Abstract
This work is an extension of previous studies on vibrations of non-homogeneous structures. It also explores the use of logistic functions. In the studies, frequency response functions (FRFs) were determined for segmented structures, using analytic and numerical approaches. The structures are composed of stacked cells, which are made of different materials and may have different geometric properties. Here the steady state response, due to harmonic forcing, of a segmented damped Euler-Bernoulli beam is investigated. FRFs for the system are sought via two methods. The first uses the displacement differential equations for each segment. 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. In addition, determining particular integrals can become cumbersome for arbitrary spatial variations. The second approach uses logistic functions to model the segment discontinuities. The result is a single partial differential equation with variable coefficients. Approaches for numerical solutions are then 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 the FRF for the case of a spatially varying force.
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Notes
Abbreviations
- A :
-
Cross-section area (A i, cross-section area for i-th material)
- a i :
-
Non-dimensional parameters related to beam properties
- B i :
-
Constants of integration
- C i :
-
Viscous damping coefficient per unit length
- CD i :
-
Non-dimensional damping coefficient
- E :
-
Youngās modulus (E i, Youngās modulus for i-th material)
- f i :
-
Non-dimensional logistic functions
- F i :
-
Forcing functions (force per unit length)
- G i :
-
Non-dimensional spatial forcing functions
- g i :
-
Non-dimensional forcing functions
- I :
-
Area moment of inertia of the beam cross section (I i, moment of inertia of i-cell)
- K :
-
Non-dimensional logistic function parameter
- K 1, i :
-
Non-dimensional ODE parameters related to beam properties
- L :
-
Length of beam (L i, length of i-th cell)
- R i :
-
Non-dimensional spatial functions
- t :
-
Time
- u :
-
Non-dimensional transverse displacement of the beam,Ā uĀ =Ā w/L
- x :
-
Longitudinal co-ordinate
- w :
-
Transverse displacement of the beam
- Ī±, Ī² :
-
Non-dimensional ODE coefficients
- Ī³ :
-
Non-dimensional viscosity coefficient
- Ī· :
-
Non-dimensional structural damping coefficient
- Ī» :
-
Complex frequency,Ā Ī»Ā =Ā (aĀ +Ā bI)
- Ī¾ :
-
Non-dimensional spatial co-ordinate, Ī¾Ā =Ā x/L
- Ļ :
-
Mass density (Ļ i, density value for i-th material)
- Ļ :
-
Non-dimensional time, ĻĀ =Ā Ī©0t
- Ī©0 :
-
Reference frequency
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Mazzei, A.J., Scott, R.A. (2021). Harmonic Forcing of Damped Non-homogeneous Euler-Bernoulli Beams. In: Epp, D.S. (eds) Special Topics in Structural Dynamics & Experimental Techniques, Volume 5. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-47709-7_2
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