17.7. Spectrum Analysis

Theory Reference Page: 117.7. Spectrum Analysis Two types of spectrum analyses (ANTYPE,SPECTR) are supported: the deterministic response spectrum and the nondeterministic random vibration method. Both excitation at the support and excitation away from support are allowed. The three response spectrum methods are the single-point, multiple-point and dynam analysis method. The random vibration method uses the power spectral density (PSD) approach. The following spectrum analysis topics are available: Assumptions and Restrictions Description of Analysis Single-Point Response Spectrum Damping Participation Factors and Mode Coefficients Combination of Modes Reduced Mass Summary Effective Mass and Cumulative Mass Fraction Dynamic Design Analysis Method Random Vibration Method Description of Method Response Power Spectral Densities and Mean Square Response Cross Spectral Terms for Partially Correlated Input PSDs Spatial Correlation Wave Propagation Multi-Point Response Spectrum Method Missing Mass Response Rigid 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 meth Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates Theory Reference Page: 2 DDAM), the structure is excited by a spectrum of known direction and frequency components, ac 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 (SPOPT structure may be excited by different input spectra at different support points or unsupported nod 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 frequency analysis. If mode combinations are needed, the required modes must also be expanded, as des Mode-Frequency Analysis. The four options available are the single-point response spectrum method (SPOPT,SPRS), the dynamic d analysis method (SPOPT ,DDAM), the random vibration method (SPOPT ,PSD) and the multiple-point resp 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 the single-point response spectrum analysis (SPOPT ,SPRS). The table below summarizes these options the input associated with each. Table 17.3 Types of Spectrum Loading Excitation Option Excitation at Support Spectrum input Response spectrum table (FREQ and SV commands) Direction vector (input on SED and ROCK commands) Constant on all support points Excitation Away From Suppor Amplitude multiplier table (FREQ commands) Orientation of load X, Y, Z direction at each node (selec by FX, FY, or FZ on F command) Distribution of loads Amplitude in X, Y, or Z directions (selected by VALUE on F command Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates Theory Reference Page: 3 Type of input Response spectrum type (KSV on SVTYP command) Velocity 0 Acceleration 2 Displacement 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) N m = number of materials {φ i} = displacement vector for mode i Contains proprietary and confidential information of ANSYS, Inc. and its subsidiaries and affiliates and its subsidiaries and affiliates . Inc.Theory Reference Page: 4 [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.5. so tha damping contribution must be included there. 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 {D} = vector describing the excitation direction (see {F} = input force vector The vector describing the excitation direction has the form: (Nrmkey on the MODOPT command has no effect) ) where: Contains proprietary and confidential information of ANSYS. 17.7. and its subsidiaries and affiliates . Z = global Cartesian coordinates of a point on the geometry Xo . the D values may be determined in one of two ways: 1. Y o .])).Theory Reference Page: 5 X. For spectrum analysis. Y. 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 po the concept of translation of axes [T] (Weaver and Johnston([279. Inc. For D values with rocking not included (based on the SED command): a where: Contains proprietary and confidential information of ANSYS. L Z X . LY . or. CY. respectively command) x = vector cross product operator rX = X . and SEDZ SED command) 2.Theory Reference Page: 6 SX.L X rY = Y . Y . and its subsidiaries and affiliates n n n n n n . LZ = location of center of rotation (input as CGX. S Z = components of excitation direction (input as SEDX. Z = coordinate of node n LX.L Y rZ = Z . OMY . SEDY. Inc. for D values with rocking included (based on the SED and ROCK command): R is defined by: where: C X. and OMZ . SY. and CGZ on ROCK command) Contains proprietary and confidential information of ANSYS. CGY. CZ = components of angular velocity components (input as OMX . the ratio of each participation factor to the largest participation factor (output as RATI printed out. are selecte label. NRLSUM Ai = mode coefficient (see below) The mode coefficient is computed in five different ways. GRP. For force excitation (SVTYP. 1) th th Contains proprietary and confidential information of ANSYS. or accelerations. 1. or 2.Theory Reference Page: 7 In a modal analysis. For velocity excitation of base (SVTYP. DSUM. respectively. 0) where: Svi = spectral velocity for the i mode (obtained from the input velocity spectrum at frequency f effective damping ratio ) th fi = i natural frequency (cycles per unit time = ω i = i natural circular frequency (radians per unit time) 2. velocities. where: m = 0. based on whether the displacements. the third field on the mode combination commands SRSS. Inc. and its subsidiaries and affiliates . CQC . depending on the type of excitation (SVTYP 1. and its subsidiaries and affiliates . For acceleration excitation of base (SVTYP.])) Contains proprietary and confidential information of ANSYS. 3. 4. 3) where: th Sui = spectral displacement for the i mode (obtained from the input displacement response spe frequency fi and effective damping ratio ). 5. Inc. 4) (Vanmarcke([34. For power spectral density (PSD) (SVTYP. For displacement excitation of base ( SVTYP. 2) where: th Sai = spectral acceleration for the i mode (obtained from the input acceleration response spec frequency fi and effective damping ratio ).Theory Reference Page: 8 where: th Sfi = spectral force for the i mode (obtained from the input amplitude multiplier table at frequen effective damping ratio ). For all excitations but the PSD this would be the maximum response. When Svi. S fi . and its subsidiaries and affiliates .7. Combination of Modes The modal displacements.Theory Reference Page: 9 where: th Spi = power spectral density for the i mode (obtained from the input PSD spectrum at frequenc effective damping ratio ) ξ = damping ratio (input as RATIO. The response includes DOF respo well as element results and reaction forces if computed in the expansion operations (Elcalc = YES on the Contains proprietary and confidential information of ANSYS. this would be the 1-σ (standard deviation) relative response. or Spi are needed between input frequencies. S ui. and fo excitation. not at the effective damping ratio 17. Inc. DMPRAT command.6.01) 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 f i = 1. log-log interpolation is done in the s defined. Sai. table are evaluated at the input curve with the lowest damping ratio. defaults to . velocity and acceleration ( ) may be combined in different way the response of the structure. Only those modes having a significant amplitude (mode coefficient) are chosen for mode combination. ROSE and PSDCOM) of the maximum mode coefficient (all modes are scanned) considered significant. The value of εij = 0. A m having a coefficient of greater than a given value (input as SIGNIF on the mode combination commands GRP.DDAM) options of the spectrum analysis . and its subsidiaries and affiliates . 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: R a = total modal response N = total number of expanded modes εij = coupling coefficient. it is possible to expand only those modes whose signif factor exceeds the significant threshold value (SIGNIF value on MXPAND command). The spectrum option provides six options for the combination of modes. Note that the mode must be available at the time the modes are expanded.0 implies modes i and j are independent and approaches 1.SPRS) or the dynamic-design analysis (SPOPT . 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.Theory Reference Page: 10 command). In the case of the single-point response spectrum method (SPOPT .0 dependency increases Contains proprietary and confidential information of ANSYS. DSUM. NRLSUM. Inc. ROSE or NRLSUM). The mode combination instructions are written to File.6. Inc.7. reactions. where: Contains proprietary and confidential information of ANSYS. GRP. et al.MCOM by the mode combination command.]). velocity or acceleration depending on the user request (Label on the mode combination co SRSS. or stresses m displacement.Theory Reference Page: 11 R i = A iΨ i = modal response in the i mode ( R j = A jΨ j = modal response in the j mode Ai = mode coefficient for the i mode Aj = mode coefficient for the j mode Ψ i = the i mode shape Ψ j = the j mode shape th th th th th th ) Ψi and Ψ j may be the DOF response. and its subsidiaries and affiliates . CQC . Inputti in POST1 automatically performs the mode combination. DSUM.1. The DOF response. or stresses.([65. is based on Wilson. reactions. 17. Complete Quadratic Combination Method This method (accessed with the CQC command). is from the NRC Regulatory Guide([41. and its subsidiaries and affiliates . Inc.2.6.]). No one frequency is to be in more than one group. For this case.7.Theory Reference Page: 12 r = ω j / ωi 17.6.7. specializes to: where: Closely spaced modes are divided into groups that include all modes having frequencies lying between th frequency in the group and a frequency 10% higher. specializes to: Contains proprietary and confidential information of ANSYS. Double Sum Method The Double Sum Method (accessed with the DSUM command) also is from the NRC Regulatory Guide([ case. Grouping Method This method (accessed with the GRP command). 17.3. and its subsidiaries and affiliates .Theory Reference Page: 13 where: = damped natural circular frequency of the ith mode ω i= undamped natural circular frequency of the i mode = modified damping ratio of the i mode The damped natural frequency is computed as: th th The modified damping ratio is defined to account for the earthquake duration time: where: td = earthquake duration time. Inc. fixed at 10 units of time Contains proprietary and confidential information of ANSYS. reduces to: 17.6.6. Rosenblueth Method The Rosenblueth Method ( 1. NRL-SUM Method The NRL-SUM (Naval Research Laboratory Sum) method (O'Hara and Belsheim([107. is from Regulatory Guide([41.7. and its subsidiaries and affiliates .6. stress or reaction at the point R ai = displacement. calculates the maximum modal response as: where: |R a1| = absolute value of the largest modal displacement. except for Contains proprietary and confidential information of ANSYS.4.5.7. The equations for the Double Sum method (above) apply.]). For this case. Revision 2July 2006] [ ) is accessed with the ROSE command.Theory Reference Page: 14 17.6.7. stress or reaction contributions of the same point from other modes. Inc. For the Rosenblue . 17. SRSS Method The SRSS (Square Root of the Sum of the Squares) Method (accessed with the SRSS command).])) (accessed with t NRLSUM command).92. First. probably a large part of the mass is relatively clo reaction points. Reduced Mass Summary For the reduced modal analysis. . If any of the three is more or significantly less.])): Note from that so that the effective mass reduces to . This does not apply to the force spectrum. Inc.7. In other words. Rotational master DOFs are not su . each row of the reduced m summed and designated .Theory Reference Page: 15 the sign of the modal responses is retained: 17. the master DOFs either are insufficient or are poorly located. for which the excitat Contains proprietary and confidential information of ANSYS.8.7. and its subsidiaries and affiliates . UY and UZ terms are handled similarly. rather than close to master DOFs. 17. and structure.7. Effective Mass and Cumulative Mass Fraction th The effective mass (output as EFFECTIVE MASS) for the i mode (which is a function of excitation directi (Clough and Penzien([80. a study of the mass distribution is made. Inc.Theory Reference Page: 16 independent of the mass distribution. Dynamic Design Analysis Method For the DDAM (Dynamic Design Analysis Method) procedure (SPOPT . 17.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 Contains proprietary and confidential information of ANSYS.7.9. and its subsidiaries and affiliates . The cumulative mass fraction for the ith mode is: where N is the total number of modes. PSD) allows multiple power spectral density (PSD) inputs (up to te these inputs can be: 1. Va . Vc = velocity spectrum computation constants (input as VF. or Contains proprietary and confidential information of ANSYS. VA. uncorrelated. VB. AD on the command) Vf. VC on the VDDAM DDAM procedure is normally used with the NRL-SUM method of mode combination. AA. A b .0) Sx = the lesser of g or ω iV g = acceleration due to gravity (386 in/sec2) A = spectral acceleration A V = spectral velocity Af. full correlated. which was described section on the single-point response spectrum. As a result. Note that unlike . AC.10. 17.Theory Reference Page: 17 Am = minimum acceleration (input as AMIN on the ADDAM command) defaults to 6g = 2316.7. Random Vibration Method The random vibration method (SPOPT . Aa . AB. and its subsidiaries and affiliates . O'Hara and Belsheim([ normalize the mode shapes to the largest modal displacements. 2. Inc. the NRL-1396 participation fa mode coefficients Ai will be different. V b . Ac. A d = acceleration spectrum computation constants (input as AF. Inc. It is as that the excitations are stationary random processes.11. and its subsidiaries and affiliates th . 17. The procedure is based on computing statistics of each modal response and then combining them. Description of Method For partially correlated nodal and base excitations. the elements along the i column of [A] are the pseudo-static displa Contains proprietary and confidential information of ANSYS.[Kff]-1[Kfr].Theory Reference Page: 18 3. {F} is the nodal force excitation activated by a nonzero value of force command). partially correlated. Physically. The free displacements can be decomposed into pseudo-static and dynamic parts as: The pseudo-static displacements may be obtained from hand side of the equation and by replacing {uf} by {us}: by excluding the first two terms in which [A] = . Note that the restrained DOF that are not excited are not included in (zero displacement on D command). The value of force can be other than unity. the complete equations of motions are segregated into 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 valu displacement on D command). allowing for scaling of the participation factors.7. Theory Reference Page: 19 due to a unit displacement of the support DOFs excited by the ith base PSD. and its subsidiaries and affiliates . Substituting and into light damping yields: The second term on the right-hand side of the above equation represents the equivalent forces due to sup excitations. Inc.rst file. These displacements are wr load step 2 on the . 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: Contains proprietary and confidential information of ANSYS. allowed in the program.12.Theory Reference Page: 20 and for nodal excitation: Note that. Inc. and its subsidiaries and affiliates . These factors are calculated (as a result of the PFACT action command) when defining base or nodal exc cases and are written to the . the response PSD's can be computed from the input PSD's with the transfer functions for single DOF systems H(ω) and by using mode superposition techniques ( RPSD POST26).7. equations for nodal excitation problems are developed for a single PSD table.7. Response Power Spectral Densities and Mean Square Response Using the theory of random vibrations. Mu excitation PSD tables are. however.12.psd file. The response PSD's for ith DOF are given by: 17. for simplicity. Dynamic Part Contains proprietary and confidential information of ANSYS.1. Mode shapes {φ j} should be normalized with respect to the mass ma . 17. 12.Theory Reference Page: 21 17. ACEL.7. Pseudo-Static Part 17. 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. and its subsidiaries and affiliates . Inc.2. The forms of the transfer functions for displacement as the output are listed below for differen 1.12. 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).7.3. Input = force or acceleration (FORC. or ACCG on PSDUNIT command): Contains proprietary and confidential information of ANSYS. and its subsidiaries and affiliates .Theory Reference Page: 22 2. 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 free displacement (ABS option on the PSDRES command) is: where: Contains proprietary and confidential information of ANSYS. Inc. Input = displacement (DISP on PSDUNIT command): 3. terms within large parentheses of Contains proprietary and confidential information of ANSYS. ≠ m) are zero. When only nodal excitations exist. and its subsidiaries and affiliates thru . the cross PSD's (i.e. udi) = covariance between the static and dynamic displacements The general formulation described above gives simplified equations for several situations commonly enco practice. the subscripts and m wou from the thru . For fully correlated nodal excitations and identical support motions.Theory Reference Page: 23 | |Re = denotes the real part of the argument C v (usi . and only the first term within the large parentheses in needs to be eva uncorrelated nodal force and base excitations. and only the terms f = m in thru need to be considered. Inc. the last two terms in 167 do not apply. thru can be rewritten as: where: = modal PSD's. Furthermore. If the stress variance is desi the mode shapes (φ ij ) and static displacements with mode stresses and static stresses the node force variance is desired. Subsequently. the variances of the first Contains proprietary and confidential information of ANSYS. replace the mode shapes and sta and static reactions displacements with mode reaction . replace the mode shapes and static displacements with mode nodal fo and static nodal forces . Finally.]) and Harichandran([194. nodal forces or reactions can be computed (Elcalc = YES on SPOPT MXPAND)) from equations similar to thru .])) .psd file. if reaction variances are desired. and its subsidiaries and affiliates . The variance for stresses. the variances become: The modal covariance matrices are available in the .Theory Reference Page: 24 Closed-form solutions for piecewise linear PSD in log-log scale are employed to compute each integration (Chen and Ali([193. Inc. Note that represent mode combination (PSDCOM command) for random vibration analysis. these values are set to zero. the"3-σ" rule (multiplying the RMS value by 3) yields a conservative estimate on the upper boun equivalent stress (Reese et al([355.7. S3.7. S2.12. 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 valu However.4. Inc. Cross Spectral Terms for Partially Correlated Input PS Contains proprietary and confidential information of ANSYS.])). and its subsidiaries and affiliates . 17. and SINT) is known.Theory Reference Page: 25 time derivatives (VELO and ACEL options respectively on the PSDRES command) of all the quantities me above can be computed using the following relations: 17. Since no information on the distribution of the principal stresses or intensity (S1.13. this matrix may range from 2 x 2 to 10 x 10 (i. and its subsidiaries and affiliates .e. the input spectra are said to be uncorrelated. (Defined by the COVAL command w 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. For the special case in which all cross terms are zero..Theory Reference Page: 26 For excitation defined by more than a single input PSD.14. maximum number of tables is 10).7. Inc. cross terms which determine the degree of correla between the various PSDs are defined as: where: Snn(ω) = input PSD spectra which are related. (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. (Defined by the PSDVAL command and located as table n (TBLNO) n) C nm(ω) = cospectra which make up the real part of the cross terms. Spatial Correlation Contains proprietary and confidential information of ANSYS. Note correlation between nodal and base excitations is not allowed. 17. Depending upon the distance be excited nodes and the values of R and R (input as RMIN and RMAX on the PSDSPL command). MIN MAX excitation PSD can be constructed such that excitation at the nodes may be uncorrelated. The following figure indicate R .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 Contains proprietary and confidential information of ANSYS. and its subsidiaries and affiliates .R and the correlation are related. Figure 17. if the distance lies between R MAX excitation is partially correlated based on the actual distance between nodes. Inc. If the distance between excited nodes is less than R . partially correla correlated.Theory Reference Page: 27 The degree of correlation between excited nodes may also be controlled. then the two nodes are uncorrelated. Spatial correlation between excited nodes is not allowed for MIN MAX PSD analysis (PSDUNIT . then the two nodes are fully correlate MIN distance is greater than R .PRES). Wave Propagation To include wave propagation effects of a random loading. Inc. VY and VZ on PSDWAV command) = nodal coordinates of excitation point Contains proprietary and confidential information of ANSYS. the excitation PSD is constructed as: where: = separation vector between excitations points and m {V} = velocity of propagation of the wave (input as VX. and its subsidiaries and affiliates .15. the PSD would be: where: D 12 = distance between the two excitation points 1 and 2 So (ω) = basic input PSD ( PSDVAL and PSDFRQ commands) 17.7.Theory Reference Page: 28 For two excitation points 1 and 2. MPRS) allows u hundred different excitations (PFACT commands). Grouping. mode stresses. CQC. and its subsidiaries and affiliates . are multiplied by the mode coefficients to modal quantities. Most of the ingredients for performing multi-point response spectrum analysis are already developed in th subsection of the random vibration method. etc.7.16. NRL-SUM. The input spectrum are assumed to be unrelated (unco each other. partial correlation amo basic input PSD's is not currently permitted. Wave propagation effects are not allowed for a pressure PSD (PSDUNIT . Assuming that the parti factors. the static shapes corresponding to for base excitation are written as load step #2 on the *. or Rosenblueth method). As with the PSD analysis. for the th input spectrum table have already been computed (by th mode coefficients for the table are obtained as: where: = interpolated input response spectrum for the PSDVAL and PSDUNIT commands) th table at the j natural frequency (defined by the th For each input spectrum.PRES). the response of the structure is obtained by combining the responses to each spectrum using the method. 17. Inc. Multi-Point Response Spectrum Method The response spectrum analysis due to multi-point support and nodal excitations (SPOPT. as described in the previous s the single-point response spectrum method. . In this case. which can then be combined with the help of any of the available mode combination tec (SRSS. Contains proprietary and confidential information of ANSYS. Finally. in which case the inpu [S(ω)] reflects the influence of two or more uncorrelated input spectra. Double Sum.Theory Reference Page: 29 More than one simultaneous wave or spatially correlated PSD inputs are permitted. the mode shapes.rst file. For mode j. Missing Mass Response The spectrum analysis is based on a mode superposition approach where the responses of the higher mo neglected. the inertia force due to ground acceleration is: where: {F T} = total inertia force vector Sa0 = spectrum acceleration at zero period (also called the ZPA value).]) permits inclusion of the missing mass effect in a single point response spectrum ( multiple point response spectrum analysis (SPOPT . The missing mass r method ([373. the modal inertia force is: where: {F j} = modal inertia force for mode j.MCOM by the mode combination comm Inputting the file in POST1 (/INPUT command) automatically performs the mode combination.Theory Reference Page: 30 The mode combination instructions are written to the file Jobname. Using equations and . Hence part of the mass of the structure is missing in the dynamic analysis. and its subsidiaries and affiliates . input as ZPA on the MMASS Mode superposition can be used to determine the inertia force. Inc.7.17. 17. this force can be rewritten: Contains proprietary and confidential information of ANSYS.MPRS) when base excitation is considered Considering a rigid structure. Inc. it is written as load step 3. Combination Method Since the missing mass response is a pseudo-static response. Hence the missing mass response and the modal responses defined in Contains proprietary and confidential information of ANSYS. the missing mass response is written as load step 2 in the Multiple Point Response Spectrum analysis.Theory Reference Page: 31 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 m 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 m supposed to be mostly rigid and exhibit pseudo-static responses to an acceleration base excitation. and its subsidiaries and affiliates . it is in phase with the imposed acceleration phase with the modal responses. In Single Point Response Spectrum Analysis. The rigid components are considered separately because the correspondi responses are all in phase. and its subsidiaries and affiliates .18.Theory Reference Page: 32 combined using the Square Root of Sum of the Squares (SRSS) method. The total response including the missing mass effect is: 17. 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 1.7. See the Gupta and Lindley-Yow method Ri = modal response of mode i The corresponding periodic component is then: where: Contains proprietary and confidential information of ANSYS. the modal responses consi periodic and rigid components. Rigid Responses For frequencies higher than the amplified acceleration region of the spectrum. Inc. Theory Reference Page: 33 Rpi = periodic component of the modal response of mode i Two methods ([374. It is input as ZPA on RIGRESP command with Method Contains proprietary and confidential information of ANSYS. F1 and F2 = key frequencies. Lindley-Yow Method where: Sa0 = spectrum acceleration at zero period (ZPA). Inc. Gupta Method αi = 0 for Fi αi = 1 for Fi where: F1 F2 Fi = ith frequency value. F1 is input as Val1 and F2 is input as Val2 on RIGRESP command with Me GUPTA.]) can be used to separate the periodic and the rigid components in each modal respo one has a different definition of the rigid response coefficients αi. and its subsidiaries and affiliates . periodic and rigid responses are combined using the SRSS method. Inc. When the missing mass resp (accessed with MMASS command) is included in the analysis. since it is a rigid response as well.0 . it is sum those components. and its subsidiaries and affiliates . Finally. the Complete (CQC) or the Rosenblueth (ROSE) combination methods. The total response with the rigid responses and the missing mass response included is expressed as: Release 12. Inc. Contains proprietary and confidential information of ANSYS. they are summed algebraically.Theory Reference Page: 34 Sai = spectrum acceleration corresponding to the ith frequency Combination Method The periodic components are combined using the Square Root of Sum of Squares (SRSS).© 2009 SAS IP. All rights reserved. Since the rigid components are all in phase.