The rolling of an aircraft about its fuselage produces centrifugal forces which affect the stiffness of the wings. A number of previous studies explain the effect of centrifugal stiffening in rotating beams and consequently on the frequencies of the beam. Multiple cases of the rotating beam are explored in this thesis to investigate effects of mass distribution and boundary conditions on the frequencies of centrifugally stiffened beams. It is found that for a uniform beam with all degrees of freedom free on both ends, the rigid modes of the beam are affected and are no longer zero when it is stiffened from centrifugal forces. This thesis aims to set up a model to investigate the stiffening effects using the mAEWing2 aircraft. A preliminary analysis is done for the mAEWing2 aircraft and the roll rate, control surface deflection and angle of attack are identified as the parameters to be studied. For a given angle of attack and control surface deflection, the centrifugal forces in the aircraft in steady roll are determined using trim analysis. These are used to pre-stress the model for modal analysis. It is found that in mAEWing2 aircraft in steady roll maneuvers, the centrifugal stiffening effect on the natural frequencies is not significant. It emphasizes the need to conduct a sensitivity analysis to include centrifugal stiffening in the dynamic analysis while designing an aircraft. This, along with some de-stiffening due to gravity loads might be important for the future N+3 aircraft with their high aspect ratio large wingspans. / MS / Structural analysis is mainly concerned with determining the behavior of a structure when subjected to a disturbance. The natural response of a structure to some disturbance is termed as free vibration of the structure. The term vibration describes repetitive motion that can be observed in a structure and is influenced by its material and structural properties. These vibrations may cause fatigue in the structure and the performance of the structure may be adversely affected. Consequently it becomes necessary to study and eliminate these vibrations. The vibration characteristics of a system are described by its natural frequencies and mode shapes. Natural frequencies of a structure are the frequencies at which the structure naturally tends to vibrate if it is subjected to a disturbance. The deformed shape of the structure vibrating at one of its specific natural frequencies of vibration is termed its normal mode shape. In the case of a rotating beam, the centrifugal force acts axially along the length of the beam. When the rotating beam deflects upwards, the centrifugal force creates a downward bending moment, reducing its net deflection. The ratio of force to displacement increases, increasing the stiffness of the rotating beam. This effect is called the stiffening effect. There is a large volume of literature that presents the effect of stiffening on the natural frequencies of a rotating beam model, for various boundary conditions. Such a stiffening analysis has also been done for the blades of a turbine and turbo fans. In addition, there are models available for analyzing the aerodynamic model of an aircraft in roll, considering stability derivatives of the aircraft. However, there are gaps in the available literature in analyzing an aircraft in roll from the perspective of structural analysis. The rolling of an aircraft about its fuselage produces centrifugal forces which affect the stiffness of the wings. A number of previous studies explain the effect of centrifugal stiffening in rotating beams and consequently on the frequencies of the beam. Multiple cases of the rotating beam are explored in this thesis to investigate effects of mass distribution and boundary conditions on the frequencies of centrifugally stiffened beams. It is found that for a uniform beam with all degrees of freedom free on both ends, the rigid modes of the beam are affected and are no longer zero when it is stiffened from centrifugal forces. This further motivates the need for investigating the effect of centrifugal stiffening in spinning spacecraft and aircraft in rapid roll maneuvers. This thesis further aims to set up a model to investigate the stiffening effects using the mAEWing2 aircraft. A preliminary analysis is done for the mAEWing2 aircraft and the roll rate, control surface deflection and angle of attack are identified as the parameters to be studied. For a given angle of attack and control surface deflection, the centrifugal forces in the aircraft in steady roll are determined using trim analysis. These are used to pre-stress the model for modal analysis. It is found that in the mAEWing2 aircraft in steady roll maneuvers the stiffening effect on the frequencies is not significant. It emphasizes the need to conduct a sensitivity analysis to include centrifugal stiffening in the dynamic analysis while designing an aircraft. This, along with some de-stiffening due to gravity loads might be important for the future N+3 aircraft with their high aspect ratio large wingspans.

The McTronX Research group at the Potchefstroom campus of the North-West University, aims to establish a knowledge base on active magnetic bearing (AMB) systems. Up to date, the group has established a firm knowledge base on various topics related to AMB systems. A recent focus was the design and development of a high speed AMB supported rotor system called the rotor delevitation system (RDS) to analyse rotor drops. During the testing phase of the RDS, the machine exhibited vibrations, of which the origins were unknown. The research presented in this dissertation sets out to characterize the vibrations of the RDS, which is the group’s first attempt to fulfil the need for characterizing vibrations in an AMB supported rotor. Emphasis is placed on characterizing the natural response of the RDS rotor, stator and integrated system. The research project is defined in terms of four main objectives: rotor and stator characterization, modelling, system characterization and rotor dynamic diagnostics. A comprehensive literature study introduces the fundamental concepts regarding vibrations of single and multiple degree of freedom systems. These concepts include; natural frequencies, damping, machine vibrations, rotor dynamics and modelling techniques. These modelling techniques are introduced to verify the experimental methodology used to determine the natural frequencies. A critical overview of the literature contextualises the theory with the research investigation. For the RDS rotor and stator characterization, a modal analysis process also known as the “bump test” is implemented in order to validate the bending natural frequencies of the rotor and stator. A simulation model of the RDS is constructed in the finite element (FE) package DyRoBeS®. The model is verified with a numerical and an analytical model and validated with the measured bending natural frequencies of the RDS rotor. For the system characterization, a number of modal analysis processes are implemented, which validates the rigid body natural frequencies of the RDS. These frequencies are also used to validate the FE simulation. The origins of the synchronous vibration harmonics are verified by formulating and evaluating hypotheses according to different modal analysis processes. From the RDS rotor modal analysis it was identified that a bending natural frequency of the rotor is situated at approximately 443.33 Hz. This was verified using the FE simulation model. During the system modal analyses, it was identified that only one rigid body natural frequency, situated at approximately 62 Hz, is excited. This frequency increases with the differential gain control parameter of the system up to approximately 140 Hz. After evaluating two hypotheses regarding the origins of the synchronous vibrations harmonics, it was verified that non-circularity of the rotor at the measuring positions is the cause. Overall the objectives of the study were addressed by characterizing the natural frequencies of the rotor, stator and RDS system. This include the mode forms of the rigid body and bending natural frequencies of the system. The results of the verification and validation methods correlated, which imply these methods are reliable to identify the origins of vibrations in rotor-bearing systems. The differential gain control parameter of the AMBs control the equivalent damping in the RDS. An increase in this parameter should lead to a decrease in amplitude and frequency of the maximum vibration, and vice versa. However, it was noted that an increase in this parameter caused a linear increase in the rigid body natural frequency. The literature indicates that this effect can only be caused by an increase in system stiffness. It is therefore recommended to evaluate the stiffness of the system as a function of the differential gain control parameter. / Thesis (M.Ing. (Mechanical Engineering))--North-West University, Potchefstroom Campus, 2012.

The McTronX Research group at the Potchefstroom campus of the North-West University, aims to establish a knowledge base on active magnetic bearing (AMB) systems. Up to date, the group has established a firm knowledge base on various topics related to AMB systems. A recent focus was the design and development of a high speed AMB supported rotor system called the rotor delevitation system (RDS) to analyse rotor drops. During the testing phase of the RDS, the machine exhibited vibrations, of which the origins were unknown. The research presented in this dissertation sets out to characterize the vibrations of the RDS, which is the group’s first attempt to fulfil the need for characterizing vibrations in an AMB supported rotor. Emphasis is placed on characterizing the natural response of the RDS rotor, stator and integrated system. The research project is defined in terms of four main objectives: rotor and stator characterization, modelling, system characterization and rotor dynamic diagnostics. A comprehensive literature study introduces the fundamental concepts regarding vibrations of single and multiple degree of freedom systems. These concepts include; natural frequencies, damping, machine vibrations, rotor dynamics and modelling techniques. These modelling techniques are introduced to verify the experimental methodology used to determine the natural frequencies. A critical overview of the literature contextualises the theory with the research investigation. For the RDS rotor and stator characterization, a modal analysis process also known as the “bump test” is implemented in order to validate the bending natural frequencies of the rotor and stator. A simulation model of the RDS is constructed in the finite element (FE) package DyRoBeS®. The model is verified with a numerical and an analytical model and validated with the measured bending natural frequencies of the RDS rotor. For the system characterization, a number of modal analysis processes are implemented, which validates the rigid body natural frequencies of the RDS. These frequencies are also used to validate the FE simulation. The origins of the synchronous vibration harmonics are verified by formulating and evaluating hypotheses according to different modal analysis processes. From the RDS rotor modal analysis it was identified that a bending natural frequency of the rotor is situated at approximately 443.33 Hz. This was verified using the FE simulation model. During the system modal analyses, it was identified that only one rigid body natural frequency, situated at approximately 62 Hz, is excited. This frequency increases with the differential gain control parameter of the system up to approximately 140 Hz. After evaluating two hypotheses regarding the origins of the synchronous vibrations harmonics, it was verified that non-circularity of the rotor at the measuring positions is the cause. Overall the objectives of the study were addressed by characterizing the natural frequencies of the rotor, stator and RDS system. This include the mode forms of the rigid body and bending natural frequencies of the system. The results of the verification and validation methods correlated, which imply these methods are reliable to identify the origins of vibrations in rotor-bearing systems. The differential gain control parameter of the AMBs control the equivalent damping in the RDS. An increase in this parameter should lead to a decrease in amplitude and frequency of the maximum vibration, and vice versa. However, it was noted that an increase in this parameter caused a linear increase in the rigid body natural frequency. The literature indicates that this effect can only be caused by an increase in system stiffness. It is therefore recommended to evaluate the stiffness of the system as a function of the differential gain control parameter. / Thesis (M.Ing. (Mechanical Engineering))--North-West University, Potchefstroom Campus, 2012.

<p> The present investigation deals with Gas Turbine Blade Vibrations. Literature on the techniques employed for experimental investigation of gas turbine blade vibration characteristics has been summarised. Various steps have been explained by reviewing the different techniques. Several causes for possible excitation of blades as well as damping methods to suppress the resulting vibrations are also included. Attempts were made to determine experimentally the natural frequencies of cantilever plates of thin uniform rectangular cross section, with and without pretwist. First natural frequency_ -of the plate without twist was in good agreement with the one calculated from the plate formula. Free vibration analysis of cantilever plates of thin uniform rectangular cross section is made. Finite Element Technique is used to determine the elastic and inertial properties of a fully compatible triangular element. Computed values of natural freqencies and mode shapes are compared with other analytical results. </p> / Thesis / Master of Engineering (ME)

ABSTRACT Forced Vibration Testing and Analysis of Pre- and Post- Retrofit Buildings Erica Dawn Jacobsen The primary goal of the thesis was to detect the retrofit through vibration testing of both buildings. The secondary goal focused on correctly identifying the behavior of the building through FVT, comparing that behavior to computational model predictions, and determining the necessary level of detail to include in the computational modeling. Forced vibration testing (FVT) of two stiff-wall/flexible-diaphragm buildings yielded natural frequencies and mode shapes for the two buildings. The buildings were nearly identical with the exception that one had been retrofitted. Both buildings were comprised of concrete shearwalls and steel moment frames in the north/south direction and moment frames in the east/west direction. The retrofit strengthened the moment connections and added braces to the perimeter walls in the east/west direction. The natural frequencies were found through FVT by setting a 30-lb shaker on the roof of both buildings and sweeping through a range of frequencies in both the east/west and north/south directions. Accelerometers were placed on the building to detect the accelerations. The peaks on the Fast Fourier Transform (FFT) graphs indicated the frequencies at which the structure resonated. Mode shapes were tested for by placing the shaker in a position ideal for exciting the mode and setting the shaker to the natural frequency detected from the FFT graphs. The accelerometers were placed around the roof of the building to record the mode shape. After testing, computational models were created to determine if the models could accurately predict the frequencies and mode shapes of the buildings as well as the effect of the retrofit. A series of increasingly complex computational models, ranging from hand calculations to 3D models, were created to determine the level of detail necessary to predict the building behavior. Natural frequencies were the primary criteria used to determine whether the model accurately predicted the building behavior. The mid-diaphragm deflection and base shear from spectral analysis were the final criteria used to compare these select models. It was determined that in order to properly capture the modal behavior of the building, the sawtooth framing, major beams, and the lateral-force-resisting-system (LFRS) must be modeled. Though the mode shape of the building is dominated by the flexible diaphragm, the LFRS is necessary to model to accurately predict both the natural frequency of the building as well as the diaphragm deflection.

Structural health monitoring (SHM) using ambient vibration has become a tool in evaluating and assessing the condition of civil structures. For bridge structures, a vibration-based SHM system uses the dynamic response of a bridge to measure modal parameters. A change in a structure’s modal parameters can indicate a physical change in the system, such as damage or a boundary condition change. These same modal parameters are sensitive to environmental factors, mainly temperature. Statistical models have been utilized to filter out modal parameter changes influenced by temperature and those caused by physical changes. Statistical models also help describe the relationship between modal parameters and environmental conditions. The Lambert Road Bridge is a concrete integral abutment bridge located south of Sacramento, California, and is studied through this paper. A SHM system has been installed and has been recorded for 3 years. Three months of SHM records will be used to understand how the bridge’s natural frequencies typically change due to temperature. First, temperature was observed to be the driving force behind many of the SHM records. A linear relationship was found between the structure’s natural frequency and temperature. Collinearities between potential predictor variables were noticed. Initial linear regression analyses were completed with a bridge average temperature. Certain strain gauge regression models were used as “base” models to eliminate other regression models that potentially were altered by aliasing. These base models, and the other seven corresponding models, showed a direct linear relationship between temperature and natural frequency. It was concluded that changes in boundary conditions due to bridge expansion have a greater effect on global dynamic properties than material property changes due to temperature. Stepwise linear regression followed the initial regression modeling. Eight thermocouple readings were consistently being selected in the stepwise process and were chosen to be the main predictor variables. Due to collinearities among the predictor variables, ridge regression was completed to eliminate any unstable variables. The final six sensors’ locations indicate that longitudinal, transverse, and depth gradients are all important factors in the linear regression models of this relationship. Comparing the multiple linear regression models to single-variable regression models with the highest averaged adjusted R2 values, a minimum percent difference of 21% and 19% was seen for the first and second natural frequencies, respectively. It was also concluded that these multiple linear regression models explained more of the variability in the natural frequencies and would be a better model to use to filter out temperature effects.

Exact solution to the vibration of rectangular plates is available only for plates with two opposite edges subject to simply supported conditions. Otherwise, they are analysed by using approximate methods. There are several approximate methods to conduct a vibration analysis, such as the Rayleigh-Ritz method, the Finite Element Method, the Finite Difference Method, and the Superposition Method. The Rayleigh-Ritz method and the finite element method give upper bound results for the natural frequencies of plates. However, there is a disadvantage in using this method in that the error due to discretisation cannot be calculated easily. Therefore, it would be good to find a suitable method that gives lower bound results for the natural frequencies to complement the results from the Rayleigh-Ritz method. The superposition method is also a convenient and efficient method but it gives lower bound solution only in some cases. Whether it gives upper bound or lower bound results for the natural frequencies depends on the boundary conditions. It is also known that the finite difference method always gives lower bound results. This thesis presents bounded eigenvalues, which are dimensionless form of natural frequencies, calculated using the superposition method and the finite difference method. All computations were done using the MATLAB software package. The convergence tests show that the superposition method gives a lower bound for the eigenvalues of fully clamped plates, and an upper bound for the completely free plates. It is also shown that the finite difference method gives a lower bound for the eigenvalues of completely free plates. Finally, the upper bounds and lower bounds for the eigenvalues of fully clamped and completely free plates are given.

Dynamic instrumentation and a series of ambient vibration tests were performed on a four storey strengthened R/C building within the scope of this study. Traffic load and wind load were accepted as natural dynamic loads and the vibrations were recorded by sensitive accelerometers.For that study, 12 uniaxial, 1 triaxial accelerometers and a 15 channel data logger system were used. Four sets of dynamic measurements were recorded over a period of 6 months. Recorded readings were analyzed using UPC, PC and CVA algorithms and Artemis software. The natural freqeuncies, mode shape of the tested building were determined. The experimental results were compared against each other. A 3D-FE model of the building was prepared and analytical results were also compared against experimental results.The calibration (updating) of the analytical model was carried out using the experimentally obtained mode shapes and freqeunices. The results of the study indicate that first few mode shapes and freqeuncies of the building can be obtained successfully within zero to 10 Hz range using ambient monitoring. Field calibrated FE models can effectively simulate the first translational and torsional modes of the building. Calibration studies indicate that the upper floor is more flexible than the nominal model and there are weaknesses between the shear wall and roof slab connections.

To deal with a rapid development of high-speed trains and high-speed railways, constant improvement of the railway infrastructure is necessary and engineers are continuously facing challenges in order to design efficient and optimized structures. Nowadays, more and more railway bridges are built and thus, they require the engineers’ attention both regarding their design and their maintenance. A comprehensive knowledge of the infrastructures and the trains is crucial: their behaviours need to be well known. However, today, the ballast - the granular material disposed on the track and on which the rails lie – is not well known and its effect in dynamic analyses are rarely accounted for. Engineers are still investigating the role played by the ballast in the dynamic behaviour of bridges.  This master thesis aims at quantifying the influence of the ballast on the dynamic properties of a bridge. Is the ballast just an additional mass on the structure or does it introduce any additional stiffness? Thus, this project investigates different alternatives and parameters to propose a realistic and reliable model for the ballast superstructure and the track. For the purpose of this study, a simply supported steel truss bridge located in Poland is studied. The bridge was excited by a harmonic force and the interesting point regarding the experiments is that acceleration measurements were collected before and after the ballasted track setting up on the bridge deck. Then, these data are processed through MATLAB in order to obtain the natural frequencies of the bridge at two different times during its construction. The determined natural frequencies for the un-ballasted case are then compared with analytical values obtained with a 3D finite element model implemented in the software LUSAS. This step aims at calibrating the un-ballasted finite element model so that the bridge is represented as realistically as possible. Once it has been done, a model both for the ballast and the track is proposed using solid elements for the ballast superstructure and beam elements for the rails, the guard rails and the sleepers. Different parameters influencing the natural frequencies and modes shapes of the bridge are testing and it appears that the ballast introduces an additional stiffness through a bending stiffness in the ballast and a change in the support conditions. Finally, the contribution of these parameters is assessed and discussed: the stiffness of the ballast increases the stiffness of the bridge by more than 20% for the 2nd vertical bending vibration mode and the support conditions increase the bridge’s stiffness by more than 15% and 30% respectively for the 1stvertical bending the 1st torsional vibration modes.

Under the direction of the Federal Highway Administration the Long Term Bridge Performance Program (LTBP) selected Minnesota Bridge number 5718 as a pilot bridge for evaluation. This program focuses on the monitoring of bridges for a 20-year period to understand the structural behavior over time due to the various loads and weathering. In monitoring this bridge a better understanding can be acquired for the maintenance issues related to the nation's deteriorating bridge infrastructure. Bridge Number 5718, which is located just outside of Sandstone Minnesota, is a steel truss bridge that spans the Kettle River. Constant monitoring of the bridge along with periodic testing of the bridge will allow for the collection of data over a 20-year period. The focus of this work is to establish a baseline for the bridges characteristics through nondestructive dynamic testing. Later tests will be compared to these results and changes can then be tracked. In order to perform the required testing, two electromagnetic shakers were used to produce the excitation. The bridge was also outfitted with an array of velocity transducers to allow for the response to be recorded. The data was then used to extract the resonant frequencies, mode shapes, and damping ratios. A modal assurance criterion was also performed to solidify the findings. These parameters define the structural identity of the bridge. Through performing these tests the database that is being collected under the Long Term Bridge Performance Program will be used to better the overall health and safety of the nation's bridges.