Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution
Abstract
:1. Introduction
2. Theory and Formulation
2.1. Material Constitutions and Distributions
2.2. Kinematics Relation
2.3. Constitutive Relations
3. Formulation of Governing Equations
3.1. Moving Load Formulations
3.2. Thermal Environment Formulations
- Linear temperature rise (LTR) for
- Nonlinear temperature rise (NTR) for
4. Analytical Solution
5. Model Validation
6. Numerical Results
6.1. Influence of the Gradation Indices
6.2. Influence of Temperature Distribution
6.3. Influence of the Moving Load Velocity
7. Conclusions
- ✓
- Fundamental frequencies of SBDFG microbeams reduce with a rise in temperature until it reaches the critical frequency temperature. This is because the geometrical stiffness decreases when the temperature rises without any variation in equivalent mass.
- ✓
- By increasing the gradation index through the thickness, the dynamic deflection increases for temperature-independent material. However, in the case of temperature-dependent material, increasing in from 0 to 0.2, the dynamic deflection increased significantly, and after that, dynamic deflection reduced with the increasing gradation index
- ✓
- The effect of gradation index in the axial direction on the dynamic deflection is increased significantly in the case of LTR relative to temperature-independent material.
- ✓
- The temperature for the LTR type is larger than that for both the NTR type and temperature-independent material. As seen, by increasing the velocity, the profile of dynamic deflection vs. time is changed completely from oscillatory to parabolic to exponential functions. Therefore, the dynamic deflection profile response is dependent on the velocity of the moving mass.
- ✓
- The shapes of the time history curves are strongly affected by the moving velocity. The number of vibration cycles of the microbeam is enlarged at low velocities of the moving load because the ratio of moving load velocity to critical velocity becomes low.
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Source | Pure SUS304 | kz = 1.0 | Pure Al2O3 | |
---|---|---|---|---|
Present, SBDFG | 1.7475 | 1.2641 | 0.9433 | |
Ref. [29] (RBT) | 1.7384 | 1.2575 | 0.9384 | |
Ref. [70] (TBT) | 1.7379 | 1.2287 | 0.9382 | |
Ref. [69] (TBT) | 1.7420 | 1.2566 | 0.9380 | |
Ref. [35] (EBT) | 1.7324 | 1.2503 | 0.9328 | |
vp | Present, SBDFG | 130 | 177 | 249 |
Ref. [29] (RBT) | 131 | 178 | 252 | |
Ref. [70] (TBT) | 132 | 179 | 252 | |
Ref. [69] (TBT) | 131 | 178 | 251 | |
Ref. [35] (EBT) | 132 | 179 | 252 |
Material | Properties | |||||
---|---|---|---|---|---|---|
SUS304 (Metal) | 0 | 201.04 × 109 | 3.079 × 10−4 | −6.534 × 10−7 | 0 | |
0 | 8166 | 0 | 0 | 0 | ||
0 | 0.3262 | 0 | 0 | 0 | ||
0 | 12.330 × 10−6 | 8.086 × 10−4 | 0 | 0 | ||
Si3N4 (Ceramic) | 0 | 348.43 × 109 | −3.070 × 10−4 | 2.160 × 10−7 | −8.946 × 10−11 | |
0 | 2170 | 0 | 0 | 0 | ||
0 | 0.24 | 0 | 0 | 0 | ||
0 | 5.8723 × 10−6 | 9.095 × 10−4 | 0 | 0 |
Source | ΔT | kz = 0.0 | kz = 1.0 |
---|---|---|---|
Present, SBDFG | 10 | 9.7643 | 5.7294 |
Ref. [71] (TBT) | 9.6461 | 5.7717 | |
Present, SBDFG | 30 | 9.6074 | 5.5964 |
Ref. [71] (TBT) | 9.4538 | 5.6105 | |
Present, SBDFG | 60 | 9.3682 | 5.3889 |
Ref. [71] (TBT) | 9.1475 | 5.3537 |
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Attia, M.A.; Melaibari, A.; Shanab, R.A.; Eltaher, M.A. Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution. Mathematics 2022, 10, 4797. https://doi.org/10.3390/math10244797
Attia MA, Melaibari A, Shanab RA, Eltaher MA. Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution. Mathematics. 2022; 10(24):4797. https://doi.org/10.3390/math10244797
Chicago/Turabian StyleAttia, Mohamed A., Ammar Melaibari, Rabab A. Shanab, and Mohamed A. Eltaher. 2022. "Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution" Mathematics 10, no. 24: 4797. https://doi.org/10.3390/math10244797
APA StyleAttia, M. A., Melaibari, A., Shanab, R. A., & Eltaher, M. A. (2022). Dynamic Analysis of Sigmoid Bidirectional FG Microbeams under Moving Load and Thermal Load: Analytical Laplace Solution. Mathematics, 10(24), 4797. https://doi.org/10.3390/math10244797