Non-linear Dynamic Analysis on a Continuum Suspension Bridge Model with Spatial Layout of Main Cables

This study proposes an approach to set up a continuum full bridge model with spatially inclined cables based on the Hamilton principle. The dynamic governing functions, considering the geometric non-linearities of cables and deck, represent simultaneously the vertical motion of deck and vertical–horizontal motion of cable. With the comparison of the modal properties obtained from the model to those from the accurate model, results show that the proposed model is capable of accurately simulating the modal properties. The primary resonance responses and corresponding frequency-response curves are obtained through the multiple-scale-method. A finite element (FE) model is established, and the corresponding non-linear dynamic analysis in time domain is conducted. Comparing the results from two models, it can be checked that the proposed model is reliable. According to the results of the proposed model, it is found that the second-order shape functions (SOSFs) play a significant role in the system response. Once the non-linear vibration of the bridge becomes significant only considering the excited mode with using the classical Galerkin decomposition cannot correctly predict the structure response. The SOSFs can be classified into stationary and vibrating components. The vibrating component can deviate the time-series of response from the harmonic wave, and the stationary component directly determines the mean value of the time-series.

An algorithm to calibrate analytical models of multi-story buildings from vibration records

The common approach to develop analytical models of multi-story buildings from their vibration records is to match the modal properties identified from the records. However, the models developed by matching only the modal properties do not necessarily represent the real structure. In other words, more than one model can match the recorded motions. Moreover, modal properties do not give information on the distribution of stiffness and damping along the height of the building. In this study, an algorithm is developed to identify the dynamic characteristics of each story of multi-story buildings using the transfer-matrix formulation of the response. The building is considered as the superposition of 1-story structures, put one on top of the other. Starting from the top story and going downward, each story’s natural frequency and damping ratio are identified as it were a 1-story building. A key requirement for this approach is to have vibration records from every story. Since this is not typically the case, we utilize the so-called Mode Shape-Based Estimation (MSBE) method to estimate the vibration time histories at non-instrumented floors. Once vibration records are available at every floor, and starting from the top story, we can calculate the individual frequency and damping ratio of each story (i.e. as if it were a 1-story building) by minimizing the error between the recorded and estimated Fourier amplitude spectra (FAS) of the vibration records in that story. The analytical models calibrated in this way are more accurate, and the system identified is unique. Numerical examples are provided to show the application of the methodology.

The effect of modal properties of crash box structures with trigger mechanisms towards the crashworthiness by using finite element analysis

In the automotive structure, there are different components that utilise aluminium alloy (AA) sheets and it is used widely in the car body-in-white which comprise bumpers and the crash box structure at the front end of the car which specifically designed to withstand the event of collision. As the structures are also experiencing dynamic loading, it were also a concern for the structures to show satisfied modal properties. In this study, the modal properties of the crash box structures are investigated along with the effect of the modal properties towards the crashworthiness behaviour of the structure itself with the approach of finite element analysis. Experimental modal analysis was also done to further validating the finite element analysis of the modal properties. Three different designs of trigger mechanisms are applied towards the crash box structure to observe on both findings. For the connector element, equivalent nodes of both parts of the crash box structures are used. For the results, the correlation from both findings did show that the presence of trigger mechanism did decreased the magnitude of natural frequencies as well as the mode shape as shown by crash box type 3 by 9.50% and for the crashworthiness output, the crashworthiness behaviour of the crash box with trigger mechanisms were better in term of the collisions phases indicated by the primary peak force and the secondary peak force from the force-displacement curve as also shown by crash box structure type 3 with the percentage of 22.59%. The study does shows that the stiffness and mass distribution due to the presence of trigger mechanism do affect the modal properties of a structure as well as its crashworthiness output.

Image Motion Extraction of Structures Using Computer Vision Techniques: A Comparative Study

Vibrational measurements play an important role for structural health monitoring, e.g., modal extraction and damage diagnosis. Moreover, conditions of civil structures can be mostly assessed by displacement responses. However, installing displacement transducers between the ground and floors in real-world buildings is unrealistic due to lack of reference points and structural scales and complexity. Alternatively, structural displacements can be acquired using computer vision-based motion extraction techniques. These extracted motions not only provide vibrational responses but are also useful for identifying the modal properties. In this study, three methods, including the optical flow with the Lucas–Kanade method, the digital image correlation (DIC) with bilinear interpolation, and the in-plane phase-based motion magnification using the Riesz pyramid, are introduced and experimentally verified using a four-story steel-frame building with a commercially available camera. First, the three displacement acquiring methods are introduced in detail. Next, the displacements are experimentally obtained from these methods and compared to those sensed from linear variable displacement transducers. Moreover, these displacement responses are converted into modal properties by system identification. As seen in the experimental results, the DIC method has the lowest average root mean squared error (RMSE) of 1.2371 mm among these three methods. Although the phase-based motion magnification method has a larger RMSE of 1.4132 mm due to variations in edge detection, this method is capable of providing full-field mode shapes over the building.

Noumenal Freedom and Kant’s Modal Antinomy

Kant states in §76 of the third Critique that the divine intuitive intellect would not represent modal distinctions. Kohl (2015) and Stang (2016) claim that this statement entails that noumena lack modal properties, which, in turn, conflicts with Kant’s attribution of contingency to human noumenal wills. They both propose resolutions to this conflict based on conjectures regarding how God might non-modally represent what our discursive intellects represent as modally determined. I argue that (i) these proposals fail; (ii) the viable resolution consists in recognizing that we modalize human noumenal wills as a merely regulative-practical principle in our judgements of imputation.