TRANSIENT RESPONSE OF MECHANICAL STRUCTURES USING

Page5.15Time-histories from a linear system.1505.16Time-histories from systems with cubic stiffness.1525.17 Time-histories from a system with backlash.1545.18 Time-histories from a system with bi-linear stiffness.1565.19 Time-histories from a system with friction.1585.20Time-histories from a system with quadratic viscous damping.5.21Trends in the frequency domain from systems with cubicstiffness subjected to impulse excitation.I5.22162Trends in the frequency domain from the first harmonic163resonance of fig (5.21 a).5.23160Trends in the frequency domain from systems with backlash164subjected to impulse excitation.5.24 Trends in the frequency domain from the first harmonic165resonance of fig (5.23a).5.25Bode plot from a system with bi-linear stiffness (ratio 4:l)166subjected to an impulse.5.26Nyquist, 3-D damping and reciprocal-of-receptance plots of the167resonances of fig (5.25).5.27 Trends in the frequency domain from systems with friction172subjected to impulse excitation.5.28 Trends in reciprocal-of-receptance plots from systems withfriction subjected to impulse excitation.1735.29 Trends in the frequency domain from systems with quadraticviscous damping subjected to impulse excitation.5.30Comparison of time-histories from a linear system using datafrom an impulse test and transformed sine test data.5.31174175Comparison of time-histories from a system with cubicstiffness using data from an impulse test and transformed sine176test data.5.32Comparison of time-histories from a system with backlashusing data from an impulse test and transformed sine test data.12177

Page5.33Comparison of time-histories from a system with bi-linearstiffness using data from an impulse test and transformed sine178test data.5.34Comparison of time-histories from a system with frictionusing data from an impulse test and transformed sine test data.5.35179Comparison of time-histories from a system with quadraticviscous damping using data from an impulse test and180transformed sine test data.6.1Solution routes for impulse response functions (IRFs) of198non-linear systems.6.2Exact and predicted transient response of a system with cubic199stiffness; v(O) 1 .O m/s.6.3Exact and predicted transient response of a system with cubic200stiffness.6.4Exact and predicted transient response of a system withbacklash; gap is approx. 1% of max. disp.6.5Exact and predicted transient response of a system withbacklash; gap is approx. 10% of max. disp.6.6204Exact and predicted transient response of a system withfriction; using a linear model; v(O) 1 .O m/s.6.9206Exact and predicted transient response of a system with207friction: using a non-linear model.6.11Exact and predicted transient response of a system with208quadratic viscous damping.6.12205Exact and predicted transient response of a system withfriction; using a linear model.6.10203Examples of the IRF of a system with bi-linear stiffness;Spring ratio 1 :16.6.8202Exact and predicted transient response of a system withbacklash: gap is approx. 5% of max. disp.6.7201Exact and predicted transient response of a system withquadratic viscous damping; using a linear model.13209

Page6.13 Solution routes for transient response predictions of variousnon-linear systems.2107.1FRFs from a linear system.2327.2FRFs from a system with cubic stiffness subjected to aconstant-force sine test (force amplitude 75).7.3FRFs from a system with backlash subjected to a constant-forcesine test (force amplitude 1).7.4I234FRFs from a system with bi-linear stiffness subjected to aconstant-force sine test.7.5235FRFs from a system with friction subjected to a constant-forcesine test (force amplitude 0.1).7.6236FRFs from a system with quadratic viscous damping subjectedto a constant-force sine test (force amplitude 2).7.7238IRFs from a system with cubic stiffness and a linear system;v(O) 5.0 m/s.7.9239IRFs from a system with cubic stiffness and a linear system;v(O) 1 0.0 m/s.7.10240IRFs from a system with backlash and a linear system:v(O) 1 .o m/s.7.11241IRFs from a system with backlash and a linear system;v(O) 50 m/s.7.12242IRFs from a system with backlash and a linear system;v(O) O.l m/s.7.13237IRFs from a system with cubic stiffness and a linear system;v(O) 1 .o m/s.7.8233243IRFs from a system with backlash and a linear system;v(O) O.O5 m/s.2447.14IRF from a system with bi-linear stiffness.2457.15Predicted IRF for a system with bi-linear stiffness using alinear model with the softer of the two springs.7.16245Predicted IRF for a system with bi-linear stiffness using alinear model with the stiffer of the two springs.14246

Page7.17Predicted IRF for a system with bi-linear stiffness using alinear model with the average of the two springs.7.18IRFs from a system with friction and a linear system;v(O) 05 m/s.7.19247IRFs from a system with friction and a linear system;v(O) O.i m/s.7.20248IRFs from a system with friction and a linear system;v(O) O.O5 m/s.7.21249IRFs from a system with quadratic viscous damping and a linearsystem; v(O) 1 .O m/s.7.22246250IRFs from a system with quadratic viscous damping and a linearsystem; v(O) 50 m/s.25115

ACKNOWLEDGEMENTSThe work in this thesis was carried out with the support of the Ministry ofDefence, who entirely funded this research project.The author wishes to thank her supervisors, Professor D. J. Ewins (ImperialCollege) and Dr. D. A. C. Parkes (Admiralty Research Establishment, Portland),for their continued help and enthusiasm throughout the duration of this project.Thanks are also due to colleagues in the Imperial College Dynamics Group, bothpast and present, and at the Admiralty Research Establishment, Portland,particularly Dr. R. M. Thompson and Mr Ft. W. Windell, for their advice andencouragement over the years. The author also thanks Mrs P. J. Thompson forher help in typing this thesis.16

. 1 INTRODUCTION1.1 Background and overviewShock loading of a submarine may cause damage to important on-board systems.This damage ranges in severity depending on the level of shock but in extremecases may render the submarine unserviceable, or unable to return safely tobase. Shock protection of critical on-board systems is therefore essential, andit is necessary to be able to predict the transient response of these systems tothe expected shock loading in order to determine the shock protectionrequirements.One obvious approach to this requirement is shock testing of the systems ofinterest to a standard specified by relevant interested parties.There areseveral problems in the shock testing of structures, including the expense andavailability of full-size models and the logistics of running such a test. Usingscale models is an alternative, but there are still problems in scaling therequired parameters and in extrapolating the results to the required shockstandard on a full-size system. Also, it is usual to have only a few of the fullsize components to test, with scale models of some of the other systems andanalytical models of the remaining components, and the subsequentmathematical models all require combining in some form to provide a completedescription of the total structure.Several techniques exist for the transient response prediction of structures,but all the methods require a mathematical description of the structure. Insituations where mathematical models do not exist there is a requirement toconstruct such a model from experimental tests. Experimental modal analysismethods are an alternative route for evaluating mathematical models.Experimental modal analysis covers the testing of. the structure - eg bystepped-sine excitation, random excitation or impact testing - and thesubsequent analysis of the data - in either the frequency or the time domain - toobtain a frequency response model of a structure.As some of thesub-structures of a submarine are too complex to be modelled theoretically, andmodels for these will need to be obtained from experimental data, the use of17

transient response prediction methods with experimental data is an importanttopic.All systems are to some degree non-linear; therefore, the use of transientresponse prediction methods for non-linear structures with mathematicalmodels derived by modal analysis techniques is an important area forinvestigation as it is reasonable to assume that most of the on-board systems ofa submarine are non-linear.1.2 ObjectivesAs experimental modal analysis methods can be used to determine mathematicalmodels of structures far more easily, and at less expense, than by calculating amodel from shock tests, a primary objective of this research is to explore thepotential for using experimental modal analysis data to evaluate suitableparameters for the prediction of the response of the structure to a shock input.An essential component of this study is to determine the limitations andrestrictions placed on the experimental analysis techniques and the transientresponse prediction methods used, and the subsequent accuracy of the results.In particular, the special problems in predicting the transient response ofstructures with non-linear components using models derived fromexperimental modal analysis are to be addressed as these conditions representthe closest to real-life applications.1.3 Contents of the thesisInitially, methods available for transient response prediction are reviewed. Allthe techniques, whether working in the time or in the frequency domains,require a mathematical model of the structure.The Fourier transformapproach uses a frequency response model of the structure, and as such isperhaps the most appropriate method for use with experimental data.Frequency response models are often determined using modal analysis methods,and experimental modal analysis techniques are reviewed in chapter 3,including both the testing and the data analysis procedures. Also included in that18.I.,.,‘Yb,.,.

chapter is a section on the process of coupling together frequency responsemeasurements on separate components in order to evaluate the responsecharacteristics of the complete structure formed by their assembly.The Fourier transform method is considered in detail in chapter 4, with a briefreview of Fourier theory followed by an examination of the requirements of thefrequency response data for accurate transient response predictions. Theserequirements have implications for the experimental and analytical proceduresused to obtain the frequency response data as it is found that errors in the modalparameters can result in large deviations of the predicted transient responsefrom the true response.In terms of the data used in the transform, the usualcriterion of a sample rate of twice the maximum frequency content avoidfrequency aliasing is mentioned, and also a maximum frequency spacing for anygiven system to avoid time aliasing when transforming from the frequency tothe time domain has been developed. Both forms of aliasing are considered. Thistechnique for transient response prediction is suitable for linear systemsprovided that certain constraints are observed in the quality of data, and in thedigitisation of the frequency response for the transform.Attention is then turned to the problems associated with non-linear structures.In chapter 5 the application of modal analysis techniques to non-linearstructures is examined in terms of the ability of the techniques to identify andquantify a non-linearity.Of particular interest is the reciprocal-of-receptance method which is found to be a powerful tool in distinguishingbetween damping and stiffness type non-linearities, and the method is easilyadapted for some non-linearities to enable specific non-linear parameters to beevaluated.Examination of the results in the frequency response data from transientexcitation again shows distinct trends for each non-linearity considered, butthese are not generally the same as the trends seen in frequency response datafor the same non-linearity but measured via a constant-force sine test. As theresults of the frequency response data vary considerably, the impulse responsefunctions from the two excitation techniques are compared, and again there isfound to be very little agreement.19

Predictions for the impulse response functions of non-linear elements areexamined in chapter 6. Two approaches are considered; using the frequencyresponse functions from modal analysis, or developing a non-linear model fromexperimental data specifically for use in transient response predictions. Thelinear approach is suitable for several applications, but generally using aspecific set of modal parameters (several different sets of modal parameterscan be evaluated from a single measurement of a non-linear structure). Themodal parameters recommended so as to ensure acceptable results are derivedfor various non-linearities. The limitations on the validity and accuracy of anyprediction is examined and it is found that for some non-linearities the linearapproach generates good predictions, whilst for others a non-linear model mustbe developed if accurate predictions are required. However, non-linear modelscan be very complex: a separate model needs to be developed for each type ofnon-linearity, and the resulting non-linear model will only be valid fortransient response predictions and would generate large errors if used forsteady-state response predictions. Therefore, non-linear models should only beused where necessary, and not as standard practice.In chapter 7 the extension of these techniques for applicationtomulti-degree-of-freedom systems is considered. The trends for identifying thepresence and type of non-linearity remain unchanged from the correspondingsingle-degree-of-freedom element.The transient response predictions usingthe linear parameters evaluated as recommended from thesingle-degree-of-freedom elements are also examined.Finally, the results of the work are summarised in the form ofrecommendations of how to use experimental modal analysis in transientresponse predictions, for both linear and non-linear structures.For linearstructures, future developments in modal analysis techniques may enableparameters to be evaluated that are a closer approximation to those of the realstructure and hence result in a better transient response prediction. However,further research is envisaged into many aspects of the transient responseprediction of non-linear structures using experimental data.20

2 A REVIEW OF TRANSIENT RESPONSE ANALYSISMETHODS2.0IntroductionThe requirements for a transient structural response analysis vary from oneapplication to another. For some analyses a single-degree-of-freedom (SDOF)model is sufficient,whilst inother cases amore detailedmulti-degree-of-freedom (MDOF) model is necessary. Further, the need for anapproach to non-linearities in the system ranges from ignoring them altogetherto attempting to model the non-linearity in detail. To satisfy this range ofrequirements, many different transient response analysis methods have beendeveloped, working in both the time and the frequency domains. Thepresentation of results also differs with the analysis and application fromshowing only the maximum response to displaying the full time history. Theaim of this chapter is to review the methods available for transient responseanalysis of theoretical and real structures.In considering the variousalternatives, the primary concern is the applicability of each method for theprediction of transient responses of complex structures using experimental dataas a means of describing their properties.The first part of the chapter is concerned with time domain methods, includingtime-marching solutions, Duhamel’s integral method and graphical techniques.Frequency domain methods are then examined. Shock spectra are used whenonly the , maximum values of response are of interest, and the Fourier transformapproach is examined in detail for use with frequency response data.Many engineering structures that are subjected to transient loading are oftentoo complex to be theoretically modelled in full. For these structures, some orall of the components may need to be described by experimental data. In thesesituations the experimental description of the structure is usually in thefrequency domain as frequency response functions or modal data, in which casethe most appropriate technique for transient response analysis is to use aFourier transform-based approach.21

2.1 Time domain methodsThis section reviews the techniques available for transient response analysis inthe time domain. The first group of methods considered solve the equations ofmotion of the structure numerically.These types of analysis are suitable forSDOF or MDOF components, and can also be adapted for non-linear systems.Duhamel’s integral method is considered next: this is based on the principle ofconvolution and is’ primarily applicable to linear models whose impulseresponse functions can be expressed analytically. Finally in this section,graphical approaches are mentioned: these analyses are performed on SDOFelements or individual modes of vibration of a structure and, when thenon-linearity in a system is known, graphical analysis can easily be adapted toinclude the effect.All of the methods in this section require a knowledge of the properties of thesystem. This makes the time domain techniques better suited to the transientresponse analysis of structures that can easily be modelled theoretically thanthose that need to be described by experimental data. Duhamel’s integral andgraphical approaches are developed for SDOF systems: these techniques can beapplied to MDOF systems but the accuracy of the solution is less than for theSDOF example.Numerical solutions are attractive for real structures as thesolutions are for MDOF systems and several of the methods can be adapted fornon-linear components - the problem being that in our case not all thestructures are analytic and therefore the form of any non-linearity is notknown.This means that the equations of motion can not be formulatedtheoretically.2.1.1 Numerical methodsMany differen

Summary of essential modal theory. Experimental testing for frequency response of structures. Extracting modal properties from experimental frequency response data. 3.3.1 Single-degree-of-freedom(SDOF) analysis. 3.3.2 Multi-degree-of-freedom(MDOF) analysis. 3.3.3 Time domain analysis. Application of frequency response data and modal parameters.