Transient dynamic analysis of functionally graded micro-beams considering small-scale effects

Domain-boundary element method for elastodynamics of functionally graded Timoshenko beams

A domain-boundary element method for forced vibration analysis of fiber-reinforced laminated composite beams is introduced. Utilizing static fundamental solutions as weight functions in weighted residual statements, governing partial differential equations of motion are reduced to a system of four coupled integral equations. Domain discretization leads to a system of ordinary differential equations in time, which is solved by the Houbolt method. Developed procedures are verified through comparisons to analytical solution for isotropic beams. Parametric results illustrate elastodynamic responses of composite beams subjected to various loading types. It is shown that angle-ply laminates undergo higher displacements compared to cross-ply laminates.

Computational elastodynamics of functionally graded thick-walled cylinders and annular coatings subjected to pressure shocks

The superharmonic resonance of second order of microelectro-mechanical system (MEMS) circular plate resonator under electrostatic actuation is investigated. The MEMS resonator consists of a clamped circular plate suspended over a parallel ground plate under an applied Alternating Current (AC) voltage. The AC voltage is characterized as hard excitation, i.e. the magnitude is large enough, and the operating frequency is near one-fourth of the natural frequency of the resonator. Reduced Order Model (ROM), based on the Galerkin procedure, transforms the partial differential equation of motion into a system of ordinary differential equations in time using mode shapes of vibration of the circular plate resonator. Three numerical methods are used to predict the voltage-amplitude response of the MEMS plate resonator. First, the Method of Multiple Scales (MMS) is directly applied to the partial differential equation of motion which is this way transformed into zero-order and first-order problems. Second, ROM using two modes of vibration is numerical integrated using MATLAB to predict time responses, and third, the AUTO 07P software for continuation and bifurcation to predict the voltage-amplitude response. The nonlinear behavior (i.e. bifurcation and pull-in instability) of the system is attributed to the inclusion of viscous air damping and electrostatic force in the model. The influences of various parameters (i.e. detuning frequency and damping) are also investigated in this work.

Panel flutter analysis of general laminated composite plates

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

1 - INTRODUCTION

Solving multivariate ordinary differential equations (ODE) systems and partial differential equations (PDE) systems is the key to many complex physics and chemistry problems, such as the combustion in process of reacting flow. However, the traditional numerical methods in solving multivariate ODE and PDE systems are limited by computational cost, and sometimes its impossible to obtain the solution due to the high stiffness of ODE or PDE. Coincident with the development of machine learning has been a growing appreciation of applying neural networks in solving physics models. DeepM&M net was proposed to address complicated problems in fluid mechanics based on another neural network: DeepONet, which is used to predict functional nonlinear operators. Inspired by these two nets, a machine learning way of solving certain ODE and PDE systems is proposed with a similar framework to the DeepM&M net, which takes inputs of the initial conditions and outputs the corresponding solutions. The main ideas of this framework are first to explore the relations among solutions of the system by DeepONets and then to train a deep neural network with the assistance of trained DeepONets. The implicit operators between variables in certain ODE systems are verified to have existed and are well predicted by the DeepONet. The feasibility of the proposed framework is implied by the success in building blocks.

Casimir effect on superharmonic resonance of electrostatically actuated bio-nano-electro-mechanical system (Bio-NEMS) circular plate resonator sensor is investigated. The plate sensor resonator is clamped at the outer end and suspended over a parallel ground plate. The sensor can be used for detecting human viruses. Superharmonic resonance of the second order, frequency near one-fourth the natural frequency of the resonator, is induced using Alternating Current (AC) voltage. The magnitude of the AC voltage is also large enough to be consider hard excitation acting on the resonator. Beside Casimir effect, other external forces (i.e. electrostatic force and viscous air damping) acting on the MEMS resonator create a nonlinear behaviors such as bifurcation and pull-in instability. Hence, numerical models, such as Method of Multiple Scales (MMS) and Reduced Order Model (ROM), are used to predict the frequency-amplitude response for MEMS resonator. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. While, ROM, based on the Galerkin procedure which uses the mode shapes of vibration of the resonator as a basis of functions, transforms the nonlinear partial differential equation of motion into a system of ordinary differential equation with respect to dimensionless time. The frequency-amplitude response allows one to observe the behavior of the system for a range of frequencies near the superharmonic resonance. The effects of parameters such as Casimir effect, voltage, and damping on the frequency-amplitude response are reported.

This paper proposes a higher order implicit numerical scheme to approximate the solution of the nonlinear partial differential equation (PDE). This equation is a simplified form of Navier–Stoke’s equation also known as Burgers’ equation. It is an important nonlinear PDE which arises frequently in mathematical modeling of turbulence in fluid dynamics. In order to handle nonlinearity a nonlinear transformation is used which converts the nonlinear PDE into a linear PDE. The linear PDE is semi-discretized in space by method of lines to yield a system of ordinary differential equations in time. The resulting system of differential equations is investigated and found to be a stiff system. A system of stiff differential equations is further discretized by a low-dispersion and low-dissipation implicit Runge–Kutta method and solved by using MATLAB 8.0. The proposed scheme is unconditionally stable. Moreover it is simple, easy to implement and requires less computational time. Finally, the adaptability of the scheme is demonstrated by means of numerical computations by taking three test problems. The present implicit scheme have been compared with existing schemes in literature which shows that the proposed scheme offers more accuracy with less computational time than the numerical schemes given in Jiwari (Comput Phys Comm 183:2413–2423, 2012), Kutluay et al. (J Comput Appl Math 103:251–261, 1998), Kutluay et al. (J Comput Appl Math 167:21–33, 2004).

Dynamic response of an elastic plate on a cross-anisotropic poroelastic half-plane to a load moving on its surface

In this chapter, we provide a method for solving systems of linear ordinary differential equations by using techniques associated with the calculation of eigenvalues, eigenvectors and generalized eigenvectors of matrices. We learn in calculus how to solve differential equations and the system of differential equations. Here, we firstly show how to represent a system of differential equations in a matrix formulation. Then, using the Jordan canonical form and, whenever possible, the diagonal canonical form of matrices, we will describe a process aimed at solving systems of linear differential equations in a very efficient way.

8 - Equation of Heat Transfer

The design an optimal numerical method for solving a system of ordinary differential equations simultaneously is described in this paper. System of differential equations was represented by a system of linear ordinary differential equations of Euler’s parameters called quaternions. The components of angular velocity were obtained by the experimental way. The angular velocity of the centre of gravity was determined from sensors of acceleration located in the plane of the centre of gravity of the machine. The used numerical method for solving was a fourth-order Runge-Kutta method. The stability of solving was based on the orthogonality of a direct cosine matrix. The numerical process was controlled on every step in numerical integration. The algorithm was designed in the C# programming language.

Systems of ordinary differential equations with a small parameter at the derivative and specific features of the construction of their periodic solution are considered. Sufficient conditions of existence and uniqueness of the periodic solution are presented. An iterative procedure of construction of the steady-state solution of a system of differential equations with a small parameter at the derivative is proposed. This procedure is reduced to the solution of a system of nonlinear algebraic equations and does not involve the integration of the system of differential equations. Problems of numerical calculation of the solution are considered based on the procedure proposed. Some sources of its divergence are found, and the sufficient conditions of its convergence are obtained. The results of numerical experiments are presented and compared with theoretical ones.