Variational Principle and Approximate Solution for the Fractal Vibration Equation in a Microgravity Space
- PDF / 1,081,812 Bytes
- 5 Pages / 595.276 x 790.866 pts Page_size
- 11 Downloads / 155 Views
RESEARCH PAPER
Variational Principle and Approximate Solution for the Fractal Vibration Equation in a Microgravity Space Kang‑Jia Wang1 Received: 3 October 2020 / Accepted: 10 November 2020 © Shiraz University 2020
Abstract Under the microgravity space, many theories derived from the earth’s surface become untenable, so a modified vibration equation with fractal derivative is presented in this work. With the help of the semi-inverse method, we successfully develop the fractal variational principle, which not only provides conservation laws in an energy form but also provides physical insight into the nature structures of the solutions. Finally, the variational iteration method, together with the two-scale transform, is applied to find the solution of the fractal vibration equation. The obtained results show that the method is powerful and accurate. Keywords Fractal derivative · Two-scale transform · Variational principle · Variational iteration method · Semi-inverse method · Microgravity space
1 Introduction Microgravity environment that usually occurs in tower falling, aircraft, rocket and spacecraft, is an extreme physical condition, which breeds the discovery of new phenomena and laws in physical and chemical processes, material preparation and biological processes, as well as the inspection and verification of basic physical laws with higher precision (Abdel-Aty et al. 2020; Baleanu et al. 2020; Brinkert et al. 2018; Das 2008, 2009). Gravity acceleration is caused by the gravity of the earth, which can be expressed as:
g∝
1 , R2
(1.1)
where R is the distance or radius from the center of the earth to the calculated point. On the earth’s surface, we have g = 9.8 m/s2 . And the microgravity environment refers to the environment in which the apparent weight of the system is far less than its actual weight under the action of gravity. In microgravity environment, the value of microgravity is usually one thousandth of the gravity on the ground, so many * Kang‑Jia Wang [email protected]; [email protected] 1
theoretical results on the vibration based on ground gravity become invalid, and the fractal calculus has to be adopted. Recently, the fractal calculus and the fractional calculus have been widely used to describe many complex phenomenon arise in biological (Günerhan et al. 2020; He 1997a, b, 1999), filter (He 2014, 2018, 2019), physics (He 2020a, b; He and Ain 2020), microstructured solid (He and Ji 2019), circuit (He and Sun 2019) and so on (Khater et al. 2020a, b, c; Kumar et al. 2020; Lawley et al. 2017). In this work, we mainly study the vibration equation of very large membrane which is given as (Li et al. 2020; McIntyre et al. 2016):
1 𝜕2u 𝜕 2 u 1 𝜕u , = + r 𝜕r c2 𝜕 2 t2 𝜕r2
(1.2)
where u(r, t) represents the displacement of finding a particle at the point r at time instant t and c is the wave velocity of free vibration. Under the microgravity condition, a modify is needed for Eq. (1.2) by the fractal calculus, and the fractal vibration equation (FVE) of very large membrane in a microgravity
Data Loading...
