Search Results (172)

Reductions in the vibration of a continuum system via a nonlinear energy sink have been widely investigated. It is usually assumed that weight effects can be ignored if the vibration is measured from the static equilibrium configuration. The present investigation reveals the dynamic effects of weight on the vertical transverse vibrations of a Euler–Bernoulli beam coupled with a nonlinear energy sink. The governing equations considering and neglecting weights were derived. The equations were discretized with some numerical support. The discretized equations were analytically solved via the harmonic balance method. The harmonic balance solutions were compared with the numerical solution via the Runge–Kutta method. Finite element simulations were performed via ANSYS software (version number: 2.2.1). Free and forced vibrations, predicted by equations considering or neglecting the weights, were compared with the finite element solutions. For the forced vibrations, the amplitude–frequency responses determined by the harmonic balance method agree well with those calculated by the Runge–Kutta method. The free and forced vibration responses predicted by the equations considering the weights are closer to those computed by the finite element method than the responses predicted by the equation neglecting the weights. The assumption that weights can be balanced by static deflections leads to errors in the analysis of the vertical transverse vibrations of a Euler–Bernoulli beam with a nonlinear energy sink. Full article

In order to enhance orchard agricultural efficiency and lower fruit production expenses, we propose a BL-DBO (Beetle Optimization Algorithm introducing Bernoulli mapping and Lévy flights) to solve the agricultural machinery dispatching model within the orchard area. First, we analyze the agricultural machinery dispatching problem in the orchard area and establish its mathematical model with the objective of minimizing dispatching costs as a constraint. To tackle the problems of uneven individual position distribution and the risk of becoming stuck in local optimal solutions in the traditional DBO algorithm, we introduce Bernoulli mapping during the initialization phase of the DBO. This method ensures a uniform distribution of the initialized population. Furthermore, during the iterative process of the algorithm, we incorporated the Lévy flight approach into the positional update equations for beetles involved in breeding, foraging, and theft activities within the DBO. This helps the beetles escape from local optimal solutions. Finally, we conduct experiments based on location information of Shunping Shunnong Orchard and fruit trees in Shijiazhuang. The results indicate that, compared to dispatching using human experience and the traditional DBO algorithm, the dispatching results generated by the BL-DBO not only reduce the number of agricultural machinery purchases but also decrease the energy loss from non-working distances of the machinery, effectively saving fruit production costs. Full article

Numerous studies on the number pi ($\pi$) explore its properties, including normality and applicability. This research, grounded in two hypotheses, proposes and proves a theorem that employs a Bernoulli experiment to demonstrate the high probability of encountering any finite bit string within a sequence of consecutive bits in the decimal part of $\pi$. This aligns with findings related to its normality. To support the hypotheses, we present experimental evidence about the equiprobable and independent properties of bits of $\pi$, analyzing their distribution, and measuring correlations between bit strings. Additionally, from a cryptographic perspective, we evaluate the chaotic properties of two images generated using bits of $\pi$. These properties are evaluated similarly to those of encrypted images, using measures of correlation and entropy, along with two hypothesis tests to confirm the uniform distribution of bits and the absence of periodic patterns. Unlike previous works that solely examine the presence of sequences, this study provides, as a corollary, a formula to calculate an upper bound N. This bound represents the length of the sequence from $\pi$ required to ensure the location of any n-bit string at least once, with an adjustable probability p that can be set arbitrarily close to one. To validate the formula, we identify sequences of up to $n=$ 40 consecutive zeros and ones within the first N bits of $\pi$. This work has potential applications in Cryptography that use the number $\pi$ for random sequence generation, offering insights into the number of bits of $\pi$ required to ensure good randomness properties. Full article

In the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function $\ln \sec x=-\ln \cos x$; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function $(2^x-1)\zeta \left(x\right)$ on $(1,\infty )$, they prove the logarithmic convexity of Qi’s normalized remainder; with the aid of a monotonicity rule for the ratio of two Maclaurin power series, the authors present the monotonic property of the ratio between two Qi’s normalized remainders. Full article

Certain methods for implementing chaotic maps can lead to dynamic degradation of the generated number sequences. To solve such a problem, we develop a method for generating pseudorandom number sequences based on multiple one-dimensional chaotic maps. In particular, we introduce a Bernoulli chaotic map that utilizes function transformations and constraints on its control parameter, covering complementary regions of the phase space. This approach allows the generation of chaotic number sequences with a wide coverage of phase space, thereby increasing the uncertainty in the number sequence generation process. Moreover, by incorporating a scaling factor and a sine function, we develop a robust chaotic map, called the Sine-Multiple Modified Bernoulli Chaotic Map (SM-MBCM), which ensures a high degree of randomness, validated through statistical mechanics analysis tools. Using the SM-MBCM, we propose a chaotic PRNG (CPRNG) and evaluate its quality through correlation coefficient analysis, key sensitivity tests, statistical and entropy analysis, key space evaluation, linear complexity analysis, and performance tests. Furthermore, we present an FPGA-based implementation scheme that leverages equivalent MBCM variants to optimize the electronic implementation process. Finally, we compare the proposed system with existing designs in terms of throughput and key space. Full article

Cracks, common indicators of deterioration in bridge frameworks, frequently stem from wear and rust, leading to increased local flexibility and changes in the structure’s dynamic behavior. This study examines how these cracks affect the dynamics of footbridges when subjected to loads generated by walking individuals. The pedestrian is modeled as a linear oscillator, while the cracked bridge is represented by a simply supported beam following Euler–Bernoulli’s theory. The use of the Dirac delta function allows for the precise representation of the localized stiffness reduction at the crack location, facilitating the calculation of analytical expressions for the beam’s vibration modes. The research suggests that the presence of cracks minimally affects the bridge’s mid-span displacement. However, with a limited depth of cracks, the appearance of cracks notably amplifies the mid-span acceleration amplitude of the bridge, leading to a pronounced concentration of energy at the third natural frequency of the bridge in the acceleration spectrum. As the depth and number of cracks increase, the acceleration amplitude continues to decrease, but the corresponding spectrum remains almost unchanged. The study’s outcomes enhance the comprehension of how cracks affect the performance of bridge structures when subjected to loads from pedestrians, offering insights for the monitoring and evaluation of the condition of cracked footbridges. Full article

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials. Full article

This study presents a numerical investigation into the adhesion strength of micro fibrillar structures, incorporating statistical analysis and the effects of excessive pre–load leading to fibril buckling. Fibrils are modeled as soft cylinders using the Euler–Bernoulli beam theory, with buckling conditions described across three distinct states, each affecting the adhesive properties of the fibrils. Iterative simulations analyze how adhesion strength varies with pre–load, roughness, number of fibrils, and the work of adhesion. Roughness is modeled both in fibril heights and in the texture of a rigid counter surface, following a normal distribution with a single variance parameter. Results indicate that roughness and pre–load significantly influence adhesion strength, with excessive pre–load causing substantial buckling and a dramatic reduction in adhesion. This study also finds that adhesion strength decreases exponentially with increasing roughness, in line with theoretical expectations. The findings highlight the importance of buckling and roughness parameters in determining adhesion strength. This study offers valuable insights into the complex adhesive interactions of fibrillar structures, offering a scalable solution for rapid assessment of adhesion in various rough surface and loading scenarios. Full article

In the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the tail of the power series expansion of the exponential function, and (5) on account of Čebyšev’s integral inequality, the authors (i) expand the logarithm of the normalized tail of the power series expansion of the exponential function into a power series whose coefficients are expressed in terms of specific Hessenberg determinants whose elements are quotients of combinatorial numbers, (ii) prove the logarithmic convexity of the normalized tail of the power series expansion of the exponential function, (iii) derive a new determinantal expression of the Bernoulli numbers, deduce a determinantal expression for Howard’s numbers, (iv) confirm the increasing monotonicity of a function related to the logarithm of the normalized tail of the power series expansion of the exponential function, (v) present an inequality among three power series whose coefficients are reciprocals of combinatorial numbers, and (vi) generalize the logarithmic convexity of an extensively applied function involving the exponential function. Full article

Since the constructions of p-adic q-integrals, these integrals as well as particular cases have been used not only as integral representations of many special functions, polynomials, and numbers, but they also allow for deep examinations of many families of special numbers and polynomials, such as central Fubini, Bernoulli, central Bell, and Changhee numbers and polynomials. One of the key applications of these integrals is for obtaining the symmetric identities of certain special polynomials. In this study, we focus on a novel generalization of degenerate central Fubini polynomials. First, we introduce two variable degenerate w-torsion central Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. Through this representation, we investigate several symmetric identities for these polynomials using special p-adic integral techniques. Also, using series manipulation methods, we obtain an identity of symmetry for the two variable degenerate w-torsion central Fubini polynomials. Finally, we provide a representation of the degenerate differential operator on the two variable degenerate w-torsion central Fubini polynomials related to the degenerate central factorial polynomials of the second kind. Full article

The ground vibration caused by the operation of high-speed trains has become a key challenge in the development of high-speed railways. In order to study the train-induced ground vibration affected by geotechnical heterogeneity, an efficient frequency–wave-number method coupled with the random variable theory model is proposed to quickly obtain the numerical results without losing accuracy. The track is regarded as a composite Euler–Bernoulli beam resting on the layered ground, and the spatial heterogeneity of the ground soil is considered. The ground dynamic characteristics of an elastic, layered, non-uniform foundation are investigated, and numerical results at three typical train speeds are reported based on the developed Fortran computer programs. The results show that as the soil homogeneity coefficient increases, the peak acceleration continuously decreases in the transonic case, while it gradually increases in the supersonic case, and the ground acceleration spectrum at a far distance obviously decreases; the maximum acceleration occurs at the track edge, and a local rebound in vibration attenuation occurs in the supersonic case. Full article

After constructions of p-adic q-integrals, in recent years, these integrals with some of their special cases have not only been utilized as integral representations of many special numbers, polynomials, and functions but have also given the chance for deep analysis of many families of special polynomials and numbers, such as Bernoulli, Fubini, Bell, and Changhee polynomials and numbers. One of the main applications of these integrals is to obtain symmetric identities for the special polynomials. In this study, we focus on a novel extension of the degenerate Fubini polynomials and on obtaining some symmetric identities for them. First, we introduce the two-variable degenerate w-torsion Fubini polynomials by means of their exponential generating function. Then, we provide a fermionic p-adic integral representation of these polynomials. By this representation, we derive some new symmetric identities for these polynomials, using some special p-adic integral techniques. Lastly, by using some series manipulation techniques, we obtain more identities of symmetry for the two variable degenerate w-torsion Fubini polynomials. Full article

This paper presents an overview of cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials, as well as several identities that are associated with these polynomials. By applying a partial derivative operator to the generating functions, the authors obtain derivative formulae and finite combinatorial sums involving these polynomials and numbers. Additionally, the paper establishes connections between cosine and sine Apostol-type Frobenius–Euler–Fibonacci polynomials of order $\alpha$ and several other polynomial sequences, such as the Apostol-type Bernoulli–Fibonacci polynomials, the Apostol-type Euler–Fibonacci polynomials, the Apostol-type Genocchi–Fibonacci polynomials, and the Stirling–Fibonacci numbers of the second kind. The authors also provide computational formulae and graphical representations of these polynomials using the Mathematica program. Full article

Most multi-target movements are nonlinear in the process of movement. The common multi-target tracking filtering methods directly act on the multi-target tracking system of nonlinear targets, and the fusion effect is worse under the influence of different perspectives. Aiming to determine the influence of different perspectives on the fusion accuracy of multi-sensor tracking in the process of target tracking, this paper studies the multi-target tracking fusion strategy of a nonlinear system with different perspectives. A GM-JMNS-CPHD fusion technique is introduced for random outlier selection in multi-target tracking, leveraging sensors with limited views. By employing boundary segmentation from distinct perspectives, the posterior intensity function undergoes decomposition into multiple sub-intensities through SOS clustering. The distribution of target numbers within the respective regions is then characterized by the multi-Bernoulli reconstruction cardinal distribution. Simulation outcomes demonstrate the robustness and efficacy of this approach. In comparison to other algorithms, this method exhibits enhanced robustness even amidst a decreased detection probability and heightened clutter rates. Full article

In practical radar systems, changes in the target aspect toward the radar will result in glint noise disturbances in the measurement data. The glint noise has heavy-tailed characteristics and cannot be perfectly modeled by the Gaussian distribution widely used in conventional tracking algorithms. In this article, we investigate the challenging problem of tracking a time-varying number of maneuvering targets in the context of glint noise with unknown statistics. By assuming a set of models for the possible motion modes of each single-target hypothesis and leveraging the multivariate Laplace distribution to model measurement noise, we propose a robust interacting multi-model multi-Bernoulli mixture filter based on the variational Bayesian method. Within this filter, the unknown noise statistics are adaptively learned while filtering and the predictive likelihood is approximately calculated by means of the variational lower bound. The effectiveness and superiority of our proposed filter is verified via computer simulations. Full article