The Nonlinear Dynamics and Vibration Laboratory: Research

Analyzing the dynamics of PWL nonlinear systems is critical in multiple applications, including structures like bridges, which have gaps and contacts, biological systems, systems with cracks, cutting tools, and train carriage shock absorbers. Computationally predicting the dynamics of these systems requires proper modeling of the nonlinearity, which eliminates the use of most linear analysis tools developed to efficiently compute the dynamics of complex systems. Timestepping methods or direct numerical integration can be used to solve for the dynamics of these systems when they are simple, low-dimensional systems. However, as these systems become more complex, numerical integration can become prohibitively expensive when maintaining a sufficiently small error tolerance. For high dimensional systems typically linear reduction techniques are used to lower the dimensionality of the linear portion of the system while treating the localized nonlinearity as master degrees of freedom (DOFs). The nonlinear DOFs are then typically modeled using alternating frequency/time domain methods. While these methods work for certain excitations and system dynamics, they still require time-consuming iterative calculations of many nonlinear algebraic equations.

My research in this field is focused on providing new understanding and analysis tools for systems that contain nonlinearities due to intermittent contact, friction contact, cracks, large displacement at joints, and stiffness changes. These new methods are based on splitting the motion of the system into distinct linear regimes and then applying the appropriate initial conditions/compatibility conditions when the system switches its state. An important advantage of these new methods is that they use linear and user-independent methods for the analysis and control of high-dimensional PWL nonlinear systems. The main goal of this research is to create a new class of effective and robust techniques to enable the computation of large PWL nonlinear systems under any forcing conditions in order to analyze, monitor and control their system dynamics. This research has led to several publications and is currently being funded by an NSF grant.

This research is focused on the creation of a specially controlled piecewise linear (PWL) harvester. This harvester differs significantly  from previous linear, PWL, and nonlinear energy harvesters. The controlled PWL harvester takes advantage of the NDVL's previous research for efficiently computing the steady state response of bilinear systems by adjusting the gap to match the frequency input to ensure resonance in the bilinear system. Assuring resonance in the bilinear system maximizes the response and can achieve similar harvesting efficiency as a linear system acting at resonance. Also, since the gap adjusts based on the excitation frequency and amplitude, the controlled PWL harvester has the key benefit of nonlinear harvesters by effectively operating over a wide frequency range. Moreover, the PWL harvester can handle slowly drifting excitation as well as excitations with large shifts. 

Most current methods are based on the source of the excitation being either broadband or at a non-drifting frequency. If the excitation signal is a constant frequency, then linear harvesters are the most effective way to extract vibration energy. However, the effectiveness of linear systems significantly decreases when the systems are excited away from resonance. Typically, nonlinear devices and array-harvesters have been used to broaden the operating range of these systems. For a large set of systems, the excitation slowly drifts over time. The benefit of the PWL nonlinear system with gap control is most evident for these systems since it can operate over a wide frequency range where the system drifts while still gaining the benefits of a linear system at resonance. These frequency-tunable devices will substantially increase power generation from energy harvesters.

Short duration intensely nonlinear phenomena such as foreign object ingestion, blade out events, and tip rubs are well suited for finite element modeling with explicit time integration. The NDVL & GTL has ongoing work modeling the ingestion of UAVs into aircraft engines with partially validated LS-DYNA models.

The use of UAVs has increased dramatically in recent years. As the number of UAVs sold continues to increase, proper integration of UAVs into the airspace is a major safety concern due to the potential for a UAV-airplane collision. Recreational users are the highest safety concern since they may be unaware or unconcerned with regulations and rules concerning restricted operation of their devices in certain airspaces. These UAVs tend to be relatively small and have the potential to be ingested into an engine of a commercial aircraft operating nearby. Currently there are regulations on aircraft engines that require full scale tests on new engines to demonstrate safe operation after certain bird and ice ingestions; however, the FAA is unlikely to place additional requirements on engine certifications for UAV ingestions.  Moreover, the current tests and regulations cannot be transferred from birds to UAVs since key components (motor, battery, camera) of these UAVs contain materials that are much denser and stiffer than ice and birds (which are typically modeled as a fluid since they are over 70% water). Preliminary work on this topic has been published and shows that UAVs can cause significantly more damage than birds. Our current studies are being conducted with models that have been specifically developed and validated for a UAV engine ingestion.

A great deal of research has been devoted to the vibration analysis of bladed disks found in turbomachinery, but there are still many challenges. The complicated geometry and small variation in each blade (mistuning) make the modeling of these systems very challenging. In order to efficiently analyze their dynamic and vibratory properties, several reduced-order modeling techniques that use cyclic symmetry and modal projections have been created for these systems. Although these methods are efficient for linear systems, they are not able to capture the nonlinearity in turbomachinery due to friction damping. Friction damping is being used to dissipate unwanted vibrations in turbomachinery via shrouds, snubbers, underplatform dampers, and ring dampers, but their design is being limited by current computational tools. Better design of these damping components can address serious high-cycle fatigue (HCF) problems. The U.S. Air Force reported in 1998 that approximately one third of maintenance costs and half of Class A mishaps (each Class A mishap results in over $1 million in damage or loss of the aircraft) are due to HCF. Furthermore, several civilian fleet accidents have also been caused by HCF.

The NDVL/GTL is currently conducting damping/mistuning experiments in bladed disks and blisks for Siemens and the Guide consortium.  These experiments are focused on characterizing the dynamic response of bladed disks and blisks for various excitations, when they are being operated at design speed in the GTL's LSTF. The method chosen to excite the blades while spinning is an air-jet system that can be operated in a constant on and pulsed mode depending on the application. This system has been developed to excite both synchronous and non-synchronous vibrations in bladed disks.  Measurements of blade responses are being taken using strain gauges mounted on the blades, and an eight probe tip timing system from Agilis. 

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