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.
A previous study in the UK demonstrated that vibration response on a scaled-down model of a missile structure in a wind tunnel could be replicated in a laboratory setting with multiple shakers using an approach dubbed as impedance matching. Here we demonstrate on a full scale industrial structure that the random vibration induced from a laboratory acoustic environment can be nearly replicated at 37 internal accelerometers using six shakers. The voltage input to the shaker amplifiers is calculated using a regularized inverse of the square of the amplitude of the frequency response function matrix and the power spectral density responses of the 37 internal accelerometers. No cross power spectral density responses are utilized. The structure has hundreds of modes and the simulation is performed out to 4000 Hz.
Some initial investigations have been published which simulate nonlinear response with almost traditional modal models: instead of connecting the modal mass to ground through the traditional spring and damper, a nonlinear Iwan element was added. This assumes that the mode shapes do not change with amplitude and there are no interactions between modal degrees of freedom. This work expands on these previous studies. An impact experiment is performed on a structure which exhibits typical structural dynamic nonlinear response, i.e. weak frequency dependence and strong damping dependence on the amplitude of vibration. Use of low level modal test results in combination with high level impacts are processed using various combinations of modal filtering, the Hilbert Transform and band-pass filtering to develop response data that are then fit with various nonlinear elements to create a nonlinear pseudo-modal model. Simulations of forced response are compared with high level experimental data for various nonlinear element assumptions.
Experimental dynamic substructuring is a means whereby a mathematical model for a substructure can be obtained experimentally and then coupled to a model for the rest of the assembly to predict the response. Recently, various methods have been proposed that use a transmission simulator to overcome sensitivity to measurement errors and to exercise the interface between the substructures; including the Craig-Bampton, Dual Craig-Bampton, and Craig-Mayes methods. This work compares the advantages and disadvantages of these reduced order modeling strategies for two dynamic substructuring problems. The methods are first used on an analytical beam model to validate the methodologies. Then they are used to obtain an experimental model for structure consisting of a cylinder with several components inside connected to the outside case by foam with uncertain properties. This represents an exceedingly difficult structure to model and so experimental substructuring could be an attractive way to obtain a model of the system.
Conference Proceedings of the Society for Experimental Mechanics Series
Allen, Matthew S.; Roettgen, Daniel; Kammer, Daniel; Mayes, R.L.
This work proposes a means whereby weak nonlinearity in a substructure, as typically arises due to friction in bolted interfaces, can be captured experimentally on a mode-by-mode basis and then used to predict the nonlinear response of an assembly. 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. Recent experiments on industrial hardware with bolted joints has shown that this type of model can be quite effective, and that 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. Once the modal Iwan models have been identified for each mode of the subcomponent(s) of interest, they can be assembled using standard techniques and used with a numerical integration routine to compute the nonlinear transient response of the assembled structure. The proposed methods are demonstrated by coupling a modal model of a 3DOF system with three discrete Iwan joints to a linear model for a 2DOF system.
Qualification of complex systems typically involves testing the components individually in shock and vibration environments before assembling them into the system. When the components are secured to a fixture on the shaker table, the mechanical impedance of the boundary condition is quite different from that of the next level of assembly. Thus the modes of the component under test are not excited in the same way that they are excited in the system using the typical methods for defining input specifications. Here, the boundary condition impedance is investigated and quantified using substructuring techniques. Also, fixture inputs are derived to overcome the impedance differences and excite a component in the same way it is excited in the next level of assembly.
Experimental-analytical substructuring is attractive when there is motivation to replace one or more system subcomponents with an experimental model. This experimentally derived substructure can then be coupled to finite element models of the rest of the structure to predict the system response. The transmission simulator method couples a fixture to the component of interest during a vibration test in order to improve the experimental model for the component. The transmission simulator is then subtracted from the tested system to produce the experimental component. The method reduces ill-conditioning by imposing a least squares fit of constraints between substructure modal coordinates to connect substructures, instead of directly connecting physical interface degrees of freedom. This paper presents an alternative means of deriving the experimental substructure model, in which a Craig-Bampton representation of the transmission simulator is created and subtracted from the experimental measurements. The corresponding modal basis of the transmission simulator is described by the fixed-interface modes, rather than free modes that were used in the original approach. These modes do a better job of representing the shape of the transmission simulator as it responds within the experimental system, leading to more accurate results using fewer modes. The new approach is demonstrated using a simple finite element model based example with a redundant interface.
Experimental dynamic substructuring is a means whereby a mathematical model for a substructure can be obtained experimentally and then coupled to a model for the rest of the assembly to predict the response. Recently, various methods have been proposed that use a transmission simulator to overcome sensitivity to measurement errors and to exercise the interface between the substructures; including the Craig-Bampton, Dual Craig-Bampton, and Craig-Mayes methods. This work compares the advantages and disadvantages of these reduced order modeling strategies for two dynamic substructuring problems. The methods are first used on an analytical beam model to validate the methodologies. Then they are used to obtain an experimental model for structure consisting of a cylinder with several components inside connected to the outside case by foam with uncertain properties. This represents an exceedingly difficult structure to model and so experimental substructuring could be an attractive way to obtain a model of the system.