Analytical Sensitivity Analysis of Dynamic Problems with Direct Differentiation of Generalized-α Time Integration
Abstract
:1. Introduction
2. Sensitivity Analysis with Generalized-α Time Integration
2.1. Generalized- Time Integration
2.2. Structural Dynamics
2.3. Flexible Multibody Dynamics
3. Numerical Results
3.1. Structural Dynamics
- sensitivities of position 2 with respect to stiffness 1 and mass 2 (Figure 4);
- sensitivities of velocity 2 with respect to stiffness 1 and mass 2 (Figure 5);
- sensitivities of acceleration 2 with respect to stiffness 1, stiffness 2 and mass 2 (Figure 6);
- sensitivities of acceleration 3 with respect to stiffness 1 and mass 2 (Figure 6).
- sensitivity of velocity 2 with respect to stiffness 2 (Figure 5);
- sensitivity of velocity 2 with respect to mass 3 (Figure 5);
- sensitivity of acceleration 2 with respect to stiffness 2 (Figure 6);
- sensitivity of acceleration 2 with respect to mass 3 (Figure 6).
3.2. Flexible Slider–Crank Mechanism
- sensitivities of with respect to and ;
- sensitivities of with respect to , , and .
- sensitivities of with respect to and ;
- sensitivities of with respect to ;
- sensitivities of with respect to , and ;
- sensitivities of with respect to ; , and ;
- sensitivities of with respect to , , and .
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Derivation of Effective Equations of Motion for Structural Dynamics
Appendix A.1. Primal Analysis
Appendix A.2. Sensitivity Analysis
Appendix B. Derivation of Effective Equations of Motion for Flexible Multibody Dynamics
Appendix B.1. Primal Analysis
Appendix B.2. Nonlinear Solver
Appendix B.3. Sensitivity Analysis
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Time Integration | Sensitivity Analysis | Calculation Time | Speedup |
---|---|---|---|
Newmark-β | None | 0.0555 | – |
Newmark-β | Finite differencing | 0.2088 | – |
Newmark-β | Analytical | 0.1023 | 2.0410 |
Generalized-α | None | 0.0458 | – |
Generalized-α | Finite differencing | 0.2122 | – |
Generalized-α | Analytical | 0.0993 | 2.1370 |
Property | Symbol | Crank | Rod | Slider | Units |
---|---|---|---|---|---|
Width | w | 20 | 20 | 30 | mm |
Height | h | 30 | 30 | 30 | mm |
Length | ℓ | 120 | 180 | 40 | mm |
Property | Symbol | Value | Units |
---|---|---|---|
Density | |||
Elastic modulus | E | 210,000 | |
Poisson ratio | 0.3 | − |
Time Integration | Sensitivity Analysis | Calculation Time [mm:ss] | Speedup |
---|---|---|---|
Newmark-β | None | 03:44 | – |
Newmark-β | Finite differencing | 18:12 | – |
Newmark-β | Analytical | 04:29 | 4.0595 |
Generalized-α | None | 02:42 | – |
Generalized-α | Finite differencing | 13:22 | – |
Generalized-α | Analytical | 03:30 | 3.8190 |
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Wehrle, E.; Gufler, V. Analytical Sensitivity Analysis of Dynamic Problems with Direct Differentiation of Generalized-α Time Integration. Machines 2024, 12, 128. https://doi.org/10.3390/machines12020128
Wehrle E, Gufler V. Analytical Sensitivity Analysis of Dynamic Problems with Direct Differentiation of Generalized-α Time Integration. Machines. 2024; 12(2):128. https://doi.org/10.3390/machines12020128
Chicago/Turabian StyleWehrle, Erich, and Veit Gufler. 2024. "Analytical Sensitivity Analysis of Dynamic Problems with Direct Differentiation of Generalized-α Time Integration" Machines 12, no. 2: 128. https://doi.org/10.3390/machines12020128
APA StyleWehrle, E., & Gufler, V. (2024). Analytical Sensitivity Analysis of Dynamic Problems with Direct Differentiation of Generalized-α Time Integration. Machines, 12(2), 128. https://doi.org/10.3390/machines12020128