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An analytical and experimental study of vibrating equilateral triangular plates

Paper presents the application of trilinear coordinates to vibrating triangular plates with experimental verification

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Abstract

A general description and a historical discussion of trilinear coordinates is given. These coordinates are then used to develop the solution of a simple supported freely vibrating equilaterial triangular plate. The solution gives fundamental frequencies, mode shapes, nodal lines and a general expression for displacements.

An experimental program is described to validate the analytical solution. Nodal lines, mode shapes and displacements are determined by holographic interferometry and compared to theoretical results. Excellent agreement is obtained. A second experimental technique, a fiber-optic device, is used to determine displacements with the results being compared to the displacements obtained by both theoretical and holographic analysis.

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Abbreviations

A i :

points referring to vertices of a triangle

a i :

length of sides of triangle

C ni :

cos\(n\pi \zeta _i \)

D :

flexural rigidity,\(D = {{Eh^3 } \mathord{\left/ {\vphantom {{Eh^3 } {12{\text{ }}(1 - v^2 )}}} \right. \kern-\nulldelimiterspace} {12{\text{ }}(1 - v^2 )}}\)

E :

modulus of elasticity

H i :

altitudes of a triangle,H i =Λ/m i

i :

dummy index, i.e.,i=1,2,3

l i :

cos\(\alpha _i \)

m i :

sin\(\alpha _i \)

m, n :

integers

R :

radius of circumscribed circle

S ni :

sin\(n\pi \zeta _i \)

w :

transverse plate displacements

(x,y), (x i ,y i ):

rectangular cartesian coordinates

(x 1,x 2,x 3):

triaxial coordinates

\((\alpha _1 ,\alpha _2 ,\alpha _3 )\) :

angles referring to the triangle of reference

β,γ:

angle of plate normal to incident and reflected laser beam

Δ:

area of the triangle of reference

δ:

experimental plate displacement

\(\zeta _i \) :

areal coordinates,\(\zeta _i = \eta _i /H_i \)

\((\eta _1 ,\eta _2 ,\eta _3 )\) :

trilinear coordinates

Λ:

2Rm 1 m 2 m 3

λ:

wavelength of laser light

μ:

mass density

ν:

Poisson’s ratio

ρ:

radius of inscribed circle\(\rho = 4R{\text{ sin }}\alpha _1 /2 \cdot \alpha _2 /2 \cdot {\text{sin }}\alpha _3 /2\)

ω:

frequency

References

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Author information

Authors and Affiliations

  1. Pratt & Whitneu Aircraft, 06108, Hartford, CT

    R. Williams (Senior Research Engineer)

  2. Virginia Polytechnic Institute and State University, 24061, Blacksburg, VA

    Y. T. Yeow (Graduate Research Assistant) & H. F. Brinson (Associate Professor of Engineering Science and Mechanics)

Authors
  1. R. Williams
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  2. Y. T. Yeow
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  3. H. F. Brinson
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Williams, R., Yeow, Y.T. & Brinson, H.F. An analytical and experimental study of vibrating equilateral triangular plates. Experimental Mechanics 15, 339–346 (1975). https://doi.org/10.1007/BF02318874

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  • DOI: https://doi.org/10.1007/BF02318874

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