Second-order Dynamic Equations Research Articles

New Oscillation Criteria for Sturm–Liouville Dynamic Equations with Deviating Arguments

New Oscillation Criteria for Sturm–Liouville Dynamic Equations with Deviating Arguments

Oscillatory Behavior of Noncanonical Quasilinear Second-Order Dynamic Equations on Time Scales

The objective of this article is to examine the oscillatory behavior of a class of quasilinear second-order dynamic equations on time scales. Our focus will be on the noncanonical case, which has received relatively less attention compared to the more commonly studied canonical dynamic equations. Our approach involves transforming the noncanonical equation into a corresponding canonical equation. By utilizing this transformation and a range of techniques, we develop new, more efficient, and precise oscillation criteria. Finally, we demonstrate the significance and usefulness of our results by applying them to specific cases within the equation being studied.

Optimal Active Vibration Control of Tensegrity Structures Using Fast Model Predictive Control Strategy

Active vibration control of tensegrity structures is often challenging due to the geometrical nonlinearity, assemblage uncertainties of connections, and actuator saturation of controllers. To tackle these technical difficulties, a fast model predictive control (FMPC) strategy is herein implemented to effectively mitigate the structural vibration. Specifically, based on the explicit expression form of the Newmark- β method, the computation of the matrix exponential is avoided and replaced by one online and two offline transient analyses at each sampling instant on the structure, and the optimal control input is attainted from the second-order dynamic equation without forming an expanded state-space equation. Meanwhile, the artificial fish swarm algorithm (AFSA) is embedded to automatically derive optimal arrangement of actuators with the selection of a reasonable objective function. Two illustrative examples, including two standard and clustered tensegrity beams and a clustered tensegrity tower, have been fully investigated. The outcomes from illustrative examples prove the effectiveness and feasibility of the proposed method in optimal active vibration control of tensegrity structures, implying a promising prospect of the investigated approach in analyzing and solving relevant engineering problems.

Variational approach of p-Laplacian impulsive differential equations with periodic conditions

Abstract In this paper, we study a boundary value problem for a second-order impulsive p-Laplacian dynamic equation on a time scale. The approach is based on a critical point theorem. Finally, some examples are given to illustrate the results.

Second-Order Dynamic Equations with Noncanonical Operator: Oscillatory Behavior

The present article aims to study the oscillatory properties of a class of second-order dynamic equations on time scales. We consider during this study the noncanonical case, which did not receive much attention compared to the canonical dynamic equations. The approach adopted depends on converting the noncanonical equation to a corresponding canonical equation. By using this transformation and based on several techniques, we create new, more effective, and sharp oscillation criteria. Finally, we explain the effectiveness and importance of the results by applying them to some special cases of the studied equation.

Oscillation criteria for second-order delay dynamic equations with a sub-linear neutral term on time scales

This paper addresses the oscillatory behavior of the solutions of second-order dynamic equations with a sublinear neutral term. Using Riccati transformation and comparison principles, we obtain new oscillation criteria. The obtained results essentially improve, complement, and simplify some of the previous ones in the literature. Some examples have been provided herein to illustrate our main results.

OSCILLATION OF SECOND-ORDER HALF-LINEAR NEUTRAL NONCANONICAL DYNAMIC EQUATIONS

In this paper, we shall establish some new criteria for the oscillation of certain second-order noncanonical dynamic equations with a sublinear neutral term. This task is accomplished by reducing the involved nonlinear dynamic equation to a second-order linear dynamic inequality. We also establish some new oscillation theorems involving certain integral conditions. Three examples, illustrating our results, are presented. Our results generalize results for corresponding differential and difference equations.

New oscillation criteria for second-order delay dynamic equations with a sub-linear negative neutral term on time scales

New oscillation criteria for second-order delay dynamic equations with a sub-linear negative neutral term on time scales

Improved Hille-Type Oscillation Criteria for Second-Order Quasilinear Dynamic Equations

In this work, we develop enhanced Hille-type oscillation conditions for arbitrary-time, second-order quasilinear functional dynamic equations. These findings extend and improve previous research that has been published in the literature. Some examples are given to demonstrate the importance of the obtained results.

New oscillation criteria and some refinements for second-order neutral delay dynamic equations on time scales

In this paper, we present more effective criteria for oscillation of second-order half-linear neutral dynamic equations with delayed arguments. Our results essentially improve, complement, and simplify several related ones in the literature. Some examples are given to illustrate our main results.

Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term

Oscillation of second-order non-canonical non-linear dynamic equations with a sub-linear neutral term

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

Oscillation of second-order nonlinear delay dynamic equations on time scales with a nonpositive neutral term

In this paper, we establish some new oscillation criteria for second- order nonlinear neutral delay dynamic equation of the form (r(t)((m(t)y(t) ? p(t)y(τ(t)))∆)γ )∆ + ƒ (t, y(?(t))) = 0 on a time scale

New oscillation criteria for second-order neutral delay dynamic equations with a non-positive neutral term on time scales

In this paper, we investigate the oscillatory behavior of solutions of a class of secondorder nonlinear neutral delay dynamic equations with non-positive coefficient where α ≥ β are ratios of odd positive integers and z(t) := x(t) - p(t)x(τ (t)). The obtained results guarantee that all solutions are oscillatory. Illustrative examples are provided.

Modeling of Pogo Vibration in Rockets with Staged Combustion Cycle Engine

The modeling method of the pogo system of launch vehicles with a staged combustion cycle engine is mainly limited to the transfer matrix modeling method, which contains transcendental functions that are difficult to solve. A state-space modeling method for the pogo system of launch vehicles with a staged combustion cycle engine is proposed in this paper. First, the second-order differential dynamic equation of the gas path is constructed by approximating the exponential term in each element of the gas path with Taylor expansion and inverse Laplace transformation. Second, the liquid and gas paths are combined in the form of mass flow to establish the second-order dynamic equation of the propulsion system. Third, the state-space equation of the coupled system is obtained by combining the structural and propulsion systems. Finally, an example of a propulsion system is modeled using the transfer matrix modeling method and the proposed method to verify the correctness of the proposed method, and the stability of the model is analyzed when the parameters of the accumulator and gas path change. The major contributions are as follows: 1) the proposed method is a systematic method that is universally applicable to the pogo analysis of a launch vehicle with a staged combustion cycle engine; 2) the proposed method inherits the advantages of Rubin’s state-space method for liquid rockets.

Earthquake Response Spectra Analysis of Bridges considering Pounding at Bilateral Beam Ends Based on an Improved Precise Pounding Algorithm

Asynchronous vibration was generated between the main bridge and approach spans or abutments due to differences in stiffness and mass during an earthquake, thus further leading to pounding at the bilateral beam ends. By taking a T-shaped rigid frame bridge as an example, the bilateral pounding model was abstracted, and the earthquake response spectra considering pounding at the bilateral beam ends were studied, including the maximum displacement spectrum, the acceleration dynamic coefficient spectrum, the pounding force response spectrum, and the response spectrum for the number of pounding events. An improved precise pounding algorithm was proposed to solve the dynamic equation of the bilateral pounding model. This algorithm is based on the precise integration method for solving the second-order dynamic differential equation and reduces the order thereof by introducing a new velocity vector and uses the series method to find the nonhomogeneous term. The system matrix is simpler, and the inversion of the system matrix can be avoided. On this basis, a multipoint earthquake-induced pounding response spectrum program was developed. A total of 18 seismic waves from Class II sites were selected, and the response spectra of 18 waves were analyzed using this new program. Furthermore, the effects of structural stiffness, mass, stiffness of contact element, pounding recovery coefficient, and peak ground acceleration (PGA) on the earthquake response spectrum were studied. Through the analysis of earthquake response spectra and a parametric study, the phenomenon of earthquake-induced pounding of bridges was clarified to the benefit of the analysis and engineering control of earthquake-induced pounding of bridges.

Oscillation Criteria of Second-Order Dynamic Equations on Time Scales

In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. λ(s)Ψ1φΔ(s)y(φ(s))ΔΔ+η(s)Φ(y(τ(s)))=0,s∈[s0,∞)T. By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.

Hyers—Ulam Stability of Second-Order Linear Dynamic Equations on Time Scales

We investigate the Hyers—Ulam stability (HUS) of certain second-order linear constant coefficient dynamic equations on time scales, building on recent results for first-order constant coefficient time-scale equations. In particular, for the case where the roots of the characteristic equation are non-zero real numbers that are positively regressive on the time scale, we establish that the best HUS constant in this case is the reciprocal of the absolute product of these two roots. Conditions for instability are also given.

On Hyers–Ulam and Hyers–Ulam–Rassias Stability of a Nonlinear Second-Order Dynamic Equation on Time Scales

In this paper, we obtain sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of an abstract second–order nonlinear dynamic equation on bounded time scales. An illustrative example is given to show the applicability of the theoretical results.

Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order

In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.