Analyses, Model Tests and Design Modifications for Vibration/Noise Mitigation K. C. Park Center for Aerospace Structures Department of Aerospace Engineering Sciences University of Colorado at Boulder, CO, USA Presented at Lund University October 28, 2013
General procedures for structural dynamics analysis Step 0: Pre-analysis. If the structure consists of many dissimilar components, then partition them into several substructures while mindful of its performance requirements; Step 1: Why free-free? Construct free-free models of each substructure; Step 2: Sequential BC applications. By introducing a point, line or surface boundary conditions in a step-by-step sequence, observe how each boundary condition modifies the dynamic behavior of substructures;
General procedures for structural dynamics analysis - Continued Step 3: BC Uncertainties. Include boundary uncertainties(e.g., not fully fixed, etc.) and carry out vibration analysis of each substructure to see how realistic boundary conditions would influence the dynamic characteristics of each substructure; Step 04: CMS. Carry out component mode synthesis of the total structure, while mindful of several aspects expected frequency contents of loads, instrumentation limitations (sensors and actuator locations), visualizing the energy flow pattern, and frequency range of interest of each substructures. A separate lecture will follow; Step 05: Analysis. Carry out vibration analysis utilizing the reduced-order model, and finalize the vibration test setup based on the analysis results;
General procedures for structural dynamics analysis - Continued Step 06: Pre-test analysis. Carry out simulation by placing the actuators (loads) and accelerometers and strain gages (sampling points). Adjust their locations, if necessary, to maximize the data richness; Step 07: Pretest setup: After initial instrumentation is completed, run preliminary tests and adjust instrumentation locations, if necessary. Carry out preliminary signal processing to see whether expected frequency contents are captured; Step 08: Modal Test. Carry out a multitude set of vibration tests, and perform system realization to obtain modes, mode shapes and damping. A separate lecture will follow.
General procedures for structural dynamics analysis - Continued Step 09: Reconciliation. Based on the experimental data, adjust boundary uncertainties. If necessary, modify or regenerate the reduced-order model such that the reduced-order model matches the experimental modes, mode shapes and damping; Step 10: Energy Map. Now use the experiment-updated model to study energy flow patterns. This is an important exercise to assess how noise/vibration energy would propagate through the structure. A separate lecture would follow if time permits. Step 11: Synthesis. Based on the knowledge gained so far, synthesize noise/vibration mitigation strategy and/or modify the design. Step 12: Iterations. Repeat Steps 1-11 as many times as necessary to arrive at the design to meet the noise/vibration requirements.
Step 0: Pre-analysis partitioning the total structure into substructures. This is perhaps the most crucial step as the actions one takes at this step will affect all the subsequent analysis, test and design improvement, etc. Understand thoroughly the performance requirements, e.g., max. vibration amplitudes, noise levels, frequency response curves, and other related aspects. Take time and iterate as many times as necessary. Here are a sample of to-do list (not comprehensive): Identify boundaries, interfaces, joints, constraints, experienced vibration sources and noise origins; If possible, partition the structure into substructures along these boundaries, interfaces, etc., and NOT by the size of each partitions (computational efficiency should not dictate partitioning) Categorize each partition according to its dominant structural behavior, e.g., beam, plate, shell, solids, shaft, discrete elements, etc.
Step 1: Why free-free structure? Construct free-free models of each substructure Vibration modes, mode shapes of free-free states yield the most of each substructure as well as the total structure. The mode shape correlation properties between the global mode shapes and local mode shapes offer critical physical insight into how the dynamic characteristics influence those of the total structures. It is well known that free-free mode shape correlations reveal uninhibited local-global interactions than utilizing constrained local modes (the Craig-Bampton CMS method relies on constrained local modes! We will talk more about this aspect later.) In reality, realistic boundary conditions and interfaces between substructures are not completely fixed. By varying the boundary/ interface constraints, one can identify the influence of partial constraints. An important modeling insight.
Step 2: Apply BCs sequentially! The Sturm sequence theorem offers the upper and lower bounds when the matrix sizes are decreased or increased. However, it says nothing about the mode shape changes! By sequentially applying one boundary condition at a time, one can obtain a visual picture of how each boundary conditions impact the modes and mode shapes. This knowledge plays an important role for the design of continuous beams, for example, with many intermediate supports. Or structures with many appendages. Conversely, assemble one substructure at a time to see how an addition of a substructure changes modes and mode shapes. In a most simplistic view, assembling one substructure to an existing partially built structure amounts to adding a spring-mass boundary constraint from the corresponding free-free boundary.
Step 3: BC Uncertainties. Whenever feasible, consider boundary uncertainties (e.g., not fully fixed, etc.) and carry out vibration analysis of each substructure to see how realistic boundary conditions would influence the dynamic characteristics of each substructure. Experience indicates that often the deviations of analytical model from test results are due to imperfect boundary conditions/ substructure-tosubstructure interface conditions. Hence, sensitivity analysis of reasonable boundary uncertainties should be carried out in advance. It should be pointed out that the interior modes are almost unaffected by variations of boundary uncertainties. However, mode shapes for regions near the boundaries are strongly affected, which in turn play dominant roles in energy flow paths.
Step 04: Component Mode Synthesis (CMS) (A separate lecture will follow) A personal bias this lecturer prefers Flexibility-based CMS to Stiffnessbased CMS (cf., Craig-Bampton method). FCMS methods utilize free-free substructural modes, whereas SCMS methods utilize fixed interior modes. To date, SCMS methods lack mode selection guides, thus mode selections are made based on heuristic or experienced-based mode selection. On the other hand, FCMS methods have as part of the theory a mode selection criterion. From a computational viewpoint, FCNMS is parallelizable whereas SCMS is not.
Step 05: Vibration Analysis Using the CMS-generated reduced-order model, carry out vibration analysis and map out the necessary vibration test setup based on the analysis results. Once the global vibration analysis is completed, carry out correlations studies, viz., projecting the local substructural modes unto the global modes. This will provide a ranking of local modes in terms of their contributions to the global modes. From an energy transmission viewpoint, this offers a picture of interaction of local modes with the global modes, or how local modes contribute to the overall vibration patterns. Map global and local vibration nodal lines and observe their overlaps, if any, and significant interval differences. This can be utilized in vibration control algorithm strategies, and/or applying internal constraints, etc. Note these are not quantitative, but rather qualitative concepts.
Step 06: Pre-Test Analysis or Experiment Design Carry out simulation, that is transient analysis, by placing the actuators (loads) and accelerometers and strain gages (sampling points). Adjust their locations, if necessary, to maximize the data richness. It is important that one apply the actuation loads whose frequency contents are rich enough to excite as many local and global modes as possible. In so doing, first, perform test simulation for each of the substructures, and assess if the substructural modes you have selected. After going over one by one all the substructures, then perform experiment simulations. Based on the simulated input and output, perform preliminary system realization. This process will offer instrumentation placement and signal processing requirements.
Step 07: Pretest Setup After initial instrumentation is completed, run preliminary tests and adjust instrumentation locations, if necessary. Carry out preliminary signal processing to see whether expected frequency contents are captured. Concurrently, rerun your system realization simulation and see whether the preliminary processed signals from the pre-tests are captured in the simulation results. If not, better identify the sources of deviations at this stage, and re-examine the pre-test setup as well as the fidelity of the simulation results. Remember we are adopting model-based design synthesis eventually. Based on the preceding information, make sure that the boundary conditions of the test substructures (most likely scale models!) and the total structure are realistic enough; if not, account for the BCs of test articles and the real systems.
Step 08: Modal Test (A separate lecture will follow). A distinct feature with the aerospace practice in modal testing from civil engineering, is the time-domain approach. While theoretically equivalent, frequency-domain processing of test data are in effect heavily filtered data. Sometimes, one may through out the baby with bath water. Avoid the sinusoidal excitations or sine dwell sweeps, if possible. Apply burst random excitations so that the identified modes and mode shapes are as close to natural vibration states as possible. A minor detail: do not store down-sampled data; store all data and then let the realization algorithm take over down-sampling along the any ensembles If deemed needed. Finally, perform system realization to obtain modes, mode shapes and damping utilizing the test data.
Step 09: Reconciliation. Identify the differences between the simulated realization results and the identified models from test data. Trace each difference not only in model errors (including BCs) but also possible test setup peculiarities (plus BCs on the test arrangements). After resolving all the possible deviations, if there are still differences between test and simulated results, then update the model incorporating the test data such that the realization of both simulated case and the test case would be reconciled. Experience indicates that often boundary uncertainties emerge as major sources of errors in the model. Hence, modify or regenerate the reducedorder model such that the reduced-order model faithfully captures the testsetup BCs. Others such as ambient damping, ground vibrations should be identified at this stage.
Step 10: Energy Map (A separate lecture would follow if time permits.) Now use the experiment-updated reduced-order model (if possible, as well as the full model of the total structure) to study energy flow patterns. This is an important exercise to assess how noise/vibration energy would propagate through the structure. In doing so, pay attention to substructural interfaces, structure-to-ground, and one substructure-to-another energy flow patterns. Additionally, obtain energy flow rate and energy flow directions. Identify critical regions such as ``excessive energy accumulation or excessive energy flow rate parts of the structure. This will be considered as potential design modifications, passive/active vibration control targets.
Step 11: Synthesis (This is where the rubber meets the road!). Set the synthesis goal(s) in terms of operational frequency ranges, maximum vibration amplitudes, maximum noise levels, and weight an weight distributions In aerospace practice, it is often easier and effective to utilize active control schemes to design vibration and noise mitigation, then try to realize the synthesis outcome by means of passive control and design modifications. For long-span beam-like structures that need multi-span supports to mitigate vibration and noses, partially constrained boundary conditions are preferred practice of assessing their effects. Based on the knowledge gained so far, synthesize noise/vibration mitigation strategy and/or modify the design.
Step 12: Iterations. If all the goals are met, then that is it! If not, Go to Step 0 and start all over again, if necessary or if time /budget permit. Review all the actions taken, results obtained, design requirements both met and unmet. Rank which of the 11 steps are least achieved, and plan strategies or methods by which goals at each step can be met. Make a virtual iteration and identify bottlenecks (if any) so that maximum improvements can be realized. Now, repeat Steps 1-11 to arrive at the design to meet the noise/vibration requirements. Deliver your design, analysis, experiment, system identification, and design synthesis to your customer. The project is finished.
Miscellaneous Considerations (perhaps more important to academics) Identify any lack of tools, theories, instrumentation needs that would have materially eased analysis, experiment and design synthesis. Match the research needs with talents available (or recruit new students / postdoctoral fellows) and embark on new projects! Go and get the necessary funding! And have fun carrying out research related to the research needs identified. Good luck! And have fun!