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Dynamics of Coupled Structures, Volume 5: Proceedings of the 39th IMAC, A Conference and Exposition on Structural Dynamics, 2021, the fourth volume of nine from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of the Dynamics of Coupled Structures, including papers on:

Methods for Dynamic Substructures

Applications for Dynamic Substructures

Interfaces & Substructuring

Frequency Based Substructuring

Transfer Path Analysis



Chapter 1. A Comparative Study of Joint Modeling Methods and Analysis of Fasteners

One of the more crucial aspects of any mechanical design is the joining methodology of parts. During structural dynamic environments, the ability to analyze the joint and fasteners in a system for structural integrity is fundamental, especially early in a system design during design trade studies. Different modeling representations of fasteners include spring, beam, and solid elements. In this work, we compare the various methods for a linear system to help the analyst decide which method is appropriate for a design study. Ultimately, if stresses of the parts being connected are of interest, then we recommend the use of the Ring Method for modeling the joint. If the structural integrity of the fastener is of interest, then we recommend the Spring Method.
Ricardo Garcia, Michael Ross, Benjamin Pacini, Daniel Roettgen

Chapter 2. Historical Perspective of the Development of Digital Twins

With modern advances in high-performance computing, design engineers have put a large focus on digital testing and simulations to inform new systems. In addition, recent market tendencies show a desire to reduce waste and for longer designed life. One major strategy used to meet these trends is the utilization of a digital twin. Digital twins are numerical analogues to physical systems such as aircraft, auto-mobiles, and power generation systems. With the wide applicability of the digital twin, an understanding of their development can give insight into the impact and direction of recent research. Understanding these advancements can also give confidence in both the technique of using a digital twin and the simulated predictions to various loading conditions. This chapter focuses on detailing the historical development of digital twins to the state-of-the-art research being done and specifically how it is relevant to the structural dynamics community.
Matthew S. Bonney, David Wagg

Chapter 3. Distributed Home Labs at the Time of the Covid

The well-known difficulties with the recent pandemic have forced people to find new communication means, especially concerning educational methods. University courses have been dramatically changed in their structure, just in a few weeks. While traditional lectures have been more easily switched to “on-line” methods and tools, classes mainly based on experimental lab activities have suffered more from the new forced approaches: a new and stronger effort had to be produced to guarantee the proper knowledge transfer. Many ways have been tried to export experimental labs into students’ houses, preserving and stimulating their curiosity. However, there was a risk to foster a more passive role; students watching a movie or listening to a faraway teacher could not have direct interaction with the instrumentation locked in not accessible labs, nor had the important chance to develop “hands-on” sessions.
This paper deals with the ideas and attempts to preserve the value of the experimental activities during the COVID period, in which experimentation has also meant experimenting a new way of teaching; early attempts will be described up to a final proposal, which has been successfully tested with students of both the bachelor and the master of science.
A. Cigada, S. Manzoni

Chapter 4. Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed

The equations of motion (EOM) for the heavy symmetrical top with one point fixed are highly nonlinear. The literature describes the numerical methods that are used to resolve this classical system, including modern tools, such as the Runge−Kutta fourth−order method. Finding the derivate of closed-form solutions for the EOM is more difficult and, as mentioned in the literature, discovering the solution is not always possible for all the EOM. Fortunately, a few examples are available that serve as a guide to move further in this topic. The purpose of this paper is to find a methodology that will produce the solutions for a given subset of EOMs that fulfill certain requisites. This paper summarizes the literature available on this topic and then follows with the derivation of the EOM using the Euler−Lagrange method. The Routhian method will be used to reduce the size of the expression, and it continues with the formulation of the classical cubic function, f(u), through a novel process. The roots of f(u) are of the utmost importance in finding the EOM closed-form solution, and once the final roots are selected, the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process, and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.
Hector Laos

Chapter 5. Equations of Motion for the Vertical Rigid-Body Rotor: Linear and Nonlinear Cases

Centuries ago, the prolific mathematician Leonhard Euler (1707–1783) wrote down the equations of motion (EOM) for the heavy symmetrical top with one point fixed. The resulting set of equations turned out to be nonlinear and had a limited number of closed-form solutions.
Today, tools such as transfer matrix and finite elements enable the calculation of the rotordynamic properties for rotor-bearing systems. Some of these tools rely on the “linearized” version of the EOM to calculate the eigenvalues, unbalance response, or transients in these systems.
In fact, industry standards mandate that rotors be precisely balanced to have safe operational characteristics. However, in some cases, the nonlinear aspect of the EOM should be considered.
The purpose of this chapter is to show examples of how the linear vs. nonlinear formulations differ. This chapter also shows how excessive unbalance is capable of dramatically altering the behavior of the system and can produce chaotic motions associated with the “jump” phenomenon.
Hector Laos

Chapter 6. Vibration Control in Meta-Structures Using Reinforcement Learning

This chapter considers using reinforcement learning (RL) to adaptively tune frequency response functions of meta-structures. RL algorithm tunes the stiffness of the spring of the lumped multi-DOF system, as the lumped mass is varied. As some of the lumped masses are modified by 10%, the spring’s stiffness is tuned to maintain the original bandgap. A Q-Learning algorithm is used for RL, wherein the Q-value is updated based on Bellman’s equation. The results compare the frequency response functions of the terminal masses of the baseline and varied mass structure.
D. Mehta, Vijaya V. N. Sriram Malladi

Chapter 7. Using Steady-State Ultrasonic Direct-Part Measurements for Defect Detection in Additively Manufactured Metal Parts

With the increasing availability and implementation of additive manufacturing in a variety of safety-critical implementations, there is a demand for non-destructive in situ quality control processes that ensure part-to-part and build-to-build repeatability. Traditional destructive evaluation methods are not financially sustainable for most additive manufacturing applications due to the long build times and the expense of manufacturing parts specifically for destruction.
Acoustic wavenumber spectroscopy, a rapid steady-state non-destructive evaluation technique that has successfully identified defects such as delamination in carbon fiber reinforced panels and weld cracking, has been modified to collect in situ direct-part measurements in a laser powder bed fusion machine for 304L stainless steel builds. During each layer of the build, a laser Doppler vibrometer records the part surface response to an ultrasonic steady-state excitation. The result is a 3D inspection volume constructed alongside the part, offering detail of the potential faults and defects within.
Various signal processing techniques were used to identify features sensitive to lack-of-fusion defects within the part. Statistical and machine learning methods were used to identify whether a direct-part response measurement is “nominal” or “abnormal.” X-ray computed tomography was used to verify the defect regions post-build and allow for accurate truth data generation.
These validated features will enable simultaneous in situ data collection and processing. The operator may be alerted of potential defects and their locations as defect-indicative features form, allowing the operator to pause the build or adjust the print parameters depending on part specifications.
Erica M. Jacobson, Ian T. Cummings, Peter H. Fickenwirth, Eric B. Flynn, Adam J. Wachtor

Chapter 8. Toward Developing Arrays of Active Artificial Hair Cells

The human cochlea perceives frequencies over a range of 20 Hz to 20 kHz, while a section of the organ of Corti in the cochlea transduces a particular frequency. The cochlear amplifier then amplifies or compresses the signal based on the stimulus level. An individual artificial hair cell (AHC) made of a piezoelectric beam with a feedback controller replicates the cochlear amplifier at a particular frequency. However, to capture a wider frequency range and mimic the tonotopic basilar membrane, an array of AHCs is required. Thus, numerical modeling of an array of active beams with different geometries is the focus of this chapter. With the dynamics of a single active artificial hair cell established, an array of AHCs with self-sensing characteristics is developed. A sample array is modeled using a few sensors to transduce a small set of frequencies in the human speech frequency range. The AHC array is simulated in Simulink and its response is controlled using a nonlinear cubic damping feedback control law that was presented in the authors’ previous work (Davaria, S., Malladi, V.V.S., Motaharibidgoli, S., Tarazaga, P.A.: Cochlear amplifier inspired two-channel active artificial hair cells. Mechanical Systems and Signal Processing, 129, 568–589 (2019)). The response of the active system to complex stimuli with multiple frequencies is analyzed. The design, modeling, and feedback control techniques developed in the current work will be applicable to future sensor arrays and cochlear implants with more beam elements. Because each sensor will utilize the active AHC technology, the total array will offer advantages over a passive series of cantilevers or traditional sensors.
Sheyda Davaria, Pablo A. Tarazaga

Chapter 9. Challenges Associated with In Situ Calibration of Load Cells in Force-Limited Vibration Testing

The difference in mounting configuration between flight and test can significantly impact the effectiveness of the test in environmental vibration testing. Many tests are performed with large electrodynamic shakers, which utilize interfaces that seek to replicate a fixed base, such as slip tables and head expanders. This fixed base configuration is rarely seen in flight configurations; rather a more realistic configuration would include a flexible mounting structure with its own compliance and dynamics. This causes significant over- and under-tests in various frequency bands depending on the differences between the test article and fixture dynamics.
The traditional way of avoiding these high loads is to limit the acceleration responses at multiple locations on the test article. However, the effectiveness of this approach is highly dependent upon the validity of the test article’s analytical in order to derive accurate acceleration response limit specifications. Also, this technique requires limiting the acceleration responses at many locations throughout the test article, which may not be implementable due to such things as access issues and cleanliness issues. An improved environmental vibration testing technique known as force limiting incorporates measurements of the forces between the test article and shaker system interface and limiting them to a specification that more accurately replicates the interface impedance of the structure the test article will be mounted to in flight. In effect, this transforms the high mechanical impedance at the test article to shaker interface to more closely match the mechanical impedance of the flight interface, which avoids producing the unrealistically high interface loads.
Typically, force gauges or load cells are used to measure these interface forces. However, utilizing load cells can present a multitude of challenges depending upon such things as their installation method, geometric layout, and test fixture setup. Regardless, it is important to perform an in situ calibration of the load cells prior to vibration testing at any significant levels. This chapter will discuss the challenges associated with utilizing load cells during the NASA Evolutionary Xenon Thruster – Commercial (NEXT-C) gridded ion thruster proto-flight vibration test performed at the NASA Glenn Research Center’s Structural Dynamics Laboratory.
Kenneth J. Pederson, Vicente J. Suarez, Emma L. Pierson, Kim D. Otten, James C. Akers, James P. Winkel
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