# Simplification of Complex Structural Dynamic Models: A Case Study Related to a Cantilever Beam and a Large Mass Attachment

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Descriptions

#### 2.1. Test Samples

#### 2.2. Finite Element Models

**1**) has no simplifications and ideal free–free boundary conditions. In the next step, three simplified finite element models are investigated (see Figure 1c–f).

**2**, c.f. Figure 1c, the effect of the tip–mass was modelled as an ideal rigid clamping condition, which defines zero motion at this point. This is possibly the most common implementation in ABAQUS/CAE, which removes the complexity of modelling a tip-mass. This compares to an approximately infinite mass and infinite rotary inertia at the end-point. For more realistic models, two additional finite element models were investigated. Modelling approach

**3**, c.f. Figure 1d, approximates the tip-mass by an ideal free-tip mass, coupling a point mass to the beam’s end.

**4**additionally takes the rotary inertia into account. For both cases, the point mass is located at the centre of gravity of the 3D tip mass and is connected to the front surface of the beam via kinematic coupling. The rotary inertia ${I}_{\mathrm{t}}$ for the tip–mass is calculated by

**2**,

**3**and

**4**include simplifications to reduce the degrees of freedom and, thus, the complexity of the numerical modal analysis. In addition to the simplifications, the finite element mesh for the beam was left unchanged and, hence, matched the reference model. In addition to the finite element models, Figure 1f shows an analytical Euler–Bernoulli beam model considering an ideal free-tip mass and the rotary inertia. This corresponds to modelling approach

**5**. The theoretical background is explained, and can be found in Section 2.4 for reference.

#### 2.3. Experimental Modal Analysis

#### 2.4. Analytical Theory

- (i)
- One of the spatial dimensions is significantly larger than the other two.
- (ii)
- The material behaves according to Hooke’s law.
- (iii)
- The Poisson effect is neglected.
- (iv)
- The angle of rotation is small; hence, the small angle assumption holds. The transverse displacements and cross-section rotations are small. Hence, the formulations are geometrically linear.
- (v)
- Cross-sections remain perpendicular to the neutral axes after deformation.
- (vi)
- The rotational inertia of the cross-sections is neglected.

## 3. Results and Discussion

**1**, i.e., ideal free–free boundary conditions as shown in Figure 1b, and modelling approach

**2**, i.e., ideal rigid–clamped conditions, c.f. Figure 1c.

**3**with a tied coupling of a point mass as illustrated in Figure 1d and a modelling approach

**4**with a tied coupling of both a point mass and a rotary inertia as illustrated in Figure 1e. The corresponding results are compared to results based on the modelling approach

**5**, i.e., using the analytical solution of the simplified beam model shown in Figure 1e, as well as to the experimental results of the corresponding specimen. For the comparisons, an averaged relative error $\epsilon $ was calculated for the first three eigenfrequencies over m specimens using

**2**, i.e., replacing the tip-mass at the end of the specimen with ideal clamped boundary conditions, yielded insufficient results. Even within the bounds of experimental uncertainty, which are largely acceptable for the second and third bending mode, the first bending mode was not correctly identified by this reduced numerical model ${\mathrm{FE}}_{\mathrm{free}-\mathrm{clamped}}$. The arithmetically averaged relative deviations for the eigenfrequencies of the first three bending modes were $-28.2\%$, $-1.8\%$, and $1.1\%$, respectively.

**2**(blue) did not overlap the uncertainty range of modelling approach

**1**(red) for the first bending eigenfrequency. Hence, there is little likelihood of predicting the first “true” bending mode from a simplified free–clamped model. However, the uncertainty ranges for the two modelling approaches do overlap for eigenfrequencies of the second and third bending mode. The consequence is that, especially for low frequencies, it is not recommended to use a free–clamped boundary condition to approximate the effect of an attached, large mass.

**2**as insufficient, the focus is set on the approaches

**3**,

**4**and

**5**, c.f. Figure 1. In this case, only one specimen is taken into account. Figure 6 shows that modelling approach

**3**, i.e., replacing the tip-mass at the end of the beam by a coupling of a point mass, yielded the highest deviation over the whole frequency range. For the first natural frequency, the deviation even exceeded $100\%$. Thus, taking only the tip mass into account and neglecting the rotary inertia is not a valid modelling approach. However, when using modelling approach

**4**, i.e., adding the rotary inertia of the tip-mass, far better results were achieved.

**5**, i.e., the analytic model ${\mathrm{Analytic}}_{\mathrm{free}-\mathrm{free}}$ assuming free–free boundary conditions, the highest deviation was found at the first eigenfrequency with $2.9\%$. For the second and third eigenfrequency, the deviations to the experiment were less than $1\%$. Thus, both the finite element model with the coupling of the point mass and rotary inertia as well as the analytical model yielded sufficiently accurate results.

_{free-tip mass & inertia}, these uncertainties were included into the model by changing the point mass and rotary inertia values accordingly. Based on the values given in Table 1, a relative deviation from the mean value, or expected value, of the determined eigenfrequencies of less than $\pm 0.8\%$ was found. Therefore, we conclude that the effect of uncertainties, which describe the tip mass in the model, can be neglected in the modelling process.

## 4. Conclusions

- A full finite element model for all the components, including large heavy masses, using ideal-free conditions for the model is better suited where comparisons to experimental results are necessary, given that sufficient computer memory resources are available.
- For the simplification of a large attached mass by an ideal clamped condition in a finite element model, a very high mass ratio is necessary, although the end results can still be quite poor even for the primary first bending mode.
- It is commonly known, and was demonstrated here, that an analytical model can yield satisfactory results for a beam with an attached tip–mass, when the structure is beam–like.
- Reduced size modelling by a finite element approach using only an offset point-mass term for simplification can be easily improved upon by adding a rotary inertia term. This increases the accuracy and reduces computer resources; however, choosing a suitable rotary inertia term has its own challenges.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

FEM | Finite Element Method |

UQ | Uncertainty Quantification |

EMA | Experimental Modal Analysis |

FRF | Frequency Response Function |

BC | Boundary Conditions |

## References

- Simulia, D.S. ABAQUS V6.10 Documentation; Dassault Systèmes: Providence, RI, USA, 2010. [Google Scholar]
- Bathe, K.J. Finite Element Procedures; Prentice-Hall: New Jersey, NJ, USA, 1996. [Google Scholar]
- Zienkiewicz, O.C.; Taylor, R.L. The Finite Element Method Volume 1: Basic Formulation and Linear Problems; MacGraw-Hill Book Company: New York, NY, USA, 1989. [Google Scholar]
- Zienkiewicz, O.C.; Taylor, R.L. The Finite Element Method Volume 2: Solid and Fluid Mechanics, Dynamics and Non-Linearity; MacGraw-Hill Book Company: New York, NY, USA, 1991. [Google Scholar]
- Szabo, B.; Babuška, I. Finite Element Analysis; John Wiley & Sons, Inc.: New York, NY, USA, 1991. [Google Scholar]
- Ghanem, R.G. Uncertainty Quantification in Computational and Prediction Science. Int. J. Numer. Methods Eng.
**2009**, 80, 671–672. [Google Scholar] [CrossRef] - Sargent, G. Verification and Validation of Simulation Models. In Proceedings of the Winter Simulation Conference, Orlando, FL, USA, 4 December 2005; pp. 53–59. [Google Scholar]
- Li, R.; Ghanem, R. Adaptive polynomial chaos expansions applied to statistics of extremes in nonlinear random vibration. Probab. Eng. Mech.
**1998**, 13, 125–136. [Google Scholar] [CrossRef] - Lucor, D.; Su, C.H.; Karniadakis, G.E. Generalized polynomial chaos and random oscillators. Int. J. Numer. Methods Eng.
**2004**, 60, 571–596. [Google Scholar] [CrossRef] - Soize, C. A comprehensive overview of a non–parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. J. Sound Vib.
**2005**, 288, 623–652. [Google Scholar] [CrossRef] [Green Version] - Sepahvand, K.; Marburg, S.; Hardtke, H.J. Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion. J. Sound Vib.
**2012**, 331, 167–179. [Google Scholar] [CrossRef] - Marburg, S.; Beer, H.J.; Gier, J.; Hardtke, H.J.; Rennert, R.; Perret, F. Experimental verification of structural–acoustic modelling and design optimization. J. Sound Vib.
**2002**, 252, 591–615. [Google Scholar] [CrossRef] - Marc, M.; Soize, C.; Avalos, J.D. Nonparametric stochastic modeling of structures with uncertain boundary, conditions/coupling between substructures. AIAA J.
**2013**, 51, 1298–1308. [Google Scholar] - Ritto, T.G.; Sampaio, R.; Aguiar, R.R. Uncertain boundary condition Bayesian identification from experimental data: A case study on a cantilever beam. Mech. Syst. Signal Process.
**2016**, 68–69, 176–188. [Google Scholar] [CrossRef] - Ewins, D. Exciting vibrations: The role of testing in an era of supercomputers and uncertainties. Meccanica
**2016**, 51, 3241–3258. [Google Scholar] [CrossRef] [Green Version] - Alvin, K.; Oberkampf, W.; Diegert, K.; Rutherford, B. Uncertainty quantification in computational structural dynamics: A new paradigm for model validation. In Proceedings of the 16th International Modal Analysis Conference, Santa Barabara, CA, USA, 2–5 February 1998; Volume 2, pp. 1191–1198. [Google Scholar]
- Oberkampf, W.L.; DeLand, S.M.; Rutherford, B.M.; Diegert, K.V.; Alvin, K.F. Error and uncertainty in modeling and simulation. Reliab. Eng. Syst. Saf.
**2002**, 75, 333–357. [Google Scholar] [CrossRef] - Smith, R.C. Uncertainty Quantification: Theory, Implementation, and Applications; Siam: Philadelphia, PA, USA, 2013; Volume 12. [Google Scholar]
- Sullivan, T.J. Introduction to Uncertainty Quantification; Springer: Cham/Heidelberg, Germany, 2015; Volume 63. [Google Scholar]
- Marburg, S.; Dienerowitz, F.; Fritze, D.; Hardtke, H.J. Case studies on structural-acoustic optimization of a finite beam. Acta Acust. United Acust.
**2006**, 92, 427–439. [Google Scholar] - Inman, D.J.; Erturk, A. Piezoelectric Energy Harvesting; John Wiley & Sons: New York, NY, USA, 2011. [Google Scholar]
- Kirk, C.L.; Wiedemann, S.M. Natural Frequencies and Mode Shapes of a Free-Free Beam with Large End Masses. J. Sound Vib.
**2002**, 254, 939–949. [Google Scholar] [CrossRef] - Langer, P.; Sepahvand, K.; Guist, C.; Bär, J.; Peplow, A.; Marburg, S. Matching experimental and three dimensional numerical models for structural vibration problems with uncertainties. J. Sound Vib.
**2018**, 417, 294–305. [Google Scholar] [CrossRef] - Neumaier, A. Interval Methods for Systems of Equations; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Moore, R.E. Methods and Applications of Interval Analysis; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1987. [Google Scholar]
- Elishakoff, I.; Soret, C. Remedy to overestimation of classical interval analysis: Analysis of beams with uncertain boundary conditions. Shock Vib.
**2013**, 20, 143–156. [Google Scholar] [CrossRef] - Langer, P.; Sepahvand, K.; Marburg, S. Uncertainty quantification in analytical and finite element beam models using experimental data. In Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014, Porto, Portugal, 30 June–2 July 2014. [Google Scholar]
- Langer, P.; Maeder, M.; Guist, C.; Krause, M.; Marburg, S. More Than Six Elements Per Wavelength: The Practical Use of Structural Finite Element Models and Their Accuracy in Comparison with Experimental Results. J. Comput. Acoust.
**2017**, 25, 1750025. [Google Scholar] [CrossRef] - Ewins, D.J. Modal Testing: Theory and Practice; Research Studies Press: Letchworth, Hertfordshire, UK, 1984. [Google Scholar]
- ISO 7626-1:2011: Methods for the Experimental Determination of Mechanical Mobility; Parts 1–5. Technical Report; International Organisation for Standardization: Geneva, Switzerland, 2011.
- Han, S.M.; Benaroya, H.; Wei, T. Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib.
**1999**, 225, 935–988. [Google Scholar] [CrossRef] [Green Version] - Benaroya, H.; Nagurka, M.L. Mechanical Vibration: Analysis, Uncertainties, and Control; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Inman, D. Engineering Vibration; Pearson Education: Upper Saddle River, NJ, USA, 2008. [Google Scholar]
- Meirovitch, L. Elements of Vibration Analysis, 2nd ed.; McGrawHill: New York, NY, USA, 1986. [Google Scholar]
- Andrews, K.; Shillor, M. Vibrations of a beam with a damping tip body. Math. Comput. Model.
**2002**, 35, 1033–1042. [Google Scholar] [CrossRef]

**Figure 1.**Investigated structure and finite element models. (

**a**) Geometry of the specimens. Coordinate axes follow: length of beam (l) x–axis, beam thickness (h) z–axis, and beam width (w) y–axis. (

**b**) Finite element model

**1**of the real structure with ideal free–free boundary condition—reference model. (

**c**) Simplified finite element model

**2**with ideal rigid clamping. (

**d**) Simplified finite element model

**3**with ideal free-tip mass modelling. (

**e**) Simplified finite element model

**4**with ideal free-tip mass and rotary inertia modelling. (

**f**) Euler–Bernoulli beam model

**5**with ideal free-tip mass and rotary inertia modelling.

**Figure 2.**Image and sketch of the experimental setup showing: elastic strings (1), specimen mass (2a), specimen beam (2b), acoustic excitation (3), and laser Doppler vibrometer (4).

**Figure 3.**Parameters of the one-dimensional model. ${m}_{\mathrm{t}}$: mass of the tip mass. ${I}_{\mathrm{t}}$: area moment of inertia of the tip mass about the axis of bending. ${I}_{\mathrm{b}}$: area moment of inertia of the beam about the axis of bending. A: cross-section area. ${\theta}_{\mathrm{t}}$: mass moment of inertia of the tip mass about the axis of bending. The nominal geometrical and material parameters are the same as in Section 2.1.

**Figure 4.**Two numerical model finite element results compared to experiments for the eigenfrequencies related to the first three bending modes. Experiment: contact-free measurement setup Figure 1a.

**Figure 5.**Mode shapes of the cantilever beam and the simple beam with a tip mass attached to the free end. (

**a**) First bending mode. (

**b**) Second bending mode. (

**c**) Third bending mode.

**Figure 6.**Experimental and further lumped-parameter finite element and analytic simulation results for the eigenfrequencies related to the first three bending modes of the first specimen. Experiment: contact-free measurement setup. Modelling approach

**3**: tied coupling of a point mass. Modelling approach

**4**: tied coupling of a point mass and a rotary inertia. Modelling approach

**5**: Analytical description of a simplified model according to the Euler–Bernoulli theory.

**Table 1.**The nominal values of the geometrical and material parameters with the averaged deviation $\epsilon $ over six samples.

Parameters (Steel Specimens) | 1 | 2 | 3 | 4 | 5 | 6 | Mean | Standard Deviation $\mathit{\epsilon}$ |
---|---|---|---|---|---|---|---|---|

Length, l [m] | 0.2001 | 0.2001 | 0.1999 | 0.2002 | 0.2000 | 0.2002 | 0.2001 | $\pm 0.02\times {10}^{-3}$ |

Width, w [m] | 0.0399 | 0.0399 | 0.0400 | 0.0400 | 0.0399 | 0.0400 | 0.0399 | $\pm 0.02\times {10}^{-3}$ |

Thickness, h [m] | 0.0040 | 0.0040 | 0.0041 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | $\pm 0.02\times {10}^{-3}$ |

Density, $\rho $ [kgm${}^{-3}$] | 7700 | 7700 | 7700 | 7700 | 7700 | 7700 | 7700 | $\pm 0.8$ |

Young’s modulus, E [GPa] | 203.1 | 202.6 | 203.1 | 203.3 | 205.2 | 204.1 | 203.6 | $\pm 4.7$ |

Poisson’s ratio, $\nu $ [-] | 0.29 | 0.29 | 0.29 | 0.29 | 0.28 | 0.28 | 0.29 | $\pm 0.01$ |

**Table 2.**Averaged relative deviation $\epsilon $ from the finite element models to the experiment for the eigenfrequencies of the first three bending modes. ${\mathrm{FE}}_{\mathrm{free}-\mathrm{free}}$ (

**1**): reference model without simplifications. ${\mathrm{FE}}_{\mathrm{free}-\mathrm{clamped}}$ and (

**2**): simplified finite element model of the beam with ideal-rigid clamping.

Mode | Experiment [Hz] | ${\mathbf{FE}}_{\mathbf{free}-\mathbf{free}}$ (1) | ${\mathbf{FE}}_{\mathbf{free}-\mathbf{clamped}}$ (2) |
---|---|---|---|

$1\mathrm{st}$ | 118 | $0.6\%$ | $-28.2\%$ |

$2\mathrm{nd}$ | 540 | $0.4\%$ | $-1.8\%$ |

$3\mathrm{rd}$ | 1462 | $0.3\%$ | $1.1\%$ |

**Table 3.**Relative deviation of the results determined by the simplified finite element models and the analytical solution to the experimental results of the eigenfrequencies of the first three bending modes. Only the first specimen is taken into account.

Mode | Experiment [Hz] | FE${}_{\mathbf{free}-\mathbf{tip}\phantom{\rule{4.pt}{0ex}}\mathbf{mass}}$ (3) | FE_{free-tip mass & inertia} (4) | ${\mathbf{Analytic}}_{\mathbf{free}-\mathbf{free}}$ (5) |
---|---|---|---|---|

$1\mathrm{st}$ | 118 | $112.7\%$ | $0.8\%$ | $2.9\%$ |

$2\mathrm{nd}$ | 540 | $63.1\%$ | $1.5\%$ | $0.6\%$ |

$3\mathrm{rd}$ | 1462 | $33.7\%$ | $1.7\%$ | $0.7\%$ |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Langer, P.; Jelich, C.; Guist, C.; Peplow, A.; Marburg, S.
Simplification of Complex Structural Dynamic Models: A Case Study Related to a Cantilever Beam and a Large Mass Attachment. *Appl. Sci.* **2021**, *11*, 5428.
https://doi.org/10.3390/app11125428

**AMA Style**

Langer P, Jelich C, Guist C, Peplow A, Marburg S.
Simplification of Complex Structural Dynamic Models: A Case Study Related to a Cantilever Beam and a Large Mass Attachment. *Applied Sciences*. 2021; 11(12):5428.
https://doi.org/10.3390/app11125428

**Chicago/Turabian Style**

Langer, Patrick, Christopher Jelich, Christian Guist, Andrew Peplow, and Steffen Marburg.
2021. "Simplification of Complex Structural Dynamic Models: A Case Study Related to a Cantilever Beam and a Large Mass Attachment" *Applied Sciences* 11, no. 12: 5428.
https://doi.org/10.3390/app11125428