Abstract
This work is on the motions of non-homogeneous elastic rods. In a previous work the natural frequencies and associated mode shapes were determined for a two-segment rod, in which the geometric and material properties were constant in each segment. Here the steady state response due to harmonic forcing is investigated using two strategies. The first employs the exact displacement equations. For harmonic forcing in time, the response is periodic and general solutions to the resulting differential equations can, in principle, be found for each segment. The constants involved are found from boundary and interface conditions and then response, as a function of forcing frequency, can be obtained. The procedure is cumbersome and problematic if the forces vary spatially, due to difficulties in finding “particular integrals”. An alternative method is developed in which geometric and material discontinuities are modeled by continuously varying functions (here logistic functions). This leads to a single differential equation with variable coefficients, which is solved numerically using MAPLE®’s PDE solver. For free-fixed boundary conditions and spatially constant force good agreement is found between the two methods, lending confidence to the continuous varying approach, which is then used to obtain response for spatially varying forces.
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Abbreviations
- A :
-
Cross-section area (A i , cross-section area for i-th material)
- A r :
-
Non-dimensional parameter \( {A}_r={A}_2/{A}_1 \)
- B i :
-
Constants of integration
- c i :
-
Wave speed \( {c}_i=\sqrt{E/{\rho}_i} \)
- c r :
-
Non-dimensional parameter \( {c}_r={c}_1/{c}_2 \)
- E :
-
Young’s modulus (E i Young’s modulus for i-th material)
- E r :
-
Non-dimensional parameter \( {E}_r={E}_2/{E}_1 \)
- F i :
-
Spatial functions
- f i :
-
Forcing functions (force per unit length q i in the continuously varying approach)
- g i :
-
Non-dimensional functions for material/geometrical properties
- L :
-
Length of rod, \( L={L}_1+{L}_2 \)
- Q 0 :
-
Non-dimensional parameter \( {Q}_0={F}_0L/{A}_1{E}_1 \)
- R i :
-
Spatial functions
- t :
-
Time
- u :
-
Longitudinal displacement of the rod
- w :
-
Non-dimensional longitudinal displacement of the rod, \( w=u/L \)
- x :
-
Longitudinal coordinate
- x D :
-
Non-dimensional longitudinal coordinate, \( {x}_D=x/L \)
- α :
-
Numerical parameter for length of individual cell components of the layered rod, \( L=\left(1+\alpha \right){L}_1 \)
- β :
-
Non-dimensional parameter, \( \beta =\sqrt{L^2{\varOmega}_0^2{\rho}_1/{E}_1} \)
- ρ :
-
Mass density (ρ i , density value for i-th material)
- τ :
-
Non-dimensional time, \( \tau ={\varOmega}_0t \)
- ω :
-
Natural frequency of longitudinal vibrations for the rod
- ω d :
-
Non dimensional natural frequency of longitudinal vibrations for the rod (ν), \( \nu =\omega /{\varOmega}_0 \)
- Ω 0 :
-
Reference frequency
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Mazzei, A.J., Scott, R.A. (2016). Harmonic Forcing of a Two-Segment Elastic Rod. In: Di Miao, D., Tarazaga, P., Castellini, P. (eds) Special Topics in Structural Dynamics, Volume 6. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-29910-5_23
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DOI: https://doi.org/10.1007/978-3-319-29910-5_23
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