Twenty Years of Structural Dynamic Modification A Review

Transcription

1 went Years of Structural Dnamic A Review Peter Avitabile, Universit of Massachusetts Lowell, Lowell, Massachusetts Structural dnamic modification techniques have been available as a design tool for several decades. his article reviews the methodolog of the technique including both proportional and comple mode formulations. Issues pertaining to limitations on their use and truncation effects of unmeasured modes are discussed. Also discussed is the use of simplistic elements through higher order elements. Discussion on tpical commercial implementations is presented. Finall, the ever-constant problems with lack of rotational DOF and truncation are presented, as well as attempts to overcome those issues. Based on a paper presented at IMAC-XX the th International Modal Analsis Conference, Los Angeles, CA, Februar. Notation Matri [M] = phsical mass matri [C] = phsical damping matri [K] = phsical stiffness matri [A] = phsical state space matri [B] = phsical state space matri [U] = modal matri real normal modes [Φ] = modal matri comple modes [I] = diagonal modal mass matri [Λ ] = diagonal modal stiffness matri unmodified [ω ] = diagonal modal stiffness matri unmodified [Ω ] = diagonal modal stiffness matri modified [ M ] = modal mass matri [ K ] = modal stiffness matri Superscript [ ] = a modal quantit Vector { ẋ } = acceleration { ẋ } = velocit { } = displacement { Ẏ} = derivative of state space variable of displacement and velocit {Y} = state space variable of displacement and velocit {F} = force real normal mode equations {Q} = force comple mode equations {p} = modal displacement {u} = modal vector Subscript n = full set of finite element DOF a = tested set of eperimental DOF d = deleted (omitted) set of DOF = original set of modes = final set of modes = transformation from state to state Structural Dnamic (SDM) became a popular modeling tool in the late 97s following its development and implementation in commerciall available software running on desktop computers (albeit predecessors to the IBM PC with limited memor and computation capabilities compared to toda s standards). Initial efforts were directed towards implementation of an efficient modeling technique to determine the effect of structural changes using modal data, either test or finite element data, as the basis of the prediction method. Initial eigenvalue modification techniques required processing matrices that taed the most powerful of earl desktop computers. he introduction of the Local Eigenvalue Procedure (LEMP) -3 reduced the more computationall involved eigensolution into a set of second order equations that could easil be handled b almost an computational device. his paved the wa to commercial implementation of the Structural Dnamic Process. SDM software 4,5 offered a powerful analtical tool for the test engineer to make quick and simple predictions of the effects of structural changes to a test article before actuall implementing the changes. Animation of the modified modes of the sstem provided a tremendous opportunit to understand the dnamics of the modified sstem prior to implementing an actual changes. he first structural elements utilized simple mass, spring and dashpot elements to eplore the effects of change to a structural sstem. Initial approaches to this technique utilized a proportional mode approimation for the solution of sstem equations. While this approach contained some approimations, the benefit of the computational tool far out-weighed the errors associated with the proportional mode assumption. pical studies included effects of simplistic structural changes (mass, damping and stiffness) on predicted frequencies and mode shapes, determination of structural characteristic changes to shift a given resonance, and the effects of the addition of a tuned absorber on the modal characteristics of a sstem. hese tools provided a great improvement to the abilit to fine-tune and adjust structural dnamic sstems, especiall in a troubleshooting environment. he use of comple modes and the stud of damping changes to the sstem required an enhancement to the equations used for the proportional mode approach. he state space formulation 6,7 for the equations was developed to handle an tpe of structural change without incurring errors associated with the proportional mode approach. Although the LEMP process was also used for this formulation, man software packages continue to use the proportional mode approach. While ver practical engineering solutions can often be obtained from this approach, some solutions are plagued b this approimation. 8 he LEMP process was the solution scheme of choice until the computational power of desktop computers achieved reasonable speeds. hen the Eigenvalue echnique (EM) became the tpical solution algorithm. 9 Again, these equations can be solved using the proportional mode approimation or b using a full state-space comple mode solution. An assortment of commerciall available software packages were developed -5 as well as other in-house implementations of the technique. Once desktop computer sstems became more powerful and the EM became more popular, the introduction of structural elements (tpical of the elemental tpes found in finite element modeling software) became commonplace for the modification of structural sstems using modal data as the basis of the computation technique. In addition to component changes on individual components, sstem models were also developed using this technique but tpicall were found onl on main- 4 SOUND AND VIBRAION/JANUARY 3

2 frame computers. As the desktop computer became more powerful, these sstem modeling tools also found their wa into test engineer design software packages.,3 he first useful structural element in the SDM process to be considered was a general 3D beam element (useful for beam and rib tpes of modification studies). 6,7 However, there was an inherent problem when these realistic structural elements were used in conjunction with modal data obtained from a modal test. he test data lacked the rotational degrees of freedom (RDOF) that are necessar for coupling the elements that contain both translation and rotation information. his lack of rotational DOF was a major obstacle at the introduction of the SDM technique and still presents unique obstacles to efficient implementation. A great deal of research effort was epended in the 98s to develop techniques for the estimation of rotational DOF 8- as well as the development of structural elements that could approimate realistic structural changes b using onl the available measured translational DOF. Man efforts were devoted to the epansion of measured mode shapes for the development of rotational DOF. Some of the techniques utilized shape function approimations while others emploed mathematical spline fitting or finite element approimations., All of these efforts were undertaken as an intermediate step while an economical rotar accelerometer was developed. Since rotational measurements ma not eist or be available, efforts were focused on approimations of general 3D structural elements using onl available translational information. Beam approimations using 3-point bending equations 3,4,3 were the first estimates to be used. hese provided reasonabl good results for sstems that behaved with beam-like responses in the modified sstem characteristics. In addition, a Guan reduced beam with onl translational DOF also proved to be a ver good approimation for sstems with beam-like modified characteristics. hese two approaches are still used toda due to continued difficulties of estimating rotational DOF from measured test data. However, of all the errors associated with the stud of SDM, the most important effect has alwas been due to the truncation of the modes that describe the modal database. 4-6 he lack of all the modes to adequatel describe the dnamic space of the model is b far the most significant error associated with the SDM process. Attempts have been made to estimate residual effects to compensate for some of the out-of-band effects, but these tpicall are significant onl if translational DOF are associated with errors of the model s truncation. However, truncating rotational DOF often causes the most error in all SDM studies and is still a major concern toda. Impedance based techniques were being developed almost parallel to the SDM process and sstem modeling tools. 7 he modification of a sstem using onl frequenc response functions 8 allowed the test engineer to investigate sstem changes using onl measured functions of the unmodified sstem. he main perceived advantage was that a measurement has no truncation associated with its formulation but rotational DOF have a considerable effect in almost an modeling stud. However, impedance based methods onl produce frequenc response functions the mode shapes are not a result of the process. Another ke drawback is the rarit that rotational DOF are ever available from a modal test. herefore, an realistic structural changes or sstem models are alwas deficient in this regard. Still, the approach has gained significant popularit during the last ten ears. Research efforts are directed towards the development of the necessar DOF to produce accurate sstem models. Implementation of the technique for finite element modeling applications tpicall results in frequenc response information that is affected b truncation of the modes associated with the modeling process. As in the SDM procedures, impedance based modeling techniques are most affected b truncation for analtical approimations and lack of rotational DOF for test applications. In general, the first ten ears of the International Modal Analsis Conference (IMAC) were seen to be the birth and development of the Structural Dnamic echnique. he development of the proportional and comple mode eigenvalue modification technique with the computationall efficient Local Eigenvalue echnique was the subject of man papers in the earl ears of IMAC. his was followed b the development of more realistic structural elements for component modification studies as well as sstem models from component modes. he development of tools to estimate rotational DOF was evident during this same period. After the first decade of IMAC, efforts tended to lean towards the utilization of SDM tools and development of sstem modeling tools. here was also a trend towards using frequenc response based tools as a mechanism to develop modifications to components and sstem models. While the measured FRF ma appear to have all the sstem characteristics (and not be affected b truncation), the effects of rotational DOF (and lack thereof) are important contributors to errors inherent in the results of these modeling techniques. Even toda, the two most important errors associated with SDM can often be identified as a lack of rotational DOF to describe the modal database of the sstem and the truncation of that database. his article is intended to summarize pertinent development of structural dnamic modeling techniques. Inherent problems that are encountered using the techniques are discussed with appropriate eamples to illustrate ke concerns and considerations when emploing SDM. he general theor of the modification process is presented followed b some of the practical limitations and restrictions that eist in utilizing the techniques. heor A brief theoretical review of the structural modification process is presented herein. Onl basic fundamental equations are presented more in-depth coverage can be found in the references. he basic equations of motion are presented first to identif the nomenclature used. Both the eigenvalue modification and local eigenvalue modification techniques for undamped sstems are presented; higher order structural elements are also discussed. his is followed b a general formulation of the equations of motion including arbitraril damped sstem using a state space formulation of the equations. All pertinent nomenclature is defined. Basic Equations of Motion. he equation of motion for a multiple degree of freedom sstem can be written in matri form as [ ]{ } [ ]{ } [ ]{ } { ( )} M + C + K = F t hese equations are of size (n n) depending on the size of the sstem matrices. Assuming that the damping matri is proportional to either the mass or stiffness matri, the eigen solution at the full space of the phsical model can be written as [ ] [ ] { } { } K λ M = he resulting eigenvalue and eigenvector are noted as ω, {u i }. he eigenvectors can be arranged in column fashion to form the modal matri [U]. picall, all of the eigenvectors are not used to describe the sstem. he modal matri is therefore rectangular and of size (n m). Using this notation and noting the eigenvalues can be assembled into a diagonal matri, the eigen problem can be restated as K U = M U ω (3) [ ][ ] [ ][ ] Using the modal matri, a transformation can be made from phsical space to modal space using the relationship { } = [ U ]{ p } Substituting this into the equation of motion and pre-multipling b the transpose of the projection operator to put the equations into normal form gives the standard modal space representation () () (4) SOUND AND VIBRAION/JANUARY 3 5

3 [ ][ ]{ } [ ] [ ][ ]{ } [ ] { ( ) } U M U p + U K U p = U F t When the mode shapes are scaled to unit modal mass, then this relationship reduces to [ I]{ p } + ω p = U n F t (6) { } { ( )} It is important to note that the diagonal modal mass matri is [ U] [ Μ][ U ] = [ Ι] (7) and the diagonal modal stiffness matri is U K U = ω [ ] [ ][ ] Epressing the modal space representation of the sstem in general terms of the modal mass and modal stiffness gives M p + K { } { p } = [ U] { F( t) } (9) (he bar overscore denotes a modal quantit.) It is ver important to note that all of the eigenvalues of the sstem are generall not available. his implies that the equation size is determined b the number of modes retained in the description of the sstem. herefore, these equations are of size (m m) and are generall much smaller than the phsical matrices describing the sstem. Provided that a sufficient number of modes is retained, not having all the modes does not pose a problem. However, this often is a problem. When all the modes of the sstem are not available, then the sstem is said to be truncated. Phsical Changes to Sstem Mass and Stiffness. An changes to the phsical sstem can be written in terms of the modified mass and stiffness as [ ]{ } [ ]{ } { ( )} M + K = F t Another eigensolution at the full space of the phsical model can be performed using While this eigensolution can be performed, the use of the modal space representation provides numerical efficiencies due to the reduced size of the modal matrices compared with the phsical matrices. he phsical matrices are tpicall man orders of magnitude larger those from modal space equations for most engineering applications of structural sstems. Structural Dnamic. An changes to the phsical sstem can be projected from phsical space to modal space through the modal vectors obtained from the original unmodified sstem. While this modal projection will never produce uncoupled equations, there is tremendous computation benefit due to the significant reduction in size of the modal space equations when compared to the phsical set of equations. picall once the structural changes are applied in modal space, these modal space equations are full populated and not diagonal. wo approaches are commonl used for the solution of these equations: the eigenvalue modification (EM) technique and the local eigenvalue modification procedure (LEMP). Eigenvalue echnique. Considering a change to the sstem matrices and using the modal space solution as a starting point for the modification, the modification equation can be written as M + M p + K { } + K { p } = [ ] (3) where (5) (8) () he modified mass and stiffness of the sstem is comprised of the original mass and stiffness plus the change in mass and stiffness as [ M ] = [ M ] + [ M ] and [ K ] = [ K ] + [ K ] () [ ] [ ] [ ] [ ] { } { } K + K -λ M + M = () DM = U M U and DK = U K U [ ] [ ][ ] [ ] [ ][ ] It is important to note that while part of this equation contains a diagonal representation of the modal mass and modal stiffness of the original sstem, the remaining change to the modal sstem is far from diagonal. Solving for the eigenvalues of this new sstem using K + [ U] [ K][ U] p = λ M + [ U] [ M][ U] will ield the new frequencies and another transformation given as { p } = [ U]{ p} he updated mode shapes can then be obtained from [ U ] = [ U][ U] { } { } (4) (5) (6) his procedure can be used to approimate the modified modal characteristics due to structural changes using onl the unmodified modal characteristics of the sstem. In general, the sstem will alwas produce higher frequencies than the eact answer. he more serious the truncation error, the higher the frequenc that will be produced from the modification approach. Figure shows the modification process schematicall. When transforming the final modified sstem to the original phsical coordinate sstem, the relationship of the initial modes to the final modes becomes ver apparent through Eq. 6, which provides ver revealing information upon epansion of the terms of this matri. he i th mode, for instance, can be written as () i ( ) u = u U ( i) ( ) i 3 + u U ( ) ( ) 3i { } { } { } + { u} U ( ) + (7) Eq. 7 implies that the final sstem models are obtained from linear combinations of the starting modes. If a mode that has a significant contribution is not included in [U ], then the final sstem mode in [U ] can be seriousl in error. his truncation effect is discussed later in more detail. Local Eigenvalue echnique. he eigenvalue modification technique requires an eigensolution to be performed for matrices that are the size of the number of modes in the modal space solution. he modification equations can be written in a more efficient form using a singular value decomposition of the change matrices in modal space. Without suppling those details here, another formulation of these equations provides tremendous computational advantages and is referred to as the local eigenvalue modification technique. For instance, a general change of the phsical sstem of stiffness can be spectrall decomposed as r K = α [ ] [ ] [ ] = { ti} αi{ ti} (8) i= he importance of this formulation lies in the fact that the singular value decomposition provides vectors that identif the linearl independent pieces that comprise the entire change matri. hese can be written in normal SVD format or as the summation of all the linearl independent pieces that form the change matri. he later formulation allows for the individual manipulation of each change to the sstem independentl of all other changes. Assuming that onl one tpe of change (i.e., mass, damping or stiffness) and onl one phsical change are to occur, then the equations can be re-written into an alternate form. he SVD format of the change to the sstem can be included in the eigenvalue modification technique. With some manipulation, the resulting equation can be used for the determination of the modified frequencies of the sstem for stiffness and mass changes, respectivel: 6 SOUND AND VIBRAION/JANUARY 3

4 { } { } u () i m tk = αk i= Ω ωi u () i m { } { tm} = Ωα m i= Ω ωi (9) () where the α coefficient results from the singular values of the decomposed change matri for the stiffness or mass and the {t} are the so called tie vectors that result from the decomposition for the stiffness or mass. he LEMP was a ver important step in the numerical implementation of the modification equations at a time when computational resources were etremel limited. One advantage of the LEMP is that rather than solving a m order set of polnomial equations, the problem was reduced to m second order polnomials that were much easier to solve. Another important feature of the procedure was that the eigenvalues of the modified sstem were alwas bounded b the unmodified eigenvalues of the original sstem. For an numerical procedure used to find the roots of the second order polnomial, the bounds of the root were clearl known, thereb minimizing computational effort. However, the drawback of the LEMP was that onl a single modification could be performed. After the modification was made, the sstem equations had to be updated and additional modifications had to be eplored; this tended to be a tedious process. As faster machines became available, the LEMP was eventuall replaced b the eigenvalue modification technique. he LEMP was originall implemented for simple structural elements and was later etended to include higher order elements such as beams, generalized beams and plates. At the time of the LEMP s development, computation capabilities were etremel limited. his procedure provided a viable approach for investigating structural changes to a sstem using onl modal data as the basis of prediction. oda, the LEMP approach is rarel used due to the significantl vast amount of computational resources available even on the slowest of machines. his technique is widel used in man commerciall available software packages. An etension of this method allows for the development of a sstem model from the component modes of substructures. he basic equations are etended in modal space to address two components, A and B, as A A M { p } + [ U] [ M][ U] B M B { p } A A K { p } + + [ U] [ K][ U] = {} B K B { p } () where the modal matrices of the two components, A and B, are written in stacked form as: A U [ U ] = B () U Figure. Schematic of the modification process. he phsical changes to the sstem are projected to modal space using the modal matri. he resulting set of equations are then solved to find updated frequencies and mode shapes. his modal modeling approach for SDM and Sstem Modeling can be found in man commerciall available software packages. -5 Comple Mode Solution. he equations presented above were developed in the absence of damping. In some structural dnamic modification approaches, a proportional model approimation is used for the development of the modification equations. At times, ver good modification predictions can be obtained especiall when considering onl mass and stiffness changes for a proportionall based set of equations. However, when the starting mode shapes are comple in nature or the modifications to be investigated include damping, then the state space solution procedure must be used to ensure accurate results. hese equations are presented below to show the basic formulation. In order to develop the proper set of equations, a supplementar equation is required to describe the equation of motion in state space form. his is presented as or where Phsical Space Modal ransformation Modal Space Original State [ M ], [ K ] Modified State [ M ], [ K ] [ M ], [ K ] {} = [U ]{p } {} = [U ]{p } [ω ], [U ] {p [Ω } = [U ]{p } ], [U ] [ ] [ ] [ ] [ ] M [ M] [ C] M = [ ] [ K] F [ ]{ } [ ]{ } { } B Y A Y = Q [ ] [ ] [ ] [ ] M B = A = M [ ] and [ ] [ M] [ C] [ ] [ K] 'N' Phsical DOF M<<N 'M' Modal DOF (3) (4) (5) (Note that some papers and other references use [A] and [B] notation that is interchanged from this notation.) An eigensolution can be performed for this set of equations [ ] [ ] A λ B { Y } = { } (6) his solution will ield comple frequencies and mode shapes. It is important to note that the matrices are now (n n) and that the mode shapes obtained are (n m) and are comple valued. he modal transformation can be written as: { Y } = [ Φ]{ p} (7) where the mode shapes are comple valued with both displacement and velocit components as () () [ Φ ] = Y Y () () { u } λ { u } λ = () () { u } { u } [ ] = U Λ [ U] (8) he transformation can be substituted into the equation of motion and premultiplied b the transform of the projection operator (to put the equations into normal form) to obtain [ ] [ ][ ]{ } [ ] [ ][ ]{ } [ ] { ( )} Φ B Φ p + Φ A Φ p = Φ Q t (9) SOUND AND VIBRAION/JANUARY 3 7

5 which is analogous to the proportional based solution without an damping included. (It is ver important to note that if the damping is zero or proportional to the mass and stiffness matrices, these equations will reduce to those previousl developed.) Now the equation of motion in modal space can be written as As before, structural changes can now be considered as where and must be written in state space form as where with M M [ B ] = and [ A ] = [ M] [ C] [ ] [ K ] (35) his modified equation in modal space can be written as the original unmodified modal quantities plus the change in modal quantities as where B p A { } { p } = Q( t) [ ]{ } [ ]{ } [ ]{ } { ( )} M + C + K = F t [ M ] = [ M ] + [ M] [ C ] = [ C ] + [ C] [ K ] = [ K ] + [ K] [ ]{ } [ ]{ } { ( )} B Y A Y = Q t [ A ] = [ A ] + [ A ]; [ B ] = [ B ] + [ B ] [ ] [ ] [ ] [ ] I + B p - { } Λ + A A = [ Φ ] [ A][ Φ ] B = [ Φ ] [ B][ Φ ] (3) (3) (3) (33) (34) { p } = { Q(t) } (36) (37) he eigensolution results in the modified sstem. In the equations above, an changes of mass, damping and stiffness can be handled. In addition, the LEMP can be easil included to add computational efficiences. 7 Again, the LEMP restrictions are that each individual modification needs to be made one at a time. In this set of equations using the full state space solution, the proper mathematical solution is obtained for arbitrar damping. If the damping is in fact proportional, then these equations reduce to the proportional mode approach. However, it is ver important to note that the structural modifications studied can in fact disrupt an proportionalit that ma eist in the original set of modes. If this is the case, then the full comple mode solution in state space needs to be utilized. Structural Using Response Functions. An interesting parallel method that uses frequenc response functions with an impedance approach is referred to as the Structural Using Response Functions (SMURF) technique. 7 It is presented here as an interesting alternate approach that affords some better understanding of the results obtained from the SDM approach and is included for completeness. For reference, recall that the frequenc responses are made up of the mode shapes of the sstem depending on the particular point to point frequenc response function considered using m * * qu k ikujk qu k ik ujk Hij( jω) = + (38) k ( j pk ) * = ω ( jω pk ) Consider the beam shown in Figure. For a cantilever beam, Figure. Cantilever beam eample. C B A the frequenc response at H cb when a = is desired; this corresponds to a pinned beam at the end of the cantilever. he response at a is related to the force at a and b through a = HabFb + HaaFa With the constraint a =, the force at a is Fa = Haa HabFb and the response at c due to an ecitation at b must include the effects of the reaction force such that With some simple substitutions using the equations above, one can see that the frequenc response function between point c and b with the tip of the cantilever beam restrained can be obtained from the frequenc response measurements of the unconstrained original sstem as c H cb = = Hcb HcaHaaHab Fb (39) (4) c = HcaFa + HcbFb (4) (4) In its simplest form, this is the basic equation for impedance modeling. his technique has been etended to handle multiple modifications simultaneousl and used as a sstem modeling technique. (his impedance modeling approach will be used in the eample section to illustrate truncation effects with the SDM approach.) Structural Dnamic Utilization Issues he theor was presented that is applicable to all structural modification techniques. here are man issues pertaining to the success of the modification process, with the major items discussed in this section. he major issues pertain to truncating modes that define the modal data base and the lack of rotational DOF. However, other issues of scaling mode shapes, rigid bod modes, higher frequenc modes, and comple vs. proportional modifications are also important considerations. hese are discussed in the following sections. runcation of Modes. he modes that are used to describe the modal data base will inevitabl suffer from modal truncation. his is a severe limitation that is encountered in an eperimental modal test. Onl the lower order modes of the sstem will be obtained from the test data. Even the finite element model will have a limited number of modes to describe the modal database. However, the finite element model can etract additional modes, if necessar, when performing structural dnamic modifications. Actuall, the higher frequenc modes are not the onl modes of concern. Man times an eperimental test is conducted to obtain the lower order fleible modes of the structure in a freefree configuration. All too often the rigid bod modes are not etracted from the test data. Man times this is due to the poor qualit of the data collected at such a low frequenc with the instrumentation used to address the fleible modes of the sstem. In addition, man test engineers do not necessaril associate the rigid bod modes as part of the description of the modal database, believing that onl the fleible modes are of concern. he rigid bod modes are a critical set of modes for structural dnamic modifications. Approimations have been attempted to estimate these necessar rigid bod modes from a variet of different measurement and analtical techniques. Rigid bod modes are discussed in a separate section. Efforts have also been epended to estimate residual com- 8 SOUND AND VIBRAION/JANUARY 3

6 pensation terms to account for the higher frequenc terms that have been ecluded from the modal database due to truncation. Static correction terms, residual shapes and other approimations have been used to assist in the modification process to improve results. Man of these generall improve the modification results but often are limited because most structural sstems require DOF that relate phsical rotational information in almost an real structural change that is investigated. Rotational effects are covered in the net section. Modal truncation is likel the most critical of all the modification errors that are encountered. Reference [6] contains several models to illustrate some of these truncation effects, while some of the more important models are presented in the Appendices. Appendi A presents a simple eample to illustrate the effects of truncation for a simple 6 DOF sstem. Appendi B presents a modification of a free-free beam to approimate a simple support beam and a cantilever beam using the SDM process to illustrate some additional truncation effects. A sstem model is presented in Appendi C to show how sstem modes ma be affected b truncation. Rotational DOF. he eperimental modal database is largel comprised of onl translational DOF. he measurements obtained are almost eclusivel translational DOF. he rotational measuring device has not found its place into affordable, economical, accurate use for eperimental modal testing. went ears ago this was a major obstacle in all structural dnamic modification and sstem modeling studies, and it still plagues the analsis process toda. At that time, man different approaches were investigated to estimate rotational DOF. Approaches that use spline fits of measured data, analtical representations of curvature, and analtical epansion methods were numerical approaches to estimate these necessar DOF. From the eperimental side of the problem, it is ver epensive to measure rotational DOF for most routine modal tests, which has led to man attempts to estimate rotational DOF. his began with closel-spaced accelerometers being added to the structure; later a housing was designed that incorporates two accelerometers to estimate rotational DOF. hese approaches provide some information but suffer from measurement inaccuracies such as cross ais sensitivit. Others have attempted to intentionall mass-load the test structure with calibrated mass adjustments to produce known effects, allowing for the ultimate estimation of rotational DOF. Further research and effort continues toda to attain an accurate, economical device for measuring rotational DOF. In order to illustrate the rotational DOF truncation effect, the cantilever modification of Appendi B is etended for rotational DOF truncation. he modal database contains different sets of translational DOF and rotational DOF to show the effects of truncation, shown in Appendi D. o further illustrate some additional points, the truncation of rotational DOF on the snthesis of FRFs is illustrated with discussion on the effect of modification of the cantilever using either the modal based approach or the impedance based approach. his model is discussed in Reference [9] and some of the pertinent results are shown in Appendi E. Mode Shape Scaling. he eperimental modal database must be obtained from accurate calibrated measurements. Mode shape scaling must be clearl identified and considered prior to performing an structural dnamic modification. Since the modification equations use the phsical properties of mass, damping and stiffness in the modal equations of motion, the equations must utilize consistent units. For most eperimental modal test results, the animation is simpl a displa of the amplified shape characteristics and does not necessaril have an phsical units. However, when the mode shapes are scaled in a particular fashion, such as unit modal mass, then there is a direct relationship between the mode shapes and phsical quantities. If improper calibration units are used for the collection of eperimental data, the mode shapes will be scaled with a bias error. his will have a direct impact on the modal change matri in modal space (the projection of the phsical change in the sstem using the mode shapes of the sstem). Obviousl, if the mode shapes are scaled incorrectl, then the projected change to the sstem in modal space will be directl affected. Drive Point Measurements. During the collection of modal data, the drive point frequenc response function is necessar in order to provide the proper calibration and scaling factors for the measured modes. he residues obtained from the modal parameter estimation process are sufficient for the definition of the sstem characteristics. However, the mode shapes form the frequenc response function from information related to both the input and response mode shapes for each of the modes of the sstems. Onl the drive point measurement contains information that allows the residue to be related to the mode shapes. his important point is the main reason wh operating data cannot be used to determine the effects of structural dnamic modifications. Operating data does not have an scale information relative to the mode shapes of the sstem. While operating data is etremel useful in studing structural dnamic sstems, this data contains no scale information. Drive point measurements are required in order to obtain accuratel scaled mode shapes. Rigid Bod Modes. For man structural modification studies involving either the constraint of a structure to ground or the connection of one substructure to another substructure, rigid bod modes are required. hese modes are necessar in order to define the inertial characteristics of the component. If these rigid bod modes are not included in the modal database for these tpes of structural changes, then significant errors will result. he rigid bod modes are a required portion of the adequate description of an free-free modal component. In man cases, the rigid bod modes comprise a necessar part of the modal description of the sstem and cannot be ignored. If these modes are not included, then for all practical purposes the modal data base is truncated (even though the rigid bod modes are the lowest modes of the database). Care must be eercised when incorporating rigid modes into the measured eperimental fleible modes. hese rigid bod modes must be scaled in a manner that is consistent with the scaling of the measured fleible modes. In addition, the rigid bod modes obtained from a finite element model will have a frequenc of Hz and the modes will be repeated. his is generall not true of the rigid bod modes obtained from testing. he tested rigid bod modes will not have a frequenc of Hz and will most likel not be repeated. his is an important point when performing structural dnamic modifications. First, the complete mode set used to describe the sstem (rigid bod modes and fleible modes) must be scaled in a consistent fashion. Second, man commerciall available implementations do not properl handle repeated roots and often do not handle modes that have a frequenc at Hz. Since most tested structures do not have Hz rigid bod modes, this is tpicall not a problem. However, caution needs to be eercised when augmenting a modal data base with analticall derived rigid bod modes or when implementing a structural dnamic modification procedure using a free-free component described b the finite element modes of the structure. Care must be taken to assure that the software can adequatel handle both Hz rigid bod modes as well as repeated roots. Comple vs. Proportional Modes. Man times, proportional modes are used for the approimation of structural dnamic sstems. picall, this is a good approimation for most structural dnamic studies. However, when dealing with the structural dnamic modification process, a comple mode solution is required to obtain accurate modified sstem characteristics. Reference [8] contains several models that illustrate some of the effects of using proportional mode approimations when performing modifications that trul require a comple mode solution scheme. he results of those studies can be summa- SOUND AND VIBRAION/JANUARY 3 9

7 rized with the following statements regarding proportional mode and comple mode solutions. When in doubt, a comple mode solution will alwas produce the proper mathematical solution. However, a proportional mode solution generall will produce a reasonabl good solution when the starting modes are indeed proportional and the changes involve onl mass and stiffness. hese tpes of changes on a proportional mode approimation generall do not cause the solution to become ver comple the modes tend to remain close to proportional. However, a proportional mode solution will often contain errors when damping modifications are investigated. As each individual modification is investigated, errors will accumulate because the proportional approimation ignores the comple phasing that results from the modified sstem. In this case, a comple mode solution scheme is necessar. When a proportional mode solution is used and the initial unmodified modes are not real normal modes or proportional modes, but actuall contain some phasing in the description of the modal database, then an changes of mass, damping or stiffness will alwas tend to accumulate error as modifications are performed. In this case, when the starting modes are not real normal modes, then one of two approaches are necessar. One approach is to emplo the full state space solution scheme and utilize the comple mode structural modification procedure. his will be necessar especiall if the unmodified modes are believed to be trul comple. However, if the modes are believed to be proportional but the measurements or modal parameter estimation process caused the modes to have phasing that is believed to not trul eist, then the mode shapes should be rescaled to obtain a real normal mode approimation of the shapes. his second approach is needed if the modes are actuall believed to be more appropriatel real normal modes. his will tend to minimize errors from the modification process. Other Issues. Of course, there are a variet of other issues that cause problems for the structural dnamic modification process. hese relate to simple and often obvious matters that are overlooked, such as geometr definition, accurate calibration, consistenc of units (weight vs. mass descriptions and feet vs. inches), lack of measured DOF for connection of structural modification elements, and a mriad of user blunders that cause errors in the modification procedure. Careful eecution of this approach will ield accurate results (ecluding the effects of truncation and rotational DOF that have been discussed). Summar Structural dnamic modification has been available for over two decades. he technique is useful for the stud of structural changes to a sstem to determine optimum design changes. Issues of truncation and lack of rotational DOF have caused difficulties with the technique since its inception. Some of these problems were discussed along with some of the other effects that need to be considered. References. Weissenberger, J.., he Effect of Local s on the Eigenvalues and Eigenvectors of Linear Sstems, D.Sc. Dissertation, Washington Universit, St Louis, MO, Pomazal, R. J., he Effect of Local s on the Eigenvalues and Eigenvectors of Damped Linear Sstems, Ph.D. Dissertation, Michigan echnological Universit, Houghton, MI, Hallquist, J. O., and Snthesis of Large Dnamic Structural Sstems, Ph.D. Dissertation, Michigan echnological Universit, Houghton, MI, Structural Measurement Sstems, An Introduction to the Structural Dnamics Sstem, echnical Note No., Formenti, D., Welaratna, S., Structural Dnamic An Etension to Modal Analsis, SAE Paper no. 843, Snder, V., Pal,., Structural and Comple Modes, 6th International Modal Analsis Conference, Orlando, FL, Februar O Callahan, J., Avitabile, P., A Structural Dnamic Procedure using Comple Modes, st International Modal Analsis Conference, Orlando, FL, November Avitabile, P., O Callahan, J., Milani, J., Errors Associated with Structural Dnamic using Assumed Proportional Modes, 6th International Modal Analsis Conference, Orlando, FL, Februar Johnson, C., Snder, V., Simultaneous vs. Sequential s, 7th International Modal Analsis Conference, Las Vegas, NV, Februar SDRC IDEAS Sstan Sstem Modeling Software, Structural Dnamic Research Corp., Milford, OH.. LMS Modal Design Software, Leuven Measurement Sstems, Leuven, Belgium.. MS/SDRC I-DEAS est Software, Mechanical esting & Simulation, Minneapolis, MN. 3. Vibrant echnolog ME scope Modal Analsis Software, Jamestown, CA. 4. SAR Modal Analsis Software, Spectral Dnamics, San Jose, CA. 5. FEMools Analsis Software, Dnamic Engineering, Leuven, Belgium. 6. O Callahan, J., Chou, C-M., Stud of Structural Procedure with hree Dimensional Beam Elements using a Local Eigenvalue Procedure, nd International Modal Analsis Conference, Orlando, FL, Februar Elliott, K., Mitchell, L. D., Realistic Structural s - Part heoretical Development, 3rd International Modal Analsis Conference, Orlando, FL, Februar Yasuda, C., Riehle, P., Brown, D. L., Allemang, R. J., An Estimation Method for Rotational Degrees of Freedom using a Mass Additive echnique, nd International Modal Analsis Conference, Orlando, FL, Februar Smile, R., Brinkman, B., Rotational Degrees of Freedom in Structural Dnamic, nd International Modal Analsis Conference, Orlando, FL, Februar Avitabile, P., O Callahan, J., Chou, C-M., Kalkunte, V., Epansion of Rotational Degrees of Freedom for Structural Dnamic, 5th International Modal Analsis Conference, London, England, April Lipkens, J, Vandeurzen, U., Smoothing of Mode Shapes and Application to Structural Analsis, 6th International Modal Analsis Conference, Orlando, FL, Februar Mitchell-Dignan, M., Paredon, G., he Estimation of Rotational Degrees of Freedom using Shape Functions, 6th International Modal Analsis Conference, Orlando, FL, Februar Wallack, P., Richardson, M., Comparison of Analtical and Eperiemntal Rib Stiffener s to a Structure, 7th International Modal Analsis Conference, Las Vegas, NV, Februar Wang, J., Helen, W., Sas, P., Accurac of Structural echniques, 5th International Modal Analsis Conference, London, England, April Elliott, K., Mitchell, L. D., he Effect of Modal runcation on Modal, 5th International Modal Analsis Conference, London, England, April Avitabile, P., O Callahan, J., Pechinsk, F., Understanding Structural Dnamic runcation, 8th International Modal Analsis Conference, Orlando, FL, Februar Otte, D., Van der Auweraer, H., Substructuring b Means of FRFs: Some Investigations on the Significance of Rotational DOF, 4th International Modal Analsis Conference, Detroit, MI, Februar Crowle, J., Klosterman, A., Rocklin, G., Vold, H., Direct Structural using Frequenc Response Functions, st International Modal Analsis Conference, Orlando, FL, November Avitabile, P., Piergentili, F., Consideration of the Effects of Rotational DOF for Hbrid Modeling Applications, 5th International Modal Analsis Conference, Orlando, FL, Februar Snder, V., Structural and Modal Analsis A Surve, International Journal of Analtical and Eperimental Modal Analsis, Vol., Januar 986. he author ma be contacted at: peter_avitabile@uml.edu. Appendi A: 6 DOF Eample A simple 6 DOF model is used to illustrate some ke points pertaining to truncation and the matrices involved in the SDM process. 6 he model is shown in Figure A. Note that the phsical set of equations involve matrices that have coupling of the various DOF of the model. he eigensolution uncouples those equations to create a set of linearl independent SDOF sstems, illustrated in Figure A. Notice that all the modes are not retained in the description of the sstem (as shown b several modes in the cross-hatched area). hese modes are not necessaril needed in order to obtain an acceptable solution regarding the sstem response. o illustrate some of the effects of the structural dnamic modification process, a simple structural change is shown in SOUND AND VIBRAION/JANUARY 3

8 Phsical Model Modal Model k m ω c k m k ω m k ω I p+ ( Ω + U K U) p = Modal Stiffness Change Matri c 7 c m k 7 c m 5 c 8 k 8 c m 6 c 9 k 9 k m k ω m 3 k ω 3 3 m 4 m 3 k ω 3 3 m 4 k ω 4 4 m 5 k ω 5 5 U KU = m k 4 ω 4 m 6 k ω 6 6 c 4 k 4 c 3 k 3 m 5 k ω 5 5 Figure A3. 6 DOF model with modal change matri. m 4 m 3 c 6 k 6 c 5 k 5 k 6 m 6 ω 6 k m ω k m ω Phsical Mass Matri Unmodified m k ω m 3 k ω 3 3 m 4 k ω 4 4 m k ω m 3 k ω 3 3 m 4 k ω 4 4 Modified Phsical Stiffness Matri Figure A. 6 DOF phsical and modal model. 5 K Phsical Change Matri K = Modified Stiffness Matri K = m 5 k ω 5 5 k 6 m 6 ω 6 m 5 k ω 5 5 Figure A4. 6 DOF model truncation for an unmodified and a modified sstem. stiffness is far from diagonal. he effect of the phsical modification projected in modal space is to couple ever one of the unmodified modal SDOF sstems to each other and to reconnect them all with additional stiffness back to ground. With an understanding of this effect, the modal truncation problem can be clearl seen. For the unmodified sstem, the truncation effect is related to the inclusion or removal of modes to describe the database. However, for the modified sstem, there is an effect of the projected change in modal stiffness due to phsical modifications. he effects of truncation can be conceptuall stated as follows: If the change in stiffness that crosses the imaginar boundar between the kept modes and the unkept modes is ver small, then the effects of truncation are minimal. But if the change in stiffness between the kept modes and unkept modes is ver large, then the effects of truncation are significant. Unfortunatel, the unkept modes are not available to ascertain their effects. k 6 m 6 ω 6 Figure A. 6 DOF model with phsical modification. Figure A. his change appears ver discrete in the matrices of the phsical model. However, when a structural modification is performed, the projection of the phsical change matri to modal space produces a ver significant amount of coupling to the original unmodified modes of the sstem, shown in Figure A3. While the original modal stiffness is clearl a diagonal representation in modal space, the change in modal Appendi B: Free-Free Beam for Simple Support and Cantilever Beam s A simple free-free beam is used to describe a modal database with 5 modes ( rigid bod and 3 fleible) for a structural dnamic modification involving onl two stiffness changes: one to simulate a simple support and one to simulate a cantilever. 6 he various beam configurations are shown in Figure B and the results of the structural modification (which involves onl two stiffness modifications) are shown in able B. he frequencies from the eact solution are shown for reference and SOUND AND VIBRAION/JANUARY 3

9 Free-Free Beam Mode Reference FEM Model.6 Modal 4.8 Simple Support Cantilever Figure B. Modal beam and proposed modifications Mode Frequenc Figure B4. Cantilever beam modal data U = U = Figure B. Free-free beam modal data. Figure B5. U projection matri for the simple support beam and cantilever beam modifications. Mode Reference FEM Model 7.9 Modal Frequenc 7. u u u Σ u 4. Figure B3. Simple support beam modal data. able B. Frequencies of different modifications. No Free Simple Support Ref. SDM Cantilever Ref. SDM Note: Frequencies in italics are approimations of constraint modes. u 5.79 () () U u = u Figure B6. Free-free mode combinations for simple support beam modification for mode. to assist in the evaluation of the accurac of the structural modification solution. he original frequencies and mode shapes for the free-free beam are shown in Figure B. he modified frequencies and mode shapes for the simple support modification stud are shown in Figure B3. he modified frequencies and mode shapes for the cantilever modification stud are shown in Figure B4. Before the lower frequencies are discussed, it is important to note that the last two frequencies of each set for the modified modes are ver high in frequenc. his is due to the fact that onl 5 modes are used for the modification of the modal sstem. In both cases, these two higher frequenc modes are approimations of the constraint modes of the sstem. hese are approimations of the modes necessar to constrain the sstem to ground. Discussion of these constraint modes are beond the scope of the information presented here but can be found as part of the description of the mode sets available for component mode snthesis. For the simple support modification, the results compare ver well to the eact solution. he results are ver acceptable, SOUND AND VIBRAION/JANUARY 3