# Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Rigid-Body Model of an Earthquake Excited Tower-Like Structure

## 3. Oscillations under Harmonic Earthquake Excitation in Both Directions

## 4. Numerical Results for Real Earthquakes

## 5. Double Pendulum

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Technical Details for Calculation with Harmonic Excitation

#### Appendix A.1. Algorithm

- Prescribe initial data.
- Define ${\mathrm{\Omega}}_{V}$ using Equation (9).
- Prescribe general initial conditions of the IVP.
- Prescribe the time step length and the number of time steps.
- Define the Runge–Kutta coefficients and formulas using Equation (15).
- Define the function that symbolically integrate polynomials.
- Symbolically integrate the truncated version of the Runge–Kutta formula.
- Add initial velocity ${\omega}_{0}$ to obtain the first iteration of $\omega $.
- Symbolically integrate $\omega $ and add initial angle ${\phi}_{0}$ to obtain the first iteration of $\phi $.
- Repeat integration to obtain a more accurate formula; in the present case, we use 3 iterations with (4, 6, 9) Taylor terms.
- Truncate the obtained formula by the third order of the Taylor series in initial conditions ${\omega}_{0}$ and ${\phi}_{0}$ of each time step.
- Compile the explicit formulas of time integration.
- Perform the iterative time-stepping method.
- Prepare graphs.

#### Appendix A.2. Effects of the Number of Terms of Taylor Expansion and of the Time Step

Numbers of Taylor Terms | Computation Time | Maximum Deviation |
---|---|---|

3, 5, 8 | 0.20 | $7.3\times {10}^{-3}$ |

4, 6, 9 | 0.20 | $6.3\times {10}^{-4}$ |

5, 7, 10 | 0.26 | $6.9\times {10}^{-4}$ |

6, 8, 11 | 0.28 | $6.8\times {10}^{-4}$ |

7, 9, 12 | 0.30 | $6.7\times {10}^{-4}$ |

Time Step Length | Number of Time Steps | Computation Time | Max. Deviation |
---|---|---|---|

0.2 | 5000 | 0.096 | $0.013$ |

0.1 | 10,000 | 0.20 | $6.3\times {10}^{-4}$ |

0.05 | 20,000 | 0.45 | $2.4\times {10}^{-5}$ |

0.025 | 40,000 | 0.80 | $2.0\times {10}^{-5}$ |

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**Figure 2.**(

**a**) Angle $\phi $ over physical time t for harmonic earthquake excitation in both directions ${a}_{\mathrm{H}0}=9.81$ m/s${}^{2}$: our method, the computation time is 0.20 s (three iterations with (4, 6, 9) Taylor terms (for NDSolve of Mathematica the time history is visually indistinguishable and the computation time is 0.94 s). (

**b**) Absolute deviation between solutions with our method and NDSolve.

**Figure 3.**Horizontal acceleration over physical time t for Kobe earthquake data interpolated with Mathematica (an initial data segment with largest accelerations is shown). The orange points denote the Kobe earthquake data and the blue curve is for the interpolated function.

**Figure 4.**(

**a**) Angle $\phi $ over time t for Kobe earthquake data with proposed method, the computation time is 0.40 s (for NDSolve of Mathematica the time history is visually indistinguishable and the computation time is 1.45 s). (

**b**) Absolute deviation between solutions with our method and NDSolve.

**Figure 6.**Error $\epsilon =({\phi}_{i}-{\phi}_{i}^{\mathrm{analytic}})/{\phi}_{20}$ of Greenspan I method for (

**a**) ${\phi}_{1}$ and (

**b**) ${\phi}_{2}$ angles for a single time step. $\tau =1$ means $\approx 5$ largest periods and ≈12 smallest periods.

**Figure 7.**Error $\epsilon =({\phi}_{i}-{\phi}_{i}^{\mathrm{analytic}})/{\phi}_{20}$ of Greenspan II method for (

**a**) ${\phi}_{1}$ and (

**b**) ${\phi}_{2}$ angles for a single time step. $\tau =1$ means $\approx 5$ largest periods and ≈12 smallest periods.

**Figure 8.**Error $\epsilon =({\phi}_{i}-{\phi}_{i}^{\mathrm{analytic}})/{\phi}_{20}$ of Greenspan II method over timestep index for (

**a**) ${\phi}_{1}$ and (

**b**) ${\phi}_{2}$ angles obtained in a multistep procedure with the fixed time step $T=0.02$ (the dimensionless periods of the natural oscillations are 0.0861 and 0.208).

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**MDPI and ACS Style**

Oborin, E.; Irschik, H.
Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study. *Appl. Sci.* **2021**, *11*, 3742.
https://doi.org/10.3390/app11093742

**AMA Style**

Oborin E, Irschik H.
Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study. *Applied Sciences*. 2021; 11(9):3742.
https://doi.org/10.3390/app11093742

**Chicago/Turabian Style**

Oborin, Evgenii, and Hans Irschik.
2021. "Application of a Novel Picard-Type Time-Integration Technique to the Linear and Non-Linear Dynamics of Mechanical Structures: An Exemplary Study" *Applied Sciences* 11, no. 9: 3742.
https://doi.org/10.3390/app11093742