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Canonical equations play a pivotal role in various sub-fields of physics and mathematics. However, for complex systems and systems without first principles, deriving canonical equations analytically is quite laborious or might even be impossible. This work is devoted to automatedly distilling the canonical equations solely from random state data. The random state data are collected from stochastically excited, dissipative dynamical systems either experimentally or numerically, while other information, such as the system characterization itself and the excitations, is not needed. The identification procedure comes down to a nested optimization problem, and the explicit expressions of the momentum (density) functions and energy (density) functions are identified simultaneously. Three representative examples are investigated to illustrate its high accuracy of identification, the small requirement for data amount, and high robustness to excitations and dissipation. The identification procedure serves as a filter, filtering out nonconservative information while retaining conservative information, which is especially suitable for systems with unobtainable excitations.

The simplified governing equations of applied mechanics play a pivotal role and were derived based on ingenious assumptions or hypotheses regarding the displacement fields for specific problems. In this paper, we introduce a data-driven method by the name AI-Timoshenko in honor of Timoshenko, father of applied solid mechanics, to automatically discover simplified governing equations for applied mechanics problems directly from discrete data simulated by the three-dimensional (3D) finite element method. The simplified governing equations are in variational form, which is compact and advantageous for data-driven discovery. AI-Timoshenko liberates applied mechanicians from burdensome labor, including assumptions, derivation, and trial and error. The simplified governing equations for Euler–Bernoulli and Timoshenko beam theories are successfully rediscovered using the present AI-Timoshenko method, which shows that this method is capable of discovering simplified governing equations for applied mechanics problems.

The response control of nonlinear random dynamical systems is an important but also difficult subject in scientific and industrial fields. This work merges the decomposition technique of feedback control and the data-driven identification method of stationary response probability density, converts the constrained functional extreme value problem associated with optimal control to an unconstrained optimization problem of multivariable function, and determines the optimal coefficients of preselected control terms by an optimization algorithm. This data-driven method avoids the difficulty of solving the stochastic dynamic programming equation or forward–backward stochastic differential equations encountered in classical control theories, the miss of the conservative mechanism in the nonlinear stochastic optimal control strategy, and the difficulty of judging the integrability and resonance of the controlled Hamiltonian systems encountered in the direct-control method. The application and efficacy of the data-driven method are illustrated by the random response control problems of the Duffing oscillator, van der Pol system, and a two degrees-of-freedom nonlinear system.

The impacting and rebounding behaviors of straight elastic components are investigated and a unified approach is proposed to analytically predict the whole process of the collision and rebounding of straight elastic bars and beams after each of them impinges on ideal (massless) elastic spring(s). The mathematical problems with definitive solution are formulated, respectively, for both the constrained-motion and free-motion stages, and the method of mode superposition, which is concise and straightforward especially for long-time interaction and multiple collision cases, is successfully utilized by repeatedly altering boundary and initial conditions for these successive stages. These two stages happen alternatively and the collision process terminates when the constrained motion no longer occurs. In particular, three examples are investigated in detail; they are: a straight bar impinges on an ideal elastic spring along its axis, a straight beam vertically impinges on an ideal elastic spring at the beam's midpoint, and a straight beam vertically impinges on two ideal springs with the same stiffness at the beam's two ends. Numerical results show that the coefficient of restitution (COR) and the nondimensional rebounding time (NRT) only depend on the stiffness ratio between the ideal spring(s) and the elastic bar/beam. Collision happens only once for the straight bar impinging on spring, while multiple collisions occur for the straight beam impinging on springs in the cases with large stiffness ratio. Once multiple collisions occur, COR undergoes complicated fluctuation with the increase of stiffness ratio. Approximate analytical solutions (AASs) for COR and NRT under the cases of small stiffness ratio are all derived. Finally, to validate the proposed approach in practical collision problems, the influence of the springs' mass on the collision behavior is demonstrated through numerical simulation.

Variable-mass systems become more and more important with the explosive development of micro- and nanotechnologies, and it is crucial to evaluate the influence of mass disturbances on system random responses. This manuscript generalizes the stochastic averaging technique from quasi-integrable Hamiltonian systems to stochastic variable-mass systems. The Hamiltonian equations for variable-mass systems are firstly derived in classical mechanics formulation and are approximately replaced by the associated conservative Hamiltonian equations with disturbances in each equation. The averaged Itô equations with respect to the integrals of motion as slowly variable processes are derived through the stochastic averaging technique. Solving the associated Fokker–Plank–Kolmogorov equation yields the joint probability densities of the integrals of motion. A representative variable-mass oscillator is worked out to demonstrate the application and effectiveness of the generalized stochastic averaging technique; also, the sensitivity of random responses to pivotal system parameters is illustrated.

The buckling of thin films with natural nonlinearity can provide a useful tool in many applications. In the present paper, the mechanical properties of controllable buckling of thin films are investigated by accounting for both geometric nonlinearity and surface effects at nanoscale. The effects of surface elasticity and residual surface tension on both static and dynamic behaviors of buckled thin films are discussed based on the surfacelayer-based model. The dynamic design strategy for buckled thin films as interconnects in flexible electronics is proposed to avoid resonance in a given noise environment based on the above analysis. Further discussion shows that the thermal and piezoelectric effects on mechanical behavior of buckled thin film are equivalent to that of residual surface tension. [DOI: 10.1115/1.4007681]