Prerpints.org logo
Preprint
Article

An Explicit Form of Ramp Function

Altmetrics

Downloads

112

Views

30

Comments

0

A peer-reviewed article of this preprint also exists.

This version is not peer-reviewed

Submitted:

20 November 2023

Posted:

20 November 2023

You are already at the latest version

Alerts
Abstract
In this paper, an analytical form of Rump Function is presented. This seminal function constitutes a fundamental concept of digital signal processing theory and is also involved in many other areas of applied mathematics. In particular, Rump Function is performed in a simple manner as the limit of a sequence of real functions letting ? tend to infinity. This limit is zero for strictly negative values of the real variable ? whereas it coincides with the independent variable ? for strictly positive values of the variable ?. The novelty of this work compared to other research studies concerning analytical expressions of the Ramp Function, is that the proposed formula is not exhibited in terms of miscellaneous special functions, e.g. Gamma Function, Biexponential Function or any other special functions such as Error Function, Hyperbolic Function, Orthogonal polynomials etc. Hence, this formula may be much more practical, flexible and useful in the computational procedures which are inserted into digital signal processing techniques and other engineering practices.
Keywords: 
Subject: Computer Science and Mathematics  -   Applied Mathematics

1. Introduction

The Ramp Function, the notation of which is R( x ), is a discontinuous single - valued function of a real variable with a point discontinuity located at zero. For negative arguments, R( x ) vanishes whilst for positive arguments, R( x ) is simply x [1]. In addition, its first derivative is the Heaviside Step function, also known as the Unit Step Function, whereas its second derivative is the Dirac delta distribution (or δ distribution), also known as the unit impulse [2]. Step, ramp and parabolic functions are called singularity functions [2,3]. In fact, the ramp function has many applications in applied sciences/engineering and is mainly involved in digital signal processing and electrical engineering. Actually it constitutes a signal the amplitude of which varies linearly with respect to time and can be expressed by several definitions [4,5]. In digital signal processing, the unit ramp function is a discrete time signal that starts from zero and increases linearly. Here one may emphasize that the basic Continuous - Time (CT) and Discrete - Time (DT) signals include impulse, step, ramp, parabolic, rectangular pulse, triangular pulse, signum function, sinc function, sinusoid and finally real along with complex exponentials [4,5,6]. Ramp Function states that the signal will start from time zero and instantly will take a slant shape and depending upon given time characteristics (i.e. either positive or negative, here positive) the signal will follow the straight slant path either towards right or left, here towards right [6,7]. In this context, the ramp function constitutes a type of elementary function which exists only for the positive side and is zero for negative [8,9,10]. Moreover, the impulse function is obtained by differentiating the ramp function twice [9,10]. On the other hand, it is well - known that an electrical network consists of passive elements like resistors, capacitors and inductors. They are connected in series, parallel and series parallel combinations [2,11,12]. The currents through and voltages across these elements are obtained by solving integro-differential equations. Alternatively, the elements in the network are transformed from the time domain and an algebraic equation is obtained which is expressed in terms of input and output [11,12]. The commonly used inputs are impulse, step, ramp, sinusoids, exponentials etc. In addition to the aforementioned above that demonstrate the central role of this function in digital signal processing and electrical engineering, we have to mention the fact that the Ramp Function has many other applications in finance as well as in applied statistics (e.g regression models) etc [2,4,12]. There are many explicit forms of this significant function that can be found in the literature.
In particular in Ref. [4] an elegant explicit representation of this function was proposed by means of the following representation:
R ( x ) = x 2 + x π ⋅ ( a r c t a n ( x ) + a r c t a n ( 1 x ) )
In Ref. [13] the following exact form of this function was performed
  R ( x ) = x 2 + i     l n   ( x ) − l n   ( − x )       2 π
Moreover, on the basis of Ref. [14] where Heaviside Step Function was analytically treated, another closed form of Ramp Function can be obtained as the next formula states
R ( x ) = 3 x 4 + x π ⋅ ( a r c t a n ( x − 1 ) + a r c t a n ( x − 2 x ) )
Further, on the basis of Ref. [15] the Ramp Function can be calculated as
R ( x ) = x π ⋅ a r c t a n ( x n ) + 2   a r c t a n ( x n x 2 n + 1 ) + a r c t a n ( 1 x n ) + 2   a r c t a n ( x 2 n − x n + 1 x 2 n + x n + 1 )
In the meanwhile, there are many smooth analytical approximations to the Ramp Function as it can be seen in the literature [16,17,18,19]. One of the simplest approximations to this function is the following [16]
R ( x ) = x 2 + 1 + x x 2 + ε 2
where ε   ∈ ( 0,1 )   such that ε   ≪ 1 .
On the other hand, in Ref. [20] an analytical form of the Unit Step Function was proposed and besides a qualitative study on Ramp and Signum Function was carried out.
Concurrently, as it was signified beforehand, there are numerous applications of Ramp Function in applied sciences and engineering as it can be observed in the literature.
In Ref. [21] a detailed study on neural networks operators by the aid of ramp functions was carried out, whilst an analogous valuable investigation took place in Ref. [22] where an interpolation by neural network operators was activated by means of ramp functions.
In Ref. [23] a remarkable study on the application of fixed-point neuron models with threshold and the role of ramp and sigmoid activation functions was presented, whereas for a thorough study on approximate solutions of Volterra integral equations via an interpolation method based on ramp functions one may refer to Ref. [24].
In Ref. [25] the role of smooth ramp functions on the activation of network interpolation operators was examined.
In Ref. [26] quadratic programming with ramp functions and fast online QP-MPC solutions was performed.
In Ref. [27] a new implementation of the Simplex method for solving linear programming problems is developed, and its application for solving Model Predictive Control (MPC) problems on the basis of ramp functions, is described.
In Ref. [28] an implementation of Ramp Function to a fracture mechanics problem concerning multi-cracked simply supported beams is carried out where the determination of the response of beams is addressed under static loads and in presence of multiple cracks, whilst in Ref. [29] a substructure elimination method for evaluating bending vibration of beams is performed. In this valuable work, a vibration analysis method is presented on the basis of the substructure elimination method, for a general class of Bernoulli-Euler beams. Here, discontinuities were treated by the use of the Heaviside Step Function, whereas the non-smooth points were approached by means of Ramp Function. In fact, referring to Euler-Bernoulli beams and Timoshenko beams there are many remarkable investigations in the literature, where singularity functions such as Unit Step Function and Ramp function have been taken into consideration to carry out analytical treatments to discontinuity problems.
In Refs. [30,31,32] the jump discontinuities on Euler-Bernoulli beams and Timoshenko beams were analytically treated by the aid of singularity functions whereas in Refs. [33,34] some basic concepts of the well - known Timoshenko beam theory were revisited and discussed in depth.
In Ref. [35] a considerable study on the dynamics of viscoelastic discontinuous beams is accomplished. This investigation deals with the dynamics of beams with an arbitrary number of Kelvin–Voigt viscoelastic rotational joints, translational supports, and attached lumped masses.
In Ref. [36] Heaviside Step Function was implemented to approximate the discontinuities in Euler-Bernoulli discontinuous beams where the analytical solution is finally carried out by means of uniform-beam Green’s functions.
In Ref. [37] an analytical treatment for Euler - Bernoulli vibrating discontinuous beams was carried out. Heaviside Step Function and Dirac’s delta distribution (also known as the unit impulse), were taken into account towards the analytical approach of the beam discontinuities.
In Ref. [38] a remarkable study concerning the achievement of closed-form solutions for stochastic Euler-Bernoulli discontinuous beams was carried out, whilst in Ref. [39] a valuable theoretical investigation on a general category of Euler - Bernoulli simply supported discontinuous beams was performed.
In Ref. [40] an Euler-Bernoulli-like Finite Element Method (FEM) for Timoshenko beams was presented and discussed whereas in Ref. [41] an exact stochastic solution for a general class of linear elastic beams subjected to delta-correlated loads was accomplished. In addition, in Ref. [42] a considerable analytical study on the effect of axial load and thermal heating on the dynamic characteristics of axially moving Timoshenko Beams was presented. On the other hand, there are many other engineering problems where Ramp Function (along with other singularity functions) are involved and play key roles. For instance, the wavemaker problem is a fundamental and important issue in the study of marine and coastal engineering. In this context, in Ref. [43] the transient waves were generated by a vertical flexible wavemaker plate by means of a general ramp function. In addition, in Ref. [44] a numerical modeling framework based on complex analysis meshless methods, which can accurately and efficiently track arbitrary crack paths in two-dimensional linear elastic solids, is presented and discussed. In this investigation, the Ramp Function is applied in compatibility conditions in order to warrant that the deformations will leave the elastic continuum body in a compatible state.
Now, in the present study, which constitutes a theoretical investigation on this special function, Rump Function is exhibited as the limit of a sequence of real functions letting n tend to infinity. This limit is proved to be zero for strictly negative values of the real variable x whereas it is proved to be simply x for strictly positive values of x . Here, one may also emphasize that the proposed exact formula is not expressed in terms of miscellaneous special functions, (elliptic integrals etc) a fact that may render this formula much more practical and helpful in the computational procedures which are inserted into digital signal processing techniques along with other engineering practices.

2. Towards an explicit form of Ramp Function

Let us introduce the following single – valued function f :   R → R + with
  f ( x ) = l i m n → + ∞ x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n ) )       2 n x + 1 + x ⋅ ( e x p ( n ⋅ x ) − 1 ) 2 e x p ( n ⋅ x ) + 2
where n ∈ N .

3. Claim

The function f coincides with Ramp Function over the set ( − ∞ , 0 ) ∪ [ 0 , + ∞ ) .

4. Proof

We shall prove that the values of the single - valued function f vanish for strictly negative arguments and coincide with the values of the real variable x for strictly positive arguments as well as at x = 0 . To this end, let us distinguish the following three cases concerning the independent variable x .
i)
x ∈ ( 0 , + ∞ )
In this context, one may deduce that
l i m n → + ∞ ( n ⋅ x ) = + ∞   ⇒ l i m n → + ∞ e x p   ( n ⋅ x ) = + ∞    
and therefore
l i m n → + ∞ 1 e x p   ( n ⋅ x ) = 0
Next, to calculate the infinitesimal quantity l i m n → + ∞ x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n ) )       2 n x + 1 , one may observe that the above fraction which the limiting operation is applied to, can be equivalently expanded as follows:
x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n ) )       2 n x + 1   = x 2 +   l n   ( n ) n x + 1       +   l n   (   x + 2 n ) 2 n x + 1       ⇔ x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n   ) )       2 n x + 1   = x 2 +   l n   ( n ) n x + 1       +   l n   ( n ) 2 n x + 1       +   l n   ( x n + 2 ) 2 n x + 1       ⇔ x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n ) )       2 n x + 1   = x 2 + 3 2 ⋅   l n   ( n ) n x + 1       +   l n   ( x n + 2 ) 2 n x + 1      
Now, since the exponent (   x + 1 ) which appears on the denominator of the fractions above is always a strictly positive quantity, i.e.   x + 1 > 0 ∀   x ∈ ( − ∞ , 0 ) ∪ [   0 , + ∞ )   one also may deduce that
l i m n → + ∞   l n   ( n ) n x + 1       = 0
In addition, since the positive term x does not vary with respect to the integer variable n , which is a natural number as it was signified beforehand, the quotient x n vanishes, letting n tend to infinity, as it is known from Calculus [45] that every sequence in the general form: a(n) = c n , c ∈ R is a convergent sequence. In this framework, one may also infer:
l i m n → + ∞   l n   ( x n + 2 ) 2 n x + 1       = 0
Hence, one obtains
l i m n → + ∞ x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n ) )       2 n x + 1 = x 2
Eqn. (6) can be combined with eqns. (7), (8) and (12) respectively, to yield
f ( x ) = x 2 + 1 2 l i m n → + ∞ x − x e x p   ( n ⋅ x ) 1 + 1 e x p   ( n ⋅ x )
Moreover, since the real variable x does not vary with respect to the integer variable n one may also deduce that
l i m n → + ∞ x − x e x p   ( n ⋅ x ) 1 + 1 e x p   ( n ⋅ x ) = x
Eqn. (13) can be combined with eqn. (14) to yield
f x = x 2 + 1 2 ⋅ x − 0 1 + 0   ⇒ f ( x ) = x
Thus, it was proved that the value of the function f ( x ) is simply x for strictly positive arguments, i.e. when x ∈   ( 0 , + ∞ ) .
ii)
x ∈ ( − ∞ , 0 )
In this context, since the real variable x as well as the natural number n (which evidently is an integer variable) do not agree in sign, one may deduce that
l i m n → + ∞ ( n ⋅ x ) = − ∞   ⇒ l i m n → + ∞ e x p   ( n ⋅ x ) = 0  
Here, one may also pinpoint that eqn. (12) which was previously derived when we considered the variable x to be strictly positive, still holds. This significant observation is attributed to a fact that we shall discuss just below. By focusing on the fraction x ⋅   n x + 1 + l n   ( n 2 ⋅   ( x + 2 n ) )       2 n x + 1 which appears on eqn. (12) one may pinpoint that the real variable x occurs in the denominator of this fraction only by its absolute value i.e. x .
Thus the sign of this real variable, cannot influence the sign of the limit of the denominator in this aforementioned fraction, i.e. the term 2 n x + 1 , letting the integer variable n tend to infinity.
Indeed, the limit of the real quantity 2 n x + 1 letting n tend to infinity, always is equal to + ∞ , regardless of the sign and the values of the real variable x even if it equals zero.
In continuing, eqn. (6) can be combined with eqns. (12) and (16) respectively to yield
f x = x 2 + 1 2 ⋅ l i m n → + ∞ x ⋅ exp ⁡ n ⋅ x − x l i m n → + ∞ exp ⁡ n ⋅ x + 1 ⇒ f ( x ) = x 2 + 1 2 ⋅ x ⋅ l i m n → + ∞ ( e x p ( n ⋅ x ) ) − x l i m n → + ∞ ( e x p ( n ⋅ x ) ) + 1
At this point, we have to elucidate that we have taken into account the fact that the real variable x does not vary with respect to the integer variable n . In this framework, we were able to pull it out of the limiting operation, letting n tend to infinity.
Thus, on the basis of eqn. (17), it implies that
f x = x 2 + 1 2 ⋅ 0 ⋅ x − x 0 + 1   ⇒ f ( x ) = x 2 − x 2 = 0
Hence, it was proved that the values of f ( x ) vanish for strictly negative arguments i.e. when the real variable x   ∈   ( − ∞ , 0 ) . In addition, we have to emphasize that eqn. (12) which was derived in the first case of the problem we studied, i.e. when x ∈ ( 0 , + ∞ ) , is always valid over the set of real numbers: ( − ∞ , 0 ) ∪ [   0 , + ∞ )   , as we have previously shown. Moreover, we have taken into consideration the fact that the real variable x and the integer variable n are always independent of each other. In this framework, the real variable x can be pulled out of the limiting operations letting n tend to infinity, since it can be roughly said that it practically behaves as a real constant during the limiting operation.
ii)
x = 0
Then, one obtains
𝑙𝑖𝑚𝑛→+∞𝑒𝑥𝑝 (𝑛⋅𝑥)= 𝑙𝑖𝑚𝑛→+∞𝑒𝑥𝑝 (𝑛⋅0)
and therefore
𝑙𝑖𝑚𝑛→+∞𝑒𝑥𝑝 (𝑛⋅𝑥)=𝑙𝑖𝑚𝑛→+∞𝑒𝑥𝑝 (0)=1
Moreover, one may note that eqn. (12) still holds.
Now, eqn. (6) can be combined with eqns. (20) and (12) respectively to yield
f ( x ) = l i m n → + ∞   l n   ( 2 n 3 )           2 n + 0 2 + 2 ⇒ f ( x ) = l i m n → + ∞   l n   ( 2 n 3 )           2 n + 0 4  
and therefore
f ( x ) = 0 + 0 4 = 0
Thus, it was proved that the value of f ( x ) vanish at   x = 0   as the Rump Function also does. Afterall, one may come to the conclusion that the single - valued real function introduced by eqn. (6) is identical to the Ramp Function over the set of real numbers.

5. Discussion

In Section 2, an explicit form of Rump Function was proposed, as the limit of a sequence of real functions letting n tend to infinity. This limit was proved to be zero for strictly negative values of the real variable x whereas it was proved to be simply x for strictly positive values of x . In fact, the proposed single - valued function coincides with Rump Function over the set ( − ∞ , 0 ) ∪ ( 0 , + ∞ ) . In addition, one may also observe that the single – valued function f introduced by eqn. (6) vanishes at   x = 0. Indeed, the Ramp Function (by its definition) also vanishes at x = 0 since it is just x for positive arguments. In this framework, one may also conclude that the single - valued function introduced by eqn. (6) coincides with Ramp Function over the set of real numbers ( − ∞ , 0 ) ∪ [   0 , + ∞ ) .
In addition, by focusing on the infinitesimal quantity l i m n → + ∞ x ⋅ ( e x p ( n ⋅ x ) − 1 ) 2 e x p ( n ⋅ x ) + 2 which can be equivalently written as: 1 2 l i m n → + ∞ x ⋅ ( e x p ( n ⋅ x ) − 1 ) e x p ( n ⋅ x ) + 1 one may observe that the following relationship holds:
l i m n → + ∞ x ⋅ ( e x p ( n ⋅ x ) − 1 ) e x p ( n ⋅ x ) + 1 = x
As a matter of fact, the infinitesimal quantity l i m n → + ∞ x ⋅ ( e x p ( n ⋅ x ) − 1 ) e x p ( n ⋅ x ) + 1 coincides with the absolute value of the real variable x over the set ( − ∞ , 0 ) ∪   ( 0 , + ∞ ) and also at x = 0, since it is obvious that the numerator of the above fraction vanishes at x = 0 whereas its denominator equals to 2.
Actually, the validity of eqn. (23) is attributed to the fact that the value of the infinitesimal quantity l i m n → + ∞ e x p   ( n ⋅ x )     depends on the sign of the variable x which is definitely independent of the integer variable n .
In this context, the infinitesimal quantity l i m n → + ∞ x ⋅ ( e x p ( n ⋅ x ) − 1 ) e x p ( n ⋅ x ) + 1 equals x for strictly positive arguments, whilst it equals −   x for strictly negative arguments and finally vanishes at x=0.

6. Conclusions

The objective of this theoretical investigation was to introduce an analytical representation of the Rump Function, which evidently is a very useful mathematical tool and it participates in many areas of applied and engineering mathematics and physics. The novelty of this work, when compared to other analytical treatments to this significant function, is that the proposed exact mathematical formula is not exhibited in terms of any miscellaneous special functions, or any other special functions such as Error Function, Hyperbolic Function, Orthogonal polynomials etc.
Nevertheless, one may also observe that an advantage of the proposed closed - form expression of this special function, is that it coincides with Ramp Function over the set ( − ∞ , 0 ) ∪ [   0 , + ∞ ) since it was proved to vanish at x = 0, as Ramp Function also does.
In closing, as a future work, one may also propose similar analytical representations to other singularity functions [46] e.g. Heaviside Step Function, Signum Function etc by taking into consideration this theoretical approach.

Funding

Not applicable

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The author declares no conflict of interest.

References

  1. M. Abramowitz, I.A. and Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications INC (1972).
  2. R.P. Kanwal, Generalized Functions Theory and Technique: Theory and Technique. 2nd ed. Boston, MA: Birkhäuser (1998).
  3. Peled, A. , and Liu, B. Digital signal processing: theory, design, and implementation. New York, (1976).
  4. J. Spanier, J. K.B. and Oldham The Unit-Step u(x-a) and Related Functions Ch. 8 from: An Atlas of Functions. Washington, DC: Hemisphere, pp. 63-69, 1987.
  5. Palani, S. The z-Transform Analysis of Discrete Time Signals and Systems, Chapter 9 in Signals and Systems, 2nd ed.; Springer Nature Switzerland AG: Cham, Switzerland, 2022; pp. 921–1055. [Google Scholar]
  6. Oppenheim, Alan V., et al. Signals and systems. Vol. 2. Upper Saddle River, NJ: Prentice hall, 1997.
  7. Simon Haykin, Barry Van Veen, Signals and systems John Wiley & Sons (Asia) Pte. Ltd., Second Edition, 2004.
  8. Siebert, W. M. (1986). Circuits, signals, and systems. MIT press.
  9. Singh, S. Proposed Concept of Signals for Ramp Functions. In Proceedings of the World Congress on Engineering and Computer Science 2010 Vol I WCECS 2010, San Francisco, CA, USA, 20–22 October 2010. [Google Scholar]
  10. Sayood, K. (2021). Special Functions. In: Signals and Systems. Synthesis Lectures on Signal Processing. Springer, Cham. [CrossRef]
  11. Chen, Wai Kai. The electrical engineering handbook. Elsevier, 2004.
  12. Bell, W. W. (2004). Special functions for scientists and engineers. Courier Corporation.
  13. K. Murphy, Explicit Forms of Discontinuous Functions the Dirac Delta and Irreducible Forms (preprint) https://www.academia.edu/11704122/Explicit_Forms_of_Discontinuous_Functions_the_Dirac_Delta_and _Irreducible_Forms.
  14. Venetis J.C., An analytic exact form of the unit step function, Mathematics and Statistics Vol. 2, Number 7, pp. 235-237, 2014.
  15. Venetis, J.C. An Explicit Form of Heaviside Step Function. Preprints 2021, 2021060132. [Google Scholar] [CrossRef]
  16. Loring, W. Tu (2008). Smooth Functions on a Euclidean Space. In: An Introduction to Manifolds. Universitext. Springer, New York, NY. [CrossRef]
  17. Sobot, R. (2021). Special Functions. In: Engineering Mathematics by Example. Springer, Cham. [CrossRef]
  18. X. Xu, Nonlinear trigonometric approximation and the Dirac delta function, Journal of Computational and Applied Mathematics 209 (2006) 234 – 245.
  19. Y.T. Li and R. Wong, Integral and Series Representations of the Dirac Delta Function, Communications on Pure and Applied Analysis, (2013). arXiv:1303. 1943.
  20. Venetis, J.C. An Explicit Expression of the Unit Step Function, International Review of Electrical Engineering, Vol. 18, Number 1. 83–87, 2023.
  21. Cao, F. and Chen Z. The construction and approximation of a class of neural networks operators with ramp functions. J. Comput Anal Appl Vol. 14, Number 1. pp. 101–112, 2012.
  22. Costarelli D Interpolation by neural network operators activated by ramp functions. J Math Anal Appl Vol. 419, Number 5. pp. 574–582, 2014.
  23. Zhang, L. Implementation of fixed-point neuron models with threshold, ramp and sigmoid activation functions. In IOP Conference Series: Materials Science and Engineering, Vol. 224, Number 1 pp. 12-54, IOP Publishing, 2017.
  24. Costarelli, D. Approximate solutions of Volterra integral equations by an interpolation method based on ramp functions. Comp. Appl. Math. Vol. 38, Article Number 159 (2019). [CrossRef]
  25. Yunyou Qian and Dansheng Yu Neural network interpolation operators activated by smooth ramp functions Analysis and Applications Vol. 20, Number 4. 791-813 (2022). [CrossRef]
  26. Valmorbida, Giorgio, and Morten Hovd. Quadratic programming with ramp functions and fast online QPMPC solutions. Automatica Vol. 153, Article Number 111011, 2023. [CrossRef]
  27. Hovd, Morten, and Giorgio Valmorbida. Solving LP-MPC problems using ramp functions 2023 IEEE Conference on Control Technology and Applications (CCTA). IEEE, 2023.
  28. Laudani, R. , & Falsone, G. (2021). Response probability density function for multi-cracked beams with uncertain amplitude and position of cracks. Applied Mathematical Modelling, 99, 14-26. [CrossRef]
  29. Yamada, Keisuke; JI, Jinchen. Substructure elimination method for evaluating bending vibration of beams. Mechanical Engineering Journal, 2023, 23-00293. [CrossRef]
  30. Yavari, A. , Sarkani, S. and Moyer, E.T. (2000), On applications of generalized functions to beam bending problems, Int. J. Solid. Struct., 37, 5675-5705. [CrossRef]
  31. Yavari, A. , Sarkani, S. and Reddy, J.N. (2001), On nonuniform Euler-Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory”, Int. J. Solid. Struct., 38, 8389-8406. [CrossRef]
  32. Yavari, A. , Sarkani, S. and Reddy, J.N. (2001), Generalized solutions for beams with jump discontinuities on elastic foundations”, Arch. Appl. Mech., 71(9), 625-639. [CrossRef]
  33. Ecsedi, I. and Dluhi, K. (2004), Use of discontinuity functions to obtain the frequency equation, J. Comput. Appl. Mech., 5(1), 193-199.
  34. Elishakoff, I., Kaplunov, J. and Nolde, E. (2015), Celebrating the centenary of Timoshenko’s study of effects of shear deformation and rotary inertia, Appl. Mech. Rev., 67(6), 1067.
  35. Failla, G. (2014), On the dynamics of viscoelastic discontinuous beams, J. Sound Vib., 60, 52-63. [CrossRef]
  36. Failla, G. and Santini, A. (2007), On Euler-Bernoulli discontinuous beam solutions via uniform-beam Green’s functions, Int. J. Solid. Struct., 44, 7666-7687. [CrossRef]
  37. Failla, G. and Santini, A. (2008), A solution method for Euler - Bernoulli vibrating discontinuous beams, Mech. Res. Commun., 35, 517-529. [CrossRef]
  38. Failla, G, Santini, A. and Spanos, P.D. (2007), “Closed-form solutions for stochastic Euler-Bernoulli discontinuous beams”, Congresso XVIII Congresso Nazionale AIMETA, Brescia, September.
  39. Falsone, G. (2002), The use of generalized functions in the discontinuous beam bending differential equations”, Int. J. Eng. Educ., 18(3), 337-343.
  40. Falsone, G. and Settineri, D. (2011), An Euler-Bernoulli-like finite element method for Timoshenko beams”, Mech. Res. Commun., 38, 12-16. [CrossRef]
  41. Falsone, G. and Settineri, D. (2013), “Exact stochastic solution of beams subjected to delta-correlated loads, Struct. Eng. Mech., 47(3), 307-329.
  42. Chen, H. Y., Wang, Y. C., Wang, D., and Xie, K. (2023). Effect of Axial Load and Thermal Heating on Dynamic Characteristics of Axially Moving Timoshenko Beam. International Journal of Structural Stability and Dynamics, 2350191. [CrossRef]
  43. Chen, Q., Chen, X., Ma, Y., and Dong, G. (2020). Transient waves generated by a vertical flexible wavemaker plate with a general ramp function. Applied Ocean Research, 103, 102335. [CrossRef]
  44. Li, D. M., Liu, J. H., Nie, F. H., Featherston, C. A., and Wu, Z. (2022). On tracking arbitrary crack path with complex variable meshless methods. Computer Methods in Applied Mechanics and Engineering, 399, 115402. [CrossRef]
  45. Borwein, J.M., Skerritt, M.P. (2011). Calculus. In: An Introduction to Modern Mathematical Computing. Springer Undergraduate Texts in Mathematics and Technology . Springer, New York, NY. [CrossRef]
  46. A. M. Mathai, Hans J. Haubold Special Functions for Applied Scientists, Springer New York, NY 2008. [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.
Copyright: This open access article is published under a Creative Commons CC BY 4.0 license, which permit the free download, distribution, and reuse, provided that the author and preprint are cited in any reuse.
Prerpints.org logo

Preprints.org is a free preprint server supported by MDPI in Basel, Switzerland.

facebook logotwitter logolinkedin logo
MDPI Initiatives
Important Links
Subscribe

Disclaimer

Terms of Use

Privacy Policy

© 2024 MDPI (Basel, Switzerland) unless otherwise stated