17.7. Spectrum Analysis

Theory Reference

Contains proprietary and confidential information of ANSYS, Inc.and its subsidiaries and affiliates

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17.7. Spectrum Analysis

Two types of spectrum analyses (ANTYPE,SPECTR) are supported: the deterministic response spectrum method and the nondeterministic random vibration method. Both excitation at the support and excitation away from the support are allowed. The three response spectrum methods are the single-point, multiple-point and dynamic design analysis method. The random vibration method uses the power spectral density (PSD) approach.

The following spectrum analysis topics are available:

Assumptions and RestrictionsDescription of AnalysisSingle-Point Response SpectrumDampingParticipation Factors and Mode CoefficientsCombination of ModesReduced Mass SummaryEffective Mass and Cumulative Mass FractionDynamic Design Analysis MethodRandom Vibration MethodDescription of MethodResponse Power Spectral Densities and Mean Square ResponseCross Spectral Terms for Partially Correlated Input PSDsSpatial CorrelationWave PropagationMulti-Point Response Spectrum MethodMissing Mass ResponseRigid Responses

17.7.1. Assumptions and Restrictions

1. The structure is linear.2. For single-point response spectrum analysis (SPOPT,SPRS) and dynamic design analysis method (

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Page: 2DDAM), the structure is excited by a spectrum of known direction and frequency components, acting uniformly on all support points or on specified unsupported master degrees of freedom (DOFs).

3. For multi-point response spectrum (SPOPT,MPRS) and power spectral density (SPOPTstructure may be excited by different input spectra at different support points or unsupported nodes. Up to ten different simultaneous input spectra are allowed.

17.7.2. Description of Analysis

The spectrum analysis capability is a separate analysis type (ANTYPE,SPECTR) and it must be preceded by a mode-frequency analysis. If mode combinations are needed, the required modes must also be expanded, as described in Mode-Frequency Analysis.

The four options available are the single-point response spectrum method (SPOPT,SPRS), the dynamic design analysis method (SPOPT,DDAM), the random vibration method (SPOPT,PSD) and the multiple-point response spectrum method (SPOPT,MPRS). Each option is discussed in detail subsequently.

17.7.3. Single-Point Response Spectrum

Both excitation at the support (base excitation) and excitation away from the support (force excitation) are allowed for the single-point response spectrum analysis (SPOPT,SPRS). The table below summarizes these options as well as the input associated with each.

Table 17.3 Types of Spectrum Loading

Excitation Option

Excitation at Support Excitation Away From Support

Spectrum input Response spectrum table (FREQ and SV commands)

Amplitude multiplier table (FREQcommands)

Orientation of load Direction vector (input on SED and ROCK commands)

X, Y, Z direction at each node (selected by FX, FY, or FZ on F command)

Distribution of loads Constant on all support points Amplitude in X, Y, or Z directions (selected by VALUE on F command)

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Type of input Velocity Acceleration Displacement

Response spectrum type (KSV on SVTYP command)

0 2 3,4

17.7.4. Damping

Damping is evaluated for each mode and is defined as:

where:

β = beta damping (input as VALUE, BETAD command)

ω i = undamped natural circular frequency of the ith mode

ξc = damping ratio (input as RATIO, DMPRAT command)

Nm = number of materials

{φ i} = displacement vector for mode i

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[Kj] = stiffness matrix of part of structure of material j

= modal damping ratio of mode i (MDAMP command)

Note that the material dependent damping contribution is computed in the modal expansion phase, so that this damping contribution must be included there.

17.7.5. Participation Factors and Mode Coefficients

The participation factors for the given excitation direction are defined as:

where:γ i = participation factor for the ith mode

{φ} i = eigenvector normalized using (Nrmkey on the MODOPT command has no effect)

{D} = vector describing the excitation direction (see )

{F} = input force vector

The vector describing the excitation direction has the form:

where:

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X, Y, Z = global Cartesian coordinates of a point on the geometry

Xo, Yo, Zo = global Cartesian coordinates of point about which rotations are done (reference point)

{e} = six possible unit vectors

We can calculate the statically equivalent actions at j due to rigid-body displacements of the reference point using the concept of translation of axes [T] (Weaver and Johnston([279.])).

For spectrum analysis, the Da values may be determined in one of two ways:

1. For D values with rocking not included (based on the SED command):

where:

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SX, SY, SZ = components of excitation direction (input as SEDX, SEDY, and SEDZSED command)

2. or, for D values with rocking included (based on the SED and ROCK command):

R is defined by:

where:

CX, CY, CZ = components of angular velocity components (input as OMX, OMY, and OMZ, respectively, on

command)

x = vector cross product operator

rX = Xn - LX

rY = Yn - LY

rZ = Zn - LZ

Xn, Yn, Zn = coordinate of node n

LX, LY, LZ = location of center of rotation (input as CGX, CGY, and CGZ on ROCK command)

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In a modal analysis, the ratio of each participation factor to the largest participation factor (output as RATIO) is printed out.

where:m = 0, 1, or 2, based on whether the displacements, velocities, or accelerations, respectively, are selected (using label, the third field on the mode combination commands SRSS, CQC, GRP, DSUM, NRLSUM

Ai = mode coefficient (see below)

The mode coefficient is computed in five different ways, depending on the type of excitation (SVTYP

1. For velocity excitation of base (SVTYP, 0)

where:

Svi = spectral velocity for the ith mode (obtained from the input velocity spectrum at frequency f

effective damping ratio )

fi = ith natural frequency (cycles per unit time =

ω i = ith natural circular frequency (radians per unit time)

2. For force excitation (SVTYP, 1)

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where:

Sfi = spectral force for the ith mode (obtained from the input amplitude multiplier table at frequency f

effective damping ratio ).3. For acceleration excitation of base (SVTYP, 2)

where:

Sai = spectral acceleration for the ith mode (obtained from the input acceleration response spectrum at

frequency fi and effective damping ratio ).

4. For displacement excitation of base (SVTYP, 3)

where:

Sui = spectral displacement for the ith mode (obtained from the input displacement response spectrum at

frequency fi and effective damping ratio ).

5. For power spectral density (PSD) (SVTYP, 4) (Vanmarcke([34.]))

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Page: 9where:

Spi = power spectral density for the ith mode (obtained from the input PSD spectrum at frequency f

effective damping ratio )

ξ = damping ratio (input as RATIO, DMPRAT command, defaults to .01)

The integral in is approximated as:

where:Li = fi (in integer form)

Spj = power spectral density evaluated at frequency (f) equal to j (in real form)

∆f = effective frequency band for fi = 1.

When Svi, Sfi, Sai, Sui, or Spi are needed between input frequencies, log-log interpolation is done in the space as

defined.

table are evaluated at the input curve with the lowest damping ratio, not at the effective damping ratio

17.7.6. Combination of Modes

The modal displacements, velocity and acceleration ( ) may be combined in different ways to obtain the response of the structure. For all excitations but the PSD this would be the maximum response, and for the PSD excitation, this would be the 1-σ (standard deviation) relative response. The response includes DOF response as well as element results and reaction forces if computed in the expansion operations (Elcalc = YES on the

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In the case of the single-point response spectrum method (SPOPT,SPRS) or the dynamic-design analysis method (SPOPT,DDAM) options of the spectrum analysis , it is possible to expand only those modes whose significance factor exceeds the significant threshold value (SIGNIF value on MXPAND command). Note that the mode coefficients must be available at the time the modes are expanded.

Only those modes having a significant amplitude (mode coefficient) are chosen for mode combination. A mode having a coefficient of greater than a given value (input as SIGNIF on the mode combination commands GRP, DSUM, NRLSUM, ROSE and PSDCOM) of the maximum mode coefficient (all modes are scanned) is considered significant.

The spectrum option provides six options for the combination of modes. They are:

Complete Quadratic Combination Method (CQC)Grouping Method (GRP)Double Sum Method (DSUM)SRSS Method (SRSS)NRL-SUM Method (NRLSUM)Rosenblueth Method (ROSE)

These methods generate coefficients for the combination of mode shapes. This combination is done by a generalization of the method of the square root of the sum of the squares which has the form:

where:Ra = total modal response

N = total number of expanded modes

εij= coupling coefficient. The value of εij = 0.0 implies modes i and j are independent and approaches 1.0 as the

dependency increases

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Ri = AiΨi = modal response in the ith mode ( )

Rj = AjΨj = modal response in the jth mode

Ai = mode coefficient for the ith mode

Aj = mode coefficient for the jth mode

Ψi = the ith mode shape

Ψj = the jth mode shape

Ψi and Ψj may be the DOF response, reactions, or stresses. The DOF response, reactions, or stresses may be

displacement, velocity or acceleration depending on the user request (Label on the mode combination commands SRSS, CQC, DSUM, GRP, ROSE or NRLSUM).

The mode combination instructions are written to File.MCOM by the mode combination command. Inputting this file in POST1 automatically performs the mode combination.

17.7.6.1. Complete Quadratic Combination Method

This method (accessed with the CQC command), is based on Wilson, et al.([65.]).

where:

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r = ω j / ωi

17.7.6.2. Grouping Method

This method (accessed with the GRP command), is from the NRC Regulatory Guide([41.]). For this case, specializes to:

where:

Closely spaced modes are divided into groups that include all modes having frequencies lying between the lowest frequency in the group and a frequency 10% higher. No one frequency is to be in more than one group.

17.7.6.3. Double Sum Method

The Double Sum Method (accessed with the DSUM command) also is from the NRC Regulatory Guide([case, specializes to:

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where:

= damped natural circular frequency of the ith mode

ω i= undamped natural circular frequency of the ith mode

= modified damping ratio of the ith mode

The damped natural frequency is computed as:

The modified damping ratio is defined to account for the earthquake duration time:

where:td = earthquake duration time, fixed at 10 units of time

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17.7.6.4. SRSS Method

The SRSS (Square Root of the Sum of the Squares) Method (accessed with the SRSS command), is from the NRC Regulatory Guide([41.]). For this case, reduces to:

17.7.6.5. NRL-SUM Method

The NRL-SUM (Naval Research Laboratory Sum) method (O'Hara and Belsheim([107.])) (accessed with the NRLSUM command), calculates the maximum modal response as:

where:|Ra1| = absolute value of the largest modal displacement, stress or reaction at the point

Rai = displacement, stress or reaction contributions of the same point from other modes.

17.7.6.6. Rosenblueth Method

The Rosenblueth Method ([

1.92, Revision 2July 2006]) is accessed with the ROSE command.

The equations for the Double Sum method (above) apply, except for . For the Rosenblueth Method,

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Page: 15the sign of the modal responses is retained:

17.7.7. Reduced Mass Summary

For the reduced modal analysis, a study of the mass distribution is made. First, each row of the reduced mass matrix

summed and designated . UY and UZ terms are handled similarly. Rotational master DOFs are not summed.

, , and structure. If any of the three is more or significantly less, probably a large part of the mass is relatively close to the reaction points, rather than close to master DOFs. In other words, the master DOFs either are insufficient in number or are poorly located.

17.7.8. Effective Mass and Cumulative Mass Fraction

The effective mass (output as EFFECTIVE MASS) for the ith mode (which is a function of excitation direction) is (Clough and Penzien([80.])):

Note from that

so that the effective mass reduces to . This does not apply to the force spectrum, for which the excitation is

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The cumulative mass fraction for the ith mode is:

where N is the total number of modes.

17.7.9. Dynamic Design Analysis Method

For the DDAM (Dynamic Design Analysis Method) procedure (SPOPT,DDAM) (O'Hara and Belsheim([weights in thousands of pounds (kips) are computed from the participation factor:

where:wi = modal weight in kips

386 = acceleration due to gravity (in/sec2)

The mode coefficients are computed by:

where:Sai = the greater of Am or Sx

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Am = minimum acceleration (input as AMIN on the ADDAM command) defaults to 6g = 2316.0)

Sx = the lesser of gA or ω iV

g = acceleration due to gravity (386 in/sec2)

A = spectral acceleration

V = spectral velocity

Af, Aa, Ab, Ac, Ad = acceleration spectrum computation constants (input as AF, AA, AB, AC, AD on the

command)

Vf, Va, Vb, Vc = velocity spectrum computation constants (input as VF, VA, VB, VC on the VDDAM

DDAM procedure is normally used with the NRL-SUM method of mode combination, which was described in the section on the single-point response spectrum. Note that unlike , O'Hara and Belsheim([normalize the mode shapes to the largest modal displacements. As a result, the NRL-1396 participation factors

mode coefficients Ai will be different.

17.7.10. Random Vibration Method

The random vibration method (SPOPT,PSD) allows multiple power spectral density (PSD) inputs (up to ten) in which these inputs can be:

1. full correlated,2. uncorrelated, or

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The procedure is based on computing statistics of each modal response and then combining them. It is assumed that the excitations are stationary random processes.

17.7.11. Description of Method

For partially correlated nodal and base excitations, the complete equations of motions are segregated into the free and the restrained (support) DOF as:

where {uf} are the free DOF and {ur} are the restrained DOF that are excited by random loading (unit value of

displacement on D command). Note that the restrained DOF that are not excited are not included in (zero displacement on D command). {F} is the nodal force excitation activated by a nonzero value of force (on the command). The value of force can be other than unity, allowing for scaling of the participation factors.

The free displacements can be decomposed into pseudo-static and dynamic parts as:

The pseudo-static displacements may be obtained from by excluding the first two terms on the left-hand side of the equation and by replacing {uf} by {us}:

in which [A] = - [Kff]-1[Kfr]. Physically, the elements along the ith column of [A] are the pseudo-static displacements

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Page: 19due to a unit displacement of the support DOFs excited by the ith base PSD. These displacements are written as load step 2 on the .rst file. Substituting and into light damping yields:

The second term on the right-hand side of the above equation represents the equivalent forces due to support excitations.

Using the mode superposition analysis of Mode Superposition Method and rewriting

the above equations are decoupled yielding:

where:n = number of mode shapes chosen for evaluation (input as NMODE on SPOPT command)

yj = generalized displacements

ω j and ξj = natural circular frequencies and modal damping ratios

The modal loads Gj are defined by:

The modal participation factors corresponding to support excitation are given by:

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and for nodal excitation:

Note that, for simplicity, equations for nodal excitation problems are developed for a single PSD table. Multiple nodal excitation PSD tables are, however, allowed in the program.

These factors are calculated (as a result of the PFACT action command) when defining base or nodal excitation cases and are written to the .psd file. Mode shapes {φ j} should be normalized with respect to the mass matrix as in

.

17.7.12. Response Power Spectral Densities and Mean Square Response

Using the theory of random vibrations, the response PSD's can be computed from the input PSD's with the help of transfer functions for single DOF systems H(ω) and by using mode superposition techniques ( RPSDPOST26). The response PSD's for ith DOF are given by:

17.7.12.1. Dynamic Part

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17.7.12.2. Pseudo-Static Part

17.7.12.3. Covariance Part

where:n = number of mode shapes chosen for evaluation (input as NMODE on SPOPT command)

r1 and r2 = number of nodal (away from support) and base PSD tables, respectively

The transfer functions for the single DOF system assume different forms depending on the type (PSDUNIT command) of the input PSD and the type of response desired (Lab and Relkey on the command). The forms of the transfer functions for displacement as the output are listed below for different inputs.

1. Input = force or acceleration (FORC, ACEL, or ACCG on PSDUNIT command):

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2. Input = displacement (DISP on PSDUNIT command):

3. Input = velocity (VELO on PSDUNIT command):

where:ω = forcing frequency

ω j = natural circular frequency for jth mode

i =

Now, random vibration analysis can be used to show that the absolute value of the mean square response of the ifree displacement (ABS option on the PSDRES command) is:

where:

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| |Re = denotes the real part of the argument

Cv (usi , udi) = covariance between the static and dynamic displacements

The general formulation described above gives simplified equations for several situations commonly encountered in practice. For fully correlated nodal excitations and identical support motions, the subscripts and m would drop out from the thru . When only nodal excitations exist, the last two terms in 167 do not apply, and only the first term within the large parentheses in needs to be evaluated. For uncorrelated nodal force and base excitations, the cross PSD's (i.e. ≠ m) are zero, and only the terms for which = m in thru need to be considered.

thru can be rewritten as:

where:

= modal PSD's, terms within large parentheses of thru

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Closed-form solutions for piecewise linear PSD in log-log scale are employed to compute each integration in (Chen and Ali([193.]) and Harichandran([194.])) .

Subsequently, the variances become:

The modal covariance matrices are available in the .psd file. Note that represent mode combination (PSDCOM command) for random vibration analysis.

The variance for stresses, nodal forces or reactions can be computed (Elcalc = YES on SPOPTMXPAND)) from equations similar to thru . If the stress variance is desired, replace

the mode shapes (φ ij) and static displacements with mode stresses and static stresses

the node force variance is desired, replace the mode shapes and static displacements with mode nodal forces

and static nodal forces . Finally, if reaction variances are desired, replace the mode shapes and static

displacements with mode reaction and static reactions . Furthermore, the variances of the first and second

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Page: 25time derivatives (VELO and ACEL options respectively on the PSDRES command) of all the quantities mentioned above can be computed using the following relations:

17.7.12.4. Equivalent Stress Mean Square Response

The equivalent stress (SEQV) mean square response is computed as suggested by Segalman et al([

where:Ψ = matrix of component "stress shapes"

Note that the the probability distribution for the equivalent stress is neither Gaussian nor is the mean value zero. However, the"3-σ" rule (multiplying the RMS value by 3) yields a conservative estimate on the upper bound of the equivalent stress (Reese et al([355.])). Since no information on the distribution of the principal stresses or stress intensity (S1, S2, S3, and SINT) is known, these values are set to zero.

17.7.13. Cross Spectral Terms for Partially Correlated Input PSDs

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For excitation defined by more than a single input PSD, cross terms which determine the degree of correlation between the various PSDs are defined as:

where:Snn(ω) = input PSD spectra which are related. (Defined by the PSDVAL command and located as table number

(TBLNO) n)

Cnm(ω) = cospectra which make up the real part of the cross terms. (Defined by the COVAL command where n and

m (TBLNO1 and TBLNO2) identify the matrix location of the cross term)

Qnm(ω) = quadspectra which make up the imaginary part of the cross terms. (Defined by the QDVAL

where n and m (TBLNO1 and TBLNO2) identify the matrix location of the cross term)

The normalized cross PSD function is called the coherence function and is defined as:

where:

Although the above example demonstrates the cross correlation for 3 input spectra, this matrix may range in size from 2 x 2 to 10 x 10 (i.e., maximum number of tables is 10).

For the special case in which all cross terms are zero, the input spectra are said to be uncorrelated. Note that correlation between nodal and base excitations is not allowed.

17.7.14. Spatial Correlation

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The degree of correlation between excited nodes may also be controlled. Depending upon the distance between excited nodes and the values of RMIN and RMAX (input as RMIN and RMAX on the PSDSPL command), an overall

excitation PSD can be constructed such that excitation at the nodes may be uncorrelated, partially correlated or fully correlated. If the distance between excited nodes is less than RMIN, then the two nodes are fully correlated; if the

distance is greater than RMAX, then the two nodes are uncorrelated; if the distance lies between R

excitation is partially correlated based on the actual distance between nodes. The following figure indicates how RMIN, RMAX and the correlation are related. Spatial correlation between excited nodes is not allowed for a pressure

PSD analysis (PSDUNIT,PRES).

Figure 17.5: Sphere of Influence Relating Spatially Correlated PSD Excitation

Node i excitation is fully correlated with node j excitation

Node i excitation is partially correlated with node k excitation

Node i excitation is uncorrelated with node excitation

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For two excitation points 1 and 2, the PSD would be:

where:

D12 = distance between the two excitation points 1 and 2

So(ω) = basic input PSD ( PSDVAL and PSDFRQ commands)

17.7.15. Wave Propagation

To include wave propagation effects of a random loading, the excitation PSD is constructed as:

where:

= separation vector between excitations points and m

{V} = velocity of propagation of the wave (input as VX, VY and VZ on PSDWAV command)

= nodal coordinates of excitation point

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More than one simultaneous wave or spatially correlated PSD inputs are permitted, in which case the input excitation [S(ω)] reflects the influence of two or more uncorrelated input spectra. In this case, partial correlation among the basic input PSD's is not currently permitted. Wave propagation effects are not allowed for a pressure PSD analysis (PSDUNIT,PRES).

17.7.16. Multi-Point Response Spectrum Method

The response spectrum analysis due to multi-point support and nodal excitations (SPOPT,MPRS) allows up to a hundred different excitations (PFACT commands). The input spectrum are assumed to be unrelated (uncorrelated) to each other.

Most of the ingredients for performing multi-point response spectrum analysis are already developed in the previous subsection of the random vibration method. As with the PSD analysis, the static shapes corresponding to equation

for base excitation are written as load step #2 on the *.rst file, Assuming that the participation

factors, , for the th input spectrum table have already been computed (by

mode coefficients for the th table are obtained as:

where:

= interpolated input response spectrum for the th table at the jth natural frequency (defined by the PSDVAL and PSDUNIT commands)

For each input spectrum, the mode shapes, mode stresses, etc. are multiplied by the mode coefficients to compute modal quantities, which can then be combined with the help of any of the available mode combination techniques (SRSS, CQC, Double Sum, Grouping, NRL-SUM, or Rosenblueth method), as described in the previous section on the single-point response spectrum method.

Finally, the response of the structure is obtained by combining the responses to each spectrum using the SRSS method.

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The mode combination instructions are written to the file Jobname.MCOM by the mode combination command. Inputting the file in POST1 (/INPUT command) automatically performs the mode combination.

17.7.17. Missing Mass Response

The spectrum analysis is based on a mode superposition approach where the responses of the higher modes are neglected. Hence part of the mass of the structure is missing in the dynamic analysis. The missing mass response method ([373.]) permits inclusion of the missing mass effect in a single point response spectrum (multiple point response spectrum analysis (SPOPT,MPRS) when base excitation is considered

Considering a rigid structure, the inertia force due to ground acceleration is:

where:

{FT} = total inertia force vector

Sa0 = spectrum acceleration at zero period (also called the ZPA value), input as ZPA on the MMASS

Mode superposition can be used to determine the inertia force. For mode j, the modal inertia force is:

where:

{Fj} = modal inertia force for mode j.

Using equations and , this force can be rewritten:

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The missing inertia force vector is then the difference between the total inertia force given by the sum of the modal inertia forces defined by :

The expression within the parentheses in the equation above is the fraction of degree of freedom mass missing:

The missing mass response is the static shape due to the inertia forces defined by equation :

where:

{RM} is the missing mass response

The application of these equations can be extended to flexible structures because the higher truncated modes are supposed to be mostly rigid and exhibit pseudo-static responses to an acceleration base excitation.

In Single Point Response Spectrum Analysis, the missing mass response is written as load step 2 in the Multiple Point Response Spectrum analysis, it is written as load step 3.

Combination Method

Since the missing mass response is a pseudo-static response, it is in phase with the imposed acceleration but out of phase with the modal responses. Hence the missing mass response and the modal responses defined in are

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The total response including the missing mass effect is:

17.7.18. Rigid Responses

For frequencies higher than the amplified acceleration region of the spectrum, the modal responses consist of both periodic and rigid components. The rigid components are considered separately because the corresponding responses are all in phase. The combination methods listed in Combination of Modes do not apply

The rigid component of a modal response is expressed as:

where:

Rri = the rigid component of the modal response of mode i

αi = rigid response coefficient in the range of values 0 through 1. See the Gupta and Lindley-Yow methods below.

Ri = modal response of mode i

The corresponding periodic component is then:

where:

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Rpi = periodic component of the modal response of mode i

Two methods ([374.]) can be used to separate the periodic and the rigid components in each modal response. Each one has a different definition of the rigid response coefficients αi.

Gupta Method

αi = 0 for Fi F1

αi = 1 for Fi F2

where:

Fi = ith frequency value.

F1 and F2 = key frequencies. F1 is input as Val1 and F2 is input as Val2 on RIGRESP command with Method = GUPTA.

Lindley-Yow Method

where:

Sa0 = spectrum acceleration at zero period (ZPA). It is input as ZPA on RIGRESP command with Method = LINDLEY

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Sai = spectrum acceleration corresponding to the ith frequency

Combination Method

The periodic components are combined using the Square Root of Sum of Squares (SRSS), the Complete Quadratic (CQC) or the Rosenblueth (ROSE) combination methods.

Since the rigid components are all in phase, they are summed algebraically. When the missing mass response (accessed with MMASS command) is included in the analysis, since it is a rigid response as well, it is summed with those components. Finally, periodic and rigid responses are combined using the SRSS method.

The total response with the rigid responses and the missing mass response included is expressed as:

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17.7. Spectrum Analysis