David A. Ehrhardt

Numerical and Experimental Determination of Nonlinear Normal Modes of a Circular Perforated Plate

It is commonly known that nonlinearities in structures can lead to large amplitude responses that are not predicted by traditional theories. Thus a linear design could lead to premature failure if the structure actually behaves nonlinearly, or, conversely, nonlinearities could potentially be exploited to reduce stresses relative to the best possible design with a purely linear structure. When examining structures that operate in environments where a nonlinear response is possible, one can gain insight into the free and forced responses of a nonlinear system by determining the structure’s nonlinear normal modes (NNMs). NNMs extend knowledge gained from established linear normal modes (LNMs) into the nonlinear response range by quantifying how the unforced vibration frequency depends on the input energy. Recent works have shown that periodic excitations can be used to isolate a single NNM, providing a means for measuring NNMs in the laboratory. An extension of the modal indicator function can be used to ensure that the measured response is on the desired NNM. The experimentally measured NNMs can then be compared to numerically calculated NNMs for model validation. In this investigation, a circular perforated plate containing a distributed geometric nonlinearity is considered. This plate has demonstrated nonlinear responses when the displacements become comparable to the plate thickness. However, the system is challenging to model because the nonlinear response is potentially sensitive to small geometric features, residual stresses within the structure, and the boundary conditions.

Mode Shape Comparison Using Continuous-Scan Laser Doppler Vibrometry and High Speed 3D Digital Image Correlation

Experimental structural dynamic measurements are traditionally obtained using discrete sensors such as accelerometers, strain gauges, displacement transducers, etc. These techniques are known for providing measurements at discrete points. Also, a majority of these sensors require contact with the structure under test which may modify the dynamic response. In contrast, a few recently developed techniques are capable of measuring the response over a wide measurement field without contacting the structure. Two techniques are considered here: continuous-scan laser Doppler vibrometry (CSLDV) and high speed three dimensional digital image correlation (3D-DIC). The large amount of measured velocities and displacements provide an unprecedented measurement resolution; however, they both require post processing to obtain measurements. In this investigation, the frequency response function of a clamped-clamped flat beam will be determined using a modal hammer test, CSLDV, and high speed 3D-DIC. The mode shapes of the beam determined by each of these experimental methods will then be compared to assess the relative merits of each measurement approach.

Experimental Investigation of Dynamic out of Plane Displacement Error in 3D Digital Image Correlation

With the development and implementation of digital image correlation (DIC) in static and dynamic experimentation, a deeper understanding of the variation of measurement error due to the experimental setup is sought. In this investigation, dynamic rigid body motion is measured using two high speed digital CMOS cameras paired with 3D digital image correlation (3D-DIC), a single point laser vibrometer, and a single axis accelerometer. To determine measurement error, a comparison is made between the measured displacement, velocity, and acceleration in the frequency domain. Prior to dynamic testing, characterization experiments with no motion were performed to determine potential sources of noise in a static environment. It was shown that the camera cooling fans and source of lighting can contribute to measurement noise. Dynamic experiments were designed to investigate the potential effect camera angle, sampling rate, and shutter speed have on measurement error. In addition to the 3D-DIC setup, speckle patterns with varying size, distribution, and randomness were investigated. Results show that the 3D-DIC setup and speckle pattern characteristics can affect the measurement error and measurement quality

Nonlinear Dynamics, Volume 1

The vibration characteristics of beams have been extensively studied due to their wide application across multiple fields (i.e. spacecraft antennae, aircraft wings, turbine blades, skyscrapers). Of particular interest, specific geometries of beams have been shown to induce coupling between the fundamental bending and torsion modes. This coupled motion can be observed in a beam’s linear normal modes can be avoided with the correct selection of geometric properties. This work investigates the coupled bending-torsion behaviour of a clamped-clamped beam that is coupled perpendicularly, mid-span to mid-span, to a second beam with tip masses within the nonlinear response regime. The first torsion mode of the beam system is tuned by modifying the mass distribution such that closely spaced bending and torsion linear normal modes can be realized. The nonlinear behaviour is presented using nonlinear normal mode backbone curves and forced responses in the vicinity of the modes of interest.

Measurement of Nonlinear Normal Modes Using Mono-harmonic Force Appropriation: Experimental Investigation

A structure undergoing large amplitude deformations can exhibit nonlinear behavior which is not predicted by traditional linear theories. Structures with some initial curvature offer an additional complication due to buckling and snap through phenomena, and can exhibit softening, hardening and, internal resonance. As a structure transitions into a region of nonlinear response, a structure’s nonlinear normal modes (NNMs) can provide insight into the forced responses of the nonlinear system. Mono-harmonic excitations can often be used to experimentally isolate a dynamic response in the neighborhood of a single NNM. This is accomplished with an extension of the modal indicator function and force appropriation to ensure the dynamic response of the structure is on the desired NNM. This work explores these methods using two structures: a nominally-flat beam and a curved axi-symmetric plate. Single-point force appropriation is used by manually tuning the excitation frequency and amplitude until the mode indicator function is satisfied for the fundamental harmonic. The results show a reasonable estimate of the NNM backbone, the occurrence of internal resonance, and couplings between the underlying linear modes along the backbone.

Nonlinear Normal Modes in Finite Element Model Validation of Geometrically Nonlinear Flat and Curved Beams

Model Validation is an important step in the design of structures operating under dynamic loading. The natural frequencies and mode shapes associated with linear normal modes (LNMs) have been traditionally used to validate and update finite element models, but their usefulness breaks down when a structure operates in a nonlinear response regime. The concept of nonlinear normal modes (NNMs) has been presented as a capable extension of LNMs into nonlinear response regimes. In this work, linear model updating is performed on one flat and one curved beam using the experimentally measured natural frequencies and mode shapes coupled with gradient based optimization. Throughout the updating process, the first NNM of these structures are numerically calculated and compared with the experimentally measured NNM. This comparison is used for model validation throughout each step of the updating procedure. Results show the importance of the definition of initial geometry and effect of the large variation in boundary conditions contributing to changes in the nonlinear behavior of the model.

Nonlinear Reduced Order Modeling of a Curved Axi-Symmetric Perforated Plate: Comparison with Experiments

Structures undergoing large amplitude deformations usually include nonlinear strain–displacement relationships defining a geometric nonlinearity. This type of nonlinearity can be observed in structures with fixed boundary conditions where mid-plane stretching occurs at large deflections. When considering the dynamic response of the structure under these circumstances, the once uncoupled linear normal modes become coupled at large response amplitudes changing the characteristic deformation shape as the fundamental frequency of vibration changes. Therefore, the development of nonlinear reduced order models to predict the dynamic response of such structures should account for any potential coupling between the linear normal modes. In this context, a structures’ nonlinear normal modes provide a compact characterization of modal coupling in the structural response in nonlinear regimes. This work uses experimentally measured and numerically calculated nonlinear normal modes for a curved axi-symmetric perforated plate to build a nonlinear reduced order model and gain insight to the underlying modal coupling. A comparison is made between two models and corresponding experimental structures to assess the model’s ability to describe the modal coupling.

System Identification to Estimate the Nonlinear Modes of a Gong

Nonlinear Normal Modes (NNMs) have proven useful in a few recent works as a basis for comparing nonlinear models during model updating. In prior works the authors have used force appropriation to measure NNMs, but this is time consuming, generally requiring hand tuning of both the frequency in question and the strength of its harmonics. This paper explores the use of system identification, using a small set of broadband response data, to estimate a model from which the NNMs can be extracted. The Frequency Domain Restoring Force Surface (RFS) method will be used to perform identification, in which the nonlinearity of the system is assumed to be a polynomial function of the modal displacements, and a least squares problem is formed to solve for the nonlinear coefficients. Existing NNM calculation approaches can then be applied to the experimentally determined model in order to calculate the NNMs of the system. This approach is evaluated by applying it to full-field measurements from a traditional Gong, obtained using Stereo 3D Digital Image Correlation (3D–DIC). The results obtained using system identification are validated with measurements of the NNMs obtained using force appropriation and a scanning laser vibrometer.

Force appropriation of nonlinear structures

Nonlinear normal modes (NNMs) are widely used as a tool for developing mathematical models of nonlinear structures and understanding their dynamics. NNMs can be identified experimentally through a phase quadrature condition between the system response and the applied excitation. This paper demonstrates that this commonly used quadrature condition can give results that are significantly different from the true NNM, in particular, when the excitation applied to the system is limited to one input force, as is frequently used in practice. The system studied is a clamped–clamped cross-beam with two closely spaced modes. This paper shows that the regions where the quadrature condition is (in)accurate can be qualitatively captured by analysing transfer of energy between the modes of the system, leading to a discussion of the appropriate number of input forces and their locations across the structure.

The Measurement of a Nonlinear Resonant Decay Using Continuous-Scan Laser Doppler Vibrometry

The nonlinear resonant decay of a structure offers much insight into the frequency-amplitude behavior of a structure’s dynamic response. The spatial deformation during this decay is especially important since nonlinear responses can cause unexpected stress concentrations necessitating full-field measurements for comparison with a model. In this context, full-field measurement techniques, such as continuous scan laser Doppler vibrometry (CSLDV) and high speed three dimensional digital image correlation (3D-DIC) provide tools to obtain the full-field dynamic response experimentally. While CSLDV has been used to measure the steady state response of linear and nonlinear structures as well as transient responses of linear structures, it is unclear whether the approach can be successful for transient nonlinear measurements where the frequency of the dynamic response is amplitude dependent. In this investigation, the capabilities of CSLDV will be utilized to measure the nonlinear resonant decay of a clamped-clamped flat beam. The response measured using CSLDV will then be compared with the decay response measured with 3D-DIC to validate the CSLDV method and to understand the advantages and disadvantages of each.