The Consistency between MLP Method and Modified Method of Multiple Scales for Strong Nonlinear Primary Resonance of Duffing Equation Subject to Harmonic Excitation in Complex Number Field

2014 ◽  
Vol 472 ◽  
pp. 62-68
Author(s):  
Zhi’an Yang ◽  
Xi Li ◽  
Jia Jia Meng
Keyword(s):  
Frequency Response ◽  
Number Field ◽  
Complex Number ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Harmonic Excitation ◽  
Duffing Equation ◽  
Modified Method ◽  
Amplitude Frequency Response ◽  
Complex Number Field

With changing strong nonlinear Duffing equation subject to harmonic excitation with damping in complex number field as an object, the amplitude frequency response equation of primary resonance of the system is obtained through parametric transformation with application of MLP method and modified method of multiple scales. In different approximation solution forms and different time scales, the two methods lead to the same amplitude frequency response equation. Thus, the two methods are mutually verifiable. Numerical analysis shows that for the strong nonlinear Duffing equation with damping in complex number field, the nonlinear stiffness coefficient is more than zero and the amplitude frequency response curve of primary resonance leans to the left, which is different from the weak nonlinear Duffing equation.Chinese books catalog: O321

scholarly journals Type II degenerations of K3 surfaces

1985 ◽  
Vol 99 ◽  
pp. 11-30 ◽  
Author(s):  
Shigeyuki Kondo
Keyword(s):  
Number Field ◽  
Complex Manifold ◽  
Complex Number ◽  
Three Dimensional ◽  
K3 Surfaces ◽  
Type Ii ◽  
Holomorphic Map ◽  
Canonical Class ◽  
Proper Holomorphic Map ◽  
Complex Number Field

A degeneration of K3 surfaces (over the complex number field) is a proper holomorphic map π: X→Δ from a three dimensional complex manifold to a disc, such that, for t ≠ 0, the fibres Xt = π-1(t) are smooth K3 surfaces (i.e. surfaces Xt with trivial canonical class KXt = 0 and dim H1(Xt, Oxt) = 0).

On the Nonemptiness of the Adjoint Linear System of Polarized Manifolds

1998 ◽  
Vol 41 (3) ◽  
pp. 267-278 ◽  
Author(s):  
Yoshiaki Fukuma
Keyword(s):  
Linear System ◽  
Number Field ◽  
Complex Number ◽  
Complex Number Field

AbstractLet (X, L) be a polarized manifold over the complex number field with dim X = n. In this paper, we consider a conjecture of M. C. Beltrametti and A. J. Sommese and we obtain that this conjecture is true if n = 3 and h0(L) ≥ 2, or dim Bs |L| ≤ 0 for any n ≥ 3. Moreover we can generalize the result of Sommese.

Phased Bessel functions

2008 ◽  
Vol 86 (7) ◽  
pp. 863-870 ◽  
Author(s):  
X Hu ◽  
H Wang ◽  
D -S Guo
Keyword(s):  
Real Number ◽  
Number Field ◽  
Complex Number ◽  
Bessel Functions ◽  
Natural Extension ◽  
State Transitions ◽  
Photon State ◽  
Analytical Extension ◽  
Complex Number Field ◽  
Real Number Field

In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

Bio-MEMS Circular Plate Sensors Under Electrostatic Hard Excitations: Frequency Response of Superharmonic Resonance of Fourth Order

2020 ◽  
Author(s):  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Frequency Response ◽  
Circular Plate ◽  
Fourth Order ◽  
Multiple Scales ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Amplitude Frequency Response ◽  
Elastic Circular Plate ◽  
Modes Of Vibration ◽  
The Difference

Abstract This paper investigates the frequency-amplitude response of electrostatically actuated Bio-MEMS clamped circular plates under superharmonic resonance of fourth order. The system consists of an elastic circular plate parallel to a ground plate. An AC voltage between the two plates will lead to vibrations of the elastic plate. Method of Multiple Scales, and Reduced Order Model with two modes of vibration are the two methods used in this work. The two methods show similar amplitude-frequency response, with an agreement in the low amplitudes. The difference between the two methods can be seen for larger amplitudes. The effects of voltage and damping on the amplitude-frequency response are reported. The steady-state amplitudes in the resonant zone increase with the increase of voltage and with the decrease of damping.

Castelnuovo-Mumford regularity bound for smooth threefolds in ℙ5 and extremal examples

1999 ◽  
Vol 1999 (509) ◽  
pp. 21-34
Author(s):  
Si-Jong Kwak
Keyword(s):  
Complete Intersection ◽  
Number Field ◽  
Complex Number ◽  
Smooth Surfaces ◽  
Veronese Surface ◽  
Optimal Bound ◽  
Integral Curves ◽  
Complex Number Field

Abstract Let X be a nondegenerate integral subscheme of dimension n and degree d in ℙN defined over the complex number field ℂ. X is said to be k-regular if Hi(ℙN, ℐX (k – i)) = 0 for all i ≧ 1, where ℐX is the sheaf of ideals of ℐℙN and Castelnuovo-Mumford regularity reg(X) of X is defined as the least such k. There is a well-known conjecture concerning k-regularity: reg(X) ≦ deg(X) – codim(X) + 1. This regularity conjecture including the classification of borderline examples was verified for integral curves (Castelnuovo, Gruson, Lazarsfeld and Peskine), and an optimal bound was also obtained for smooth surfaces (Pinkham, Lazarsfeld). It will be shown here that reg(X) ≦ deg(X) – 1 for smooth threefolds X in ℙ5 and that the only extremal cases are the rational cubic scroll and the complete intersection of two quadrics. Furthermore, every smooth threefold X in ℙ5 is k-normal for all k ≧ deg(X) – 4, which is the optimal bound as the Palatini 3-fold of degree 7 shows. The same bound also holds for smooth regular surfaces in ℙ4 other than for the Veronese surface.

scholarly journals Some inequalities in Pseudo-Hilbert spaces

2011 ◽  
Vol 42 (4) ◽  
pp. 483-492
Author(s):  
Loredana Ciurdariu
Keyword(s):  
Linear Space ◽  
Number Field ◽  
Hilbert Spaces ◽  
Complex Number ◽  
Normed Linear Space ◽  
Normed Algebra ◽  
The Real ◽  
Complex Number Field

The aim of this paper is to obtain new versions of the reverse of the generalized triangle inequalities given in \cite{SSDNA}, %[4],and \cite{SSDPR} %[5] if the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ from Theorem 1 of \cite{SSDNA} %[4] belongs to ${\mathbb C}\times\mathcal H $, where $\mathcal H$ is a Loynes $Z$-space instead of ${\mathbb K}\times X$, $X$ being a normed linear space and ${\mathbb K}$ is the field of scalars. By comparison, in \cite{SSDNA} %[4] the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ belongs to $A^2$, where $A$ is a normed algebra over the real or complex number field ${\mathbb K}.$ The results will be given in Theorem 1, Theorem 3, Remark 2 and Corollary 3 which represent other interesting variants of Theorem 2.1, Remark 2.2, Theorem 3.2 and Theorem 3.4., see \cite{SSDNA}. %[4].

Amplitude-Frequency Response of Superharmonic Resonance of Third Order of Electrostatically Actuated MEMS Cantilever Resonators

2019 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christopher Reyes
Keyword(s):  
Frequency Response ◽  
Cantilever Beam ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Beam Theory ◽  
Driving Frequency ◽  
Third Order ◽  
Reduced Order ◽  
Superharmonic Resonance ◽  
Amplitude Frequency Response

Abstract This work deals with amplitude frequency response of MEMS cantilever resonators undergoing superharmonic resonance of third order. The cantilever resonator is parallel to a ground plate and under alternating current (AC) voltage that excites the cantilever into vibrations. The driving frequency of the AC voltage is near one sixth of the first natural frequency of the cantilever beam resulting into superharmonic resonance of third order. The cantilever beam is modeled using Euler-Bernoulli beam theory. The electrostatic force is modeled using Palmer’s formula to include the fringe effect. In order to investigate the amplitude frequency behavior of the system reduced order models (ROMs) are developed. Three methods are used to solve these ROMs they are 1) the method of multiple scales (MMS) for ROM with one mode of vibration, 2) homotopy analysis method (HAM) for ROM with one mode of vibration, and 3) direct numerical integration for 2 modes of vibration Reduced Order Model (2T ROM) producing time responses of the tip of the cantilever resonator. In this work the limitations of MMS and HAM are highlighted when considering large voltage values i.e hard excitations. For large voltage values MMS and HAM cannot accurately predict the amplitude frequency response; the results from 2T ROM time responses disagree significantly with the MMS and HAM solutions. The effect of voltage on the frequency response is investigated. As the voltage values in the system increase the responses shift to lower frequencies and larger amplitudes.

scholarly journals The ring of invariants of matrices

1986 ◽  
Vol 104 ◽  
pp. 149-161 ◽  
Author(s):  
Yasuo Teranishi
Keyword(s):  
Vector Space ◽  
Number Field ◽  
Complex Number ◽  
Rational Representation ◽  
Invertible Matrices ◽  
Complex Number Field

We denote by M(n) the space of all n × n-matrices with their coefficients in the complex number field C and by G the group of invertible matrices GL(n, C). Let W = M(n)i be the vector space of l-tuples of n × ra-matrices. We denote by ρ: G → GL(W) a rational representation of G defined as follows:if S ∈ G, A(i) ∈ M(n) (i = 1, 2, …, l).

scholarly journals Analytic Log Picard Varieties

2008 ◽  
Vol 191 ◽  
pp. 149-180 ◽  
Author(s):  
Takeshi Kajiwara ◽  
Kazuya Kato ◽  
Chikara Nakayama
Keyword(s):  
Number Field ◽  
Complex Number ◽  
Basic Properties ◽  
Picard Variety ◽  
Complex Number Field

AbstractWe introduce a log Picard variety over the complex number field by the method of log geometry in the sense of Fontaine-Illusie, and study its basic properties, especially, its relationship with the group of log version of m-torsors.

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