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Vibrations of the Euler–Bernoulli Beam Under a Moving Force based on Various Versions of Gradient Nonlocal Elasticity Theory: Application in Nanomechanics

Śniady Paweł

Śniady Paweł

Faculty of Environmental Engineering and Geodesy, Wroclaw University of Environmental and Life ScienceWroclaw, Poland
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Paweł, Śniady
, 
Katarzyna Misiurek

Katarzyna Misiurek

Faculty of Civil Engineering, Wroclaw University of Science and TechnologyWroclaw, Poland
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Misiurek, Katarzyna
, 
Olga Szyłko-Bigus

Olga Szyłko-Bigus

Faculty of Civil Engineering, Wroclaw University of Science and TechnologyWroclaw, Poland
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Szyłko-Bigus, Olga
 and 
Idzikowski Rafał

Idzikowski Rafał

Faculty of Environmental Engineering and Geodesy, Wroclaw University of Environmental and Life ScienceWroclaw, Poland
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Rafał, Idzikowski
  
Jun 29, 2020
Vibrations of the Euler–Bernoulli Beam Under a Moving Force based on Various Versions of Gradient Nonlocal Elasticity Theory: Application in Nanomechanics's Cover Image
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Article Category: Research Article
Published Online: Jun 29, 2020
Page range: 306 - 318
Received: Jan 14, 2020
Accepted: Apr 16, 2020
DOI: https://doi.org/10.2478/sgem-2019-0049
Keywords
© 2020 Paweł Śniady et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Beam under moving force.
Beam under moving force.

Figure 2

Stress gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.01.
Stress gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.01.

Figure 3

Strain gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.01.
Strain gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.01.

Figure 4

Stress gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.05.
Stress gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.05.

Figure 5

Strain gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 001.
Strain gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 001.

Figure 6

Stress gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.10.
Stress gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.10.

Figure 7

Strain gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.10.
Strain gradient model. Displacement and bending moment of the beam for T = 0.5, ηg = 0.10.

Figure 8

Stress gradient model. Vibrations of the beam in the middle cross section if ηg = 0.01.
Stress gradient model. Vibrations of the beam in the middle cross section if ηg = 0.01.

Figure 9

Strain gradient model. Vibrations of the beam in the middle cross section if ηg = 0.01.
Strain gradient model. Vibrations of the beam in the middle cross section if ηg = 0.01.

Figure 10

Stress gradient model. Vibrations of the beam in the middle cross-section if ηg = 0.05.
Stress gradient model. Vibrations of the beam in the middle cross-section if ηg = 0.05.

Figure 11

Strain gradient model. Vibrations of the beam in the middle cross-section if ηg = 0.05.
Strain gradient model. Vibrations of the beam in the middle cross-section if ηg = 0.05.

Figure 12

Stress gradient model. Vibrations of the beam in the middle cross section if ηg = 0.10.
Stress gradient model. Vibrations of the beam in the middle cross section if ηg = 0.10.

Figure 13

Strain gradient model. Vibrations of the beam in the middle cross-section if ηg = 0.10.
Strain gradient model. Vibrations of the beam in the middle cross-section if ηg = 0.10.

Relation for nonlocal material properties me, ms and the ratio of the critical force velocity to wave velocity_

μe = μsηe,cr=ve,crvg\eta _{e,cr} = \frac{{v_{e,cr} }}{{v_g }}ηs,cr=vs,crvg\eta _{s,cr} = \frac{{v_{s,cr} }}{{v_g }}
0.20.08860.124
0.40.06520.168
0.60.04890.224
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