Abstract
This work is the next of a series on vibrations of non-homogeneous structures. It addresses the lateral harmonic forcing, with spatial dependencies, of a two-segment damped Timoshenko beam. In the series, frequency response functions (FRFs) were determined for segmented structures, such as rods and beams, using analytic and numerical approaches. These structures are composed of stacked cells, which are made of different materials and may have different geometric properties. The goal is the determination of frequency response functions (FRFs). Two approaches are employed. The first approach uses displacement differential equations for each segment, where boundary and interface continuity conditions are used to determine the constants involved in the solutions. Then the response, as a function of forcing frequency, can be obtained. This procedure is unwieldy, and determining particular integrals can become difficult for arbitrary spatial variations. The second approach uses logistic functions to model segment discontinuities. The result is a system of partial differential equations with variable coefficients. Numerical solutions are developed with the aid of MAPLEĀ® software. For free/fixed boundary conditions, spatially constant force, and viscous damping, excellent agreement is found between the methods. The numerical approach is then used to obtain FRFs for cases including spatially varying load.
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Abbreviations
- A :
-
Cross-section area (A i, cross-section area for i-th material)
- C Ti, C Ri :
-
Viscous damping coefficients per unit length
- C i :
-
Non-dimensional damping coefficients
- D ij :
-
Proportional damping matrix coefficients
- E :
-
Youngās modulus (E i, Youngās modulus for i-th material)
- f i :
-
Non-dimensional logistic functions
- G i :
-
Beam segment material shear modulus
- I :
-
Area moment of inertia of the beam cross-section (I i, moment of inertia of i-cell)
- k :
-
Shear coefficient (k i, shear coefficient of i-cell)
- K :
-
Non-dimensional logistic function parameter
- K ij :
-
Stiffness matrix coefficients
- L :
-
Length of beam (L i, length of i-th cell)
- M :
-
Bending moment
- M ij :
-
Mass matrix coefficients
- P i :
-
Generalized external forces
- p i :
-
Force acting on the i-segment
- Q :
-
Non-dimensional forcing function
- q :
-
External force per unit length acting on the beam
- q 1 :
-
Spatial forcing function for harmonic solution
- r 1, s 1 :
-
Shape function constants
- t :
-
Time
- \( {\hat{U}}_i \), \( {\hat{V}}_i \):
-
Generalized coordinates
- V :
-
Shear force
- x :
-
Longitudinal coordinate
- w :
-
Transverse displacement of the beam
- Y :
-
Non-dimensional transverse displacement of the beam, YĀ =Ā w/L
- Z, Ļ :
-
Spatial functions for harmonic solution
- Ī±, Ī² :
-
Constants of mass and stiffness proportionality
- Ī± 1, Ī³ 1 :
-
Non-dimensional parameters
- Ī³ :
-
Shear strain
- Ī“ j :
-
Cell properties
- Ī¶ i :
-
Modal damping ratio
- Īø :
-
Rotational angle of the beam cross-section
- \( \hat{\lambda} \) :
-
Complex frequency, \( \hat{\lambda}=\left(a+ bI\right) \)
- Ī» i, Ī· i:
-
Shape functions
- Ī½ :
-
Non-dimensional frequency, Ī½Ā =Ā Ī©/Ī©0
- Ī¾ :
-
Non-dimensional spatial coordinate, Ī¾Ā =Ā x/L
- Ļ :
-
Mass density (Ļ i, density value for i-th material)
- Ļ :
-
Non-dimensional time, ĻĀ =Ā Ī©0 t
- Ļ i :
-
Beam segment material Poissonās ratio
- Ī©0 :
-
Reference frequency
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Mazzei, A.J. (2023). Forced Vibrations of Damped Non-homogeneous Timoshenko Beams. In: Walber, C., Stefanski, M., Harvie, J. (eds) Sensors and Instrumentation, Aircraft/Aerospace and Dynamic Environments Testing, Volume 7. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-031-05415-0_2
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