# On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics

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## Abstract

**:**

## 1. Introduction

^{®}computer code in a dataset linked to this paper that implements the method. A write-up of the method in the form of post-graduate teaching notes prepared by Miskandarani can be found, e.g., in [65]. We applied the method to the construction of an explicit finite difference solution scheme for the Euler–Bernoulli equation for the application to the analysis of beams in the presence of time-varying external loads with a velocity-dependent damping term and taking into account mass, the second moment of area, and elastic modulus beam structural properties, that arbitrarily vary with the distance along the beam.

## 2. Governing Equations and Solution Methodology

#### 2.1. Development of a Numerical Approach for Solving the Inhomogeneous Euler–Bernoulli Equation

^{®}script that implements this algorithm can be found in the MDPI code database (see Appendix E).

#### 2.2. Extension of the Lax–Richtmyer Stability Criteria to the Fourth-Order Euler–Bernoulli Equation

#### 2.3. Specification of Boundary Conditions and Values of Functions F and W at the Ghost Nodes

^{®}script which is included in the dataset linked to this paper (see Appendix E). All simulations were performed using this script.

## 3. Results and Discussions

#### 3.1. Grid Sensitivity Study for the Fourth-Order Euler–Bernoulli Equation

^{®}script with the parameters presented in Table 1.

^{®}. On a PC with 8 Intel Core i7-7800X @3.50GHz processors, one iteration of the fourth case of the grid convergence study (101 nodes) takes approximately five microseconds to execute, as measured by the Matlab

^{®}

`tic`and

`toc`functions. This way, to compute one second of beam dynamics for this case takes approximately 10 s of computation time.

#### 3.2. Comparison of the Simulation Results with an Analytical Solution

#### 3.3. Application of the Finite Difference Scheme to a Realistic Case of a Commercial Aircraft Wing in a Gust Encounter

#### 3.4. Validation of the Finite Difference Approximation for the Euler–Bernoulli Equation against Experimental Data

#### 3.4.1. Damping Coefficient $\beta $

#### 3.4.2. Comparison of Static Deflection Experiment with Finite Difference Simulation

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GCI | Grid Convergence Index |

MEMS | Micro-Electronic Mechanical System |

PDE | Partial Differential Equation |

Latin Nomenclature | |

A | Beam Cross-Section Area (${m}^{2}$) |

A | Constant in Analytical Solution (-) |

B | Constant in Analytical Solution (-) |

C | Constant in Analytical Solution (-) |

C | Fraction of Beam Span Used in Richardson Analysis (−) |

D | Constant in Analytical Solution (-) |

$E\left(x\right)$ | Young’s Modulus ($Pa$) |

$F\left(x\right)$ | The Product of E and I ($kg{m}^{3}/s$) |

${F}_{n}$ | Sequence of Constants in Analytical Solution (-) |

${G}_{n}$ | Sequence of Constants in Analytical Solution (-) |

${G}_{j}$ | Richardson Grid Convergent Index (−) |

G | Beam Shear Modulus ($Pa$) |

${H}_{n}$ | Sequence of Constants in Analytical Solution (-) |

$I\left(x\right)$ | Second Moment of Area (${m}^{4}$) |

i | Beam Node Index (-) |

${i}_{j}$ | Beam Node Index Used for ${j}^{th}$ GCI (-) |

L | Total Beam Length (m) |

M | The Total Mass of the Beam ($kg$) |

M | Gust Constant ($m/s$) |

${M}_{j}$ | Beam Node Mass of ${j}^{th}$ Richardson Simulation ($kg$) |

${m}_{i}$ | Mass of ${i}^{th}$ Node ($kg$) |

N | The Number of Finite Difference Nodes along the Beam (-) |

${N}_{j}$ | Number of Beam Nodes Of ${j}^{th}$ Richardson Simulation (-) |

n | Time Step Index (-) |

${n}_{i,j}$ | Auxiliary Quantity (−) |

$q(x,t)$ | Beam External Loading Function ($kg/m{s}^{2}$) |

S | Gust Constant (-) |

S | Auxiliary Quantity in Finite Difference ($kg/{s}^{2}$) |

T | Analytic Deflection Function Dependent Only on Time (m) |

t | Time (s) |

${t}^{\star}$ | Time at which Richardson Criteria Are Applied (s) |

$\Delta t$ | Finite Difference Time Step Size (s) |

$\overrightarrow{v}$ | Gust Velocity Vector Field ($m/s$) |

$w(x,t)$ | Beam Deflection Distance Function (m) |

${w}^{a}(x,t)$ | Analytic Deflection Function (m) |

X | Analytic Deflection Function Dependent Only on Distance Along Beam (m) |

x | Distance along the beam (m) |

${x}_{i}$ | Distance Along Beam of ${i}^{th}$ Node (m) |

$\Delta x$ | Distance Between Adjacent Nodes (m) |

z | Distance Perpendicular to Beam (m) |

Greek Symbols | |

${\alpha}_{n}$ | Auxiliary Constants in the Analytical Solution (-) |

$\beta $ | Velocity-Dependent Damping Coefficient (-) |

$\kappa \left(x\right)$ | Timoshenko Shear Modulus Coefficient (-) |

$\lambda $ | Rayleigh Stiffness Damping Coefficient (-) |

$\mu \left(x\right)$ | Beam Mass per Unit Length ($kg/m$) |

$\mu $ | Rayleigh Mass Damping Coefficient (-) |

$\lambda $ | Auxiliary Constant in the Analytical Solution (-) |

$\phi $ | Angular Rotation of Beam Cross-Section ($rad$) |

## Appendix A. An Analytical Solution of the Homogeneous Euler–Bernoulli Equation for Validation Purposes

## Appendix B. Algorithm for Deriving the Coefficients of a Finite Difference of a Given Degree and Order of Accuracy

## Appendix C. Accounting for Centrifugal Force in the Euler–Bernoulli Equation

## Appendix D. Apply the Finite Difference Methods to Other PDEs Relevant to the Subject of This Paper

## Appendix E. Description of Matlab^{®} Supplementary Files

**constructCoefficients.m**: This file contains a Matlab^{®}script that generates the coefficients of a finite difference approximation of given order q, degree p, and bias b using the methods described in [65]. The order, degree and bias are provided by the user at the Matlab command line, and the coefficients are displayed in the Matlab command window in rational number format. The algorithm is only valid for p and q, chosen such that $p+q-1\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}2=0$, that is, $p+q-1$ is even.**EBFiniteDifference.m**: This file contains a Matlab^{®}script that performs calculations for the finite difference approximation to the inhomogeneous dynamical Euler–Bernoulli equation discussed in this paper. The script implements boundary conditions for a cantilevered beam with non-constant material properties and mass distribution that represents the scale model of the aircraft wing used in the experiments described in Section 3.4.2 of the paper. A “one minus cosine” time-dependent external load distribution is considered. In the interest of simplicity, just eight equally spaced beam nodes with lumped masses are considered. This simulation discretises the smoothly varying external load distribution into 317 time slices, each one lasting for ${1}^{-2}$ seconds in duration. The time step used in the finite difference calculation was computed using the Lax–Richtmyer criteria.**massR.mat**: This file contains the masses (in kg) of each of the eight nodes used in the simulation.**IyRnew.mat**: This file contains the distribution of the second moment of area perpendicular to the bending axis of the wing beam cross-section. The values are sampled at the locations of the eight nodes along the length of the beam.**IyRnew.mat**: This file contains the values of the externally applied loads (the “one minus cosine” distribution) at each node at each of the 317 time slices.**calcDt.m**: This file contains a Matlab^{®}function that is called by**EBFiniteDifference.m**and computes the maximum time step. For some runs of the simulation, the Lax–Richtmyer time steps are over-ridden and smaller time step values are used.

## Appendix F. Coupling between Bending and Twisting

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**Figure 5.**Stencil for the spatial and temporal discretisation of the function w in the finite difference scheme.

**Figure 8.**Analytical solution of the Euler–Bernoulli homogeneous equation for a 10-m cantilevered beam in free vibration.

**Figure 9.**Comparison of the tip deflection between the analytical solution and finite difference predictions for a 10-m cantilevered beam in free vibration.

**Table 1.**Physical parameters used in the grid convergence study of the Euler–Bernoulli finite difference scheme.

Symbol | Name | Vaue | Units | Notes |
---|---|---|---|---|

${t}^{\star}$ | - | 1.0 | s | |

E | Young’s modulus | $70.0\times $${10}^{9}$ | N/m^{2} | Aluminium |

I | Second moment of area | $6.67\times {10}^{-5}$ | m^{4} | Rectangular section, height 0.2 m, breadth 0.1 m |

M | Total mass | 80 | kg | Mass evenly distributed over the nodes |

$\beta $ | Damping coefficient | 10 | − | |

q | External load | 4940 | N/m | |

C | Fraction of span | 0.5 | − | |

L | Beam length | 10 | m |

j | $\mathbf{\Delta}{\mathit{x}}_{\mathit{j}}$ | $\mathbf{\Delta}{\mathit{t}}_{\mathit{j}}$ | No. of Iterations | ${\mathit{i}}_{\mathit{j}}$ | ${\mathit{w}}_{\mathit{j}}$ | No. of Nodes |
---|---|---|---|---|---|---|

1 | 0.833 | $1.0\times {10}^{-4}$ | $1.0\times {10}^{4}$ | 13 | 1.0043 | 13 |

2 | 0.4 | $3\times {10}^{-5}$ | $3.33\times {10}^{4}$ | 26 | 1.1881 | 26 |

3 | 0.2 | $8\times {10}^{-6}$ | $1.24\times {10}^{5}$ | 51 | 1.2609 | 51 |

4 | 0.1 | $2\times {10}^{-6}$ | $5\times {10}^{5}$ | 101 | 1.2957 | 101 |

j | $\mathbf{\Delta}{\mathit{x}}_{\mathit{j}}$ | $\mathbf{\Delta}{\mathit{t}}_{\mathit{j}}$ | No. of Iterations | ${\mathit{i}}_{\mathit{j}}$ | ${\mathit{w}}_{\mathit{j}}$ | No. of Nodes | No. of Oscillations |
---|---|---|---|---|---|---|---|

1 | 0.8 | $1.0\times {10}^{-4}$ | $1.0\times {10}^{4}$ | 13 | −0.7165 | 13 | 5 |

2 | 0.4 | $3\times {10}^{-5}$ | $3.33\times {10}^{4}$ | 26 | 3.1390 | 26 | 4.6 |

3 | 0.2 | $8\times {10}^{-6}$ | $1.24\times {10}^{5}$ | 51 | 3.2071 | 51 | 4.5 |

4 | 0.1 | $2\times {10}^{-6}$ | $5\times {10}^{5}$ | 101 | 2.5763 | 101 | 4.4 |

Station | Chord Length (m) | ${\mathit{I}}_{\mathit{y}}\left({\mathit{m}}^{4}\right)$ | Element Mass (kg) |
---|---|---|---|

1 | $100.0000\times {10}^{-3}$ | $48.6000\times {10}^{-12}$ | $6.8760\times {10}^{-3}$ |

2 | $100.0000\times {10}^{-3}$ | $48.6000\times {10}^{-12}$ | $11.9109\times {10}^{-3}$ |

3 | $173.2233\times {10}^{-3}$ | $84.1865\times {10}^{-12}$ | $11.4449\times {10}^{-3}$ |

4 | $166.4466\times {10}^{-3}$ | $80.8930\times {10}^{-12}$ | $10.9789\times {10}^{-3}$ |

5 | $159.6699\times {10}^{-3}$ | $77.5996\times {10}^{-12}$ | $10.5130\times {10}^{-3}$ |

6 | $152.8932\times {10}^{-3}$ | $74.3061\times {10}^{-12}$ | $10.0470\times {10}^{-3}$ |

7 | $146.1165\times {10}^{-3}$ | $71.0126\times {10}^{-12}$ | $9.5810\times {10}^{-3}$ |

8 | $139.3398\times {10}^{-3}$ | $67.7191\times {10}^{-12}$ | $9.1151\times {10}^{-3}$ |

9 | $132.5631\times {10}^{-3}$ | $64.4256\times {10}^{-12}$ | $8.6491\times {10}^{-3}$ |

10 | $125.7864\times {10}^{-3}$ | $61.1322\times {10}^{-12}$ | $8.1831\times {10}^{-3}$ |

11 | $119.0096\times {10}^{-3}$ | $57.8387\times {10}^{-12}$ | $7.7172\times {10}^{-3}$ |

12 | $112.2329\times {10}^{-3}$ | $54.5452\times {10}^{-12}$ | $7.2512\times {10}^{-3}$ |

13 | $105.4562\times {10}^{-3}$ | $51.2517\times {10}^{-12}$ | $6.7852\times {10}^{-3}$ |

14 | $98.6795\times {10}^{-3}$ | $47.9582\times {10}^{-12}$ | $6.3193\times {10}^{-3}$ |

15 | $91.9028\times {10}^{-3}$ | $44.6648\times {10}^{-12}$ | $5.8533\times {10}^{-3}$ |

16 | $85.1261\times {10}^{-3}$ | $41.3713\times {10}^{-12}$ | $5.3873\times {10}^{-3}$ |

17 | $78.3494\times {10}^{-3}$ | $38.0778\times {10}^{-12}$ | $4.9214\times {10}^{-3}$ |

18 | $71.5727\times {10}^{-3}$ | $34.7843\times {10}^{-12}$ | $4.4554\times {10}^{-3}$ |

19 | $64.7960\times {10}^{-3}$ | $31.4909\times {10}^{-12}$ | $3.9894\times {10}^{-3}$ |

20 | $58.0193\times {10}^{-3}$ | $28.1974\times {10}^{-12}$ | $3.7958\times {10}^{-3}$ |

21 | $55.2031\times {10}^{-3}$ | $26.8287\times {10}^{-12}$ | $3.7111\times {10}^{-3}$ |

22 | $53.9721\times {10}^{-3}$ | $26.2304\times {10}^{-12}$ | $3.6265\times {10}^{-3}$ |

23 | $52.7411\times {10}^{-3}$ | $25.6322\times {10}^{-12}$ | $3.5419\times {10}^{-3}$ |

24 | $51.5101\times {10}^{-3}$ | $25.0339\times {10}^{-12}$ | $3.4572\times {10}^{-3}$ |

25 | $50.2791\times {10}^{-3}$ | $24.4357\times {10}^{-12}$ | $3.3726\times {10}^{-3}$ |

26 | $49.0482\times {10}^{-3}$ | $23.8374\times {10}^{-12}$ | $3.2879\times {10}^{-3}$ |

27 | $47.8172\times {10}^{-3}$ | $23.2391\times {10}^{-12}$ | $3.2033\times {10}^{-3}$ |

28 | $46.5862\times {10}^{-3}$ | $22.6409\times {10}^{-12}$ | $3.1186\times {10}^{-3}$ |

29 | $45.3552\times {10}^{-3}$ | $22.0426\times {10}^{-12}$ | $3.0340\times {10}^{-3}$ |

30 | $44.1242\times {10}^{-3}$ | $21.4444\times {10}^{-12}$ | $2.9494\times {10}^{-3}$ |

31 | $42.8932\times {10}^{-3}$ | $20.8461\times {10}^{-12}$ | $2.8647\times {10}^{-3}$ |

32 | $41.6623\times {10}^{-3}$ | $20.2479\times {10}^{-12}$ | $2.7801\times {10}^{-3}$ |

33 | $40.4313\times {10}^{-3}$ | $19.6496\times {10}^{-12}$ | $2.6954\times {10}^{-3}$ |

34 | $39.2003\times {10}^{-3}$ | $19.0513\times {10}^{-12}$ | $2.6108\times {10}^{-3}$ |

35 | $37.9693\times {10}^{-3}$ | $18.4531\times {10}^{-12}$ | $2.5261\times {10}^{-3}$ |

36 | $36.7383\times {10}^{-3}$ | $17.8548\times {10}^{-12}$ | $2.4415\times {10}^{-3}$ |

37 | $35.5073\times {10}^{-3}$ | $17.2566\times {10}^{-12}$ | $2.3569\times {10}^{-3}$ |

38 | $34.2764\times {10}^{-3}$ | $16.6583\times {10}^{-12}$ | $2.2722\times {10}^{-3}$ |

39 | $33.0454\times {10}^{-3}$ | $16.0601\times {10}^{-12}$ | $2.1876\times {10}^{-3}$ |

40 | $31.8144\times {10}^{-3}$ | $15.4618\times {10}^{-12}$ | $2.1029\times {10}^{-3}$ |

41 | $30.5834\times {10}^{-3}$ | $14.8635\times {10}^{-12}$ | $2.0183\times {10}^{-3}$ |

42 | $29.3524\times {10}^{-3}$ | $14.2653\times {10}^{-12}$ | $1.9336\times {10}^{-3}$ |

43 | $28.1214\times {10}^{-3}$ | $13.6670\times {10}^{-12}$ | $1.8490\times {10}^{-3}$ |

44 | $26.8905\times {10}^{-3}$ | $13.0688\times {10}^{-12}$ | $1.7644\times {10}^{-3}$ |

45 | $25.6595\times {10}^{-3}$ | $12.4705\times {10}^{-12}$ | $1.6797\times {10}^{-3}$ |

46 | $24.4285\times {10}^{-3}$ | $11.8722\times {10}^{-12}$ | $1.5951\times {10}^{-3}$ |

47 | $23.1975\times {10}^{-3}$ | $11.2740\times {10}^{-12}$ | $1.5104\times {10}^{-3}$ |

48 | $21.9665\times {10}^{-3}$ | $10.6757\times {10}^{-12}$ | $1.4258\times {10}^{-3}$ |

49 | $20.7355\times {10}^{-3}$ | $10.0775\times {10}^{-12}$ | $1.3411\times {10}^{-3}$ |

50 | $19.5046\times {10}^{-3}$ | $9.4792\times {10}^{-12}$ | $1.2565\times {10}^{-3}$ |

51 | $18.2736\times {10}^{-3}$ | $8.8810\times {10}^{-12}$ | $1.1719\times {10}^{-3}$ |

52 | $17.0426\times {10}^{-3}$ | $8.2827\times {10}^{-12}$ | - |

Case | Tip Load (N) | Measured Deflection (m) | Simulated Deflection. Thickness 1.9 mm | Simulated Deflection. Thickness 1.8 mm | Simulated Deflection. Thickness 1.6 mm |
---|---|---|---|---|---|

0 | 0.000 | 0.026 | 0.0135 | 0.0150 | 0.024 |

1 | 0.098 | 0.029 | 0.0151 | 0.0169 | 0.027 |

2 | 0.196 | 0.033 | 0.0168 | 0.0189 | 0.031 |

3 | 0.294 | 0.036 | 0.0184 | 0.0208 | 0.035 |

4 | 0.392 | 0.039 | 0.0200 | 0.0227 | 0.038 |

5 | 0.490 | 0.042 | 0.0217 | 0.0246 | 0.042 |

6 | 0.687 | 0.048 | 0.0249 | 0.0285 | 0.050 |

7 | 0.981 | 0.056 | 0.0299 | 0.0343 | 0.061 |

8 | 1.177 | 0.062 | 0.0331 | 0.0381 | 0.068 |

9 | 1.373 | 0.069 | 0.0364 | 0.0420 | 0.076 |

10 | 1.471 | 0.071 | 0.0381 | 0.0439 | 0.079 |

11 | 1.668 | 0.076 | 0.0413 | 0.0478 | 0.087 |

12 | 1.962 | 0.086 | 0.0463 | 0.0536 | 0.097 |

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## Share and Cite

**MDPI and ACS Style**

Fleischmann, D.; Könözsy, L.
On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics. *Aerospace* **2021**, *8*, 356.
https://doi.org/10.3390/aerospace8110356

**AMA Style**

Fleischmann D, Könözsy L.
On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics. *Aerospace*. 2021; 8(11):356.
https://doi.org/10.3390/aerospace8110356

**Chicago/Turabian Style**

Fleischmann, Dominique, and László Könözsy.
2021. "On a Novel Approximate Solution to the Inhomogeneous Euler–Bernoulli Equation with an Application to Aeroelastics" *Aerospace* 8, no. 11: 356.
https://doi.org/10.3390/aerospace8110356