List of symbols
A A B B* C C D E E G I Ji Ki M N Q R S T U V Yi Σ a, b aij b* b c c0 c1 c2 cp cϕ* Area and constant Data matrix (spectrum estimation) Constant Complex bulk modulus Longitudinal stiffness of a plate (membrane stiffness) and constant Augmented matrix (spectrum estimation) Bending stiffness of a plate Young’s modulus Error matrix (noise) of the data matrix (spectrum estimation) Shear modulus Area moment of inertia of unit length Bessel function of order i Bessel function of order i Moment of unit length Normal stress resultant of unit length and number of samples Shear stress resultant of unit length Radius of the shell mid-plane and radius of the rod Centre of gravity and elastic compliance matrix Relaxation time and period of a harmonic cycle Matrix (spectrum estimation) Matrix (spectrum estimation) Bessel function of order i Diagonal matrix containing singular values (spectrum estimation)
Constants Matrix element Complex bulk viscosity Observation vector (spectrum estimation) Phase velocity Velocity of longitudinal waves of infinite wavelength in an elastic bar Velocity of dilatational waves in an unbounded elastic medium Velocity of transverse waves in an elastic medium Velocity of longitudinal waves in a thin plate (membrane waves) Complex velocity of the potential ϕ equals complex velocity of dilatational waves in an unbounded viscoelastic medium cψ* Complex velocity of the potential ψ equals complex velocity of transverse waves in a viscoelastic medium
List of symbols
ix
cg cgr cgi cij e e f g h i j k m mij n nt ns p q r s t u v w x y z α α β γ0 δij δ ε
Group velocity Real part of complex group velocity Imaginary part of complex group velocity Elastic constants of anisotropic material Euler’s constant Unit base vector Frequency and function and index for quantities associated with the core Function Shell wall thickness – 1 and general index General index Wavenumber Number of expected modes, prediction order (spectrum estimation) Matrix element Constant and polynomial order and order of non-axisymmetric modes Number of samples in time domain Number of samples in space domain Number of propagating modes and constant Constant Radial coordinate (core and anisotropic rod) Index for quantities associated with the shell Time variable Displacement (general) and radial displacement (anisotropic rod) Circumferential displacement (anisotropic rod) Axial displacement (anisotropic rod) Longitudinal coordinate (shell and core) Radial coordinate within the shell Longitudinal coordinate (anisotropic rod) and constant and sampling index along the specimen (spectrum estimation) Phase angle Vector (spectrum estimation) Difference between two wavenumbers Angle caused by shear effects Kronecker symbol 0.2 % dynamic penetration depth Strain
x
List of symbols
ε η* θ ϑ κ λ λ* µ* µ ν ξ ρ σ τ ϕ Ψ ω ωc ( )* ( ),i ( )’ (^)
Error vector (noise) of the observation vector (spectrum estimation) Complex shear viscosity Circumferential coordinate (anisotropic rod) Circumferential coordinate (shell and core) Timoshenko’s shear coefficient Wavelength and constant Complex Lamé constant Complex Lamé constant equals complex shear modulus Constant Poisson’s ratio Angle caused by bending Density Normal and shear stress (core) and singular value and variance of Gaussian white noise (spectrum estimation) Shear stress (anisotropic rod) Scalar displacement potential Vectorial displacement potential Angular frequency Angular cut-off frequency Complex quantity Derivative with respect to i Index for reflected waves (isotropic rod) Component of normalized vector
AT Transpose of matrix A AH Hermitian transpose of matrix A A-1 Inverse of matrix A A# Pseudoinverse of matrix A (Moore-Penrose inverse) A Matrix with less noise A F Frobenius norm of matrix A
˜ x x ∇ ∆ ) Normalized quantity x Vector x Nabla operator Laplace operator and difference between two numbers
Numbers in [ ] brackets refer to the bibliography.