Study of the Vibrations of a System Consisting …
65
a)
b)
c)
d)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
10
20
30
40
50
60
70
80
90
t [s]
P [N]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
50
100
150
200
250
300
350
t [s]
H [N]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
t [s]
h [m]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
5
10
15
20
25
30
35
40
45
50
55
t [s]
θ
f
[ 0
]
Fig. 2 Time history for: a P = P(t); b H = H (t) with n = 2 (blue), n = 5 (green), n = 10 (red)
and n = 20 (light green); c h = h(t) with n = 2 (blue), n = 5 (green), n = 10 (red) and n = 20
(light green); d θ f = θ f (t) with n = 2 (blue), n = 5 (green), n = 10 (red) and n = 20 (light green)
References
1. D.G. Fertis, Nonlinear Mechanics (CRC Press, Boca Raton, 1999)
2. X. Huang, T.X. Yu, G. Lu, H. Lippmann, Large deflection of elastoplastic beams with prescribed
moving and rotating ends. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 217, 1001–1013
(2003)
3. P. Jiao, A.H. Alavi, W. Borchani, N. Lajnef, Small and large deformation models of post-buckled
beams under lateral constraints. Math. Mech. Solids 24(2), 386–405 (2019)
4. Y.A. Kang, X.F. Li, Large deflections of a non-linear cantilever functionally graded beam. J.
Reinf. Plast. Compos. 29(12), 1761–1774 (2010)
5. A.A. Nada, B.A. Hussein, S.M. Megahed, A.A. Shabana, Use of the floating frame of reference formulation in large deformation analysis: experimental and numerical validation. Proc.
IMechE Part K J. Multi-body Dyn. 224, 45–58 (2009)
6. O. Kopmaz, Ö. Gündo˘ gdu, On the curvature of an Euler-Bernoulli beam. Int. J. Mech. Eng.
Educ. 31(2), 132–142 (2003)
7. P. Teodorescu, N.D. St˘ anescu, N. Pandrea, Numerical Analysis with Applications in Mechanics
and Engineering (Wiley, Hoboken, 2013)
8. H. Yu, C. Zhao, B. Zheng, H. Wang, A new higher-order locking-free beam element based on
the absolute nodal coordinate formulation. Proc. IMechE Part C J. Mech. Eng. Sci. 232(19),
3410–3423 (2018)
65
a)
b)
c)
d)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
10
20
30
40
50
60
70
80
90
t [s]
P [N]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
50
100
150
200
250
300
350
t [s]
H [N]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
t [s]
h [m]
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
5
10
15
20
25
30
35
40
45
50
55
t [s]
θ
f
[ 0
]
Fig. 2 Time history for: a P = P(t); b H = H (t) with n = 2 (blue), n = 5 (green), n = 10 (red)
and n = 20 (light green); c h = h(t) with n = 2 (blue), n = 5 (green), n = 10 (red) and n = 20
(light green); d θ f = θ f (t) with n = 2 (blue), n = 5 (green), n = 10 (red) and n = 20 (light green)
References
1. D.G. Fertis, Nonlinear Mechanics (CRC Press, Boca Raton, 1999)
2. X. Huang, T.X. Yu, G. Lu, H. Lippmann, Large deflection of elastoplastic beams with prescribed
moving and rotating ends. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 217, 1001–1013
(2003)
3. P. Jiao, A.H. Alavi, W. Borchani, N. Lajnef, Small and large deformation models of post-buckled
beams under lateral constraints. Math. Mech. Solids 24(2), 386–405 (2019)
4. Y.A. Kang, X.F. Li, Large deflections of a non-linear cantilever functionally graded beam. J.
Reinf. Plast. Compos. 29(12), 1761–1774 (2010)
5. A.A. Nada, B.A. Hussein, S.M. Megahed, A.A. Shabana, Use of the floating frame of reference formulation in large deformation analysis: experimental and numerical validation. Proc.
IMechE Part K J. Multi-body Dyn. 224, 45–58 (2009)
6. O. Kopmaz, Ö. Gündo˘ gdu, On the curvature of an Euler-Bernoulli beam. Int. J. Mech. Eng.
Educ. 31(2), 132–142 (2003)
7. P. Teodorescu, N.D. St˘ anescu, N. Pandrea, Numerical Analysis with Applications in Mechanics
and Engineering (Wiley, Hoboken, 2013)
8. H. Yu, C. Zhao, B. Zheng, H. Wang, A new higher-order locking-free beam element based on
the absolute nodal coordinate formulation. Proc. IMechE Part C J. Mech. Eng. Sci. 232(19),
3410–3423 (2018)