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Journals
Studia Geotechnica et Mechanica
Volume 42 (2020): Issue 4 (December 2020)
Open Access
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
,
Katarzyna Misiurek
Katarzyna Misiurek
,
Olga Szyłko-Bigus
Olga Szyłko-Bigus
and
Idzikowski Rafał
Idzikowski Rafał
| Jun 29, 2020
Studia Geotechnica et Mechanica
Volume 42 (2020): Issue 4 (December 2020)
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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
vibration
,
beam
,
moving force
,
nonlocal elasticity
© 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.
Figure 2
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.
Figure 4
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.
Figure 6
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.
Figure 8
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.
Figure 10
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.
Figure 12
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.
Relation for nonlocal material properties me, ms and the ratio of the critical force velocity to wave velocity.
μ
e
= μ
s
η
e
,
cr
=
v
e
,
cr
v
g
\eta _{e,cr} = \frac{{v_{e,cr} }}{{v_g }}
η
s
,
cr
=
v
s
,
cr
v
g
\eta _{s,cr} = \frac{{v_{s,cr} }}{{v_g }}
0.2
0.0886
0.124
0.4
0.0652
0.168
0.6
0.0489
0.224