We retrieve unknown nonlinear large space-time dependent forces burdened with the vibrating nonlinear Euler-Bernoulli beams under varied boundary data, comprising two-end fixed, cantilevered, clamped-hinged, and simply supported conditions in this study. Even though some researchers used several schemes to overcome these forward problems of Euler-Bernoulli beams; however, an effective numerical algorithm to solve these inverse problems is still not available. We cope with the homogeneous boundary conditions, initial data, and final time datum for each type of nonlinear beam by employing a variety of boundary shape functions. The unknown nonlinear large external force can be recuperated via back-substitution of the solution into the nonlinear Euler-Bernoulli beam equation when we acquire the solution by utilizing the boundary shape function scheme and deal with a small-scale linear system to gratify an additional right-side boundary data. For the robustness and accuracy, we reveal that the current schemes are substantiated by comparing the recuperated numerical results of four instances to the exact forces, even though a large level of noise up to 50% is burdened with the overspecified conditions. The current method can be employed in the online real-time computation of unknown force functions in space-time for varied boundary supports of the vibrating nonlinear beam.

As we all know, retrieving external forces of Euler-Bernoulli beams plays very important roles in many engineering and scientific areas. These equations occur in the vibration of a structure, the cutting process in engineering, the sandwiches beams, the cable-stayed beams, the rotating beams, the aircraft engineering, the design of mechanical cutting tools, the nondestructive testing and so forth.

For linear external forces of linear Euler-Bernoulli equations, Han et al. [

For the free vibration of composite beams and non-uniform beams, Liu et al. [

For the difficult nonlinear Euler-Bernoulli beams, Barari et al. [

This article is arranged as follows. Section 2 displays the nonlinear problem statement and constructs the new boundary shape function, and homogeneous boundary data of the beam. In Section 3, we acquire shape functions and introduce a free parameter into the boundary shape function, which leads to a variety of boundary shape functions. Four numerical examples of the nonlinear large external forces on vibrating nonlinear Euler-Bernoulli beams are shown in Section 4. At last, we display the conclusions in Section 5.

We deliberate an inverse source issue to reveal an unknown nonlinear force function

It cannot be resolved forthrightly to reveal

First of all, we deliberate a partial boundary shape function in the time orientation:

Assuming

Next, we search four polynomial-type shape functions

We find that those four boundary data in

We can acquire through some calculations

We can verify that the boundary shape function

The boundary shape function

Note that we do not utilize the basis of the solution since there subsists no free parameter in

We can acquire through some calculations

We can induce

These conditions are gratified can be confirmed easily as follows:

Assuming

The strain data

The linear equations can be acquired by collocating points

We can decide the

By utilizing the boundary shape functions method (BSFM), we can reveal that this present scheme is suitable to solve the inverse source issue of the beam equation. In fact,

While the maximum values of

Employing

All the computational schemes were implemented to the Fortran code on the Microsoft Developer Studio platform in OS Windows 10 (64 bit) with i3-4160 3.60 GHz CPU and 16 GB memory.

In the following four numerical examples, we usually choose the absolute error

We deliberate a complicated space-time–dependent force generated from

The noise is

Considering a large relative noise of

We show that the clamped-hinged beam owns the boundary data, shape functions, and boundary shape functions as follows:

We ponder a complicated space-time–dependent force of the clamped-hinged beam produced from

The noise is

Deliberating a large relative noise of

We demonstrate that the cantilevered beam has the boundary data, shape functions, and boundary shape functions as follows:

We contemplate a nonseparable space-time–dependent force produced from

The noise is

Pondering a large relative noise of

For the two-end fixed beam we can calculate the boundary data, shape functions, and boundary shape functions as follows:

In place of

Concurrently, we alter

Letting

The noise is

Considering a large relative noise of

We addressed the recovery issues of revealing unknown forces burdened with the nonlinear Euler-Bernoulli beams with four boundary-supported data, e.g., cantilevered, simply supported, two-end fixed, and clamped-hinged beams, by utilizing the boundary shape functions method. The innovation of this scheme is the establishment of a variety of boundary shape functions with varied orders for each beam sort, such that we have influential foundations from which to extend the solution in the whole space-time realm. The proposed scheme can be utilized in the online real-time estimation of unknown force functions in space-time for varied boundary supports of the vibrating beam. On the basis of those numerical examples, we display that the proposed algorithm is applicable to the nonlinear external forces of nonlinear Euler-Bernoulli equations and pretty excellent computational efficiency, and even for adding the large random noise up to 50%. Furthermore, to the author’s best knowledge, there has no report in the literature that the numerical schemes for those four issues can offer more accurate results than the present results. The present approach can be extended to cope with the multi-dimensional inverse nonlinear transient PDEs and will be worked out in the future.