Periodic Temperature Responses in a Thermal System Under a Periodic Heating

In this paper, the periodic temperature responses of a thermal system under a periodic heating input are studied. Using the implicit mapping method, periodic temperature responses varying with excitation frequency are predicted for different input amplitudes. The corresponding stability of the periodic responses are discussed through eigenvalue analysis. The experimental and numerical results of the periodic response are presented for comparison to the analytical results.

A predictive deep-learning approach for homogenization of auxetic kirigami metamaterials with randomly oriented cuts

This paper describes a data-driven approach to predict mechanical properties of auxetic kirigami metamaterials with randomly oriented cuts. The finite element method (FEM) was used to generate datasets, the convolutional neural network (CNN) was introduced to train these data, and an implicit mapping between the input orientations of cuts and the output Young’s modulus and Poisson’s ratio of the kirigami sheets was established. With this input–output relationship in hand, a quick estimation of auxetic behavior of kirigami metamaterials is straightforward. Our examples indicate that if the distributions of training and test datasets are close to each other, a good prediction is achievable. Our efforts provide a fast and reliable way to evaluate the homogenized properties of mechanical metamaterials with various microstructures, and thus accelerate the design of mechanical metamaterials for diverse applications.

Period-3 Motions in a Parametrically Exited Inverted Pendulum

In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

On Periodic Motions in a Periodically Driven van der Pol-Duffing Oscillator

In this paper, the symmetric and asymmetric period-1 motions on the bifurcation tree are obtained for a periodically driven van der Pol-Duffing hardening oscillator through a semi-analytical method. Such a semi-analytical method develops an implicit mapping with prescribed accuracy. Based on the implicit mapping, the mapping structures are used to determine periodic motions in the van der Pol-Duffing oscillator. The symmetry breaks of period-1 motion are determined through saddle-node bifurcations, and the corresponding asymmetric period-1 motions are generated. The bifurcation and stability of period-1 motions are determined through eigenvalue analysis. To verify the semi-analytical solutions, numerical simulations are also carried out.

Independent Asymmetric Period-3 Motions in a Periodically Forced Double-Pendulum

In this paper, the independent asymmetric period-3 motions of a periodically forced, damped, double-pendulum are predicted through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the paired asymmetric period-3 motions are determined through eigenvalue analysis. Numerical simulation of the two asymmetric period-3 motions in the double-pendulum system is completed from analytical predictions. The example presented herein can be used for the vibration reduction of the first pendulum through the motions of the second pendulum.

On the implicit and inverse many-valued functions in topological spaces

he conditions of continuity of the implicit set-valued map and the inverse setvalued map acting in topological spaces are proposed. For given mappings f∶ T ×X → Y, y∶ T → Y, where T,X,Y are topological spaces, the space Y is Hausdorff, the equation f(t,x) = y(t) with the parameter t ∈ T relative to the unknown x ∈ X is considered. It is assumed that for some multi-valued map U∶ T ⇉ X for all t ∈ T the inclusion f(t,U(t)) ∋ y(t) is satisfied. An implicit mapping R_U ∶ T ⇉ X, which associates with each value of the parameter t ∈ T the set of solutions x(t) ∈ U(t) of this equation. It is proved that R_U is upper semicontinuous at the point t_0 ∈ T, if the following conditions are satisfied: for any x ∈ X the map f is continuous at (t_0,x), the map y is continuous at t_0, a multi-valued map U is upper semicontinuous at the point t_0 and the set U(t_0) ⊂ X is compact. If, in addition, with the value of the parameter t_0 , the solution to the equation is unique, then the map R_U is continuous at t_0 and any section of this map is also continuous at t_0. The listed results are applied to the study of a multi-valued inverse mapping. Namely, for a given map g∶ X → T we consider the equation g(x) = y with respect to the unknown x ∈ X. We obtain conditions for upper semicontinuity and continuity of the map V_U ∶ T ⇉ X, V_U (t) = {x ∈ U(t)∶ g(x) = t}, t ∈ T.

Period Motions in a Periodically Forced, Damped Double Pendulum

In this paper, period motions in a periodically forced, damped, double pendulum are analytically predicted through a discrete implicit mapping method. The implicit mapping is established via the discretized differential equation. The corresponding stability and bifurcation conditions of the period motions are predicted through eigenvalue analysis. Numerical simulation of the period motions in the double pendulum is completed from analytical predictions.

Analytical Prediction of Period-1 Motions in a Time-Delayed, Softening Duffing Oscillator

In this paper, symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, softening Duffing oscillator is analytically predicted through a discrete implicit mapping method. Such a method is based on the discretization of the corresponding differential equation. The stability and bifurcations of the symmetric and asymmetric period-1 motions are determined through eigenvalue analysis. Numerical simulation of the period-1 motions in the time-delayed softening Duffing oscillator is presented for verification of the analytical prediction.

Periodic Motions in a 2-DOF Self-Excited Duffing Oscillator

In this paper, analytical solutions of periodic motions in a 2-DOF self-excited Duffing oscillator are investigated through a semi-analytical method. The semi-analytical method discretizes the self-excited Duffing oscillator for the discrete implicit mappings. Through the implicit mapping, period-1 motion varying with excitation frequency are presented, and the corresponding stability and bifurcation are discussed via the eigenvalues analysis. The Neimark and saddle-node bifurcations of the periodic motion are obtained. Initial conditions for numerical simulations are from analytical solutions. Numerical and analytical solutions of periodic motions are illustrated for comparison.