Abstract

A wave-based model that incorporates the effects of shear deformation, rotary inertia and elastic coupling due to structural anisotropy, is developed to analyze the free vibrations of elastically restrained laminated planar frames. In this work, a generalized frame structure is represented as an assemblage of laminated beam segments that act as one-dimensional waveguides. The segments are assumed to undergo only in-plane motion, which upon applying Hamilton's principle, is described by a system of coupled differential equations. Dispersion analysis is conducted, and the nature of the wavefields associated with the propagation matrix is discussed. Generally restrained boundaries and internal joints are considered, and the associated reflection and transmission matrices are derived. Using the principle of wave-train closure, the characteristic equation is obtained by systematically assembling the propagation, reflection, and transmission matrices. The wave-based model is inherently deterministic, and solving the characteristic equation offers the advantage of determining the exact natural frequencies using conventional root finding algorithms. Application of the proposed model is demonstrated by analyzing an elastically restrained inclined laminated portal frame. Extensive computational analysis is conducted to illustrate the influence of stacking sequence, frame angle, relative frame length, orthotropicity ratios, and spring stiffness on the exact natural frequencies (and in certain cases the mode shapes) of the frame. Independent finite element simulations conducted in ansys® APDL are consistently used to verify the validity of the analytical results.

References

1.
Duan
,
Z.
,
Liu
,
Y.
,
Xu
,
B.
, and
Yan
,
J.
,
2022
, “
Structural Topology Design Optimization of Fiber-Reinforced Composite Frames With Fundamental Frequency Constraints
,”
J. Struct. Eng.
,
148
(
4
), p.
04022027
.
2.
Azarov
,
A. V.
,
Antonov
,
F. K.
,
Golubev
,
M. V.
,
Khaziev
,
A. R.
, and
Ushanov
,
S. A.
,
2019
, “
Composite 3D Printing for the Small Size Unmanned Aerial Vehicle Structure
,”
Composites, Part B
,
169
, pp.
157
163
.
3.
Teh
,
K. K.
, and
Huang
,
C. C.
,
1980
, “
The Effects of Fibre Orientation on Free Vibrations of Composite Beams
,”
J. Sound Vib.
,
69
(
2
), pp.
327
337
.
4.
Weisshaar
,
T. A.
,
1981
, “
Aeroelastic Tailoring of Forward Swept Composite Wings
,”
J. Aircr.
,
18
(
8
), pp.
669
676
.
5.
Chandrashekhara
,
K.
,
Krishnamurthy
,
K.
, and
Roy
,
S.
,
1990
, “
Free Vibration of Composite Beams Including Rotary Inertia and Shear Deformation
,”
Compos. Struct.
,
14
(
4
), pp.
269
279
.
6.
Mahapatra
,
D. R.
,
Gopalakrishnan
,
S.
, and
Sankar
,
T. S.
,
2000
, “
Spectral-Element-Based Solutions for Wave Propagation Analysis of Multiply Connected Unsymmetric Laminated Composite Beams
,”
J. Sound Vib.
,
237
(
5
), pp.
819
836
.
7.
Miao
,
F.
,
Sun
,
G.
, and
Zhu
,
P.
,
2016
, “
Developed Reverberation-Ray Matrix Analysis on Transient Responses of Laminated Composite Frame Based on the First-Order Shear Deformation Theory
,”
Compos. Struct.
,
143
, pp.
255
271
.
8.
Minghini
,
F.
,
Tullini
,
N.
, and
Laudiero
,
F.
,
2008
, “
Buckling Analysis of FRP Pultruded Frames Using Locking-Free Finite Elements
,”
Thin-Wall.Struct.
,
46
(
3
), pp.
223
241
.
9.
Bachoo
,
R.
,
2022
, “
Vibration Analysis of Laminated Planar Frame Structures
,”
J. Sound Vib.
,
526
, p.
116787
.
10.
Liu
,
G.
, and
Li
,
Y.
,
2011
, “
Vibration Analysis of Liquid-Filled Pipelines With Elastic Constraints
,”
J. Sound Vib.
,
330
(
13
), pp.
3166
3181
.
11.
Abramovich
,
H.
, and
Livshits
,
A.
,
1994
, “
Free Vibrations of Non-symmetric Cross-ply Laminated Composite Beams
,”
J. Sound Vib.
,
176
(
5
), pp.
597
612
.
12.
Vinson
,
J. R.
, and
Sierakowski
,
R. L.
,
2006
,
The Behavior of Structures Composed of Composite Materials
,
Springer
,
Dordrecht
.
13.
Abramovich
,
H.
,
2019
,
Advanced Aerospace Materials: Aluminum-Based and Composite Structures
,
De Gruyter
,
Berlin
.
14.
Nickalls
,
R. W.
,
1993
, “
A new Approach to Solving the Cubic: Cardan’s Solution Revealed
,”
Math. Gaz.
,
77
(
480
), pp.
354
359
.
15.
Mei
,
C.
,
2005
, “
Effect of Material Coupling on Wave Vibration of Composite Timoshenko Beams
,”
ASME J. Vib. Acoust.
,
127
(
4
), pp.
333
340
.
16.
Wittrick
,
W.
, and
Williams
,
F.
,
1971
, “
A General Algorithm for Computing Natural Frequencies of Elastic Structures
,”
Q. J. Mech. Appl. Math.
,
24
(
3
), pp.
263
284
.
17.
Cowper
,
G. R.
,
1966
, “
The Shear Coefficient in Timoshenko’s Beam Theory
,”
ASME J. Appl. Mech.
,
33
(
2
), pp.
335
340
.
18.
Chang
,
C. H.
,
1978
, “
Vibrations of Frames With Inclined Members
,”
J. Sound Vib.
,
56
(
2
), pp.
201
214
.
19.
Chang
,
C. H.
,
2005
,
Mechanics of Elastic Structures With Inclined Members: Analysis of Vibration, Buckling and Bending of X-Braced Frames and Conical Shells
,
Springer
,
Berlin
.
20.
Thompson
,
M. K.
, and
Thompson
,
J. M.
,
2017
,
ANSYS Mechanical APDL for Finite Element Analysis
,
Butterworth-Heinemann
,
UK
.
21.
ANSYS
,
2012
,
ANSYS Mechanical APDL Element Reference
,
ANSYS Inc.
,
Canonsburg, PA
.
You do not currently have access to this content.