Forced Torsional Vibrations With Damping: An Extension of Holzer’s Method

1946 ◽  
Vol 13 (4) ◽  
pp. A276-A280
Author(s):  
J. P. Den Hartog ◽  
J. P. Li
Keyword(s):  
Distributed Systems ◽  
Imaginary Part ◽  
Complex Number ◽  
Torsional Vibrations ◽  
The Real ◽  
Damped Systems

Abstract An extension of Holzer’s method is given for the case of damped systems of discreet as well as of uniformly distributed inertias and flexibilities. For discreet systems the modification in the Holzer table consists of replacing I by I − jc0/ω, and of replacing k by k + jωci, whereby most numbers in the tables become complex. The meaning of the real part of any complex number is that quantity which is in time-phase with the motion at the free end, while the imaginary part is 90 deg out of time-phase with that motion. For distributed systems the results are given by Equations [12] and [12a] for a free forward end; by Equations [13] and [13a] for a damped forward end, while the letters a and b appearing in these results are defined by Equations [8] and [8a].

scholarly journals On the complex q-Appell polynomials

2020 ◽  
Vol 74 (1) ◽  
pp. 31
Author(s):  
Thomas Ernst
Keyword(s):  
Imaginary Part ◽  
Generating Function ◽  
Complex Number ◽  
Analytic Functions ◽  
Complex Case ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
The Real ◽  
Cauchy Riemann

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.

scholarly journals Calculating time dilation using Euler’s formula

2020 ◽  
Author(s):  
Andrea Conte
Keyword(s):  
Imaginary Part ◽  
Complex Number ◽  
Lorentz Factor ◽  
Time Dilation ◽  
Gravitational Attraction ◽  
Speed Of Light ◽  
The Real ◽  
Euler’S Formula

This paper shows how the time dilation due to motion, calculated normally using the Lorentz factor, can be encoded in the real part of a complex number by using the Euler's formula. The imaginary part of this complex number will contain the velocity. It also shows how the time dilation due to gravitational attraction can be encoded using the same formula. A combination of time dilation and gravitational time dilation is presented using this formula. The magnitude of this complex number represents the constancy of the speed of light.

scholarly journals Complex q-Rung Orthopair Fuzzy Aggregation Operators and Their Applications in Multi-Attribute Group Decision Making

Information ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 5 ◽  
Author(s):  
Liu ◽  
Mahmood ◽  
Ali
Keyword(s):  
Decision Making ◽  
Fuzzy Sets ◽  
Imaginary Part ◽  
Group Decision Making ◽  
Group Decision ◽  
Aggregation Operators ◽  
Uncertain Information ◽  
Membership Degree ◽  
The Real ◽  
Complex Valued

In this manuscript, the notions of q-rung orthopair fuzzy sets (q-ROFSs) and complex fuzzy sets (CFSs) are combined is to propose the complex q-rung orthopair fuzzy sets (Cq-ROFSs) and their fundamental laws. The Cq-ROFSs are an important way to express uncertain information, and they are superior to the complex intuitionistic fuzzy sets and the complex Pythagorean fuzzy sets. Their eminent characteristic is that the sum of the qth power of the real part (similarly for imaginary part) of complex-valued membership degree and the qth power of the real part (similarly for imaginary part) of complex-valued non‐membership degree is equal to or less than 1, so the space of uncertain information they can describe is broader. Under these environments, we develop the score function, accuracy function and comparison method for two Cq-ROFNs. Based on Cq-ROFSs, some new aggregation operators are called complex q-rung orthopair fuzzy weighted averaging (Cq-ROFWA) and complex q-rung orthopair fuzzy weighted geometric (Cq-ROFWG) operators are investigated, and their properties are described. Further, based on proposed operators, we present a new method to deal with the multi‐attribute group decision making (MAGDM) problems under the environment of fuzzy set theory. Finally, we use some practical examples to illustrate the validity and superiority of the proposed method by comparing with other existing methods.

One Method to Construct the S-Box Based on Improved Chaos Map

2014 ◽  
Vol 651-653 ◽  
pp. 2164-2167
Author(s):  
Hang Zhang ◽  
Xiao Jun Tong
Keyword(s):  
Imaginary Part ◽  
Logistic Map ◽  
Block Ciphers ◽  
Mandelbrot Set ◽  
New Method ◽  
Hénon Map ◽  
Henon Map ◽  
The Real

Many methods of constructing S-box often adopt the classical chaotic equations. Yet study found that some of the chaotic equations exists drawbacks. Based on that, this paper proposed a new method to generate S-Box by improving the Logistic map and Henon map, and combining the real and imaginary part of complex produced by the Mandelbrot set. By comparing with several other S-boxes proposed previously, the results show the S-box here has better cryptographic properties. So it has a good application prospect in block ciphers.

scholarly journals Approximate Closed-Form Formulas for the Zeros of the Bessel Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-10
Author(s):  
Rafael G. Campos ◽  
Marisol L. Calderón
Keyword(s):  
Closed Form ◽  
Complex Plane ◽  
Complex Number ◽  
Bessel Polynomials ◽  
The Real ◽  
Bessel Polynomial ◽  
Limit Points ◽  
Electrostatic Interpretation ◽  
Approximate Expressions

We find approximate expressionsx̃(k,n,a)andỹ(k,n,a)for the real and imaginary parts of thekth zerozk=xk+iykof the Bessel polynomialyn(x;a). To obtain these closed-form formulas we use the fact that the points of well-defined curves in the complex plane are limit points of the zeros of the normalized Bessel polynomials. Thus, these zeros are first computed numerically through an implementation of the electrostatic interpretation formulas and then, a fit to the real and imaginary parts as functions ofk,nandais obtained. It is shown that the resulting complex numberx̃(k,n,a)+iỹ(k,n,a)isO(1/n2)-convergent tozkfor fixedk.

scholarly journals GAP EQUATION IN SCALAR FIELD THEORY AT FINITE TEMPERATURE

1999 ◽  
Vol 14 (04) ◽  
pp. 257-266
Author(s):  
KRISHNENDU MUKHERJEE
Keyword(s):  
Field Theory ◽  
Perturbation Theory ◽  
Scalar Field ◽  
Imaginary Part ◽  
Finite Temperature ◽  
Thermal Mass ◽  
Scalar Field Theory ◽  
Gap Equation ◽  
The Real

We investigate the two-loop gap equation for the thermal mass of hot massless g2ϕ4 theory and find that the gap equation itself has a nonzero finite imaginary part. This implies that it is not possible to find the real thermal mass as a solution of the gap equation beyond g2 order in perturbation theory. We have solved the gap equation and obtained the real and imaginary parts of the thermal mass which are correct up to g4 order in perturbation theory.

FACE REPRESENTATION WITH GRADIENT ORIENTATIONS AND EULER MAPPING: APPLICATION TO FACE RECOGNITION

2014 ◽  
Vol 28 (08) ◽  
pp. 1456014 ◽  
Author(s):  
ZHAOKUI LI ◽  
LIXIN DING ◽  
YAN WANG ◽  
JINRONG HE
Keyword(s):  
Face Recognition ◽  
Imaginary Part ◽  
Feature Vector ◽  
Low Frequency ◽  
Subspace Learning ◽  
The Real ◽  
Face Representation ◽  
Central Pixel ◽  
Low Dimensional ◽  
Two Stages

This paper proposes a simple, yet very powerful local face representation, called the Gradient Orientations and Euler Mapping (GOEM). GOEM consists of two stages: gradient orientations and Euler mapping. In the first stage, we calculate gradient orientations of a central pixel and get the corresponding orientation representations by performing convolution operator. These representation results display spatial locality and orientation properties. To encompass different spatial localities and orientations, we concatenate all these representation results and derive a concatenated orientation feature vector. In the second stage, we define an explicit Euler mapping which maps the space of the concatenated orientation into a complex space. For a mapping image, we find that the imaginary part and the real part characterize the high frequency and the low frequency components, respectively. To encompass different frequencies, we concatenate the imaginary part and the real part and derive a concatenated mapping feature vector. For a given image, we use the two stages to construct a GOEM image and derive an augmented feature vector which resides in a space of very high dimensionality. In order to derive low-dimensional feature vector, we present a class of GOEM-based kernel subspace learning methods for face recognition. These methods, which are robust to changes in occlusion and illumination, apply the kernel subspace learning model with explicit Euler mapping to an augmented feature vector derived from the GOEM representation of face images. Experimental results show that our methods significantly outperform popular methods and achieve state-of-the-art performance for difficult problems such as illumination and occlusion-robust face recognition.

scholarly journals Stiffness of sphere–plate contacts at MHz frequencies: dependence on normal load, oscillation amplitude, and ambient medium

2015 ◽  
Vol 6 ◽  
pp. 845-856 ◽  
Author(s):  
Jana Vlachová ◽  
Rebekka König ◽  
Diethelm Johannsmann
Keyword(s):  
Imaginary Part ◽  
Coulomb Friction ◽  
Ambient Medium ◽  
Normal Load ◽  
Contact Stiffness ◽  
Oscillation Amplitude ◽  
Partial Slip ◽  
Squeeze Flow ◽  
Frictional Stress ◽  
The Real

The stiffness of micron-sized sphere–plate contacts was studied by employing high frequency, tangential excitation of variable amplitude (0–20 nm). The contacts were established between glass spheres and the surface of a quartz crystal microbalance (QCM), where the resonator surface had been coated with either sputtered SiO2 or a spin-cast layer of poly(methyl methacrylate) (PMMA). The results from experiments undertaken in the dry state and in water are compared. Building on the shifts in the resonance frequency and resonance bandwidth, the instrument determines the real and the imaginary part of the contact stiffness, where the imaginary part quantifies dissipative processes. The method is closely analogous to related procedures in AFM-based metrology. The real part of the contact stiffness as a function of normal load can be fitted with the Johnson–Kendall–Roberts (JKR) model. The contact stiffness was found to increase in the presence of liquid water. This finding is tentatively explained by the rocking motion of the spheres, which couples to a squeeze flow of the water close to the contact. The loss tangent of the contact stiffness is on the order of 0.1, where the energy losses are associated with interfacial processes. At high amplitudes partial slip was found to occur. The apparent contact stiffness at large amplitude depends linearly on the amplitude, as predicted by the Cattaneo–Mindlin model. This finding is remarkable insofar, as the Cattaneo–Mindlin model assumes Coulomb friction inside the sliding region. Coulomb friction is typically viewed as a macroscopic concept, related to surface roughness. An alternative model (formulated by Savkoor), which assumes a constant frictional stress in the sliding zone independent of the normal pressure, is inconsistent with the experimental data. The apparent friction coefficients slightly increase with normal force, which can be explained by nanoroughness. In other words, contact splitting (i.e., a transport of shear stress across many small contacts, rather than a few large ones) can be exploited to reduce partial slip.

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