5 Curvature Effects on Vibrational Powerflow
5.2 Out-of-Plane Curvature Effects
Curvature Effects on Vibrational Power Flow of Smooth Bent Beams 91
100 102 104
0 0.005 0.01 0.015 0.02
100 102 104
0 0.005 0.01 0.015 0.02
Fig. 6 Wave numbers for in-plane vibration of a curved beam
𝛺3=
√Iz A
R , (7)
which depends only on geometric properties and occurs at a longitudinal wave length 𝜆l= 2𝜋R.
One can categorize the first medium-low transmission region as0◦ < 𝛩y<150◦ below 𝛺3, the very low transmission region as 180◦< 𝛩y<350◦ until mid- frequency of part II (between𝛺1and𝛺2), and finally, by exclusion, all other regions as high transmission.
For the longitudinal transmission discontinuity, the following formula can be fit as follows:
𝛺d(𝛩y) ≈ 0.9𝛺1− 4.3 𝛩2y+√
𝛩y
, (8)
≈ 0.3√
EIz 𝜌A
R2 − 4.3 𝛩2y+√
𝛩y
, (9)
which depend only on the angle𝛩yand𝛺1, because the discontinuity grows asymp- totically to it, as shown in the Fig.4.
This fitted formula can be used to estimate a critical scenario of longitudinal waves, e.g., in an application which this kind of transmission should be avoided by design.
92 P. Martins and A. Lenzi
very similar to those in the previous case, all discussion about “transmission regions”
is valid by analogy (in- and out-of-plane flexural waves, as well as longitudinal and torsional waves, are similar).
A new approach is necessary to investigate 3D effects, so the following setup is proposed: Let there be a simple curve with angle𝛩y (which is the same “curves angle”𝛩yfrom before) and lengthLwith an in-plane prescribed flexural wave. This curve will be sliced in the middle (𝛩y∕2, thereforeL∕2) and a “rolling” of𝛩xwill be applied to the second section.
As this case will be tested for𝛩yfrom0◦to360◦,𝛩xfrom0◦ to180◦, and fre- quencyf from0to10kHz, the level curve approach will not suffice because of the extra variable. Therefore, a mean power over frequency is used as a parameter in the form of
W̄ = 1 nf
nf
∑
i=1
W. (10)
This parameter condenses the results, allowing another “level curve-like” plot. The only disavantage to this approach is the loss of detailed frequency analysis.
To simplify even further for preliminary analysis, the observed transmitted power is a sum of all transmission coefficients
W =WuT+WwT+WvT+W𝜃Tx, (11) so that details about a waves direction are not overwhelming. The results are shown in Fig.7, in whichxshows the variation for pitch angles andyaxis shows the variation in “rolling angles” (𝛩x), for aR= 6 mm configuration.
One may immediately detect in Fig.7some low transmission areas, which are highlighted by black contours: Region I on the left with curves from30◦to120◦and rolling angles up to about140◦(about 63% of mean power transmission); Region II, a small one with low transmission levels (about 34%), with curves from180◦to330◦ but rolling angles only going up to60◦; and Region III, the region of lowest lowest
Fig. 7 Condensed mean transmission for out-of-plane flexural vibration acting over aR= 6mm curve
Curvature Effects on Vibrational Power Flow of Smooth Bent Beams 93
Fig. 8 Power coefficients for a flexural out-of-plane vibration acting over a single R= 6mm curve (without the “rolling aspect”)
100 101 102 103 104
0 100 200 300
0 20 40 60 80 100
[%]
transmission (reaching below 12%), with curves going from180◦all the way up to a full loop, and rolling angles above120◦.
These regions can be explained by an in-plane level curve map withR= 6 mm, resulting in Fig.8. In- and out-of-plane vibrations are changed in a very similar man- ner, so it makes clear why regions above 180◦ are prone to low transmission lev- els, given that a small radius will increase the cutoff frequencies, especially the first (𝛺1≈√
EIz∕𝜌A∕3R2).
Thus, for a small radius, the best configuration possible is a series of𝛩x= 180◦ curves aligned in the same plane.
As the radius increases (nowR= 50 mm), the low transmission regions for pitch angles𝛩y>180◦vanishes, as illustrated in Fig.9.
This phenomenon is due to the low cutoff frequencies, which result from the high transmission region discussed in the previous section. Hence, for50 mm radius, the
“medium-to-low transmission” region appears as the best option, with curve angle 𝛩y from20◦ to120◦ and rolling angles higher than95◦ (highlighted by the black line).
When the radius is increased even further, the tendency is to obtain more trans- mission overall. However, starting fromR= 100 mm, low power transmission starts to concentrate between30◦ < 𝛩y<60◦and high rolling angles.
Fig. 9 Condensed mean transmission for out-of-plane flexural vibration acting over aR= 50 mm curve.
Highlighted area shows the minimum transmission region
94 P. Martins and A. Lenzi
In general, from a vibrations perspective, best case scenario is when the struc- ture have a succession of50◦curves, aligned in the same plane. Evidently, for more complex structures, the curves radius should be considered to yield better results.
6 Conclusions
The present work investigated the changes in power flow of vibrational waves, while propagating through smooth bent slender beams. This paper presented the wave propagation approach, as well as the standardization of results using power coef- ficients.
Subsequently, an FEM program was constructed to simulate the proposed approach. The power coefficients were obtained through this numerical tool. Ana- lytical validation was carried out, with all formulations for both in-plane and out-of- plane vibrations. Differences between the two were discussed.
With the validated FEM tool, the effects of in-plane and out-of-plane vibrations were discussed in detail. Moreover, different setups for obtaining results from each case were proposed. First, longitudinal and torsional waves do not transmit mean- ingful vibrational power through curves, except for curve angles nearing0◦and for one “discontinuity line” given by𝛺d(𝛩y) ≈ 0.9𝛺1− 4.3∕(𝛩2y+√
𝛩y).
Both in- and out-of-plane vibrations are similarly changed when their paths have a smooth bend. Longitudinal waves are analogous to torsional ones, and flexural waves on the same plane of a curve are analogous to the plane perpendicular to the curve.
For the effects of in-plane vibration, three major regions were identified: medium- to-low transmission/wide frequency range, very low frequency/low frequency range and high transmission.
Finally, for sequential out-of-plane curves, the radius plays a huge influence over overall transmitted power. If the cutoff frequencies are sufficiently high, two impor- tant low-transmission regions exist for curve angles above180◦. Otherwise, the pre- viously mentioned medium-to-low region is highlighted, showing that the lowest transmission values are between30◦and120◦curves.
This study aimed to enlighten the curvature effects on vibrations, for the benefit of engineers to design of new products that use curved slender structures.
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