A new nonlinear fractal vibration of the Euler–Bernoulli beams in a microgravity space

Abstract

Microgravity is an extreme physical environment, where many theories deduced on the earth’s surface become invalid. So a fractal vibration of Euler–Bernoulli beams in a microgravity space is presented in this paper via He’s fractal derivative. With help of the fractal two-scale transform, two methods, namely, the energy balance theory and He’s frequency–amplitude formulation are adopted to seek the solutions of the fractal vibration equation. As expected, the solutions obtained by the two methods are the same. Finally, the numerical example illustrates the effectiveness of the proposed method and the impact of different fractal orders on the vibration behavior is revealed in detail.

Introduction

Microgravity science is a new science developed with space exploration. It mainly studies physics, chemistry, life science, and materials science under microgravity. 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. At present, there are four common methods to generate microgravity environment: tower falling, aircraft, rocket, and spacecraft.^1–6 As known to all, gravity acceleration is caused by the gravity of the earth, which can be expressed as ⁷

$$g\propto \frac{1}{R^2},$$

(1a)

Here, $R$ is the distance or radius from the studied object to the center of the earth. On the earth’s surface, we usually take the acceleration of gravity as $g=9.8m/s^2$ . However, the value of microgravity is usually one thousandth of the gravity on the ground in the microgravity space, and many theories derived from the earth’s surface become untenable.

The study of the beam theory is always the hot topic^8,9 and among which the famous nonlinear vibration of Euler–Bernoulli beams on the earth’s surface is governed as^10,11

$$\frac{d^2\mathrm{\Xi }\left(t\right)}{d{t}^2}+\left({\gamma }_1+p{\gamma }_2\right)\mathrm{\Xi }\left(t\right)+{\gamma }_3{\mathrm{\Xi }}^3\left(t\right)=0.$$

(1b)

And the center of the beam is subjected to the following initial conditions

$$\mathrm{\Xi }\left(0\right)=\psi ,\frac{d\mathrm{\Xi }\left(0\right)}{dt}=0,$$

(1c)

In equation (1b), p is the axial force of magnitude and ${\gamma }_1$ , ${\gamma }_2$ , and ${\gamma }_3$ are constants that are defined in Ref. 10. (More details about equation (1b) can be seen in Ref. 10.) Up to now, many effective methods have been developed to solve the nonlinear vibration such as the variational method, ¹² adomian decomposition method (ADM), ¹³ variational iteration method (VIM), ¹⁴ homotopy perturbation method,^15–18 Hamiltonian-based method, ¹⁹ and so on.^20–22 The study of equation (1b) have been studied a lot recently. In Ref. 10, the variational method is adopted to examine equation (1b). In Ref. 23, the ADM is used to investigate equation (1b). In Ref. 24, the homotopy analysis method (HAM) was used for analyzing the problem. The VIM is applied to construct the analytical solution in Ref. 25. These different methods can help us understand the vibration behavior of the Euler–Bernoulli beam. However, under the microgravity condition, equation (1b) becomes invalid. In recent years, the fractional derivative and fractal derivative have attracted extensive attention and are used widely to model many complex phenomena under the extreme environment such as the non-smooth boundary,^26,27 un-smooth surface, ²⁸ fractal medium, ²⁹ porous medium,^30,31 and others^32-38 and that the integer derivative cannot be modeled. Encouraged by recent research results on the fractal calculus, we adopt the fractal calculus from equation (1b) to obtain its fractal form for the microgravity condition as

$$\frac{d_{\mathit{\hslash }}^2\mathrm{\Xi }\left(t\right)}{d{t}^{2\chi }}+\left({\gamma }_1+p{\gamma }_2\right)\mathrm{\Xi }\left(t\right)+{\gamma }_3{\mathrm{\Xi }}^3\left(t\right)=0,$$

(1d)

where $\chi$ ( $0<\chi \le 1$ ) is the fractal order, and $\frac{d_{\mathit{\hslash }}}{d{t}^{\chi }}$ is He’s fractal derivative with respect to $t$ that is defined as^39,40

$$\frac{d_{\mathit{\hslash }}\mathrm{\Xi }}{d{t}^{\chi }}=\mathrm{\Gamma }\left(1+\chi \right)\underset{\begin{array}{l}t-t_0=\mathrm{\Delta }t\\ \mathrm{\Delta }t\ne 0\end{array}}{\lim }\frac{\mathrm{\Xi }\left(t\right)-\mathrm{\Xi }\left(t_0\right)}{{\left(t-t_0\right)}^{\chi }}.$$

(1e)

and there is the following chain rule

$$\frac{d_{\mathit{\hslash }}^2}{d{t}^{2\chi }}=\frac{d_{\mathit{\hslash }}}{d{t}^{\chi }}\frac{d_{\mathit{\hslash }}}{d{t}^{\chi }}.$$

(1f)

The objective system of equation (1d) can be used in the extreme environment as the microgravity space occurring in the tower falling, aircraft, rocket, spacecraft, and so on. For the special case of $\chi =1$ , equation (1d) is the classic vibration of Euler–Bernoulli beams on the ground of equation (1d).

The solutions of the fractal model

In this section, the energy balance theory (EBT) and He’s frequency–amplitude formulation (HFAF) will be used to find the solutions of equation (1d). For this end, the following fractal two-scale transform^41,42 is introduced

$$T=t^{\chi }.$$

(2a)

By the transform, equation (1d) can be converted into the following form

$$\frac{d^2\mathrm{\Xi }\left(T\right)}{d{T}^2}+\left({\gamma }_1+p{\gamma }_2\right)\mathrm{\Xi }\left(T\right)+{\gamma }_3{\mathrm{\Xi }}^3\left(T\right)=0,$$

(2b)

Taking the initial conditions as

$$\mathrm{\Xi }\left(0\right)=\psi ,\frac{d\mathrm{\Xi }\left(0\right)}{dT}=0.$$

(2c)

The EBT

The EBT, which is based on the variational principle and Hamiltonian, is a powerful tool to study the nonlinear vibration. To use the EBT, we set up the variational principle of equation (2b) with the aid of the semi-inverse method^43–53 as

$$\begin{array}{l}J\left(\mathrm{\Xi }\right)={\int }_0^{\frac{T}{4}}\left\{\frac{1}{2}{\left({\mathrm{\Xi }}_T\right)}^2-\left[\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\mathrm{\Xi }}^2+\frac{1}{4}{\gamma }_3{\mathrm{\Xi }}^4\right]\right\}dT\\ ={\int }_0^{\frac{T}{4}}\left\{D-S\right\}dT\end{array}$$

(2d)

where $D$ and $S$ represent the kinetic energy and potential energy, respectively. And they are

$$D=\frac{1}{2}{\left({\mathrm{\Xi }}_T\right)}^2,$$

(2e)

$$S=\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\mathrm{\Xi }}^2+\frac{1}{4}{\gamma }_3{\mathrm{\Xi }}^4.$$

(2f)

So the Hamiltonian invariant can be attained as

$$\mathit{\hslash }=D+S=\frac{1}{2}{\left({\mathrm{\Xi }}_T\right)}^2+\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\mathrm{\Xi }}^2+\frac{1}{4}{\gamma }_3{\mathrm{\Xi }}^4,$$

(2g)

According to the energy conservation theory, the Hamiltonian invariant remains unchanged in the whole vibration process, which gives ⁵⁴

$$\mathit{\hslash }=D+S=\frac{1}{2}{\left({\mathrm{\Xi }}_T\right)}^2+\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\mathrm{\Xi }}^2+\frac{1}{4}{\gamma }_3{\mathrm{\Xi }}^4={\mathit{\hslash }}_0,$$

(2h)

The solution of equation (2b) can be assumed as

$$\mathrm{\Xi }\left(T\right)=\psi \cos \left(\varpi T\right),$$

(2i)

Taking the initial conditions as

$$\psi \left(0\right)=\psi ,{\mathrm{\Xi }}^{\prime }\left(0\right)=0.$$

(2j)

Substituting equation (2j) into (2h), it gives

$$\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\psi }^2+\frac{1}{4}{\gamma }_3{\psi }^4={\mathit{\hslash }}_0,$$

(2k)

Then equation (2h) reduces to

$$\frac{1}{2}{\left({\mathrm{\Xi }}_T\right)}^2+\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\mathrm{\Xi }}^2+\frac{1}{4}{\gamma }_3{\mathrm{\Xi }}^4-\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\psi }^2-\frac{1}{4}{\gamma }_3{\psi }^4=0,$$

(2l)

Now, we substitute equation (2i) into (2l) and use $\varpi T=\frac{\pi }{4}$ , and it yields ⁵⁵

$$\frac{1}{4}{\psi }^2{\varpi }^2+\frac{1}{4}\left({\gamma }_1+p{\gamma }_2\right){\psi }^2+\frac{1}{16}{\gamma }_3{\psi }^2-\frac{1}{2}\left({\gamma }_1+p{\gamma }_2\right){\psi }^2-\frac{1}{4}{\gamma }_3{\psi }^4=0,$$

(2m)

Solving it yields

$$\varpi =\sqrt{{\gamma }_1+p{\gamma }_2+\frac{3{\psi }^2{\gamma }_3}{4}}>0,$$

(2n)

Then the solution of equation (2b) is gained as

$$\mathrm{\Xi }\left(T\right)=\psi \cos \left(\sqrt{{\gamma }_1+p{\gamma }_2+\frac{3{\psi }^2{\gamma }_3}{4}}T\right).$$

(2o)

In view of equation (2a), we can get the solution of equation (1d) as

$$\mathrm{\Xi }\left(t\right)=\psi \cos \left(\sqrt{{\gamma }_1+p{\gamma }_2+\frac{3{\psi }^2{\gamma }_3}{4}}{t}^{\chi }\right).$$

(2p)

The HFAF

The HFAF is a simple but effective approach to investigate the nonlinear vibration, and it can give the amplitude–frequency relationship by one step. For applying the HFAF, we first re-write equation (2b) as the following form

$${\mathrm{\Xi }}^{″}+f\left(\mathrm{\Xi }\right)=0,$$

(2q)

There is

$$f\left(\mathrm{\Xi }\right)=\left({\gamma }_1+p{\gamma }_2\right)\mathrm{\Xi }+{\gamma }_3{\mathrm{\Xi }}^3,$$

(2r)

Based on HFAF, we can get the amplitude–frequency relationship^56,57 through one step as

$$\varpi ={\sqrt{\frac{df\left(\mathrm{\Xi }\right)}{d\mathrm{\Xi }}}|}_{\mathrm{\Xi }=\frac{\psi }{2}}=\sqrt{{\gamma }_1+p{\gamma }_2+\frac{3{\psi }^2{\gamma }_3}{4}},$$

(2s)

which is the same as equation (2n). So the solution of equation (1d) is

$$\mathrm{\Xi }\left(t\right)=\psi \cos \left(\sqrt{{\gamma }_1+p{\gamma }_2+\frac{3{\psi }^2{\gamma }_3}{4}}{t}^{\chi }\right).$$

(2t)

Note: When $\chi =1$ , equations (2p) and (2t) become the solution of the classic nonlinear vibration of Euler–Bernoulli beams on the earth’s surface as

$$\mathrm{\Xi }\left(t\right)=\psi \cos \left(\sqrt{{\gamma }_1+p{\gamma }_2+\frac{3{\psi }^2{\gamma }_3}{4}}t\right).$$

(2u)

This is consistent with equation (25) obtained in Ref. 10, which fully proves the correctness of our method.

Results and Discussions

In this section, two examples are given to verify the correctness of our method.

Example one: Here, we use $\psi =0.6$ , ${\gamma }_1=1$ , ${\gamma }_2=0$ , and ${\gamma }_3=3$ , and then the solution of equation (1c) is

$$\mathrm{\Xi }\left(t\right)=\mathrm{0.6}\cos \left( \mathrm{1.34536} t^{\chi }\right).$$

(3a)

The effect of different fractal orders $\chi$ on the vibration characteristics is illustrated through Figures 1–3. Obviously, it can be found that the value of $\chi$ has a great influence on the vibration characteristics. When $\chi =\mathit{1}$ , the vibration is periodic, which is the classic vibration of Euler–Bernoulli beams on the earth’s surface. In the case of $\chi <\mathit{1}$ , the vibration is singular periodic. We can find the vibration period gradually increases as the time goes on, that is, the vibration slows down as the time $t$ goes on. The smaller the value of $\chi$ is, the more prominent this feature is. The 3-D vibration characteristics of equation (3a) with $t$ and $\chi$ are plotted in Figure 4.

Figure 1.

Vibration characteristics of equation (3a) for $\chi =\mathrm{1}$ .

Figure 2.

Vibration characteristics of equation (3a) for $\chi =\mathrm{0.6}$ .

Figure 3.

Vibration characteristics of equation (3a) for $\chi =\mathrm{0.4}$ .

Figure 4.

3-D vibration characteristics of equation (3a) vs $t$ and $\chi$ .

Example two: If $\psi =2$ , ${\gamma }_1=1$ , ${\gamma }_2=2$ , ${\gamma }_3=0.4$ , and $p=0.5$ , by equation (3h), the solution of equation (1d) can be obtained as

$$\mathrm{\Xi }\left(t\right)=\mathrm{2}\cos \left(\mathrm{1.78885} t^{\chi }\right),$$

(3b)

We plot the behavior of equation (3b) in Figures 5–7. In this example, we can get the same conclusion as example 1. The 3-D vibration characteristics of equation (3b) with $t$ and $\chi$ are illustrated in Figure 8.

Figure 5.

Vibration characteristics of equation (3b) for $\chi =\mathrm{1}$ .

Figure 6.

Vibration characteristics of equation (3b) for $\chi =\mathrm{0.6}$ .

Figure 7.

Vibration characteristics of equation (3b) for $\chi =\mathrm{0.4}$ .

Figure 8.

3-D vibration characteristics of equation (3b) vs $t$ and $\chi$ .

Conclusion

Based on He’s fractal derivative, a new fractal vibration of Euler–Bernoulli beams in a microgravity space is proposed in this work for the first time. Aided by the fractal two-scale transform, two effective methods, the energy balance theory and He’s frequency–amplitude formulation, are employed to find the solutions of the fractal vibration equation. As predicted, the results obtained by the two methods are consistent. Finally, two examples are presented to verify the applicability and effectiveness of the method. The obtained results in this paper are expected to open some new perspectives toward the study of the fractal vibration.