Morad Nazari

Bio: Morad Nazari is an academic researcher from Embry–Riddle Aeronautical University. The author has contributed to research in topics: Spacecraft & Rigid body. The author has an hindex of 13, co-authored 48 publications receiving 489 citations. Previous affiliations of Morad Nazari include New Mexico State University & Embry-Riddle Aeronautical University, Daytona Beach.

Decentralized Consensus Control of a Rigid-Body Spacecraft Formation with Communication Delay

Abstract: The decentralized consensus control of a formation of rigid-body spacecraft is studied in the framework of geometric mechanics while accounting for a constant communication time delay between spacecraft. The relative position and attitude (relative pose) are represented on the Lie group SE(3) and the communication topology is modeled as a digraph. The consensus problem is converted into a local stabilization problem of the error dynamics associated with the Lie algebra se(3) in the form of linear time-invariant delay differential equations with a single discrete delay in the case of a circular orbit, whereas it is in the form of linear time-periodic delay differential equations in the case of an elliptic orbit, in which the stability may be assessed using infinite-dimensional Floquet theory. The proposed technique is applied to the consensus control of four spacecraft in the vicinity of a Molniya orbit.

Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate

Abstract: The nonlinear vibration of an isotropic cantilever plate with viscoelastic laminate is investigated in this article. Based on the Von Karman’s nonlinear geometry and using the methods of multiple scales and finite difference, the dimensionless nonlinear equations of motion are analyzed and solved. The solvability condition of nonlinear equations is obtained by eliminating secular terms and, finally, nonlinear natural frequencies and mode-shapes are obtained. Knowing that the linear vibration of this type of plate does not have exact solution, Ritz method is employed to obtain semi-analytical nonlinear mode-shapes of transverse vibration of this plate. Airy stress function and Galerkin method are employed to reduce nonlinear PDEs into an ODE of duffing type. Stability of plate and chaotic behavior are investigated by Runge–Kutta method. Poincare section diagrams are in good agreement with results of Lyapunov criteria.

Optimal Periodic-Gain Fractional Delayed State Feedback Control for Linear Fractional Periodic Time-Delayed Systems

Abstract: This paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional-order periodic time-delayed systems. Although there exist techniques for the state feedback control of linear periodic time-delayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for a steady-state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a closed form for FPTD systems, and hence, the short-memory principle along with the fractional Chebyshev collocation method is used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers, in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Mathieu's Equation and Its Generalizations: Overview of Stability Charts and Their Features

Abstract: This work is concerned with Mathieu’s equation a classical differential equation, which has the form of a linear second-order ordinary di¤erential equation with Cosine-type periodic forcing of the sti¤ness coe¢ cient, and its di¤erent generalisations/extensions. These extensions include: the e¤ects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it di¤ers from that of the classical Mathieu’s equation. 1 Ac ce pt ed M an us cr ip t N ot C op ye di te d Applied Mechanics Reviews. Received August 07, 2017; Accepted manuscript posted January 31, 2018. doi:10.1115/1.4039144 Copyright (c) 2018 by ASME Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use Keywords: parametric excitation; stability chart; transition curves; perturbation method; Floquet theory; harmonic balancing; geometric nonlinearity; damping nonlinearity; fractional derivative; delay; quasiperiodic excitation; elliptic-type excitation. 1 Introduction Mathieu’s equation is one of the archetypical equations of Nonlinear Vibrations Theory [1]. However, this equation is not only associated with this …eld, but due to the tools and techniques needed for its quantitative analysis and diverse applications, it appears also in Applied Mathematics [2]-[4] and in many engineering …elds [5]-[7]. The form of Mathieu’s equation is very simple it is a linear second-order ordinary di¤erential equation, which di¤ers from the one corresponding to a simple harmonic oscillator in the existence of a time-varying (periodic) forcing of the sti¤ness coe¢ cient as follows: dx dt2 + ( + cos t) x = 0; (1) where and are constant parameters, while x is a dependent variable (its mechanical interpretation will be de…ned in Section 2.1) and t is time. So, the simple harmonic oscillator is obtained for = 0, and the sti¤ness parameter corresponds then to the square of its natural frequency, i.e. !0 = p . It is well-known that this oscillator performs free vibrations around the stable equilibrium position x = 0. However, if the sti¤ness term contains the parametric excitation, i.e. 6= 0, the motion can stay bounded (this case is referred to as stable) or the motion becomes unbounded (this case is referred to as unstable). The occurrence of one of these two outcomes depends on the combination of the parameters and . When presented graphically, this gives the so-called stability chart with regions of stability and regions of instability (tongues) separated by the so-called transition curves, enabling one to clearly determine the resulting behaviour and the stability property mentioned. Historically speaking, what is now termed ’Mathieu’s equation’ is attributed to Mathieu’s investigations of vibrations in an elliptic drum from 1868 [8]. The extract from this work is presented in Appendix A. It is shown therein how the derivation of Eq. (1) stems from the Helmholtz equation 2 Ac ce pt e M an u cr ip t N ot C op ye di te d Applied Mechanics Reviews. Received August 07, 2017; Accepted manuscript posted January 31, 2018. doi:10.1115/1.4039144 Copyright (c) 2018 by ASME Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use for the vibrations of a membrane with an elliptic boundary. Mathieu also developed the power series expansion method, determining the mutual relationships between the sti¤ness parameter and the amplitude of parametric excitation and the respective solutions of motion. These solutions are called after him ’Mathieu functions’and are presented in Appendix B. The relationships between the sti¤ness parameter and the amplitude of parametric excitation can be presented graphically (but Mathieu did not do it at that time) and represent the transition curves mentioned above. A few subsequent important developments of Mathieu’s equation are listed below [3]: 1878 Heine expresses the solution as an in…nite continued fraction [9]. 1883 Floquet presents a generalized treatment of di¤erential equations with periodic coe¢ cients [10]. 1886 Hill expresses solution as an in…nite determinant [11]. 1887 Lord Rayleigh (J.W. Strutt) applies Mathieu’s equation toMelde’s problem (a tuning fork with an attached string) [12]. 1908 Sieger presents the application to di¤raction of electromagnetic waves by an elliptic cylinder [13]. 1912 Whittaker expresses solution as an integral equation [14]. 1915 Ince publishes the …rst of 18 papers on Mathieu functions, including: 1927 Ince introduces the stability chart [15]. Note that the stability chart is sometimes called ’Strutt diagram’ or ’Strutt-Ince diagram’. However, Strutt’s work (M.J.O. Strutt, not to be confused with Lord Rayleigh, who is J.W. Strutt) with this chart [16] was published later than Ince’s. Thus, it was Ince who …rst presented it graphically and his …gure from [15] is redrawn and included in Appendix B. The determination and description of the stability chart is the focus of this work. Besides this, the aim is to show how its structure and features change as the form given by Eq. (1) is modi…ed by additional or di¤erent geometric, damping and excitation terms. This work is organized as follows. First, a brief overview of mechanical models that are associated with classical Mathieu’s equation are given in Section 2. In addition, certain mathematical tools for its quantitative analysis are presented in Section 3, yielding the basic structure and features of the stability chart. Section 4 is concerned with the in‡uence of geometric and damping nonlinearities on this stability chart (it should be noted that Sections 2,3, and 4.1 and Appendix C are strongly based on Rand’s online notes [1]). Section 5 deals with the in‡uence of a delay term 3 Ac ce pt e M an us cr ip t N ot C op ye di te d Applied Mechanics Reviews. Received August 07, 2017; Accepted manuscript posted January 31, 2018. doi:10.1115/1.4039144 Copyright (c) 2018 by ASME Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use and Section 6 with the e¤ects of a fractional-derivative term. Sections 7 and 8 deal with the e¤ects of di¤erent types of excitation: quasiperiodic and elliptic-type excitations. 2 Classical Mathieu’s equation: mechanical models and applications Generally speaking and related to nonlinear vibration problems, Mathieu’s equation in its classical form (1) is associated with di¤erential equations derived in two general cases [1]: Case 1 in systems with periodic forcing, and Case 2 in stability studies of periodic motions in nonlinear autonomous systems. As illustrative example of Case 1 is a mathematical pendulum whose support moves periodically in a vertical direction (Figure 1a). The governing di¤erential equation is dx dt2 + g L A L cos t sin x = 0; (2) where x is the generalised coordinate being the angle of de‡ection, g is the acceleration of gravity, L is the pendulum’s length, while the vertical motion of the support is A cos t. Two equilibrium solutions exist: x = 0 or x = . In order to investigate their stability, one would linearize Eq. (2) about the desired equilibrium, deriving an equation of the form of Eq. (1). If the motion of the support is de…ned by A cos t, the equation of motion for small x has the form dx dt2 + !0 A 2 L cos t x = 0; (3) in which one can recognize two frequencies: the natural frequency !0 = p g=L and the excitation frequency . However, by introducing = t, one can obtain the form given by Eq. (1) with = g=(L ), = A=L. Moreover we can consider the case of the vertically forced inverted pendulum by setting y = x , whereupon the small y di¤erential equation becomes dy dt2 !0 A 2 L cos t y = 0; (4) 4 Ac ce pt ed M an us cr ip t N ot C op ye di te d Applied Mechanics Reviews. Received August 07, 2017; Accepted manuscript posted January 31, 2018. doi:10.1115/1.4039144 Copyright (c) 2018 by ASME Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 02/13/2018 Terms of Use: http://www.asme.org/about-asme/terms-of-use Note that in this case the parameter is negative, and hence the unforced equilibrium y = 0, i.e. x = , is unstable. Nevertheless we will show that the equilibrium can be made stable by an appropriate choice of parameter values. This remarkable example was evidently …rst considered by A. Stephenson in 1908 [17] and [18]. Additional examples that have the same governing equations are, for instance: a frequency-modulated tuned circuit, the Paul trap for charged particles, stability of a ‡oating body, the mirror trap for neutral particles [7], certain autoparametric vibration absorbers [19], stability of elastic systems (bars, for example) under certain time-varying loading [20], asymmetric shaft and bearings in rotor dynamics [21], torsional motions of a rotor in contact with a stator [22], etc. Additional examples from other …elds include those from aerospace engineering: for example, helicopter rotor blades in forward ‡ight, attitude stability of satellites in elliptic orbits) and biology (for instance, heart rhythms, membrane vibrations in the inner ear). As an example of Case 2, one can consider a system called “the particle in the plane”(Figure 1b), which was …rst studied in [23], [24]. It contains a particle of unit mass which is constrained to move in the x-y plane, and is restrained by two linear springs, each with spring constant k of 1=2. Each of the two springs has unstretched length L. The anchor points of the two springs are located on the x axis at x = 1 and x = 1. This autonomous two-degree-of-freedom system has the following equatio

Nonlinear oscillation of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method

Abstract: Nonlinear dynamic analysis of a cantilever functionally graded materials (FGM) rectangular plate subjected to the transversal excitation in thermal environment is presented for the first time in this paper. Material properties are assumed to be temperature dependent. The nonlinear governing equations of motion for the FGM plate are derived based on Reddy’s third-order plate theory and Hamilton’s principle. The first two vibration mode functions satisfying the boundary conditions of the cantilever FGM rectangular plates are chosen to be the admissible displacement functions. Galerkin’s method is utilized to convert the governing partial differential equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under combined external excitations. The present study focuses on resonance case with 1:1 internal resonance and subharmonic resonance of order 1/2. The asymptotic perturbation method is employed to obtain four nonlinear averaged equations which are then solved by using Runge–Kutta method to find the nonlinear dynamic responses of the plate. It is found that chaotic, periodic and quasi-periodic motions of the plate exist under certain conditions and the forcing excitations can change the form of motions for the FGM rectangular plate.

Nonlinear vibrations of FGM rectangular plates in thermal environments

Abstract: Geometrically nonlinear vibrations of FGM rectangular plates in thermal environments are investigated via multi-modal energy approach. Both nonlinear first-order shear deformation theory and von Karman theory are used to model simply supported FGM plates with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. A pseudo-arclength continuation and collocation scheme is used and it is revealed that, in order to obtain the accurate natural frequency in thermal environments, an analysis based on the full nonlinear model is unavoidable since the plate loses its original flat configuration due to thermal loads. The effect of temperature variations as well as volume fraction exponent is discussed and it is illustrated that thermally deformed FGM plates have stronger hardening behaviour; on the other hand, the effect of volume fraction exponent is not significant, but modal interactions may rise in thermally deformed FGM plates that could not be seen in their undeformed isotropic counterparts. Moreover, a bifurcation analysis is carried out using Gear’s backward differentiation formula (BDF); bifurcation diagrams of Poincare maps and maximum Lyapunov exponents are obtained in order to detect and classify bifurcations and complex nonlinear dynamics.