Coupled Torsional and Transverse Vibration Analysis of Panels Partially Supported by Elastic Beam
CC OUPLED T ORSIONAL AND T RANSVERSE V IBRATION A NALYSIS OF P ANELS P ARTIALLY S UPPORTED BY E LASTIC B EAM
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Mostafa Bagheri ∗ Department of Mechanical and Aerospace EngineeringUniversity of California San DiegoLa Jolla, CA 15213 [email protected]
Mohammad Mohammadi Aghdam
Department of Mechanical EngineeringAmirkabir University of Technology (Tehran polytechnic)Tehran 1591634311 [email protected]
Meitham Amereh
Department of Mechanical EngineeringUniversity of VictoriaVictoria, BC, V8P 5C2 [email protected]
February 11, 2021 A BSTRACT
This study presents torsional and transverse vibration analysis of a solar panel including a rectangularthin plate locally supported by an elastic beam. The plate is totally free in all boundaries, except forthe local part attached to the beam. The response of the system, which is subjected to a combination oftorsional and transverse vibration, identifies with a couple of PDEs developed by the Euler-Bernoulliassumption and classical plate theory. To calculate the system’s natural frequencies, the domain ofthe solution is discretized by zeroes of the Chebyshev polynomials to apply the Modified GeneralizedDifferential Quadrature method (MGDQ). Furthermore, governing equations along with continuityand boundary conditions are discretized. After obtaining solutions to the eigenvalue problem, severalstudies are investigated to validate the accuracy of the proposed method. As can be concluded fromthe tables, MGDQ improves the accuracy of results obtained by GDQ. Results for various case studiesreveal that MGDQ is properly devised for the vibration analysis of systems with local boundary andcontinuity conditions. K eywords Vibration; Locally Supported Panels · MGDQ · Local Boundary Conditions · Natural Frequencies
Vibration analysis of plates and panels is an interesting practical subject in both the engineering area and industrialapplications. Considerable researches have been performed on the buckling and vibration of plates, however, limitedinvestigations focused more on complicated case studies within the literature. Local suspended plates are amongexamples of practical subjects in different industries which that need more investigation.For instance, a suspended free plate on elastic beams can be seen in aerospace and nautical industries. In the area ofmicro-electromechanical systems (MEMSs), conventional sensors and actuators are among interesting application,where a proof mass attached to one or two suspended beams (e.g. cantilever accelerometers), were studied in [1, 2, 3, 4].These works only considered free transverse vibration analysis of the system while the torsional part is completely ∗ [email protected], http://flyingv.ucsd.edu/mostafa a r X i v : . [ m a t h . NA ] F e b PREPRINT - F
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11, 2021ignored, which plays important role in satellite’s solar panels. Solar panels on spacecraft are important samples inwhich a thin plate is connected to the body of the spacecraft by a thin elastic beam, as shown in Fig. 1.Figure 1: Inmarsat-5 F2 satelliteNotably, that solar panels may experience heavy vibrations due to the environmental effects [5]. Hence, different modeshapes of the vibrating system should be taken into account.Various techniques are accomplished to find the solution for vibration analysis of shells and plates under specificforms of loading and boundary conditions, series-type method [6] and integral equations method [7, 8], to name a few.Leissa also presented an inclusive and accurate analytical approach for the solution of the free vibration in rectangularplates [9, 10, 7]. Moreover, finite element (FE) and finite strip method (FSM) [11, 12, 13], Rayleigh-Ritz method[14], Galerkin method [15], domain decomposition method [16, 17], energy balance method (EBM) [18], differentialquadrature (DQ) [19], and generalized differential quadrature (GDQ) methods [20, 21] are among different techniquesproposed within literature.The DQ and GDQ methods have been developed by Bellman and Casti [22] and Shu and Richards [23]. The efficiencyin computing the weighting coefficients is the principal advantage of GDQ due to its simplicity in the choice of gridpoints. For the vibration analysis of plates and shells, various studies concerning the validity of DQ/GDQ methodsexist in the scientific literature. Most of these studies have investigated typical cases including a combination ofstandard boundary conditions and uniformly distributed loads. For rectangular plates with different shapes of holes, anindependent coordinate coupling method is employed for vibration analysis [24]. Additionally, local effects such as thevibration of rectangular plates with concentrated mass–springs [25], distinct elastic edge restraints [26], and thermalnonlinear vibration of the orthotropic DLGS [27] are investigated within the literature.This study investigates combined torsional and transverse vibration analysis of a solar panel including a completelyfree rectangular plate locally supported by an elastic beam. By applying the GDQ method, the domain of the solutiontogether with governing PDEs for the beam and plate are discretized. Based on governing equations along withboundary and continuity conditions, the final eigenvalue problem is developed. The solution to the final equationsleads to eigenvalues and eigenvectors which are respectively natural frequencies and mode shapes of the system. Toassess the reliability of the proposed method, results for various simplified case studies, in Section 5, are evaluated withavailable analytical and numerical solutions within the literature. For the complicated cases, results are also validatedby commercial FE code, due to lack of information in the literature. Finally, frequency predictions and mode shapes ofthe general case show an acceptable correlation with finite element results. It is expecting that the results of this papercan be utilized as a benchmark in future studies dealing with the combined transverse and torsional vibration analysisof suspended plates and panels.
The system comprises a rectangular isotropic thin elastic plate with dimensions of a × b and thickness of h . The plate islocally attached to a clamped beam with length l and the cross-section of t × d , as shown in Fig. 2. Considering thepossibility of resonance in the vibration of the plates, one should pay attention to the material properties and geometry2 PREPRINT - F
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11, 2021of the system. The plate and the beam are elastic with E , E as modulus of elasticity, and ν , ν as Poisson’s ratio,respectively. Figure 2: Schematic modelAlso, G and I are shear modulus and polar moment of inertia of the beam, respectively. Due to external excitation, thesystem may start to vibrate in the form of transverse and/or torsional vibration. Assuming W ( x, y, t ) as the deflectionof the plate, and U ( x (cid:48) , t ) and θ ( x (cid:48) , t ) as deflection and rotation of the beam, the governing equations of the system canbe obtained as [7]: E I ∂ U ( x (cid:48) , t ) ∂x (cid:48) + ρ A ∂ U ( x (cid:48) , t ) ∂t = 0 (1a) G J ∂ θ ( x (cid:48) , t ) ∂x (cid:48) = ρ I ∂ θ ( x (cid:48) , t ) ∂t (1b) D ∂ W ( x, y, t ) ∂x + 2 D ∂ W ( x, y, t ) ∂x ∂y + D ∂ W ( x, y, t ) ∂y + ρ h ∂ W ( x, y, t ) ∂t = 0 (1c)in which x (cid:48) = x + l and A = t × d is the cross-sectional area of the beam. ρ and ρ are respectively mass densitiesof the beam and the plate. Also, D is the flexural rigidity of the plate, which can be determined as D = E h − υ ) (2)Assuming u ( x (cid:48) , t ) = U ( x (cid:48) ) e iω zb t , θ ( x (cid:48) , t ) = Θ( x (cid:48) ) e iω φb t , and w ( x, y ,t ) = W ( x, y ) e iω p t , one may rewrite Eq. 1 asfollows ∂ U∂X (cid:48) = ω zb U (3a) ∂ Θ ∂X + ω φb Θ = 0 (3b) ∂ W∂X + 2 β ∂ W∂X ∂Y + β ∂ W∂Y = ω p W (3c)where ω zb and ω φb are natural frequencies for transverse and torsional vibration of the beam, respectively, and ω b isnatural frequency for vibration of plate. 3 PREPRINT - F
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11, 2021Also, other parameters are defined as ω z b = ρ AL E I ω z b ω φ b = ρIL GJ ω φ b ω p = ρ ha D ω p X (cid:48) = x (cid:48) Lβ = abX = xa , Y = yb Note that the length of the plate is defined as a , while b is the plate width. Generally, for a thin plate, shear forces andmoments are V x = − D (cid:16) ∂ W∂x + (2 − v ) ∂ W∂x∂y (cid:17) V y = − D (cid:16) ∂ W∂y + (2 − v ) ∂ W∂x ∂y (cid:17) (4a) (cid:34) M x M y M z (cid:35) = − D (cid:34) v v − v (cid:35) ∂ W∂x ∂ W∂y ∂ W∂x∂y (4b)Implementation of boundary conditions, which refers to the free edges of the plate and clamped side of the beam, aswell as continuity of displacements, shear forces, and bending moments at the attached points should be considered.Therefore, boundary equations for the beam section can be written as u (0 , t ) = 0 (5a) ∂u (0 , t ) ∂x (cid:48) = 0 (5b) θ (0 , t ) = 0 (5c)Also, boundary conditions at free edges of the plate are ∂ w ( a, y, t ) ∂x + ν β ∂ w ( a, y, t ) ∂y = 0 (6a) ∂ w (0 , y, t ) ∂x + ν β ∂ w (0 , y, t ) ∂y = 0 for | y | ≥ d (6b) β ∂ w ( x, , t ) ∂y + ν ∂ w ( x, , t ) ∂x = 0 (6c) β ∂ w ( x, b, t ) ∂y + ν ∂ w ( x, b, t ) ∂x = 0 (6d)4 PREPRINT - F
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11, 2021and ∂ w ( a, y, t ) ∂x + (2 − ν ) β ∂ w ( a, y, t ) ∂x∂y = 0 (7a) ∂ w (0 , y, t ) ∂x + (2 − ν ) β ∂ w (0 , y, t ) ∂x∂y = 0 for | y | ≥ d (7b) β ∂ w ( x, , t ) ∂y + (2 − ν ) ∂ w ( x, , t ) ∂x ∂y = 0 (7c) β ∂ w ( x, b, t ) ∂y + (2 − ν ) ∂ w ( x, b, t ) ∂x ∂y = 0 (7d)Continuity conditions at the attached points can be written as u ( l, t ) = w (0 , y, t ) (8a) ∂u ( l, t ) ∂x (cid:48) = ∂w (0 , y, t ) ∂x (8b) E I ∂ u ( l, t ) ∂x (cid:48) = D (cid:18) ∂ w (0 , y, t ) ∂x + ν ∂ w (0 , y, t ) ∂y (cid:19) (8c) E I ∂ u ( l, t ) ∂x (cid:48) = D (cid:18) ∂ w (0 , y, t ) ∂x + (2 − ν ) ∂ w (0 , y, t ) ∂x∂y (cid:19) (8d)in which the domain of y in Eqs. 8 is | y | ≥ d . To commence the procedure of GDQ method, the domain of the solution should be first discretized into several gridpoints. Zeros of the Chebyshev polynomials is one of the best choices to discretize the domain, see [28]. As a result,the plate is discretized into N × M grid points in x and y directions, respectively, while the beam may be divided intothree or five rows of S grid points through x direction, as shown in Fig. 3. Afterward, governing partial differentialequations are expanded over these grid points. By assuming three or five rows of grid points in the width of the beam,continuity conditions can be satisfied.Figure 3: Discretization of the system into grid pointsAccording to the GDQ method, a general form of the approximation for the calculation of higher-order derivatives is u mx ( x i , t ) = b (cid:88) j =1 τ ( m ) ij u ( x j , t ) for i = 1 , · · · , S (9)5 PREPRINT - F
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11, 2021where u mx ( x i , t ) is the m th derivative of beam deflection at point ( x i , b ) is the number of grid points and τ ( m ) ij is theweighting coefficients for the m th derivative calculation. The grid points on the beam and plate based on zeros of theChebyshev polynomials can be determined as: x (cid:48) k = L (cid:20) − cos (cid:18) k − S − π (cid:19)(cid:21) (10a) x i = a (cid:20) − cos (cid:18) i − N − π (cid:19) (cid:21) (10b) y j = b (cid:20) − cos (cid:18) j − M − π (cid:19) (cid:21) (10c)In discretization, Eq. (11) gives the weighting coefficients for higher order derivatives ζ ( m ) rk = m (cid:32) η ( m − rr η (1) rk − η ( m − rk x (cid:48) r − x (cid:48) k (cid:33) for r (cid:54) = k r, k = 1 , · · · , S (11a) c ( m ) pi = m (cid:32) c ( m − pp c (1) pi − c ( m − pi x p − x i (cid:33) for r (cid:54) = k r, k = 1 , · · · , S (11b) c ( m ) lj = m (cid:32) c ( m − ll c (1) lj − c ( m − lj y l − y j (cid:33) for r (cid:54) = k r, k = 1 , · · · , S (11c)The ζ ( m ) rk , c ( m ) pi , and c ( m ) lj can be respectively derived based on the ( m − th order coefficients ζ ( m − rk , c ( m − pi , and c ( m − lj . The ζ ( m ) rk , c ( m ) pi , and c ( m ) lj can be obtained from: ζ ( m ) rr = S (cid:88) k = 1 k (cid:54) = r ζ ( m ) rk for r = 1 , · · · , S (12a) c ( m ) pp = N (cid:88) l = 1 l (cid:54) = p c ( m ) pl for p = 1 , · · · , N (12b) c ( m ) ll = M (cid:88) j = 1 j (cid:54) = l c ( m ) ij for l = 1 , · · · , M (12c)Based on the recursive formulas 11 and 12, η (1) rk , c (1) pi and c (1) lj can be calculated as follow: η (1) rk = P( x (cid:48) r )( x (cid:48) r − x (cid:48) k )P( x (cid:48) k ) for r (cid:54) = k (13a) c (1) pi = Q ( x p )( x p − x i ) Q ( x i ) for p (cid:54) = i (13b) c (1) lj = H ( y l )( y l − y j ) H ( y j ) for l (cid:54) = j (13c)6 PREPRINT - F
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11, 2021where P ( x (cid:48) r ) = S (cid:89) k = 1 k (cid:54) = r ( x (cid:48) r − x (cid:48) k ) , Q ( x p ) = N (cid:89) i = 1 i (cid:54) = p ( x p − x i ) , H ( x l ) = M (cid:89) j = 1 j (cid:54) = l ( x l − x j ) Also, one may obtained η (1) kk , c (1) ii and c (1) jj using: η (1) kk = − S (cid:88) k = 1 k (cid:54) = r η (1) rk for r = 1 , , · · · , N (14a) c (1) ii = − N (cid:88) i = 1 i (cid:54) = p c (1) pi for p = 1 , , · · · , N (14b) c (1) jj = − M (cid:88) j = 1 j (cid:54) = l c (1) lj for l = 1 , , · · · , N (14c)According to the definition of the derivative terms and applying the GDQ approximation, Eq. (3) can be rewritten as S (cid:88) k =1 ζ (4) rk U k = ω U r (15a) S (cid:88) k =1 ζ (2) rk Θ k = − ω φb Θ r for r = 3 , · · · , S − (15b) N (cid:88) k =1 c (4) ik W kj + 2 β M (cid:88) m =1 c (2) jm N (cid:88) k =1 c (2) ik W km + β M (cid:88) m =1 c (4) jm W im = ω p W ij (15c)for i = 3 , · · · , N − and j = 3 , · · · , M − . It is worth mentioning that the aforementioned equations should beconsidered for all grid points except boundary and adjacent nodes, see [29]. Boundary conditions for the beam sectionat the fixed end, which have been written in Eq. (5), can be respectively discretized as U = 0 , (cid:80) Sk =1 ζ (1)1 k U k = 0 , and Θ = 0 .In addition to that, the discretized forms of the boundary conditions of zero normal moment at free edges in Eqs. (6)and (7) are N (cid:88) k =1 c (2) Nk W kj + ν β M (cid:88) m =1 c (2) jm W Nm = 0 (16a) N (cid:88) k =1 c (2)1 k W kj + ν β M (cid:88) m =1 c (2) jm W m = 0 for j = 3 , , · · · , M − , M − , and β M (cid:88) m =1 c (2)1 m W im + ν N (cid:88) k =1 c (2) ik W k = 0 (16b) β M (cid:88) m =1 c (2) Mm W im + ν N (cid:88) k =1 c (2) ik W kM = 0 for i = 1 , , · · · , N PREPRINT - F
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11, 2021Also, we have N (cid:88) k =1 c (3) Nk W kj + (2 − ν ) β N (cid:88) k =1 c (1) Nk M (cid:88) m =1 c (2) jl W km = 0 (17a) N (cid:88) k =1 c (3)1 k W kj + (2 − ν ) β N (cid:88) k =1 c (1)1 k M (cid:88) m =1 c (2) jl W km = 0 for j = 3 , , · · · , M − , M − , and β M (cid:88) m =1 c (3)1 m W im + (2 − ν ) N (cid:88) k =1 c (2)1 m M (cid:88) m =1 c (1) ik W km = 0 (17b) β M (cid:88) m =1 c (3) Mm W im + (2 − ν ) N (cid:88) k =1 c (2) Mm M (cid:88) m =1 c (1) ik W km = 0 for i = 1 , , · · · , N Moreover, the continuity condition for the local part (Eq. (8)) can be written as follows U S = W j (18a) S (cid:88) r =1 ζ (1) Sr U i = N (cid:88) k =1 c (1)1 k W kj (18b) E I S (cid:88) r =1 ζ (2) Sr U r = − D (cid:32) N (cid:88) k =1 c (2)1 k W kj + ν M (cid:88) m =1 c (2) jm W m (cid:33) (18c) E I S (cid:88) r =1 ζ (3) Sr U r = − D (cid:32) N (cid:88) k =1 c (2)1 k W kj + (2 − ν ) N (cid:88) k =1 c (1)1 k M (cid:88) m =1 c (2) jm W km (cid:33) (18d)in which the domain of k in Eqs. 18 is ( b − d )( N − b ) + 1 ≤ j ≤ ( b + d )( N − b ) + 1 . Considering governing Eqs. (15), boundary conditions (16) and (17) and continuity conditions (18), one can write thesesystems of equations in form of two sets of algebraic equations as following: [ A IB ] { W B } + [ A II ] { W I } = Ω { W I } (19a) [ A BB ] { W B } + [ A BI ] { W I } = 0 (19b)where { W B } and { W I } are nodes deflection inside the domain and on boundaries, respectively. Furthermore,dimensions of [ A II ] P × P , [ A BB ] Q × Q , [ A IB ] P × Q and [ A BI ] Q × P are defined such that P = ( S − M − × ( N − and Q = (4 M + 4 N − .Substitution of Eq. (19a) in (19b) gives the final system of eigenvalue equation as: (cid:110) [ A II ] − [ A IB ] [ A BB ] − [ A BI ] (cid:111) { W I } = ω { W I } (20)The standard form of eigenvalue equation can be obtained by restating Eq. (20) as (cid:104)(cid:110) [ A II ] − [ A IB ] [ A BB ] − [ A BI ] (cid:111) − I (cid:105) { W I } = ω { W I t } (21)where { I } is the identity matrix. After implementing the values in stiffness matrix in Eq. (21), by solving this standardeigenvalue equation, natural frequencies of the system together with related mode shapes could be determined. Free torsional and transverse vibration analysis of a system comprises a free plate locally supported by an elastic beamis presented. At first, three types of equations, i.e. governing equations, equations of boundary conditions and equationsof continuity conditions are developed. To solve these equations, GDQ method is applied to the domain of the solution.8
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11, 2021Accordingly, both domains of the solution and governing equations, together with boundary/continuity equationsare discretized. Thereafter, all equations are rewritten in the form of two sets of equations giving the final systemof eigenvalue equation. The system is eventually simplified into several widely-known cases of vibration analysis,e.g. completely free (FFFF) plate, cantilever (CFFF) plate, clamped-free beam and beam with concentrated mass, todetermine the accuracy of the proposed solution, the system is simplified into several. Also, the general model has beeninvestigated in the final case in which the accuracy of the results is evaluated. Table 1 shows the properties of the plateand beam. Table 1: Material and geometry properties
Properties Values a mb mL md . mt . cmν , ν cmρ , ρ kg/m E , E GP a
Case study 1
As the first case study, material properties and geometry parameters for the plate are assumed to be thesame as those of the beam which simplifies the model to a simple cantilever beam. Hence, identical width, thicknessand material properties should be considered, i.e. t = h, d = 2 b, E = E , ρ = ρ . Moreover, the Poisson ratio of theplate section should be neglected to eliminate the deformation of the plate in the y direction. Taking into account theaforementioned properties, the model is simplified to a homogeneous clamped–free beam with length L ∗ = L + a . Dueto the attachment of the plate to the end of the beam, vibration of the plate may cause torsional vibration over the beam.Therefore, 2D discretization is applied through the domain of the beam. It is worth mentioning that analytical solutionscan be found for natural frequencies of both transverse and torsional vibrations of clamped-free beams using modalanalysis as [6]: cos βL ∗ × cosh βL ∗ + 1 = 0 (22a) cos ω θj L ∗ c = 0 (22b)in which β = ρA ( ω zi ) EI , c = (cid:115) Gρ , ω θj = α j cl Note that ω zi and ω θj are respectively the i th natural frequency of transverse vibration and j th natural frequency oftorsional vibration of the beam.Tables 2 and 3 respectively demonstrate the first eight natural frequencies of the system for both transverse and torsionalvibration, computed by both GDQ and frequency Eqs. (22).Another simple case can be considered by assuming a plate with large density and small sizes which reduces the modelto a simple cantilever beam with concentrated end mass. To achieve this, the plate is assumed to be made of Lead withdensity equals to ρ = 11 . gr/cm and geometry properties as a = b = h = 0 . m . Again, the frequency equationTable 2: First eight natural frequencies of transverse vibration of clamped-free beam Method/Grid Size Ω Ω Ω Ω Ω Ω Ω Ω Exact Method (Eq. 22.a) × × × Error (%)
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11, 2021for transverse and torsional vibration of this simple case can be found analytically as [6]: βl ) cosh( βl ) − R z βl (tan( βl ) − tanh( βl )) = 0 (24a) β = ρA ( ω zi ) EI (24b) α j tan( α j ) = R θ (24c) ω θj = α j cl (24d)Table 3: First eight natural frequencies of torsional vibration of clamped-free beam Method/Grid Size Ω Ω Ω Ω Ω Ω Ω Ω Exact Method (Eq. 22.b) × × × Error (%) R z = MρAl is the ratio of the attached mass M to the mass of the beam ρAl . Also, R θ is equal to ρjlI d in which I d is the moment of inertia of the plate and ρ, j and l are properties of the beam. Based on the proposed geometry andmaterial properties, one may obtain R z = 9 . and R θ = 0 . . Similarly, Tables 4 and 5 contain first eight naturalfrequencies of the beam with concentrated mass, obtained by GDQ and Eqs. 24. Results show reasonably accuratepredictions for both cases. Case study 2
As another simple case for evaluation of the model and solution procedure, one may ignore the effects ofthe beam by assuming zero geometry and material parameters which reduces the system to a completely free plate. Asa result, the beam exerts no force and moment on the plate. Table 6 includes the results of first five natural frequenciesobtained by GDQ method. Also, table 6 shows the results of direct GDQ method obtained by Shu and Du [29], analyticalresults presented by Leissa [9, 10] and also results obtained using commercial FE code ANSYS. Predictions show goodcorrelations with other methods. It is worth mentioning that by reselecting grid points by modified formulation providedin [29], the accuracy of the results is significantly improved.
Case study 3
As the final test case to evaluate the performance of the presented method, the beam is considered to berigid with identical widths for both the beam and plate. This will reduce the system to a cantilever plate. To this end,the Young and shear modulus of elasticity of the beam should tend to infinity. Table 7 includes the GDQ predictions forthe first five natural frequencies of the plate. Also, the result of direct GDQ and the analytical method by Leissa [9, 10]are presented in Table 7. To prove the validity of the presented method, results of FE code (ANSYS) are also includedin the table. To develop the accuracy of the method, the same procedure as the previous case study may be applied. Thenew results are presented in Table 7 as well.
Case study 4:
The final case is the complicated system consists of a regionally suspended plate connected to the elasticbeam. The standard form of eigenvalue equations for the combined transverse/torsional vibration of the system issolved and consequently eigenvalues and eigenvectors are derived. To validate the calculation procedure, finite elementcode ANSYS was also used to analyze the system. Figures 4 to 8 depicts five different mode shapes of the systemtogether with associated frequencies, which are predicted by the presented GDQ technique. As shown in these figures,Table 4: First eight natural frequencies of clamped-free beam with concentrated mass R z = 9 . ; transverse vibration Method / Grid Size Transverse Frequencies: R z = 9 . Ω Ω Ω Ω Ω Ω Ω Exact Method (Eq. 23.a) × × × Error (%)
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11, 2021Table 5: First eight natural frequencies of clamped-free beam with concentrated mass R θ = 0 . ; torsional vibration Method / Grid Size Torsional Frequencies; R θ = 0 . Ω Ω Ω Ω Ω Ω Ω Exact Method (Eq. 23.b) × × × Error (%)
Method / Grid Size Ω Ω Ω Ω Ω Ω Leissa [9]
Leissa & Narita [10]
Shu and Du (15*15) [29]
Shu and Du(12*12) [29]
FEM
GDQ × × × Error (%)
Modified GDQ × × × Error (%)
Method / Grid Size Ω Ω Ω Ω Ω Ω Leissa [4]
Shu and Du (15*15) [29]
Shu and Du (12*12) [29]
FEM
GDQ × × × Error (%)
Modified GDQ × × × Error (%)
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11, 2021the system can vibrate based on various mode shapes of the beam and the plate and also their combination. For instance,Figs. 4 and 5 depict mode shapes of the system once the first modes of transverse and torsional vibrations of the beamare excited, respectively. Besides, Figs. 6 to 8 show mode shapes of the system as the modes of beam and plate aresimultaneously excited. Included in these figures are also results of FE analysis for both frequencies and mode shapesof the system, which indicates reasonable accuracy for the proposed approach. The range of errors in predictions isabout . to . , in which minimum and maximum errors belong to predicted frequencies of . Hz and . Hz, respectively. Frequency: 0.593 HzFigure 4: First mode shape of transverse vibration of the beamFrequency: 2.675 HzFigure 5: First mode shape of torsional vibration of the beam
This study is dedicated to investigate combined torsional and transverse free vibration analysis of thin plates locallysupported by an elastic beam. In particular, the performance of the GDQ method to offer solutions for practical problemswith local effects is examined. Using Euler-Bernoulli assumption and classical plate theory, two coupled governingPDEs together with continuity/boundary conditions are developed based on the GDQ method. All equations are thenrestated as two sets of algebraic equations. The solution to the final equations leads to eigenvalues and eigenvectorsthat respectively indicate the system’s natural frequencies and mode shapes. To assess the reliability of the proposedmethod, results for different well-known cases are compared by available analytical and numerical methods. This12
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11, 2021Frequency: 32.037 HzFigure 6: First mode shape of torsional vibration of the beam together with first mode shape of the plateFrequency: 118.276 HzFigure 7: First mode shape of transverse vibration of the beam together with second mode shape of the plateFrequency: 147.494 HzFigure 8: First mode shape of torsional vibration of the beam together with forth mode shape of the plate13
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11, 2021includes examples of several well-known cases of classical vibration such as cantilever beam, beam with concentratedmass, FFFF and CFFF plate. Results revealed that predictions in these case studies are in close correlation withanalytical/numerical results while the accuracy can also be remarkably increased by reselecting the grid points.Finally, predictions for the complicated system are validated by commercial FE code ANSYS, due to lack of informationwithin the literature. Comparisons for different frequencies and mode shapes with FE results are encouraging particularlyfor systems with local boundary and continuity conditions. Considering the performance of the presented modeling andsolution technique, it is expected to be used as a benchmark for vibration analysis of suspended plates/shells and panelsin future theoretical and experimental studies.
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