An exact solution of fractional Euler-Bernoulli equation for a beam with fixed-supported and fixed-free ends

Dynamic Response Analysis of a Forced Fractional Viscoelastic Beam ∗

“…e books can provide a general overview about the Euler-Bernoulli beam theory, please see [1][2][3]. Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4][5][6][7][8][9][10][11][12][13][14][15]. e beam systems in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have the integer order derivatives of the state function.…”

“…Some important studies related to beams modeled in the sense of classical beam theory are also summarized as follows, but not limited to [4][5][6][7][8][9][10][11][12][13][14][15]. e beam systems in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] have the integer order derivatives of the state function. In the beginning of 1930s, fractional derivative was introduced for describing the constitutive relation of some beam materials [16], and after 1980s, since fractional order equations have good memory and can be used to describe material properties more accurately with fewer parameters, they are considered to be good mathematical models for describing the dynamic mechanical behavior of materials [17].…”

Investigating Fractional Damping Effect on Euler–Bernoulli Beam Subjected to a Moving Load

“…Ten, the initial part of the solution in the Laplace domain is simplifed by rearranging equation (21) in terms of y 0 n (s) and then applying the direct Laplace inverse to get the initial solution in the time domain denoted by Y 0 n (t), as follows:…”

“…Te equations were formulated and solved for beams with and without tip mass. Te other two boundary conditions (namely, fxed-fxed and Cantilever-fxed) were considered by Blaszczyk et al [21] to be analyzed by the fractional Euler-Bernoulli beam equation, where the diferential equation of motion was converted into an integral one considering the mentioned boundary conditions and exact solution was obtained, which contained a composition of the left and right Reimann-Liouville integrals. Te study presented three solutions for a constant, power, and trigonometric functions.…”

Effects of Hausdorff Dimension on the Static and Free Vibration Response of Beams with Koch Snowflake-like Cross Section

“…In this context, Euler-Bernoulli beams with complex geometry have been solved using fractional calculus [16][17][18]. Meanwhile, solutions of the Euler-Bernoulli self-similar beam equation using fractal continuum calculus [19,20], a product-like fractal measure [21], and fractal calculus [22] were recently introduced.…”