Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters
Abstract
:1. Introduction
2. Finite Element Formulation of Composite Beam with Viscoelastic Layers
2.1. Elastic Layers
2.2. Viscoelastic Layer
2.3. Matrix Formulation of Equation of Motion for Composite Beam
2.4. Dynamic Characteristics of Composite Beam
3. Uncertain Design Parameters
3.1. Interval Analysis
3.2. Approximation of the Lower and Upper Bounds of Objective Function Based on Taylor Series Expansion
3.3. Vertex Method
4. Algorithm for Determining Critical Combinations of Parameters
Algorithm 1: Determining the critical combination of parameters. |
1° Input data: r—number of uncertain parameters —vector of lower bounds of uncertain parameters —vector of upper bounds of uncertain parameters —vector of middle values of uncertain parameters 2° set n:= r set ( denotes parameter vector for the critical combination) 3° for i = 1,2,…,n calculate sensitivities of the response function with respect to change in parameter (sensitivities of the first order) and (sensitivities of the second order) 4° for i = 1,2,…,n calculate the increment in response function according to Equation (37) for and 5° select the maximum value of and corresponding parameter 6° substitute 7° exclude parameter and its lower and upper bounds from futher calculations 8° update n:=n − 1 9° repeat steps 3° to 8° until n = 0 10° calculate the maximum value of the response function for the critical combination of parameters . |
5. Examples
5.1. Veryfication Example
5.2. The Influence of the Layer Height Uncertainty
5.3. The Influence of the Parameters and of the Viscoelastic Layers
5.4. The Influence of the Parameters α and τ of the Viscoelastic Layers
5.5. Comparison of Methods
6. Conclusions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Method | Characteristics/Advantages and Disadvantages |
---|---|
Probabilistic methods [1,2,3] | Parameters are assumed as random variables, and their probability distributions are estimated based on available data. Advantage:
|
Fuzzy theory methods [4] | Parameters are described as fuzzy sets. Advantage:
|
Methods based on interval analysis [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,29,30,31,32] | Parameters are modeled as intervals with uncertain-but-bounded values. Advantages:
|
Hybrid methods [27,28] | A combination of different methods, such as probabilistic, fuzzy, or interval-based. |
Dynamic Characteristics | Mode | |||
---|---|---|---|---|
1 | 2 | 3 | 4 | |
Natural frequency (present paper) [rad/s] | 404.875 | 1336.507 | 2860.876 | 4990.735 |
Natural frequency [35] [rad/s] | 404.879 | 1336.560 | 2861.150 | 4991.590 |
Nondimensional damping ratio (present paper) [-] | 0.1378 | 0.1343 | 0.1184 | 0.1042 |
Nondimensional damping ratio [35] [-] | 0.1378 | 0.1343 | 0.1184 | 0.1042 |
Static Scheme | Natural Frequency ω1 [rad/s] | Nondimensional Damping Ratio γ1 [-] |
---|---|---|
(a) Simply supported | 1107.60 | 0.0706 |
(b) Cantilever | 443.36 | 0.0879 |
(c) Fixed-pinned | 1672.05 | 0.0491 |
(d) Fixed-fixed | 2381.97 | 0.0346 |
Uncertainties of Parameters | Layer Height [m] |
---|---|
5% | |
30% |
Bounds | Methods | Natural Frequency [rad/s] | Error [%] | Combination of Parameters |
---|---|---|---|---|
lower upper | 968.63 1259.25 | Comparative method | ||
lower upper | II | 962.53 1252.67 | 0.63 0.52 | |
lower upper | III | 965.42 1255.55 | 0.33 0.29 | |
lower upper | IV | 968.83 1258.96 | 0.02 0.02 | |
lower upper | V | 968.63 1259.25 | 0.00 0.00 |
Bounds | Methods | Natural Frequency [rad/s] | Error [%] | Combination of Parameters |
---|---|---|---|---|
lower upper | 735.89 1597.05 | Comparative method | ||
lower upper | II | 672.40 1542.80 | 8.63 3.40 | |
lower upper | III | 698.38 1568.77 | 5.10 1.77 | |
lower upper | IV | 729.02 1599.42 | 0.93 0.15 | |
lower upper | V | 735.89 1597.05 | 0.00 0.00 |
Uncertainties of Parameters | Design Parameters [Pa] |
---|---|
5% | |
30% |
Uncertainties of Parameters | Design Parameters |
---|---|
5% | |
30% |
Bounds | Methods | Natural Frequency [rad/s] | Error [%] | Combination of Parameters |
---|---|---|---|---|
Lower Upper | 1096.63 1122.06 | Comparative method | ||
Lower Upper | II | 1094.99 1120.21 | 0.15 0.16 | |
Lower Upper | III | 1096.26 1121.49 | 0.03 0.05 | |
Lower Upper | IV | 1096.69 1121.91 | 0.01 0.01 | |
Lower Upper | V | 1096.63 1122.06 | 0.00 0.00 |
Bounds | Methods | Natural Frequency [rad/s] | Error [%] | Combination of Parameters |
---|---|---|---|---|
Lower Upper | 1082.70 1163.71 | Comparative method | ||
Lower Upper | II | 1069.76 1145.44 | 1.19 1.57 | |
Lower Upper | III | 1081.24 1156.91 | 0.13 0.58 | |
Lower Upper | IV | 1085.08 1160.76 | 0.22 0.25 | |
Lower Upper | V | 1082.70 1163.71 | 0.00 0.00 |
Bounds | Methods | Natural Frequency [rad/s] | Error [%] | Combination of Parameters |
---|---|---|---|---|
Lower Upper | 1075.97 1124.08 | Comparative method | ||
Lower Upper | II | 1044.54 1170.66 | 2.92 4.36 | |
Lower Upper | III | 1076.42 1202.54 | 0.04 1.76 | |
Lower Upper | IV | 1080.85 1213.22 | 0.45 0.89 | |
Lower Upper | V | 1075.97 1124.08 | 0.00 0.00 |
Bounds | Methods | Natural Frequency [rad/s] | Error [%] | Combination of Parameters |
---|---|---|---|---|
Lower Upper | 1075.87 1227.85 | Comparative method | ||
Lower Upper | II | 1043.03 1172.17 | 3.05 4.53 | |
Lower Upper | III | 1074.90 1204.05 | 0.09 1.94 | |
Lower Upper | IV | 1080.22 1216.86 | 0.40 0.89 | |
Lower Upper | V | 1075.87 1227.85 | 0.00 0.00 |
Methods | Number of Combinations | Number of Eigenproblems to Be Solved for the New Parameter Values |
---|---|---|
II | 2r | 1 |
III | 2r | 1 |
IV | 1 | |
V | r(r + 1) | r + 1 |
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Łasecka-Plura, M. Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters. Appl. Sci. 2023, 13, 6473. https://doi.org/10.3390/app13116473
Łasecka-Plura M. Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters. Applied Sciences. 2023; 13(11):6473. https://doi.org/10.3390/app13116473
Chicago/Turabian StyleŁasecka-Plura, Magdalena. 2023. "Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters" Applied Sciences 13, no. 11: 6473. https://doi.org/10.3390/app13116473
APA StyleŁasecka-Plura, M. (2023). Dynamic Characteristics of a Composite Beam with Viscoelastic Layers under Uncertain-But-Bounded Design Parameters. Applied Sciences, 13(11), 6473. https://doi.org/10.3390/app13116473