Input¬output approximation of spacecraft motion is convenient and necessary in many situations. For a rigid¬body spacecraft, this process is simple because the system is governed by a set of equations that is linear in the system parameters. However, the combination of a flexible appendage and a rigid hub adds complexity by increasing the degrees of freedom and by introducing nonlinear coupling between the hub and appendage. Assumed Modes is one technique for modeling flexible body motion. Traditional Assumed Modes uses a set of linear assumed modes, but when dealing with rotating flexible systems, a modification of this method allows for the use of quadratic assumed modes. The quadratic assumed model provides an increased level of sophistication, but the derivation still neglects a set of higher¬order terms. This work develops the nonlinear equations of motion that arise from retaining these nonlinear, higher¬order terms. Simulation results reveal that the inclusion of these terms noticeably changes the motion of the system. Once the equations of motion have been developed, focus turns to the input¬output mapping of a system that is simulated using this set of equations. Approxi¬mating linear systems is straightforward, and many methods exist that can success¬fully perform this function. On the other hand, few approximation methods exist for nonlinear systems. Researchers at Texas A&M University recently developed one such method that obtains a linear estimation and then uses an adaptive polynomial estimation method to compensate for the disparity between that estimate and the true measurements. This research includes an in¬depth investigation of this nonlinear approximation technique. Finally, these two major research thrusts are combined, and input¬output approx¬imation is performed on the nonlinear rotational spacecraft model developed herein. The results of this simulation show that the nonlinear method holds a significant advantage over a traditional linear method in certain situations. Specifically, the nonlinear algorithm provides superior approximation for systems with nonzero natu¬ral frequencies. For the algorithm to be successful when rigid¬body modes are present, the system motion must be persistently exciting. This research is an important first step toward developing a nonlinear parameter identification algorithm.

Trial and error and the use of highly time-consuming methods are often necessary for investigation and characterization of nonlinear systems. However, for the rather common case where a nonlinear system has linear relations between many of its degrees of freedom there are opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient simulation and experimental methods for characterization of mechanical systems with localized nonlinearities. The purpose is to contribute to the development of analysis tools for such systems that are useful in early phases of the product innovation process for predicting product properties and functionality. Fundamental research is combined with industrial case studies related to metal cutting. Theoretical modeling, computer simulations and experimental testing are utilized in a coordinated approach to iteratively evaluate and improve the methods. The nonlinearities are modeled as external forces acting on the underlying linear system. In this way, much of the linear theories behind forced response simulations can be utilized. The linear parts of the system are described using digital filters and modal superposition, and the response of the system is recursively solved for together with the artificial external forces. The result is an efficient simulation method, which in conjunction with experimental tests, is used to validate the proposed characterization methods. A major part of the thesis addresses a frequency domain characterization method based on broad-band excitation. This method uses the measured responses to create artificial nonlinear inputs to the parameter estimation model. Conventional multiple-input/multiple-output techniques are then used to separate the linear system from the nonlinear parameters. A specific result is a generalization of this frequency domain method, which allows for characterization of continuous systems with an arbitrary number of localized zero-memory nonlinearities in a structured way. The efficiency and robustness of this method is demonstrated by both simulations and experimental tests. A time domain simulation and characterization method intended for use on systems with hysteresis damping is also developed and its efficiency is demonstrated by the case of a dry-friction damper. Furthermore, a method for improved harmonic excitation of nonlinear systems using numerically optimized input signals is developed. Inverse filtering is utilized to remove unwanted dynamic effects in cutting force measurements, which increases the frequency range of the force dynamometer and significantly improves the experimental results compared to traditional methods. The new methods form a basis for efficient analysis and increased understanding of mechanical systems with localized nonlinearities, which in turn provides possibilities for more efficient product development as well as for continued research on analysis methods for nonlinear mechanical structures.

Aerospace structures with large aspect ratio, such as airplane wings, rotorcraft blades, wind turbine blades, and jet engine fan and compressor blades, are particularly susceptible to aeroelastic phenomena. Finite element analysis provides an effective and generalized method to model these structures; however, it is computationally expensive. Fortunately, these structures have a length appreciably larger than the largest cross-sectional diameter. This characteristic is exploitable as these potential aeroelastically unstable structures can be modeled as cantilevered beams, drastically reducing computational time. In this thesis, the nonlinear equations of motion are derived for an inextensional, non-uniform cantilevered beam with a straight elastic axis. Along the elastic axis, the cross-sectional center of mass can be o set in both dimensions, and the principal bending and centroidal axes can each be rotated uniquely. The Galerkin method is used, permitting arbitrary and abrupt variations along the length that require no knowledge of the spatial derivatives of the beam properties. Additionally, these equations consistently retain all third-order nonlinearities that account for flexural-flexural-torsional coupling and extend the validity of the equations for large deformations. Furthermore, linearly independent shape functions are substituted into these equations, providing an efficient method to determine the natural frequencies and mode shapes of the beam and to solve for time-varying deformation. This method is validated using finite element analysis and is extended to swept wings. The importance of retaining cubic terms, in addition to quadratic terms, for nonlinear analysis is demonstrated for several examples. Ultimately, these equations are coupled with a fluid dynamics solver to provide a structurally efficient aeroelastic program.

With an ongoing desire to improve product performance, in combination with the continuously growing complexity of engineering structures, there is a need for well-tested and reliable engineering tools that can aid the decision making and facilitate an efficient and effective product development. The technical assessment of the dynamic characteristics of mechanical systems often relies on linear analysis techniques which are well developed and generally accepted. However, sometimes the errors due to linearization are too large to be acceptable, making it necessary to take nonlinear effects into account. Many existing analysis techniques for nonlinear mechanical systems build on the assumption that the input excitation of the system is periodic and deterministic. This often results in highly inefficient analysis procedures when nonlinear mechanical systems are studied in a non-deterministic environment where the excitation of the system is stochastic. The aim of this thesis is to develop and validate new efficient analysis methods for the theoretical and experimental study of nonlinear mechanical systems under stochastic excitation, with emphasis on two specific problem areas; forced response simulation and system identification from measurement data. A fundamental concept in the presented methodology is to model the nonlinearities as external forces acting on an underlying linear system, and thereby making it possible to use much of the linear theories for simulation and identification. The developed simulation methods utilize a digital filter to achieve a stable and condensed representation of the linear subparts of the system which is then solved recursively at each time step together with the counteracting nonlinear forces. The result is computationally efficient simulation routines, which are particularly suitable for performance predictions when the input excitation consist of long segments of discrete data representing a realization of the stochastic excitation of the system. Similarly, the presented identification methods take advantage of linear Multiple-Input-Multiple-Output theories for random data by using the measured responses to create artificial inputs which can separate the linear system from the nonlinear parameters. The developed methods have been tested with extensive numerical simulations and with experimental test rigs with promising results. Furthermore, an industrial case study of a wave energy converter, with nonlinear characteristics, has been carried out and an analysis procedure capable of evaluating the performance of the system in non-deterministic ocean waves is presented.

Seismically vulnerable buildings constitute a major problem for the safety of human beings. In many parts of the world, reinforced concrete (RC) frame buildings designed and constructed with substandard detailing, no consideration of capacity design principles, and improper or no inclusion of the seismic actions, have been identified. Amongst those vulnerable building, one particular typology representative of the construction practice of the years previous to the 1970’s, that most likely represents the worst case scenario, has been widely investigated in the past. The deficiencies of that building typology are related to non-ductile detailing in beam column joints such as the use of plain round bars, the lack of stirrups inside the joint around the longitudinal reinforcement of the column, the use of 180° end hooks in the beams, the use of lap splices in potential ‘plastic hinge’ regions, and substandard quality of the materials. That type of detailing and the lack of a capacity design philosophy create a very fragile fuse in the structure where brittle inelastic behaviour is expected to occur, which is the panel zone region of exterior beam column joints. The non-ductile typology described above was extensively investigated at the University of Canterbury in the context of the project ‘Retrofit Solutions for New Zealand Multi-Storey Buildings’ (2004-2011), founded by the ‘Foundation for Research, Science and Technology’ Tūāpapa Rangahau Pūtaiao. The experimental campaign prior to the research carried out by the author consisted of quasi-static tests of beam column joint subassemblies subjected to lateral loading regime, with constant and varying axial load in the column. Most of those specimens were representative of a plane 2D frame (knee joint), while others represented a portion of a space 3D frame (corner joints), and only few of them had a floor slab, transverse beams, and lap splices. Using those experiments, several feasible, cost-effective, and non-invasive retrofit techniques were developed, improved, and refined. Nevertheless, the slow motion nature of those experiments did not take into account the dynamical component inherent to earthquake related problems. Amongst the set of techniques investigated, the use of FRP layers for strengthening beam column joints is of particular interest due to its versatility and the momentum that its use has gained in the current state of the practice. That particular retrofit technique was previously used to develop a strengthening scheme suitable for plane 2D and space 3D corner beam column joints, but lacking of floor slabs. In addition, a similar scheme was not developed for exterior joints of internal frames, referred here as ‘cruciform’. In this research a 2/5 scale RC frame model building comprising of two frames in parallel (external and internal) joined together by means of floor slabs and transverse beams, with non-ductile characteristics identical to those of the specimens investigated previously by others, and also including lap splices, was developed. In order to investigate the dynamic response of that building, a series of shake table tests with different ground motions were performed. After the first series of tests, the specimen was modified by connecting the spliced reinforcement in the columns in order to capture a different failure mode. Ground motions recorded during seismic events that occurred during the initial period of the experimental campaign (2010) were used in the subsequent experiments. The hierarchy of strengths and sequence of events in the panel zone region were evaluated in an extended version of the bending moment-axial load (M-N) performance domain developed by others. That extension was required due to the asymmetry in the beam cross section introduced by the floor slab. In addition, the effect of the torsion resistance provided by the spandrel (transverse beam) was included. In order to upgrade the brittle and unstable performance of the as-built/repaired specimen, a practical and suitable ad-hoc FRP retrofit intervention was developed, following a partial retrofit strategy that aimed to strengthen exterior beam column joints only (corner and cruciform). The ability of the new FRP scheme to revert the sequence of events in the panel zone region was evaluated using the extended version of the M-N performance domain as well as the guidelines for strengthening plane joints developed by others. Weakening of the floor slab in a novel configuration was also incorporated with the purpose of reducing the flexural capacity of the beam under negative bending moment (slab in tension), enabling the damage relocation from the joint into the beam. The efficacy of the developed retrofit intervention in upgrading the seismic performance of the as-built specimen was investigated using shake table tests with the input motions used in the experiments of the as-built/repaired specimen. Numerical work aimed to predict the response of the model building during the most relevant shake table tests was carried out. By using a simple numerical model with concentrated plasticity elements constructed in Ruaumoko2D, the results of blind and post-experimental predictions of the response of the specimen were addressed. Differences in the predicted response of the building using the nominal and the actual recorded motions of the shake table were investigated. The dependence of the accuracy of the numerical predictions on the assumed values of the parameters that control the hysteresis rules of key structural members was reviewed. During the execution of the experimental campaign part of this thesis, two major earthquakes affected the central part of Chile (27 of February 2010 Maule earthquake) and the Canterbury region in New Zealand (22 February 2011 Canterbury earthquake), respectively. As the author had the opportunity to experience those events and investigate their consequences in structures, the observations related to non-ductile detailing and drawbacks in the state of the practice related to reinforced concrete walls was also addressed in this research, resulting in preliminary recommendations for the refinement of current seismic code provisions and assessment guidelines. The investigations of the ground motions recorded during those and other earthquakes were used to review the procedures related to the input motions used for nonlinear dynamic analysis of buildings as required by most of the current code provisions. Inelastic displacement spectra were constructed using ground motions recorded during the earthquakes mentioned above, in order to investigate the adequacy of modification factors used to obtain reduced design spectra from elastic counterparts. Finally a simplified assessment procedure for RC walls that incorporates capacity compatible spectral demands is proposed.

Theoretical and experimental modal analysis, i.e., the computation of vibration modes from a mathematical model and from experimental data, respectively, is quite sophisticated and advanced in linear structural dynamics. However, nonlinearity is a frequent occurrence in real-world engineering structures, and the existing linear methodologies fail dramatically in the presence of nonlinear dynamical phenomena. Therefore, the present thesis focuses on the development of a practical nonlinear analog of modal analysis for properly accounting for nonlinearity in mechanical systems. The concept of nonlinear normal mode (NNM) provides solid mathematical and theoretical foundations for a rigorous, yet understandable by the practicing engineer, analysis of nonlinear dynamical behaviors. In this context, a useful framework for nonlinear modal analysis of vibrating structures, which includes the computation of NNMs from finite element models and their identification from experimental data, is proposed in this dissertation. In view of the still limited use of NNMs in structural dynamics, special attention is devoted to progress toward a practical tool that has the potential to deal with large-scale, real-world structures. Targeting an effective and exact computation of NNMs, even in strongly nonlinear regimes of motion, one original contribution of this work is to resort to numerical methods. An algorithm combining a shooting procedure and the so-called pseudo-arclength continuation method is developed. On the other hand, a nonlinear extension of phase resonance testing (also known as force appropriation) is introduced for the experimental identification of NNMs, which is another innovative aspect of the doctoral thesis. In particular, the phase lag quadrature criterion, which is used for linear experimental modal analysis, is generalized in the presence of nonlinear dynamical behavior. Academic examples are first considered to illustrate, in a simple manner, that the proposed methods form an effective and adequate framework for nonlinear modal analysis. Furthermore, more realistic structures, including a full-scale aircraft, are studied to demonstrate the potential applicability of the approach to large-scale, real-life applications.

Most modern product development utilizes computational models. With increasing demands on reducing the product development lead-time, it becomes more important to improve the accuracy and efficiency of simulations. In addition, to improve product performance, a lot of products are designed to be lighter and more flexible, thus more prone to nonlinear behaviour. Linear finite element (FE) models, which still form the basis of numerical models used to represent mechanical structures, may not be able to predict structural behaviour with necessary accuracy when nonlinear effects are significant. Nonlinearities are often localized to joints or boundary conditions. Including nonlinear behaviour in FE-models introduces more sources of uncertainty and it is often necessary to calibrate the models with the use of experimental data. This research work presents a model calibration method that is suitable for mechanical systems with structural nonlinearities. The methodology concerns pre-test planning, parameterization, simulation methods, vibrational testing and optimization. The selection of parameters for the calibration requires physical insights together with analyses of the structure; the latter can be achieved by use of simulations. Traditional simulation methods may be computationally expensive when dealing with nonlinear systems; therefore an efficient fixed-step state-space based simulation method was developed. To gain knowledge of the accuracy of different simulation methods, the bias errors for the proposed method as well as other widespread simulation methods were studied and compared. The proposed method performs well in comparison to other simulation methods. To obtain precise estimates of the parameters, the test data should be informative of the parameters chosen and the parameters should be identifiable. Test data informativeness and parameter identifiability are coupled and they can be assessed by the Fisher information matrix (FIM). To optimize the informativeness of test data, a FIM based pre-test planning method was developed and a multi-sinusoidal excitation was designed. The steady-state responses at the side harmonics were shown to contain valuable information for model calibration of FE-models representing mechanical systems with structural nonlinearities. In this work, model calibration was made by minimizing the difference between predicted and measured multi-harmonic frequency response functions using an efficient optimization routine. The steady-state responses were calculated using the extended multi-harmonic balance method. When the parameters were calibrated, a k-fold cross validation was used to obtain parameter uncertainty. The proposed model calibration method was validated using two test-rigs, one with a geometrical nonlinearity and one with a clearance type of nonlinearity. To attain high quality data efficiently, the amplitude of the forcing harmonics was controlled at each frequency step by an off-line force feedback algorithm. The applied force was then measured and used in the numerical simulations of the responses. It was shown in the validation results that the predictions from the calibrated models agree well with the experimental results. In summary, the presented methodology concerns both theoretical and experimental aspects as it includes methods for pre-test planning, simulations, testing, calibration and validation. As such, this research work offers a complete framework and contributes to more effective and efficient analyses on mechanical systems with structural nonlinearities.

Most engineering structures include nonlinearity to some degree. Depending on the dynamic conditions and level of external forcing, sometimes a linear structure assumption may be justified. However, design requirements of sophisticated structures such as satellites, stabilized weapon systems and radars may require nonlinear behavior to be considered for better performance. Therefore, it is very important to successfully detect, localize and parametrically identify nonlinearity in such cases. In engineering applications, the location of nonlinearity and its type may not be always known in advance. Furthermore, as the structure will be excited from only a few coordinates, the frequency response function matrices will not be complete. In order to parametrically identify more than one type of nonlinearity which may co-exist at the same location with the above mentioned limitations, a method is proposed where restoring force surface plots are used which are evaluated by describing function inversion. Then, by reformulating this method, a second method is proposed which can directly evaluate the total describing function of more than one type of nonlinearity which may co-exist at the same location without using any linear frequency response function matrix. It is also aimed in this study to use the nonlinearity localization formulations for damage localization purposes. The validation of the methods developed in this study is demonstrated with case studies based on simulated experiments, as well as real experiments with nonlinear structures and it is concluded that the methods are very promising to be used in engineering structures.