Abstract
This theoretical and experimental work deals with the power law evolution followed by the natural frequency of a hanging, heavy and flexible plate as a function of its length . When the plate length is small enough, it behaves as an elastic plate whose weight can be neglected: it is well known that evolves as a function of . Nevertheless, when the plate length is increased, the mass has to be taken into account, and the previous evolution is not valid anymore. In the case of long elastic plates, , just like hanging chains. These two power laws depend on the ratio , where is a critical length that writes as a function of the plate mass and the flexural rigidity. After the theory is developed and the plate motion equation is solved using a Galerkin expansion, we find the theoretical evolution of the natural frequencies as a function of length. Experiments were performed with three distinct materials and the natural frequency was systematically measured for a wide length interval. Our data points fit the above-mentioned limit cases and the intermediate case was calculated thanks to our Galerkin expansion.
Keywords:
natural frequency; elastic plate; power law
1 INTRODUCTION
Resonance in structural engineering may have disastrous consequences since it generally leads to high amplitude oscillations that severely damage or destroy the structure. A recent example is the London Millennium Footbridge which had been closed for almost two years because of its lateral oscillations, generated by the footsteps of thousands of pedestrians whose rhythm matched the bridge natural frequency (Newl, 2004Newl, D.E. (2004). Pedestrian excitation of bridges. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 218(5): 477-492.). A similar phenomenon may occur in airplanes via the so-called “aeroelastic instability” (Eloy et al., 2007Eloy, C., Souilliez, C., Schouveiler, L. (2007). Flutter of a rectangular plate. Journal of Fluids and Structures 23(6): 904-919.), which originates from a complex interaction between the wing and the airflow (Dungundji et al., 1963Dungundji, J., Dowell, E., Perkins, B. (1963). Subsonic flutter of panels on continuous elastic foundations. AIAA Journal 1: 1146-1154.). This instability was shown to come from the second natural mode of the structure (Cros and Arellano Castro, 2016Cros, A., Arellano Castro, R.F. (2016). Experimental study on the resonance frequencies of a cantilevered plate in air flow. Journal of Sound and Vibration 363: 240-246.).
The natural frequency of a flexible plate whose weight is neglected can be easily deduced from the motion equation (Turvey, 1990Turvey, K. (1990). An undergraduate experiment on the vibration of a cantilever and its application to the determination of young’s modulus. American Journal of Physics 58(5): 483-487.). In this case, depends upon the plate flexural rigidity , its mass per unit of length and evolves proportionally to , where is the plate length. Now, the case where corresponds to a hanging chain and it was shown that (Verbin, 2014Verbin, Y. (2014). Boundary conditions and modes of the vertically hanging chain. European Journal of Physics 36(1): 015005.; Levinson, 1977Levinson, D.A. (1977). Natural frequencies of a hanging chain. American Journal of Physics 45(7): 680-681.; Lamb, 1929Lamb, H. (1929). Higher Mechanics. Cambridge University Press.).
The intermediate case where both gravity and rigidity are taken into account was considered by different authors. In a chronological order, Gregory and Païdoussis (1966Gregory, R., Païdoussis, M. P. (1966) Unstable oscillation of tubular cantilevers conveying fluid ii. experiments. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 293(1435): 528-542.) used Rayleigh’s method to extract an analytical law for the evolution of as a function of . Païdoussis and Des Trois Maisons (1971) and Schäfer (1985Schäfer, B. (1985). Free vibrations of a gravity-loaded clamped-free beam. Ingenieur-archiv 55(1): 66-80.) adopted the Euler beam model in order to estimate how gravity modifies the natural frequency of a flexible beam. Yokoyama (1990Yokoyama, T. (1990). Vibrations of a hanging timoshenko beam under gravity. Journal of Sound and Vibration 141(2): 245-258.) chose to describe the plate thanks to the Timoshenko theory and solved the equation by using the finite element procedure (Xi et al., 2013Xi, L.Y., Li, X.F., Tang, G.J. (2013) Free vibration of standing and hanging gravity-loaded rayleigh cantilevers. International Journal of Mechanical Sciences 66: 233-238.).
In this study, we first recall the plate motion equation and the nondimensional numbers associated to the problem. The complete system is theoretically solved thanks to a Galerkin expansion and then the experiments are presented. The natural plate frequency is measured for a wide range of sample lengths. The last graphics show how experiments fit the theoretical law and how the nondimensional natural frequency depends only upon a nondimensional length.
2 THEORY
The system is shown in Figure 1. Axis is downwards so that the plate is clamped at and free at . Lateral oscillations are identified thanks to variable which depends upon the longitudinal coordinate and time .
Scheme of our system. The plate is hanging from its top extreme at and is the transversal deflection. is gravity. The plate is -long, -wide and -thick.
The complete motion equation is given by the following equation (Païdoussis and Des Trois Maisons, 1971Païdoussis, M.P., Des Trois Maisons, P.E. (1971). Free vibration of a heavy, damped, vertical cantilever. Journal of Applied Mechanics 38(2): 524-526.):
where is the flexural rigidity, the mass per unit of length and is gravity. The boundary conditions write as:
2.1 Limit case
In the case where the plate’s own weight can be neglected, Eq. 1 can be solved by separating variables. The boundary conditions lead to a discrete set of frequency values whose lower frequency is expressed as (Turvey, 1990Turvey, K. (1990). An undergraduate experiment on the vibration of a cantilever and its application to the determination of young’s modulus. American Journal of Physics 58(5): 483-487.):
2.2 Limit case
If the fourth spatial derivative of Eq. 1 is set to zero, the system corresponds to a hanging chain and the solution is a Bessel function. Once again, the limit conditions at , only permit a discrete set of values (Lamb, 1929Lamb, H. (1929). Higher Mechanics. Cambridge University Press.) in such a way that:
Let’s note that this limit case can be recovered for a flexible plate (for which strictly speaking) if its length is high enough. The following section shows indeed that this is possible if where is defined by Eq. 6.
2.3 Nondimensional equation
Eq. 1 may be nondimensional by defining the following parameters:
Moreover, a critical length is given by (De Langre et al., 2001De Langre, E., Doaré, O., Pellet, F. (2001). Force suiveuse critique sur une colonne pesante semi-infinie: modèle et expériences. Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics 329(3): 175-178.):
Then Eq. 1 writes as:
We see that the plate profile only depends upon one parameter which is the nondimensional length . This result is coherent with the Vaschy-Buckingham theorem since there are five independent dimensional parameters: the natural frequency , the plate length , the flexural rigidity , the mass per unit of length and gravity . As a consequence, we can affirm that the nondimensional natural frequency depends only upon a nondimensional length.
2.4 Intermediate case: our results
In order to solve the complete equation (Eq. 7) joint with the boundary conditions (Eq. 2), we use the procedure described in Schäfer (1985Schäfer, B. (1985). Free vibrations of a gravity-loaded clamped-free beam. Ingenieur-archiv 55(1): 66-80.). This method allows indeed to relate naturally the eigenfrequencies with the plate length without solving the solution . We assumed a harmonic solution of form:
where is the natural frequency and is the corresponding mode shape. Replacing Eq. 8 in Eq. 7 leads to:
with
Function is approximated with eigenfunctions ,
where the ’s are constant factors to be determined using Galerkin’s method. The functions:
with
are orthogonal and normalized with respect to the inner product , where is the Kronecker symbol. These functions satisfy the boundary value problem
having the same boundary conditions as those presented in Eq. 2. The eigenvalues are solutions of
giving , , and for the higher values .
The boundary value problem (Eq. 9) together with the homogeneous boundary conditions (Eq. 2) take of the general form:
where
Inserting Eq. 11 into Eq. 14, we obtain:
Multiplying both sides of the above equation by ,
for , and applying Galerkin's method leads to the eigenvalue problem:
where is the eigenvector, is the identity matrix and is an symmetric and positive definite matrix consisting of the entries:
Upon integration by parts and taking into account both Eq. 13 and the corresponding boundary conditions (Eq. 2):
Details for solving the above integral can be found in Schäfer (1985Schäfer, B. (1985). Free vibrations of a gravity-loaded clamped-free beam. Ingenieur-archiv 55(1): 66-80.).
Left column plots: computation of as a function of ; right-column semi-log plots: relative errors of for a) and b) , c) and d) , e) and f) .
The nondimensional natural frequency , where:
is related to the lowest eigenvalue of and is extracted from Eq. 10. It depends only upon the ratio and the size of the matrix . To investigate the accuracy of numerical solution of Eq. 15, we compute for several values while is fixed. Figure 2 shows our results for to for and . We have also calculated the relative error with respect to a reference value obtained by solving Eq.15 with . We can see that, as expected, the error is low for small values of (see Figure 2a) and 2b)) because in this case, the plate weight can be neglected and the solution corresponds to Eq. 3. For larger values, a higher order Galerkin decomposition is needed to reach more accurate results. However, we observe that from a value as small as , variations of are insignificant (Figure 2e)). Beyond this value, the relative error is lower than as it can be seen in Figure 2f).
In order to have an idea of the critical value necessary to obtain a given relative error or we plotted as a function of in Figure 3. It can be observed that is sufficient to obtain a relative error equal to for the whole explored length interval. An accuracy is reached only if for , and for . A higher accuracy requires as soon as .
Evolution of the critical value required to reach a relative error equal to (red straight line at ), (blue dotted line), (green plain line), (black dashed dotted line) and (magenta dashed line).
3 EXPERIMENTS
3.1 Plate characteristics
Experiments were performed with three different materials whose length was varied in order to access the power law of with respect to . Plate 1 corresponds to a polyester plate while plates 2 and 3 were molded with silicon. Each plate flexural rigidity was determined by preliminary experiments which consist in measuring the cantilever plate's static deflection when it is submitted to its own weight. A photography of this method is shown in the inset of Figure 4. In this case indeed, the free extreme of the plate is subjected to a transversal deflection equal to (Wilson and Lord Jr, 1973Wilson, F., Lord Jr, A. (1973). Young’s modulus determination via simple, inexpensive static and dynamic measurements. American Journal of Physics 41(5): 653-656.):
In order to have a more accurate value, is measured for different free lengths values. The plot of vs. permits to perform a least-square linear fit whose coefficient is related to . Figure 4 shows the experimental points and the linear fits for the two silicon plates.
Table 1 shows the plate dimensions, their mass, the flexural rigidity, and the critical length calculated via Eq. 6.
3.2 Methodology
The experimental scheme is shown in Figure 5. The plate is clamped from its upper extreme while its bottom extreme is taken away from its equilibrium position and left free to oscillate. A motion sensor (Keyence LK-GD500) is connected to a computer and registers the free extreme position each 2 millisecond for plate 1 and each 2 ms for plates 2 and 3. The spatial resolution is 0.3 mm. The same procedure is repeated for different hanging lengths and for the three plates.
Figure 6a) shows the example of a registered signal. The Fourier spectrum calculated in Figure 6b) presents clearly a peak whose position provides the oscillation frequency for each plate length. The width at half-height, pointed out by a red line in Figure 6b), allows to estimate an errorbar for each frequency value.
Experimental points of the static transversal deflection as a function of for (●) plate 2, (■) plate 3. Lines show the linear fits from which flexural rigidity is extracted. Insert: photography of plate 2 deflected by its own weight.
a) Oscillations of plate 1’s free extreme for cm. b) Fourier spectrum. The red star points out the peak máximum while the red line at the peak mid-height permits to estimate the errorbar.
4 RESULTS
Raw frequency values are shown in the log-log plot of Figure 7 as a function of length for the three plates. The different lines represent the two limit cases: the three parallel colored, tilted dashed lines stand for the purely flexible theory (Eq. 3, ) while the black dotted line points out the frequency evolution for a hanging chain (Eq. 4, ). These straight lines intersect for and . The experimental points of the more rigid plate (plate 3, blue triangles) are better described by Eq. 3 for the smallest lengths whereas they diverge from this law for longer plates. The most flexible plates (plates 2 and 3, green circles and red squares) clearly behave as hanging chains for . As expected, this limit case clearly does not depend upon the plate Young's modulus since the same dotted line represents a unique asymptotic behavior for all the plates. The intermediate experimental values are in agreement with the three theoretical plain colored lines.
Experimental points for the three plates superimposed with the two limit cases and with the Galerkin expansion calculated for each plate. (▲) plate 1, (●) plate 2, (■) plate 3. The three parallel colored dashed straight lines point out theoretical laws (Eq. 3) for each flexible plate, while the black dotted line denotes the hanging chain frequency (Eq. 4). The three curved colored lines stand for theory.
Finally, Figure 8 shows how all the experimental points can be superimposed by considering the critical length and critical frequency for each plate shown in Table 1. Errorbars were removed in order to get a clearer graphical visualization. The theoretical curve tends to the expected limit power laws (dashed line) for and (dotted line) for whereas the intermediate values (thick gray curve) superimpose on the solution found by the Galerkin expansion.
Experimental points for the three plates superimposed with theoretical curve where is in and in : (▲) plate 1, (●) plate 2, (■) plate 3. Dashed line: nondimensional version of the purely flexible plate (Eq. 3). Dotted line: nondimensional version of the hanging chain (Eq. 4). Plain thick gray line: Galerkin expansion.
5 CONCLUSION
This study is devoted to the modification of the flexible plate natural frequency when its weight cannot be neglected anymore. This occurs when the plate length reaches values greater than a critical length . This latter quantity is expressed as a function of the ratio between the plate flexural rigidity and the weight per unit of length. In order to solve the complete differential motion equation, a Galerkin decomposition was performed using the spatial modes which develop for a purely flexible plate. We computed the number of modes necessary to get a fixed relative error: as expected, this number increases as is greater since in this case, the plate weight gets more and more important. Moreover, laboratory experiments were performed with different plates molded in three distinct materials and for various length values. The natural frequency is measured by recording the plate's free oscillations. The systematic exploration of a large length interval allows to visualize how the power law evolves when is varied. The data points tend to the two limit power laws for and for whereas the intermediate values are correctly fitted by our Galerkin expansion. All the experimental points superimpose on a same curve by considering a nondimensional parameter frequency and a nondimensional length .
ACKNOWLEDGMENTS
J. R. I and A. C. acknowledge CONACyT and the Mexican Secretaría de la Educación Superior for supporting this project [grant number A1-S-55355].
References
- Cros, A., Arellano Castro, R.F. (2016). Experimental study on the resonance frequencies of a cantilevered plate in air flow. Journal of Sound and Vibration 363: 240-246.
- De Langre, E., Doaré, O., Pellet, F. (2001). Force suiveuse critique sur une colonne pesante semi-infinie: modèle et expériences. Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics 329(3): 175-178.
- Dungundji, J., Dowell, E., Perkins, B. (1963). Subsonic flutter of panels on continuous elastic foundations. AIAA Journal 1: 1146-1154.
- Eloy, C., Souilliez, C., Schouveiler, L. (2007). Flutter of a rectangular plate. Journal of Fluids and Structures 23(6): 904-919.
- Gregory, R., Païdoussis, M. P. (1966) Unstable oscillation of tubular cantilevers conveying fluid ii. experiments. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 293(1435): 528-542.
- Lamb, H. (1929). Higher Mechanics. Cambridge University Press.
- Levinson, D.A. (1977). Natural frequencies of a hanging chain. American Journal of Physics 45(7): 680-681.
- Newl, D.E. (2004). Pedestrian excitation of bridges. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 218(5): 477-492.
- Païdoussis, M.P., Des Trois Maisons, P.E. (1971). Free vibration of a heavy, damped, vertical cantilever. Journal of Applied Mechanics 38(2): 524-526.
- Schäfer, B. (1985). Free vibrations of a gravity-loaded clamped-free beam. Ingenieur-archiv 55(1): 66-80.
- Turvey, K. (1990). An undergraduate experiment on the vibration of a cantilever and its application to the determination of young’s modulus. American Journal of Physics 58(5): 483-487.
- Verbin, Y. (2014). Boundary conditions and modes of the vertically hanging chain. European Journal of Physics 36(1): 015005.
- Wilson, F., Lord Jr, A. (1973). Young’s modulus determination via simple, inexpensive static and dynamic measurements. American Journal of Physics 41(5): 653-656.
- Xi, L.Y., Li, X.F., Tang, G.J. (2013) Free vibration of standing and hanging gravity-loaded rayleigh cantilevers. International Journal of Mechanical Sciences 66: 233-238.
- Yokoyama, T. (1990). Vibrations of a hanging timoshenko beam under gravity. Journal of Sound and Vibration 141(2): 245-258.
Edited by
Editor:
Publication Dates
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Publication in this collection
18 June 2021 -
Date of issue
2021
History
-
Received
09 Mar 2021 -
Reviewed
20 Apr 2021 -
Accepted
21 Apr 2021 -
Published
03 May 2021