“**Method for Experimental Identification of Nonlinear Dynamic Systems of Unknown Form and Order with Application to Human Gait**,” PI, National Science Foundation, Program Manager: Eduardo Misawa, 2010-2014, $279,982.

During the course of this project we have created and evaluated new methods for testing and interpreting measurements from systems modeled by time varying systems of differential equations. These types of mathematical models are critical to modeling rotating machinery such as wind turbines and the gear trains that are found within automobiles, aircraft and industrial machinery. Time varying models have particular significance to engineers because they provide important insights into the behavior of nonlinear systems such as a biomechanical model of an elderly adult walking at steady speed on a treadmill.

One of the key outcomes of this work was a new methodology for identifying mathematical models for linear time periodic systems in cases where the forces exciting the structure cannot be measured. A wind turbine rotating at constant speed is an important example of a time periodic system, and they are driven by turbulence within the wind so that it is impossible to measure the pressure field that causes vibrations of the turbine blades and tower. Vibration is one of the key factors that limits the life of a wind turbine. Allen’s research group addressed this issue, collaborating with researchers in Denmark who had instrumented an operating wind turbine with a small number of vibration sensors on the blades and tower. The vibration of the turbine was measured over the course of several months and these measurements were then used to derive a mathematical model of the turbine including the effect of small differences between the three nominally identical blades. The methodology developed in this work could eventually be used to design longer lasting wind turbines and to use vibration measurements to detect damage in a blade before it fails catastrophically. Figure 1 shows a photograph of the wind turbine that was tested, a schematic showing where sensors (accelerometers) were located, and a sample measurement. The methods developed in this work allow us to explain how the various frequencies present in the vibration signature shown to the right are related to variations in stiffness and mass between the three blades.

The mathematical concepts developed in this work were also used to create a new means of measuring the vibration of large structures such as wind turbines. For example, wind turbines can be more than 100 meters tall and so to mount sensors it is necessary to hire climbers or use specialized cranes, making this an expensive and time consuming endeavor. We developed an alternative which uses a laser Doppler vibrometer to measure the structure’s motion. A vibrometer is an instrument which measures the velocity at a point by reflecting a laser off of the structure and collecting some of the backscattered laser light. To make this method practical for large structures whose oscillation frequencies are low, the laser was swept continuously along the length of a blade while measuring the vibration at the point where the laser was incident. This is illustrated in Figure 2. The measurement was then analyzed using the methods developed in this work in order to determine the deflection shape of the blade at various vibration frequencies. This information is helpful when developing simulation models of wind turbine blades and could also reveal structural degradation or damage (for example, due to a lightning strike) within the blade that could compromise its performance.

The methods developed in this work were also applied to many other systems, for example nonlinear structures which behave in a complicated manner such as the human musculoskeletal system. Simulations were carried out to seek to devise new means of understanding and modeling these systems, in an effort to better understand what factors increase one’s risk of suffering a fall while walking and how this can be mitigated. A series of measurements of healthy young adults and older adults walking on an instrumented treadmill were provided by a collaborator and work is ongoing to understand and model the complicated interactions between the skeletal structure, the muscles which actuate the motion and the control systems within the central nervous system.

In order to disseminate these findings within the engineering community, Allen has developed a short course that has already been offered in two venues to practicing engineers (see Figure 3), graduate students and other researchers.

Two graduate students were funded over the course of this work, one of which is now working in engineering research and development for a US firm that manufactures high power air compressors and the other is a professor at a University.