Experimental modal analysis via shaker testing introduces errors in the measured structural response that can be attributed to the force transducer assembly fixed on the vibrating structure. Previous studies developed transducer mass-cancellation techniques for systems with translational degrees of freedom; however, studies addressing this problem when rotations cannot be neglected are sparse. In situations where rotations cannot be neglected, the apparent mass of the transducer is dependent on its geometry and is not the same in all directions. This paper investigates a method for correcting the measured system response that is contaminated with the effects of the attached force transducer mass and inertia. Experimental modal substructuring facilitated estimations of the translational and rotational mode shapes at the transducer connection point, thus enabling removal of an analytical transducer model from the measured test structure resulting in the corrected response. A numerical analysis showed the feasibility of the proposed approach in estimating the correct modal frequencies and forced response. To provide further validation, an experimental analysis showed the proposed approach applied to results obtained from a shaker test more accurately reflected results obtained from a hammer test.
Many test articles exhibit slight nonlinearities which result in natural frequencies shifting between data from different references. This shifting can confound mode fitting algorithms because a single mode can appear as multiple modes when the data from multiple references are combined in a single data set. For this reason, modal test engineers at Sandia National Laboratories often fit data from each reference separately. However, this creates complexity when selecting a final set of modes, because a given mode may be fit from a number of reference data sets. The color-coded complex mode indicator function was developed as a tool that could be used to reduce a complex data set into a manageable figure that displays the number of modes in a given frequency range and also the reference that best excites the mode. The tool is wrapped in a graphical user interface that allows the test engineer to easily iterate on the selected set of modes, visualize the MAC matrix, quickly resynthesize data to check fits, and export the modes to a report-ready table. This tool has proven valuable, and has been used on very complex modal tests with hundreds of response channels and a handful of reference locations.
Several recent studies (Mayes, R.L., Pacini, B.R., Roettgen, D.R.: A modal model to simulate typical structural dynamics nonlinearity. In: Proceedings of the 34th International Modal Analysis Conference. Orlando, FL, (2016); Pacini, B.R., Mayes, R.L., Owens, B.C., Schultz, R.: Nonlinear finite element model updating, part I: experimental techniques and nonlinear modal model parameter extraction. In: Proceedings of the 35th international modal analysis conference, Garden Grove, CA, (2017)) have investigated predicting nonlinear structural vibration responses using modified modal models. In such models, a nonlinear element is added in parallel to the traditional linear spring and damping elements. This assumes that the mode shapes do not change with amplitude and there are no interactions between modal degrees of freedom. Previous studies have predominantly applied this method to idealistic structures. In this work, the nonlinear modal modeling technique is applied to a more realistic industrial aerospace structure which exhibits complex bilinear behavior. Linear natural frequencies, damping values, and mode shapes are first extracted from low level shaker testing. Subsequently, the structure is excited using high level tailored shaker inputs. The resulting response data are modally filtered and used to empirically derive the nonlinear elements which, together with their linear counterparts, comprise the nonlinear modal model. This model is then used in both modal and physical domain simulations. Comparisons to measured data are made and the performance of the nonlinear modal model to predict this complex bilinear behavior is discussed.
This work extends recent methods to calculate dynamic substructuring predictions of a weakly nonlinear structure using nonlinear pseudo-modal models. In previous works, constitutive joint models (such as the modal Iwan element) were used to capture the nonlinearity of each subcomponent on a mode-by-mode basis. This work uses simpler polynomial stiffness and damping elements to capture nonlinear dynamics from more diverse jointed connections including large continuous interfaces. The proposed method requires that the modes of the system remain distinct and uncoupled in the amplitude range of interest. A windowed sinusoidal loading is used to excite each experimental subcomponent mode in order to identify the nonlinear pseudo-modal models. This allows for a higher modal amplitude to be achieved when fitting these models and extends the applicable amplitude range of this method. Once subcomponent modal models have been experimentally extracted for each mode, the Transmission Simulator method is implemented to assemble the subcomponent models into a nonlinear assembled prediction. Numerical integration methods are used to evaluate this prediction compared to a truth test of the nonlinear assembly.
Effective mass for a particular mode in a particular direction is classically calculated using a combination of fixed base mode shapes, the mass matrix, and a rigid body mode shape from a finite element model. Recently, an experimental method was developed to calculate effective mass using free experimental mode shapes of a structure on a fixture (the base) along with the measured mass of the fixture and of the test article. The method required three steps. The first step involved constraining all the free modes of the fixture except one rigid body mode in the direction of interest. The second step involved calculating pseudo-modal participation factors for this case. The third step involved constraining the final fixture rigid body degree of freedom and utilizing the constraint matrices with pseudo-modal participation factors to obtain the estimate of the standard modal participation factors which can be converted to effective mass. This work provides a simpler formulation. After the constraint in step one above, the effective masses are calculated directly from the mass normalized mode shapes of the fixture. In most cases this method gives the same answer as the original approach, within experimental error. In some instances, it appears more robust with low signal to noise ratios. It also provides better physical insight as to which modes have significant effective mass in a particular direction. The new approach is illustrated by experimental example.
Linear structural dynamic models are often used to support system design and qualification. Overall, linear models provide an efficient means for conducting design studies and augmenting test data by recovering un-instrumented or unmeasurable quantities (e.g. stress). Nevertheless, the use of linear models often adds significant conservatism in design and qualification programs by failing to capture critical mechanisms for energy dissipation. Unfortunately, the use of explicit nonlinear models can require unacceptably large efforts in model development and experimental characterization to account for common nonlinearities such as frictional interfaces, macro-slip, and other complex material behavior. The computational requirements are also greater by orders of magnitude. Conversely, modal models are much more computationally efficient and experimentally have shown the ability to capture typical structural nonlinearity. Thus, this work will seek to use modal nonlinear identification techniques to improve the predictive capability of a finite element structural dynamics model. Part I of this paper discussed experimental aspects of this work. Part II will consider use of nonlinear modal models in finite element modeling. First, the basic theory and numerical implementation is discussed. Next, the linear structural dynamic model of a configuration of interest is presented and model updating procedures are discussed. Finally, verification exercises are presented for a high level excitation using test data and simulated predictions from a structural dynamics model augmented with models obtained in nonlinear identification efforts.
Effective mass for a particular mode in a particular direction is classically calculated using a combination of fixed base mode shapes, the mass matrix, and a rigid body mode shape from a finite element model. Recently, an experimental method was developed to calculate effective mass using free experimental mode shapes of a structure on a fixture (the base) along with the measured mass of the fixture and of the test article. The method required three steps. The first step involved constraining all the free modes of the fixture except one rigid body mode in the direction of interest. The second step involved calculating pseudo-modal participation factors for this case. The third step involved constraining the final fixture rigid body degree of freedom and utilizing the constraint matrices with pseudo-modal participation factors to obtain the estimate of the standard modal participation factors which can be converted to effective mass. This work provides a simpler formulation. After the constraint in step one above, the effective masses are calculated directly from the mass normalized mode shapes of the fixture. In most cases this method gives the same answer as the original approach, within experimental error. In some instances, it appears more robust with low signal to noise ratios. It also provides better physical insight as to which modes have significant effective mass in a particular direction. The new approach is illustrated by experimental example.
This work uses a method whereby weak nonlinearity in a substructure, as typically arises due to microslip in bolted interfaces, can be captured and modeled on a mode-by-mode basis. The method relies on the fact that the modes of a weakly nonlinear structure tend to remain uncoupled so long as their natural frequencies are distinct and higher harmonics generated by the nonlinearity do not produce significant response in other modes. A single degree-of-freedom (DOF) system with an Iwan joint, which is known as a modal Iwan model, effectively captures the way in which the stiffness and damping depend on amplitude for each mode. This work presents the experiments used to generate these modal Iwan models. In a companion paper this model is assembled to another component using dynamic substructuring techniques to estimate the amplitude dependent frequency and damping of the full assembly.
In a companion paper (Roettgen, D.R., et al.: Substructuring of a nonlinear beam using modal Iwan framework, Part 1: nonlinear modal model identification. Presented at the international modal analysis conference XXXV, Garden Grove, 2017), “Substructuring of a nonlinear beam using modal Iwan framework, Part I: Nonlinear Modal Model Identification”, nonlinear modal models are constructed for an experimental substructure that represent the dynamics using a set of uncoupled weakly nonlinear modes. This assumes that the linear modes of the structure remain uncoupled so that the nonlinearity can be described in a mode by mode fashion. These nonlinear modal models can be used to simulate the response of the experimental system. This paper demonstrates the use of these models to represent a substructure in an experimental-analytical substructuring prediction. The authors utilize the transmission simulator method on the experimentally derived models to generate predictions of a modified Brake-Reuss Beam system. The substructuring predictions are then compared to a truth test data set to validate the method. To further understand the limitations of the method and its sensitivity to measurement noise, the modal substructuring approach is also simulated on a finite element model of the beam that contains three discrete nonlinear elements to represent the joint.