Michael W. Sracic

Output-Only Modal Analysis of Linear Time Periodic Systems with Application to Wind Turbine Simulation Data

Many important systems, such as wind turbines, helicopters and turbomachinery, must be modeled with linear time-periodic equations of motion to correctly predict resonance phenomena. Time periodic effects in wind turbines might arise due to blade-to-blade manufacturing variations, stratification in the velocity of the wind with height and changes in the aerodynamics of the blades as they pass the tower. These effects may cause parametric resonance or other unexpected phenomena, so it is important to properly characterize them so that these machines can be designed to achieve high reliability, safety, and to produce economical power. This work presents a system identification methodology that can be used to identify models for linear, periodically time-varying systems when the input forces are unmeasured, broadband and random. The methodology is demonstrated for the well-known Mathieu oscillator and then used to interrogate simulated measurements from a rotating wind turbine. The measurements were simulated for a 5 MW turbine modeled in the HAWC2 simulation code, which includes both structural dynamic and aerodynamic effects. This simulated system identification provides insights into the test and measurement requirements and the potential pitfalls, and simulated experiments such as this may be useful to obtain a set of time-periodic equations of motion from a numerical model, since a closed form model is not readily available by other means due to the way in which the aeroelastic effects are treated in the simulation code.

Identifying the Modal Properties of Nonlinear Structures Using Measured Free Response Time Histories from a Scanning Laser Doppler Vibrometer

This paper explores methods that can be used to characterize weakly nonlinear systems, whose natural frequencies and damping ratios change with response amplitude. The focus is on high order systems that may have several modes although each with a distinct natural frequency. Interactions between modes are not addressed. This type of analysis may be appropriate, for example, for structural dynamic systems that exhibit damping that depends on the response amplitude due to friction in bolted joints. This causes the free-response of the system to seem to have damping ratios (and to a lesser extent natural frequencies) that change slowly with time. Several techniques have been proposed to characterize such systems. This work compares a few available methods, focusing on their applicability to real measurements from multi-degree-of-freedom systems. A beam with several small links connected by simple bolted joints was used to evaluate the available methods. The system was excited by impulse and the velocity response was measured with a scanning laser Doppler vibrometer. Several state of the art procedures were then used to process the nonlinear free responses and their features were compared. First the Zeroed Early Time FFT technique was used to qualitatively evaluate the responses. Then, the Empirical Mode Decomposition method and a simple approach based on band pass filtering were both employed to obtain mono-component signals from the measured responses. Once mono-component signals had been obtained, they were processed with the Hilbert transform approach, with several enhancements made to minimize the effects of noise.

Mass normalized mode shapes using impact excitation and continuous-scan laser Doppler vibrometry

Conventional scanning laser Doppler vibrometer (LDV) systems cannot be effectively employed with impact excitation because they typically measure a structures response at only one point at a time. This necessitates exciting the structure at multiple points to create a multi-input-single-output modal test data base, which is not only tedious, but prone to errors due to variations in the impact characteristics from one point to the next. Previous works have demonstrated that an LDV can be used to measure the mode shapes of a structure over a surface by scanning the laser spot continuously as the structures response decays. The author recently presented a procedure that allows one to post-process continuous-scan LDV (CSLDV) measurements of the free decay of a structure using standard modal parameter identification techniques. Using this approach, one can find the natural frequencies, damping ratios and mode shapes of a structure at hundreds of points simultaneously from a few free responses. The procedure employs a novel resampling approach to transform the continuous-scan measurements into pseudo-frequency response functions, fits a complex mode model, and then accounts for the time delay between samples to obtain the mode shapes. This paper extends the previous work by presenting an algorithm that uses the input force spectrum, measured by an instrumented hammer, to mass normalize the mode shapes obtained using the continuous-scan LDV process. Other issues such as the effect of the scan frequency on the procedure and on speckle noise are also briefly addressed.

Two Algorithms for Mass Normalizing Mode Shapes From Impact Excited Continuous-Scan Laser Doppler Vibrometry

Continuous-scan laser Doppler vibrometry (CSLDV), a concept where a vibrometer measures the motion of a structure as the laser measurement point sweeps over the structure, has proven to be an effective method for rapidly obtaining mode shape measurements with very high spatial detail using a completely non-contact approach. Existing CSLDV methods obtain only the operating shapes or arbitrarily scaled modes of a structure, but the mass-normalized modes are sought in many applications; for example, when the experimental modal model is to be used for substructuring predictions or to predict the effect of structural modifications. This paper extends an approach based on impact excitation and CSLDV, presenting a new least squares algorithm that can be used to estimate the mass-normalized modes of a structure from CSLDV measurements. Two formulations are derived: one based on real-modes that is appropriate when the structure is proportionally damped and a second that accommodates a complex-mode description. The latter approach also gives the algorithm further latitude to accommodate time-synchronization errors in the data acquisition system. The method is demonstrated on a free-free beam, where both CSLDV and a conventional test using an accelerometer and a roving-hammer are used to find its first seven mass normalized modes. The scale factors produced by both methods are found to agree with a tuned analytical model for the beam to within about ten percent. The results are further verified by attaching a small mass to the beam and using the model to predict the change in the structure’s natural frequencies and mode shapes due to the added mass.

Numerical Continuation of Periodic Orbits for Harmonically Forced Nonlinear Systems

A nonlinear structure will often respond periodically when it is excited with a sinusoidal force. Several methods are available that can compute the periodic response for various drive frequencies, which is analogous to the frequency response function for a linear system. The simplest approach would be to compute a sequence of simulations where the equations of motion are integrated until damping drives the system to steady state, but that approach suffers from a number of drawbacks. Recently, numerical methods have been proposed that use a solution branch continuation technique to find the free response of unforced, undamped nonlinear systems for different values of a control parameter. These are attractive because they are built around broadly applicable time-integration routines, so they are applicable to a wide range of systems. However, the continuation approach is not typically used to calculate the periodic response of a structural dynamic system to a harmonic force. This work adapts the numerical continuation approach to find the periodic, forced steady-state response of a nonlinear system. The method uses an adaptive procedure with a prediction step and a mode switching correction step based on Newton-Raphson methods. Once a branch of solutions has been computed, it explains how a full spectrum of harmonic forcing conditions affect the dynamic response of the nonlinear system. The approach is developed and applied to calculate nonlinear frequency response curves for a Duffing oscillator and a low order nonlinear cantilever beam.

A Numerical Continuation Method to Compute Nonlinear Normal Modes Using Modal Reduction

Nonlinearities in structural dynamic systems introduce behavior that cannot be described with linear vibration theory, such as frequency-energy dependence and internal resonances. The concept of nonlinear normal modes accommodates such phenomena, providing a rigorous framework to characterize and design nonlinear structures. A recently developed method has enabled the computation of nonlinear normal modes for structures with hundreds of degrees of freedom, but the formulation is not readily applicable to large scale geometrically nonlinear structures that are modeled within finite element software. This work presents a variation on that approach that can be used to extract the nonlinear normal modes of a structure using commercial finite element software. A model of the structure is created in the finite element package and the algorithm then iterates on the nonlinear transient response in a non-intrusive way to estimate the nonlinear modes. A modal coordinate transformation is used to reduce the order of the Jacobians required by the algorithm. The method is demonstrated on a fixed-fixed beam that is geometrically nonlinear due to coupling between transverse and axial displacements. An alternative procedure is also presented in which static load cases are used to compute a reduced order model of the nonlinear system and then standard continuation is used to find the nonlinear modes of the reduced order model. That approach is explored using both enforced displacements and applied loads and the results obtained are compared with those from the full-order model.

Experimental Modal Analysis on a Rotating Fan Using Tracking-CSLDV

Continuous Scan Laser Doppler Vibrometry (CSLDV) modifies the traditional mode of operation of a vibrometer by sweeping the laser measurement point continuously over the structure while measuring, enabling one to measure spatially detailed mode shapes quickly and minimizing the inconsistencies that can arise if the structure or test conditions change with time. When a periodic scan path is employed, one can decompose the measurement into the response that would have been measured at each point traversed by the laser and obtain the structure’s mode shapes and natural frequencies using conventional modal analysis software. In this paper, continuous‐scan vibrometry is performed on a rotating fan, using computer controlled mirrors to track the rotating fan blades while simultaneously sweeping the measurement point over the blades. This has the potential to circumvent the difficulty of attaching contact sensors such as strain gauges, which might modify the structure and invalidate the results. In this work, im...

Comparing Measured and Computed Nonlinear Frequency Responses to Calibrate Nonlinear System Models

Many systems of interest contain nonlinearities that are difficult to accurately model from first principles, so it would be preferable to characterize the system experimentally. For many nonlinear systems, it is now possible to measure frequency response curves with stepped sine testing and to compute frequency response curves with numerical continuation. Nonlinear frequency response curves are very sensitive to the system model and the nonlinearities and they provide a lot of insight into the response of the system to a variety of inputs. This paper explores the feasibility of a nonlinear model updating approach based on nonlinear frequency response and the experimental and analytical tools that are needed. For the experiment, a cantilever beam with an unknown nonlinearity is driven with a harmonic force at various frequencies. The steady-state response is measured and processed with the fast Fourier transform to obtain the frequency response curve. Some subtle yet important details regarding how this is implemented are discussed. An analytical model is also constructed and its frequency response computed using a recently developed technique. The measured and simulated frequencies are then compared and used to tune the analytical model.

SYSTEM IDENTIFICATION OF DYNAMIC SYSTEMS WITH CUBIC NONLINEARITIES USING LINEAR TIME-PERIODIC APPROXIMATIONS

This work develops methods to identify parametric models of nonlinear dynamic systems from response measurements using tools for Linear Time Periodic (LTP) systems. The basic approach is to drive the system periodically in a stable limit cycle and then measure deviations of the response from that limit cycle. Under certain conditions, the resulting response can be well approximated as that of a linear-time periodic system. In the analytical realm it is common to linearize a system about a periodic trajectory and then use Floquet analysis to assess the stability of the limit cycle. This work is concerned with the inverse problem, using a measured time-periodic response to derive a nonlinear dynamic model for the system. Recently, a few new methods were developed that facilitate the experimental identification of linear time periodic systems, and those methods are exploited in this work. The proposed system identification methodology is evaluated by applying it to a Duffing oscillator, demonstrating that the nonlinear forcedisplacement relationship can be identified without a priori knowledge of its functional form. The proposed methods are also applied to simulated measurements from a cantilever beam with a cubic nonlinear spring on its tip, revealing that the model order of the system and the displacement dependent stiffness can be readily identified.

Identifying parameters of nonlinear structural dynamic systems using linear time-periodic approximations

While numerous mature parametric identification methods are available for linear systems, there are only a few methods capable of identifying parametric models for multiple degree of freedom nonlinear systems. In a previous work, the authors proposed a new identification routine for nonlinear systems based on harmonically forcing a system in a periodic orbit and then recording deviations from that orbit. Under mild assumptions one can model the response about the periodic orbit using a linear time-periodic system model that is relatively easy to identify from the measurements using a variety of techniques. The method provides an estimate of the time periodic state coefficient matrix of the system which gives direct information on the order of the system and the nonlinear-parameters. A prior work explored the method in detail for a single degree-offreedom system, but it has only been applied to an MDOF system with a limited set of excitation conditions. This work explores a range of possible excitation signals using an analytical model of a cantilever beam with a cubic spring at its tip. Numerical continuation techniques are used to find the stable and unstable periodic responses of the beam and different excitation strategies are explored. Additionally, the method is validated on the analytical model with a conventional approach for nonlinear system identification. The most promising strategies are then applied to a real beam with a significant geometric nonlinearity.