Research Article Forced free vibrations of a square plate

S. J. D. D’Alessio1

Received: 6 September 2020 / Accepted: 22 December 2020 / Published online: 10 January 2021 © The Author(s) 2021 OPEN

Abstract The vibrations of a square plate with free edges that is forced at its center has been investigated experimentally and numerically. A two-step fnite volume scheme is proposed and is tested against an exact solution to a related problem. The governing equation and numerical solution procedure were constructed to closely mimic the experiment. Good agreement was found between the numerical simulations and the experiments.

Keywords Experimental · Numerical · Linear elasticity theory · Resonance

1 Introduction were later observed to settle along the anti-nodes, form- ing what are known as inverse patterns, due to air currents Owing to the numerous engineering applications there that are induced by the vibrating plate which drag the par- have been a plethora of investigations devoted to vibrat- ticles to the anti-nodes. In a vacuum it was believed that ing rectangular plates since the original mathematical all particles would migrate to the nodes regardless of their approach formulated by Euler in 1766. The documented size. However, Gerner et al. [5] showed that air currents are studies in the literature are thoroughly summarized in not the only mechanism leading to inverse patterns. They the works of Leissa [1, 2] for the various distinct combi- discovered that when the acceleration of the resonating nations of boundary conditions including clamped, sim- plate did not exceed the acceleration due to gravity, the ply-supported or free. Interest and research in vibrating particles would always roll to the anti-nodes, irrespective plates gained a lot of attention after the German physicist of their size, and even in the absence of air. and musician Ernst Chladni, who lived from 1756 to 1827, Chladni also noticed that for a circular plate conducted his famous experiments [3]. Chladni is often f ∼(m + 2n)2 where f is the frequency, m is the number referred to as the father of acoustics for his ground-break- of nodal diameters and n is the number of nodal circles. ing work on vibrating plates. He visualized the patterns This result is known as Chladni’s Law [6] and Kirchhof [7] formed when sand was sprinkled on a vibrating horizontal showed that this is an asymptotic relation valid for large plate. Chladni observed that when the plate was allowed f. The quest to uncover the theory behind the Chladni fg- to resonate the sand particles formed stationary nodal ures was ignited by Emperor Napolean when he observed lines which are now known as Chladni fgures. He would a live demonstration delivered by Chladni during a pres- demonstrate his discovery by strumming his bow along entation in Paris in the year 1809 [8]. In the same year he the edge of a plate and watching the sand dance on the commissioned the French Academy of Science to ofer surface. The sand would eventually settle along the nodal a prize to whomever could formulate the mathematical lines which are places on the plate which remained sta- theory that could explain the observed patterns. Famous tionary. The complex patterns changed as the frequency mathematicians including Lagrange, Laplace, Legendre of vibration was varied [4]. Apparently, very fne particles and Poisson served on the judging panel. In the first

* S. J. D. D’Alessio, [email protected] | 1Faculty of Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada.

SN Applied Sciences (2021) 3:60 | https://doi.org/10.1007/s42452-020-04062-6

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∞ competition only one submission was received; it was m x n y from Sophie Germain, a French mathematician who lived u = cos cos m,n=1 a a from 1776 to 1831 and made important contributions in     number theory as well as the theory of elasticity [9, 10]. Amn cos(mnt)+Bmn sin(mnt) , Unfortunately, errors were found in her derivation and no   prize was awarded. A second competition was announced. where Again, only one submission was received and it was from 1 a a m�x n�y A = U (x, y) cos cos dxdy , Sophie Germain. This time she received an honourable mn 2 0 a −a −a a a mention but still no prize because there were mistakes in a a   1 m�x n�y B = U̇ (x, y) cos cos dxdy , her analysis. A third competition then took place. Finally, mn 2 0 �mna −a −a a a in 1816 Sophie Germain received the prize and was the   frst woman to receive an award of this importance from with U0 denoting the initial membrane displacement while the French Academy. Her contributions to this problem ̇ U0 refers to the initial velocity of the membrane. The angu- have been published in the following articles [11, 12]. To lar frequencies are given by = c m2 + n2 . Note mn mn a recognize her achievement the French Academy estab- that for a specifed angular frequency there may be mn√ lished the Sophie Germain Prize in her honour and it has one or more distinct modes of vibration having the same been awarded annually since 2003. Final corrections and angular frequency. For example, 12 = 21 = 5c∕a and improvements to the mathematical formulation were the modes are made by Lagrange, Poisson and Kirchhof [7]. √ 2 x 1 y 1 x 2 y As a frst approximation one can model the plate as cos cos and cos cos . a tightly stretched thin elastic membrane. If we further a a a a     assume that the defection from equilibrium, denoted by   u, is small, the tension ( ) and membrane density ( ) are Hence, the general spatial form of the membrane defec- uniform, and there is no resistance to motion, then u(x, y, t) tion for this frequency will be given by satisfes the two-dimensional wave equation given by 2 x 1 y 1 x 2 y C cos cos + D cos cos 1 2u 2u 2u a a a a = + , −a ≤ x, y ≤ a (Square Plate) .       c2 t2 x2 y2 where C, D are constants. Because of symmetry we expect Here, c refers to the wave speed which is related to the ten- that C = D and hence the nodal lines satisfy [13] sion and mass density through c2 = ∕ . For free edges we 2 x 1 y 2 x 1 y impose the following boundary conditions cos cos + cos cos = 0. a a a a     u u   = 0 along x =±a and = 0 along y =±a . x y The nodal lines are plotted in Fig. 1 (left) and are com- pared to the observed experimental pattern (right) corre- By separation of variables the general solution is easily sponding to f = 190Hz . As explained in a later section, the found to be experiments were carried out using a square plate which

Fig. 1 Comparison of the nodal pattern 1 0.8

0.6

0.4

0.2

0 y/a

−0.2

−0.4

−0.6

−0.8

−1 190 Hz −1 −0.5 0 0.5 1 x/a

Vol:.(1234567890) SN Applied Sciences (2021) 3:60 | https://doi.org/10.1007/s42452-020-04062-6 Research Article is forced at its center to vibrate at a set frequency. Apart and ∇4 is the biharmonic operator given by from the diamond shape in the left-hand plot the agree- 2 2 2 4 4 4 ment is remarkable. Unfortunately, the wave equation ∇4 = (∇2)(∇2)= + = + 2 + . x2 y2 x4 x2 y2 y4 model is highly simplifed and cannot exactly reproduce  the observed patterns. (3) The main goal of this investigation is to bring the For free vibrations equation (1) is to be solved subject to experimental observations in harmony with theory. The the following boundary conditions [17] previous studies by Gander and Kwok [14] and Gander and 2u 2u 3u 3u Wanner [15] tackled this problem by solving an associated +  = +(2 − ) = 0 along x =±a , 2 2 3 2 eigenvalue problem using the Ritz method [16] to com- x y x x y pute the corresponding eigenpairs. As noted by Kirchhof (4) [7], the eigenvalues are related to the resonant frequen- 2u 2u 3u 3u +  = +(2 − ) = 0 along y =±a , cies while the eigenfunctions represent the nodal patterns. y2 x2 y3 x2 y The approach adopted here is signifcantly diferent and (5) new; to our knowledge the proposed solution procedure 2u has not been carried out before. As we will see in Sect. 2 = 0 at the corners (x, y)=(±a, ±a) . (6) y x the equation governing the motion of the plate and the accompanying boundary conditions are far more compli- We next cast the problem in dimensionless form. In order cated than the above wave equation model. Since an exact to achieve this we introduce H as a representative scale for solution to the equation is still out of reach a numerical the defections and apply the following transformation in solution procedure that closely mimics the experiment is terms of dimensionless quantities identifed by an asterisk outlined in Sect. 3. So rather than solving an eigenvalue a2 problem the governing equation will be solved with spe- (x, y)=a(x∗, y∗) , t = t∗ , u = Hu∗ . (7) cial attention paid to enforcing the non-standard bound- D ary conditions. The experimental and numerical results are With these scalings in place, and dropping the asterisks presented and discussed in Sect.4. Finally, in Sect. 5 the key for notational convenience, the dimensionless governing fndings of the investigation are summarized. equation and boundary conditions become

2u +∇4u = 0, (8) 2 Mathematical formulation t2

We consider the transverse defections of a square plate. 2u 2u 3u 3u +  = +(2 − ) = 0 along x =±1, The plate is taken to be composed of an isotropic, homo- x2 y2 x3 x y2 geneous material and is assumed to be thin; that is, the (9) side length, 2a, is much larger than the uniform thickness, 2u 2u 3u 3u l. The material properties of the plate include the density +  = +(2 − ) = 0 along y =±1, y2 x2 y3 x2 y (mass per unit volume), , Young’s modulus, q, and the (10) Poisson ratio, . A Cartesian coordinate system (x, y, z) is adopted whereby the centre of the plate coincides with 2u = 0 at the corners (x, y)=(±1, ±1) . (11) the origin, the transverse defections occur in the z direc- y x tion, and the edges of the plate are located at x =±a and Lastly, we also impose initial conditions which in dimen- y =±a . For small defections from the equilibrium posi- sionless form are tion z = 0 linear elasticity theory predicts that the dis- placement of the plate, denoted by u(x, y, t), will satisfy �u u = U (x, y) and = U̇ (x, y) at t = 0. (12) the following partial diferential equation given by [17] 0 �t 0 2u Here, U refers to the initial plate displacement while U̇ is + D2∇4u = 0, (1) 0 0 t2 the initial velocity of the plate. Although the governing equation (8) is linear, what where D2 is the fexural rigidity defned by makes this problem particularly difficult are the free ql2 edge boundary conditions (9–11). As noted by Rayleigh D2 = , (2) 12 (1 − 2) [13], this is why an exact solution to this problem has not yet been found. This motivated Ritz [16] to formulate a

Vol.:(0123456789) Research Article SN Applied Sciences (2021) 3:60 | https://doi.org/10.1007/s42452-020-04062-6 technique, known as the Ritz method, that allowed one 1 0, x < to construct an approximate solution. He was able to 2n � (x)= n, − 1 < x < 1 (Square Function) , produce the first numerical solution of Chladni figures n 2n 2n ⎧ 0, x > 1 on a square plate in 1909. ⎪ 2n It is worth noting that an exact solution to eq. (8) sat- ⎨n 2 2 �n(x)=⎪ exp(−n x ) (Gaussian Function) , isfying the boundary conditions ⎩ � n u w � (x)=√ (Resonance Function) , = 0, = 0 along x =±1, (13) n � 2 2 x x (1 + n x ) sin2(nx) �n(x)= (Sinc Squared Function) , u w n�x2 = 0, = 0 along y =±1, y y (14) with similar expressions for n(y) . The actual expres- can be found. Here, sion used in our simulations will be discussed later in Section 4.2. 2u 2u w =∇2u = + , x2 y2 3 Numerical solution procedure and it is easy to show that w also satisfes equation (8). The The fnite volume method was used to solve equation (16). above boundary conditions are inspired by those used for The square domain enclosed by the lines x =±1 and y =±1 solving the wave equation and the exact solution to eq. was discretized and we have adopted the notation uk to (8) subject to the boundary conditions (13–14) and initial i,j ̇ represent the numerical approximation to u(xi, yj, tk) at the conditions (12) with U0 = 0 is given by mesh points (xi, yj, tk) defned by xi =−1 + ih , yj =−1 + jh , ∞ tk = kΔt where i, j = 0, 1, 2, ⋯ , N and k = 1, 2, ⋯ . Here, u(x, y, t)= Amn cos(m x) cos(n y) cos(mnt) h = 2∕N is the uniform grid spacing in the x, y directions and m,n=1 (15) Δt is the time step. A two-step scheme was used to solve eq. 2 2 2 where mn = (m + n ) , (16). We frst solve for w where where 2u 2u w =∇2u = + , 2 2 (17) 1 1 x y Amn = U0(x, y) cos(m x) cos(n y)dxdy . −1 −1 and then solve the forced wave-type equation for u given by The solution bears a close resemblance to that of the 2u 2w 2w wave equation; the only diference lies in the frequency + + = Q . 2 2 2 (18) mn which is the square of the frequency found for the t x y wave equation when cast in dimensional form. Further, the above solution also satisfes the corner conditions When discretizing eqs. (17 and 18) we must consider three (11). This exact solution will be used to test our numerical cases: interior points, edge points and the corners. We will scheme outlined in the following section. follow the approach implemented by Gander and Kwok Since the experiment was configured so that the plate [14] in their discretization of the associated eigenvalue is forced at its center, in order to mimic this we propose problem and begin with an interior grid point. In this case solving the equation all the grid points will lie inside the domain and the fnite volume discretization will be identical to the standard 2u fnite diferencing discretization. Using second-order cen- +∇4u = Q , (16) t2 tral diferencing this leads to the following explicit scheme [18] where Q(x, y, t)= (x) (y) cos(0t) represents a forcing term concentrated at the origin which coincides with the 1 wk = uk + uk + uk + uk − 4uk , (19) center of the plate. Here, (x) and (y) denote the Dirac i,j h2 i+1,j i−1,j i,j+1 i,j−1 i,j  delta functions and 0 is the angular forcing frequency. Boundary conditions (9–11) and initial conditions (12) still (Δt)2 uk+1 = 2uk − uk−1 + apply. Approximations to the Dirac delta function with i,j i,j i,j 2 h (20) varying degrees of usefulness include: k k k k k k wi+1,j + wi−1,j + wi,j+1 + wi,j−1 − 4wi,j + Qi,j . 

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Fig. 2 The control volume for w 2u 2u 3u 3u an edge grid point = + = + x x x2 y2 x3 x y2  3u =−(1 − ) . x y2

Thus,

w 2u 2u ds = −(1 − ) − , n x y −1,y x y −1,y Γ4 j+ 1 j− 1 ⎛ �� 2 � �� 2 �⎞ ⎜ � � ⎟ � � where ⎜ � � ⎟ An advantage of using a fnite volume method is that it ⎝ ⎠ is well suited to handle the singular forcing term. At all 2u 1 ≈ (uk − uk − uk − uk ) , interior points excluding the central grid point the inte- 2 1,j −1,j 1,j−1 −1,j−1 x y (−1,y 1 ) 2h gral of Q over a control volume vanishes, while when Q j− 2 is integrated over the central control volume we obtain 2u 1 ≈ (uk − uk − uk − uk ) . 2 1,j+1 −1,j+1 1,j −1,j cos( t ). x y (−1,y ) 2h 0 k j+ 1 We next consider an edge point shown in Fig. 2. We 2 will present the details for the edge x =−1 with the The ghost points appearing in the above (i.e. having a understanding that the other edges are dealt with in a subscript−1) can be determined by making use of the frst similar manner. Integrating equation (18) over the con- condition in (9) which when discretized gives trol volume yields k k k k k k u−1,j = 2u0,j − u1,j − (u0,j+1 + u0,j−1 − 2u0,j) . 2u dxdy + ∇2wdxdy = 0, t2 (21) Vij Vij Making use of this expression the discretized form of eq. (19) along the edge becomes where the frst term is approximated as 2 k (1 − ) k k k 2 2 w = (u + u − 2u ) . u h 0,j 2 0,j+1 0,j−1 0,j dxdy ≈ uk+1 − 2uk + uk−1 . h 2 2 i,j i,j i,j V t 2(Δt) ij  To evaluate the second term we apply the two-dimen- Lastly, we consider a corner shown in Fig. 3. Again, sional divergence theorem to obtain we will present the details for the corner located at (x, y) = (−1, −1) knowing that the remaining corners are w ∇2wdxdy = ds , handled in a similar fashion. This time the first term in n Vij Vij (21) yields

2u h2 and approximate the fuxes over each segment of Vij . For dxdy ≈ uk+1 − 2uk + uk−1 , 2 2 i,j i,j i,j example, V t 4(Δt) ij  w ds ≈(wk − wk ) while the second term after repeating the calculations out- n i+1,j i,j Γ1 lined above simplifes to w 1 while ds ≈ (wk − wk ) . n 2 0,j+1 0,j Γ2 Fig. 3 The control volume at As noted in [14], along Γ4 we must incorporate the bound- a corner ary condition as follows

y 1 w j+ 2 w ds = dy , n x Γ4 y 1 j− 2

and from (9) and (17) we have that

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1 Table 1 Material properties of aluminum ∇2wdxdy ≈ (wk + wk )+(1 − ) 2 1,0 0,1 Vij Property Symbol Value

3 2u 2u Density 2.7 g∕cm + . Young’s modulus q 69 GPa xy −1,y 1 xy x 1 ,−1 2 2 ⎛ �� � �� �⎞ Poisson ratio 0.33 ⎜ � � ⎟ � � In arriving at this result⎜ we� have made use� of the⎟ corner ⎝ ⎠ condition (11) as well as the fact that w(±1, ±1, t)=0 . The mixed derivative terms above can be approximated using formulae similar to those presented earlier. We will next show that this scheme is stable provided Δt ≤ h2∕2 by performing a stability analysis [19]. We con- sider the one-dimensional version of eq. (8) given by

2u 4u + = 0, (22) t2 x4 which in discretized form using second-order central dif- ferencing becomes

k+1 k k−1 k k k k k ui − 2ui + ui + (ui+2 − 4ui+1 + 6ui − 4ui−1 + ui−2)=0, Fig. 4 The experimental setup

k 2 4 where ui = u(xi, tk) and = (Δt) ∕(Δx) . Setting k j xi −tk −tk ui = Re[e ]=cos( xi)e (with j ≡ −1 ), then 4 Results and discussion k √ 2u 4u u + ≈ i [eΔt + e−Δt − 2(1 − 3) We next present the experimental and numerical results t2 x4 (Δt)2 obtained and discuss the comparisons. + 2(cos(2Δx)−4 cos(Δx))] .

uk Thus, in order for i to satisfy (22) the quantity in the brack- 4.1 Experimental results ets must vanish; solving for e− Δt yields The experiments were conducted using the PASCO Sci- e− Δt =  ± 2 − 1 where  = 1 − 2[1 − cos(2Δx)]2 . entific Mechanical Wave Driver (model no. SF-9324) and tk k √ −tk −Δt Δt the Chladni Plates Kit (model no. WA-9607). The wave Now, ui = cos( xi)e = cos( xi)[e ] so for the solu- k → driver was a standard electromagnetic device resem- tion ui to remain bounded as tk ∞ we require that bling a speaker with a steel rod attached to it, and the −1 ≤ ± 2 − 1 ≤ 1, center of the plate was fastened to the rod. The alu- minum square plate had a side length of 2a = 24cm √ which means that −1 ≤ ≤ 1 . In order to guarantee that and a thickness of l = 0.09cm ; the material properties this holds for all and Δx we must have that ≤ 1∕4 . Thus, of aluminum are listed in Table 1. Using these values the 2 (Δx)4 (Δx)2 flexural rigidity, D , given by eq. (2) has a numerical value (Δt)2 ≤ or Δt ≤ , 4 2 2 4 2 of approximately 1.936 m ∕s and thus a ∕D ≈ 0.010s . The experimental setup is illustrated in Fig. 4. The wave which is also the convergence criterion for the one-dimen- driver was connected to a Digital Function Generator/ sional heat equation Amplifier (model no. PI-9598) which forced the center of the plate to vibrate over frequencies ranging from u 2u = . 1 Hz up to 1 kHz. Following Chladni’s technique sand t x2 was sprinkled on the plate in order to make the nodal lines visible. As the frequency of vibration was varied the plate was excited and various modes were observed revealing a variety of standing wave patterns. The observed patterns, which are readily available on the internet, are shown in Fig. 5. As expected, the patterns

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Fig. 5 The nodal patterns observed from the experiment are symmetrical about the origin and the x, y axes. The As expected, the solution did not change with time and resonant frequencies in the interval 1–1,000 Hz were the nodal pattern was identical to that shown in Fig. 1 experimentally found to be 190, 340, 490, 800 and 955 (left). Then we solved the forced eq. (16) using the bound- Hz. Based on the experimental data the dimensionless ary conditions (13 and 14) with a forcing frequency of 2 ̇ angular forcing frequency, 0 , needed in eq.(16) can be 0 = 12 = 21 = 5 and initial conditions U0 = U0 = 0 . related to the observed dimensional resonant frequency, This time we observed the nodal pattern approach that f, through the relation shown in Fig. 1 as time advanced. Figure 6 displays the computed nodal patterns at t = 1, 5, 25 . Again, this is the 2a2 0 = f ≈ 0.065f . (23) expected outcome because we are forcing the plate at D a resonant frequency, and although all the modes are excited, the modes having the same frequency will even- tually dominate. 4.2 Numerical results We also plotted the displacement of a corner of the plate and ran a long simulation. The output illustrated in In dimensionless form the problem is completely char- Fig. 7 shows a beat pattern emerging; the left panel shows acterized by the Poisson ratio and the angular forcing a blown up portion of the right panel and displays the rapid oscillations happening within the much longer oscil- frequency 0 . In our simulations we set = 0.33 for alu- minum. The uniform grid spacing in the x, y directions lations taking place in the right panel. The occurrence of was taken to be h = 0.01 and a constant time step of the beat pattern can be explained as follows. The discre- Δt = 0.000025 was used. In order to test the numerical tized eqs. (19 and 20) actually approximates the coupled scheme we made use of the exact solution given by eq. set of partial diferential equations given by (15). First we solved (8) subject to the boundary conditions 2u (Δt)2 4u 2w 2w (13 and 14) using the initial conditions + + + t2 12 t4 x2 y2 2 2 U0(x, y)=cos(�x) cos(2�y) (Δx) 4w (Δy) 4w + + = O((Δt)4 + (Δx)4 + (Δy)4) , ̇ 4 4 + cos(2�x) cos(�y) , U0 = 0 at t = 0. 12 x 12 y (24)

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1 1 1

0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2 y y 0 y 0 0

−0.2 −0.2 −0.2

−0.4 −0.4 −0.4

−0.6 −0.6 −0.6

−0.8 −0.8 −0.8

−1 −1 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x x x

Fig. 6 The nodal patterns at t = 1 (left), t = 5 (middle) and t = 25 (right)

0.15 3

0.1 2

0.05 1

0 0 Displacement u(1,1) −0.05 Displacement u(1,1) −1

−0.1 −2

−3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 500 1000 1500 time, t time, t

Fig. 7 The time evolution of the plate displacement at the corner (x, y)=(1, 1) : 0 < t < 5 (left), 0 < t < 1500 (right). Here, h = .01 and Δt = 0.000025

2u 2u (Δx)2 4u If we assume a solution of the form w = + + 2 2 x2 y2 12 x4 u = cos( 0t) cos(mx) cos(ny) and ignore the 1 and 2 (25) terms, then (Δy)2 4u + + O((Δx)4 + (Δy)4) . 12 y4 4 2 4 2 2 2 6 2 2 4 4 20 − 0 +  (m + n ) − 2 1 (m + n )(m + n )=0. Combining (24) and (25) and setting 4 Since �1 ≫�2 we can ignore the term 20 in the above, and so h2 (Δt)2 h =Δx =Δy , 1 = , 2 = , 12 12 2 4 4 2 2 2 21 (m + n ) 0 =  (m + n ) 1 − yields (m2 + n2) 2 4 4 4 4 2 2 2 4 4 4 u u u u u ≈  (m + n )−1 (m + n ) . +  + + 2 + + 2 t2 2 t4 x4 x2 y2 y4 1 Hence, the resonant angular frequency of the discretized 6 6 6 6 (26) u u u u 2 2 + + + = O( +  ) . equation, 0 , is slightly diferent from the theoretical reso- x6 y6 x2 y4 x4 y2 1 2 2 2 2  nant angular frequency, mn =  (m + n ) , of the gov- erning equation. Because of this, we should expect a beat pattern to form. Associated with the beat pattern are short

and long periods, Tshort and Tlong , respectively, where

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2 2 emerged. The agreement between the experiments and Tshort = , Tlong = . 1 ( +  ) 1  −  the simulations is quite good. For f = 190Hz the angular 2 0 mn 2 0 mn forcing frequency, 0 , needed to generate the simulated nodal pattern shown in Fig. 9 is close to that predicted by Substituting the values m = 1, n = 2 we obtain Tshort ≈ 0.13 and Tlong ≈ 911 which are in close agreement with the peri- equation (23). Here, 0 = 13 was used in the simulation ods shown in Fig. 7. Displayed in Fig. 8 are time variations while equation (23) yields 0 = 12.4 . As the frequency of the displacement of the corner of the plate using dif- increased the diference in 0 also increased. For example, ferent grid spacings and time steps. As suggested by the for f = 955Hz equation (23) gives 0 = 62.1 while in the expressions for 0 and Tlong the long period of the beat simulation 0 = 72 was needed to reproduce the nodal pattern decreases by a factor of four when the grid spacing pattern in Fig. 13. Although a full explanation for this dis- is doubled. For example, with h = 0.02, Tlong ≈ 228 while crepancy is unknown, it is partly due to the experimental with h = 0.04, Tlong ≈ 57 . On the other hand, the short error in determining the resonance frequency f. period, Tshort , remains relatively the same as h varies. We To further validate the angular forcing frequencies, 0 , also observe that as the grid gets coarser the time varia- used in the simulations the following analysis was con- tion of the corner displacement begins to deviate slightly ducted. We frst set u(x, y, t)=f (x, y) cos( 0t) and substi- from the purely oscillatory beat profle shown in Fig. 7. tuting this into the governing eq. (8) yields

Having validated the numerical scheme we are now ready 4 4 4 f f f 2 to present comparisons between the experiments and the + 2 + =  f . (27) x4 x2 y2 y4 0 simulations. In the experiment the forcing was applied by an oscillating cylindrical rod having a fnite radius. This suggests Next, eq. (27) was discretized using second-order central- using the following polar coordinate approximation for the diference approximations. Along and near the bounda- Dirac delta function ries discretized versions of the boundary conditions (9–11) were used to relate the values of f at grid points lying out- � (r) n,0< r < 1 �(x)�(y)= n where � (r)= n . side the domain (i.e. ghost points) to those lying inside the 2�r n 0, r > 1 n domain. Doing this leads to an eigenvalue problem given 4 2 by MF = F where = h 0 with h =Δx =Δy = 2∕N . In the experiment the rod diameter was 12.7mm and the Here, F is the column vector consisting of the (N + 1)2 val- side length of the plate was 24cm. Thus, the ratio of the ues fi,j = f (xi, yj) in the following order rod diameter to the side length is 0.0529, and the cross- sectional area of the rod is negligible compared to the F =[f1,1, ⋯ , fN+1,1, f1,2, ⋯ , fN+1,2, ⋯ , plate area. Because of this we have decided to model the T f1,N+1, ⋯ , fN+1,N+1] , forcing as a point source. Contrasted in Figs. 9,10,11,12,13, are nodal patterns for frequencies f = 190, 340, 490, 800 and M is the (N + 1)2 ×(N + 1)2 block pentadiagonal and 955Hz, respectively. The simulated nodal patterns matrix having the form shown are at a later time (t = 40 ) when a steady pattern

0.8 0.2

0.6 0.15

0.4 0.1

0.2 0.05

0 0

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−0.4 −0.1

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−0.8 −0.2 0 50 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 60 70 80 90 100 time, t time, t

Fig. 8 The time evolution of the plate displacement at the corner (x, y)=(1, 1) : h = 0.02, Δt = 0.001 (left), h = 0.04, Δt = 0.004 (right)

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1

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−0.6 −0.8 190 Hz −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 x

Fig. 9 Simulated (left) and observed (right) nodal pattern for f = 190Hz

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Fig. 10 Simulated (left) and observed (right) nodal pattern for f = 340Hz

Vol:.(1234567890) SN Applied Sciences (2021) 3:60 | https://doi.org/10.1007/s42452-020-04062-6 Research Article

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−1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 490 Hz x

Fig. 11 Simulated (left) and observed (right) nodal pattern for f = 490Hz

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−1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 800 Hz x

Fig. 12 Simulated (left) and observed (right) nodal pattern for f = 800Hz

Vol.:(0123456789) Research Article SN Applied Sciences (2021) 3:60 | https://doi.org/10.1007/s42452-020-04062-6

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−1 −1 −0.8800−0.6 −0.4 Hz−0.2 0 0.2 0.4 0.6 0.8 1 955 Hz x

Fig. 13 Simulated (left) and observed (right) nodal pattern for f = 955Hz

D1 B1 I1 0 ⋯⋯⋯⋯⋯⋯⋯0 5 Summary B2 D2 B3 I2 0 ⋮ 0 ⋮ ⎡ I2 B3 D3 B3 I2 ⎤ Investiagted in this paper was the free vibrations of a square ⎢ 0 I2 B3 D3 B3 I2 0 ⋮ ⎥ plate that is forced at its center. The experiments were found ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ to be in harmony with the numerical simulations that were ⎢ ⎥ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮ conducted. A numerical solution procedure which is well M = ⎢ ⎥ , ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ suited to handle the complicated boundary conditions and ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ forcing was presented; it consisted of a two-step fnite vol- ⎢ ⋮ ⎥ ⎢ 0 I2 B3 D3 B3 I2 0 ⎥ ume scheme. The numerical scheme was checked by mak- ⎢ ⋮ 0 I2 B3 D3 B3 I2 ⎥ ing comparisons with a known exact solution to a related ⎢ ⋮ 0 I B D B ⎥ problem. For the related problem the plate displacement ⎢ 2 3 2 2 ⎥ ⎢ 0 ⋯⋯⋯⋯⋯⋯⋯0 I1 B1 D1 ⎥ at the corners displayed the formation of beats which were ⎢ ⎥ (28) shown to be the result of a slight diference between the ⎢ ⎥ forcing frequency and the natural frequency of the discre- where⎣ B1, B2, B3, D1, D2, D3, I1, I2 are (N + 1)×(N⎦ + 1) sparse matrices given in the Appendix. Thus, the eigen- tized equation. values of matrix M are related to the angular forcing frequencies. The MATLAB routine eig(M) was imple- mented to compute the eigenvalues. It was found that Compliance with ethical standards using N = 10 was sufcient in size to yield eigenvalues that are consistent with the angular forcing frequen- Conflict of Interest The author wishes to report that there is no con- fict of interest. cies used in the simulations. For example, the angu- lar forcing frequencies used in the simulations were Open Access This article is licensed under a Creative Commons Attri- 0 = 13(190Hz), 26(340Hz), 40(490Hz), 61(800Hz), 72(955Hz) . bution 4.0 International License, which permits use, sharing, adap- With N = 10 the following angular forcing frequencies tation, distribution and reproduction in any medium or format, as were computed: 14.0, 26.5, 39.7, 62.6, 72.2. Of course, long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate many more frequencies were obtained. Here, we list only if changes were made. The images or other third party material in this those that are close to the values used in the simulations article are included in the article’s Creative Commons licence, unless and the agreement is good. As N was increased the values indicated otherwise in a credit line to the material. If material is not of these frequencies changed but not by much, and more included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted physically unrelevant frequencies emerged. use, you will need to obtain permission directly from the copyright

Vol:.(1234567890) SN Applied Sciences (2021) 3:60 | https://doi.org/10.1007/s42452-020-04062-6 Research Article holder. To view a copy of this licence, visit http://creat​iveco​mmons​ −b b b 0 ⋯⋯⋯⋯⋯⋯⋯ 0 .org/licenses/by/4.0/​ . 1 1 6 b1 b b b6 0 ⋮ 2 7 5 2 b6 b6 ⎡ b5 b8 b5 0 ⋮ ⎤ ⎢ 2 2 ⎥ Appendix 0 b6 b b b b6 0 ⋮ ⎢ 2 5 8 5 2 ⎥ ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮ ⎥ The (N + 1)×(N + 1) sparse matrices ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮ ⎥ ⎢ ⎥ D1 = B1, B2, B3, D1, D2, D3, I1, I2 in eq. (28) are given by ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮ ⎥ ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮ ⎥ ⋯⋯⋯⋯⋯ ⎢ ⎥ b1 b2 0 0 b6 b6 ⎢ ⋮ 0 b5 b8 b5 0 ⎥ b3 b4 b3 0 ⋮ 2 2 ⎢ b6 b6 ⎥ ⋮ ⋮ 0 b b b ⎡ 0 b3 b4 b3 0 ⎤ ⎢ 2 5 8 5 2 ⎥ ⋮ ⋱⋱⋱⋱⋱ ⋮ ⎢ ⋮ 0 b6 b b b1 ⎥ ⎢ ⎥ 2 5 7 2 ⎢ ⋮ ⋱⋱⋱⋱⋱ ⋮⎥ , ⎢ ⋯⋯⋯⋯⋯⋯⋯ ⎥ B1 = ⎢ 0 0 b6 b1 − b1 ⎥ ⎢ ⋮ ⋱⋱⋱⋱⋱⋮⎥ ⎢ ⎥ ⎢ ⎥ ⋮ ⎢ ⎥ ⎢ 0 b3 b4 b3 0 ⎥ ⎣ ⎦ ⎢ ⎥ b7 b4 20⋯⋯⋯⋯⋯⋯⋯0 ⋮ 0 b3 b4 b3 ⎢ ⎥ b4 18 − 81 0 ⋮ ⎢ 0 ⋯⋯⋯⋯⋯0 b2 b1 ⎥ 2 ⎢ ⎥ ⎡ 1 − 8 19 − 81 0 ⋮ ⎤ ⎢ ⎥ ⎢ 01− 8 19 − 81 0 ⋮ ⎥ ⎣ b ⎦ ⎢ ⎥ 1 b 0 ⋯⋯⋯⋯⋯0 ⋮⋱⋱⋱⋱⋱⋱⋱ ⋮ 2 3 ⎢ ⎥ b b b ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ 2 4 3 0 ⋮ D = 4 2 2 2 ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ ⎡ 0 b3 b4 b3 0 ⋮ ⎤ ⎢ ⎥ ⎢ 2 2 2 ⎥ ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱⋮⎥ ⎢ ⋮ ⋱⋱⋱⋱⋱ ⋮⎥ ⎢ ⋮ 01− 8 19 − 81 0⎥ B = ⎢ ⋮ ⋱⋱⋱⋱⋱ ⋮⎥ , ⎢ ⋮ 01− 8 19 − 81⎥ 2 ⎢ ⎥ ⎢ ⎥ ⋮ ⋱⋱⋱⋱⋱⋮ b4 ⎢ ⎥ ⎢ ⋮ 01− 8 18 ⎥ b b b 2 ⎢ ⋮ 0 3 4 3 0 ⎥ ⎢ 0 ⋯ ⋯ ⋯ ⋯⋯⋯⋯ 02b b ⎥ ⎢ 2 2 2 ⎥ ⎢ 4 7 ⎥ ⋮ 0 b3 b4 b2 ⎢ ⎥ ⎢ 2 2 4 ⎥ ⎢ ⎥ ⎢ 0 ⋯⋯⋯⋯⋯0 b b1 ⎥ ⎣ ⎦ ⎢ 3 2 ⎥ b8 b4 20⋯⋯⋯⋯⋯⋯⋯0 ⎢ ⎥ b4 19 − 81 0 ⋮ ⎢ ⎥ 2 ⎡ 1 − 8 20 − 81 0 ⋮ ⎤ ⎣ b b 0 ⋯⋯⋯ ⋯ ⋯⎦ 0 5 3 ⎢ 01− 8 20 − 81 0 ⋮ ⎥ b3 − 82 0 ⋮ ⎢ ⎥ 2 ⎢ ⋮⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ ⎡ 02− 82 0 ⋮ ⎤ ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ D = ⎢ ⋮ ⋱ ⋱ ⋱⋱⋱ ⋮⎥ 3 ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱ ⋮⎥ ⎢ ⎥ ⎢ ⎥ B3 = ⎢ ⋮ ⋱ ⋱⋱⋱ ⋱ ⋮⎥ ⎢ ⋮ ⋱⋱⋱⋱⋱⋱⋱⋮⎥ ⎢ ⋮ ⋱⋱⋱ ⋱ ⋱ ⋮⎥ ⎢ ⋮ 01− 8 20 − 81 0⎥ ⎢ ⎥ ⎢ ⋮ ⎥ ⋮ 02− 82 0 ⎢ 01− 8 20 − 81⎥ ⎢ ⎥ b ⎢ ⋮ 02− 8 b3 ⎥ ⎢ ⋮ 01− 8 19 4 ⎥ 2 2 ⎢ ⎥ ⎢ 0 ⋯ ⋯ ⋯ ⋯⋯⋯⋯ 02b b ⎥ 0 ⋯ ⋯ ⋯⋯⋯ 0 b3 b5 ⎢ 4 8 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

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8. Stockmann H-J (2007) Chladni meets Napoleon. Eur Phys J Spec b6 0 ⋯⋯⋯⋯0 02 0 ⋮ Top 145:15–23 9. Bucciarelli LL, Dworsky N (1980) Sophie Germain: an essay in the ⋮ 020 ⋮ ⎡ ⎤ 1 history of the theory of elasticity. D. Reidel Publishing Company, I = ⋮ ⋱⋱⋱ ⋮ , I = I Boston 1 ⎢ ⎥ 2 2 1 ⎢ ⋮ 020⋮ ⎥ 10. Hill AM (1995) “Sophie germain: A mathematical biography”, ⎢ ⋮ 0 20⎥ Master’s thesis, University of Oregon, Eugene, Oregon ⎢ ⎥ é é ⋯⋯⋯⋯ 11. Germain S (1821) Recherches sur la th orie des surfaces last- ⎢ 0 0 b6 ⎥ iques. Mme. Veuve Courcier, Paris ⎢ 4 3 1 , ⎥8 1 , 2 2 , 12. Germain S (1826) Remarques sur la nature, les bornes et l’éten- ⎢ b1 =− ( + )( − ) b2 = ⎥( − ) b3 = ( − ) ⎣ ⎦ due de la question des surfaces élastiques, et l’équation g énérale whereb4 =−4(3 − ), b5 =−4(2 + )(1 − ) , de ces surfaces. Mme. Veuve Courcier, Paris 2 2 2 13. Strutt JW (1945) The theory of sound, vol 1, 2nd edn. Dover, New b6 = 2(1 − ), b7 = 15 − 8 − 5 , b8 = 2(8 − 4 − 3 ) . York (Baron Rayleigh) 14. Gander MJ, Kwok F (2012) Chladni figures and the Tacoma References bridge: motivating PDE eigenvalue problems via vibrating plates. SIAM Rev 54:573–596 1. Leissa AW (1969) Vibration of plates (NASA SP-160). Government 15. Gander MJ, Wanner G (2012) From Euler, Ritz and Galerkin to Printing Ofce, Washington, DC modern computing. SIAM Rev 54:627–666 2. Leissa AW (1973) The free vibration of rectangular plates. J 16. Ritz W (1909) Theorie der Transversalschwingungen einer quad- Sound Vib 31:257–293 ratischen Platte mit freien R ändern. Ann Physik 18:737–807 3. Ullmann D (2007) Life and work of E. F. F. Chladni. Eur Phys J Spec 17. Szilard R (2004) Theories and applications of plate analysis: clas- Top 145:25–32 sical, numerical and engineering methods. Wiley, New York 4. Waller MD (1939) Vibrations of free square plates: Part I. Normal 18. Le Veque RJ (2002) Finite volume methods for hyperbolic prob- vibrating modes. London School of Medicine for Women, Lon- lems. Cambridge University Press, Cambridge don, pp 831–844 19. Keller HB, Isaacson E (1966) Analysis of numerical methods. 5. van Gerner HJ, van der Hoef MA, van der Meer D, van der Weele Wiley, New York K (2010) Inversion of Chladni patterns by tuning the vibrational acceleration. Phys Rev E 82:012301 Publisher’s Note Springer Nature remains neutral with regard to 6. Rossing TD (1982) Chladni’s law for vibrating plates. Am J Phys jurisdictional claims in published maps and institutional afliations. 50:271–274 7. Kirchhof GR (1850) Uber das Gleichgewicht und die Bewegung einer elastischen Schiebe. J Math 40:51–88

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