rmax ¼ lRjl þ

ffiffiffiffiffiffiffiffiffiffiffi

Xm

r¼1;r–j

R2r

vuut

ð9:37Þ

where

Rj ¼ max for all rlRr l ð9:38Þ

The above superposition rules are employed to get the response maxima due to ground excitation in one

direction. Similarly, the maximum responses due to ground motions in other directions are computed

separately. These maxima are then superimposed by using the SRSS or ABS procedures to get the total

response.

A final note is given here on the use of spectrum analysis in earthquake-resistant design. Most building

codes require the designer to consider the response of the structure due to simultaneous action of three

translational components of ground motion: two in the horizontal plane and one in the vertical

direction. Standard design practice is to calculate the peak responses of these inputs independently and

then combine these peak responses according to rules similar to those discussed above (Menun and Der

Kiureghian, 1998; Lopez et al., 2001).

Table 9.5 summarizes the above modal combination methods, their use and merits.

TABLE 9.5 Common Methods of Combining Modal Responses

Combinational Method Comments

Sum of absolute magnitudes (ABS or PEAK) Generally gives conservative results

Used if the natural frequencies are closely spaced

(within 10% of each other) and/or when

damping is large

Square root of the sum of

the squares (SRSS or RMS)

Use when natural frequencies are far apart and/or

when damping is small

May underestimate the response of structures with high

frequency modes

Complete quadratic combination (CQC) Combination of the SRSS and the ABS methods

Certain modifications exist to correct the response of

structures with high frequency modes

Combination of ABS and SRSS Another form of combination of the SRSS and the

ABS methods employed by simply adding

the response of the two methods

REMARKS

* Response spectrum and shock analysis are efficient alternatives to transient dynamic

analysis for estimating the maximum response under support excitations without regard to

the time at which the maximum occurs.

* In response spectrum analysis, the solution of the individual decoupled modal equation

(Equation 9.13) for ground motion, wðtÞ; is obtained (Equation 9.31) and the maximum

value of the response is recorded (with no reference to the time it occurs).

* Reasonable, but arbitrary, estimates of the maxima for the overall structure may be obtained

by using various methods of combining the modal maximums (Table 9.5).

9-40 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

Навигация