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Abstract Quasi-periodic motions can be numerically found in piecewise-linear systems, however, their characteristics have not been well understood. To illustrate this, an incremental harmonic balance (IHB) method with two timescales is extended in this work to analyze quasi-periodic motions of a non-smooth dynamic system, i.e., a gear transmission system with piecewise linearity stiffness. The gear transmission system is simplified to a four degree-of-freedom nonlinear dynamic model by using a lumped mass method. Nonlinear governing equations of the gear transmission system are formulated by utilizing the Newton’s second law. The IHB method with two timescales applicable to piecewise-linear systems is employed to examine quasi-periodic motions of the gear transmission system whose Fourier spectra display uniformly spaced sideband frequencies around carrier frequencies. The Floquet theory is extended to analyze quasi-periodic solutions of piecewise-linear systems based on introduction of a small perturbation on a steady-state quasi-periodic solution of the gear transmission system with piecewise linearities. Comparison with numerical results calculated using the fourth-order Runge-Kutta method confirms that excellent accuracy of the IHB method with two timescales can be achieved with an appropriate number of harmonic terms. 

This work investigates the steady-state nonlinear dynamics of a large-deformation flexible beam model under oscillatory flow. A flexible beam dynamics model combined with hydrodynamic loading is employed using large deformation beam theory. The equations of motion discretised using the high-order finite element method (FEM) are solved in the time domain using the efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method. The arc-length continuation method and Hsu’s method trace stable and unstable solutions. The numerical results are in accordance with the physical experimental results and reveal multiple resonance phenomena. Low-order resonances exhibit hardening due to geometric nonlinearity, while higher-order resonances transition from softening to hardening influenced by inertia and geometric nonlinearity. A strong coupling between tensile and bending deformation is observed. The axial deformation is dominated by inertia, while bending resonance is influenced by an interplay between inertia, structure stiffness, and fluid drag. Finally, the effects of two dimensionless parameters, Keulegan and Carpenter number (KC) and Cauchy number (Ca), on the response of the flexible beam are discussed. 

We rigorously quantify the improvement in the sample complexity of variational divergence estimations for group-invariant distributions. In the cases of the Wasserstein-1 metric and the Lipschitz-regularized $$\alpha$$-divergences, the reduction of sample complexity is proportional to an ambient-dimension-dependent power of the group size. For the maximum mean discrepancy (MMD), the improvement of sample complexity is more nuanced, as it depends on not only the group size but also the choice of kernel. Numerical simulations verify our theories. 

In 2019, University of Houston (UH) at Houston, Texas was awarded an NSF Research Experience for Teachers (RET) site grant titled “RET Site: High School Teacher Experience in Engineering Design and Manufacturing.” The goal of the project is to host 12 high school teachers each summer to participate in engineering design and manufacturing research and then convert their experience into high school curriculum. In summer of 2021, the first cohort of 12 teachers from Region 4 of Southeast Texas participated in the RET program at UH College of Technology (COT). This six-week program, open to local high school STEM teachers in Texas, sought to advance educators’ knowledge of concepts in design and manufacturing as a means of enriching high school curriculums and meeting foundational standards set by 2013’s Texas House Bill 5. These standards require enhanced STEM contents in high school curricula as a prerequisite for graduation, detailed in the Texas Essential Knowledge and Skills standard. Due to the pandemic situation, about 50% of the activities are online and the rest are face to face. About 40% of the time, teachers attended online workshops to enhance their knowledge of topics in engineering design and manufacturing before embarking on applicable research projects in the labs. Six UH COT engineering technology professors each led workshops in a week. The four tenure-track engineering mentors, assisted by student research assistants, each mentored three teachers on projects ranging from additive manufacturing to thermal/fluids, materials, and energy. The group also participated in field trips to local companies including ARC Specialties, Master Flo, Re:3D, and Forged Components. They worked with two instructional track engineering technology professors and one professor of education on applying their learnings to lesson plan design. Participants also met weekly for online Brown Bag teacher seminars to share their experiences and discuss curricula, which was organized by the RET master teacher. On the final day of the program, the teachers presented their curriculum prototype for the fall semester to the group and received completion certificates. The program assessment was led by the assessment specialist, Director of Assessment and Accreditation at UH COT. Teacher participants found the research experience with their mentors beneficial not only to them, but also to their students according to our findings from interviews. The mentors will visit their mentees’ classrooms to see the lesson plans being implemented. In the spring of 2022, the teachers will present their refined curricula at a RET symposium to be organized at UH and submit their standards-aligned plans to teachengineering.org for other K-12 educators to access. 

Incorporating group symmetry directly into the learning process has proved to be an effective guideline for model design. By producing features that are guaranteed to transform covariantly to the group actions on the inputs, group-equivariant convolutional neural net- works (G-CNNs) achieve significantly improved generalization performance in learning tasks with intrinsic symmetry. General theory and practical implementation of G-CNNs have been studied for planar images under either rotation or scaling transformation, but only individu- ally. We present, in this paper, a roto-scale-translation equivariant CNN (RST-CNN), that is guaranteed to achieve equivariance jointly over these three groups via coupled group con- volutions. Moreover, as symmetry transformations in reality are rarely perfect and typically subject to input deformation, we provide a stability analysis of the equivariance of representation to input distortion, which motivates the truncated expansion of the convolutional filters under (pre-fixed) low-frequency spatial modes. The resulting model provably achieves deformation-robust RS T equivariance, i.e., the RST symmetry is still “approximately” preserved when the transformation is “contaminated” by a nuisance data deformation, a property that is especially important for out-of-distribution generalization. Numerical experiments on MNIST, Fashion-MNIST, and STL-10 demonstrate that the proposed model yields remarkable gains over prior arts, especially in the small data regime where both rotation and scaling variations are present within the data.