,Notation
Matrices and Vectors
Matrices are in general bold upper-case Latin or Greek letters, e.g., K or Φ.
Vectors are in general bold lower-case Latin or Greek letters, e.g., q or ϕ.
diag[·] is a diagonal matrix whose content is written within the brackets.
det(·) is the determinant of the matrix within the brackets.
tr (·) is the trace of a matrix. √
x is the norm of vector x, i.e., x = x T x.
Imaginary Quantities
√
i is the imaginary unit (i.e., i = −1). Re(·) and Im(·) are the real and imaginary
parts of the variable within the brackets, respectively.
Superscripts and Bars Above Symbols
Super-script T indicates the transposed of a vector or a matrix.
Super-script * indicates the complex conjugate of a quantity.
Dots above symbols (e.g., ṙ, r̈) indicates the time derivatives, i.e., d/dt, d 2 /dt 2 .
Prime on a variable (e.g., C L or φ ) indicates the first derivative, e.g., φ = dφ/d x,
two primes are the second derivative (e.g., φ = d 2 φ/d x 2 ), and so on.
−
Bar (−) above a variable (e.g., r ) indicates its time-invariant average value.
Tilde (~) above a symbol (e.g., M̃n ) indicates a modal quantity.
Hat (∧) above a symbol (e.g., H η ) indicates a normalised quantity.
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,xii Notation
The Use of Indices and Superscript
Index x, y or z refers to structural axis. i and j are general indices on variables.
n and m are mode shapes or element numbers. p and k are node numbers.
Abbreviations
CC and SC are short for the centre of cross-sectional neutral axis and the shear centre.
tot is short for total. max, min are short for maximum and minimum.
L or A means integration over the entire length or the area of the system.
Latin Letters
A, A j Area, cross-sectional area, coefficient associated with variable j
A∗1 − A∗6 Aerodynamic derivatives associated with the motion in torsion
A, Am , An Connectivity matrix (associated with element m or n)
a, ai Distance, coefficient, amplitude
a x , a y , az Vector components associated with motion in x, y, z directions
B Cross-sectional width
b, bi Coefficient, bandwidth parameter
bc Distance between cable planes is a suspension bridge
bq Buffeting dynamic load coefficient matrix at cross-sectional level
C, C Damping coefficient, matrix containing damping coefficients
Cae , C ae Aerodynamic damping, aerodynamic damping matrix
c, ci Coefficient, damping coefficient at cross-sectional level
c, cae Damping matrix at element level, aerodynamic damping matrix
Co, C o Co-spectral density, co-spectral density matrix
C ov j Covariance matrix associated with variable j
D, d Cross-sectional depth, coefficient
d, di , dk Element displacement vector, element end displacement component
E, E i Modulus of elasticity, impedance function (i = 1, 2, . . . )
E0 Matrix containing cross-sectional elastic constants
e, ec Exponential number (≈ 2.718281828), cable sag
F, F, F n Force, force vector, element force vector
f, f n Frequency [Hz], eigenfrequency associated with mode n
f (·) Function of variable within brackets
G Modulus of elasticity in shear
G0 Matrix containing cross-sectional time invariant stress resultants
g(·), g Function of variable within brackets, gravity constant
H (t), H̄ Horizontal cable force component, its mean value
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H1∗ − H6∗ Aerodynamic derivatives associated with the across-wind motion
Hn , H r Frequency response function, frequency response matrix
Hη , H η Modal frequency response functions, matrix containing Hηn
hc, hm Length of suspension bridge hangers, hanger length at mid-span
hr Vertical distance between shear centre and hanger attachment
h0 Height (above girder) of suspension bridge tower
It , Iw St Venant torsion and warping constants
I y , Iz Moment of inertia with respect to bending about y or z axis
Ij Turbulence intensity of flow components j = u, v or w
I Identity matrix √
i The imaginary unit (i.e., i = −1)
J, J Joint acceptance function, joint acceptance matrix
j Index variable
K, K Stiffness, stiffness matrix
K ae , K ae Aerodynamic stiffness, aerodynamic stiffness matrix
k Index variable, node, or sample number
kp Peak factor
k, kae Stiffness matrix at element level, aerodynamic stiffness matrix
L , L(t, r, ṙ) Length (of structural system), Lagrange function
m
Ln Integral length scales (m = x, y or z, n = u, v or w)
e Effective length
M, Mg , M g Mass, concentrated mass, mass matrix containing Mg
Mn Cross-sectional bending moment (n = y or z)
M x , Mθ Cross-sectional torsion moment, external torsion moment
m Index variable
mx , m y, mz Mass per unit length associated with motion in x, y, z directions
M Mass matrix
mn Modally equivalent and evenly distributed mass (n = x, y or z)
m0 , m Mass matrix at a cross-sectional level, mass matrix at element level
N , Nr Number, number of elements, number of degrees of freedom
N , Nx , N y Normal force, normal force in x or y directions
n Index variable
P, PF , Pq External load energy
P1∗ − P6∗ Aerodynamic derivatives associated with the along-wind motion
p, p() Index variable, node or sample number, probability of occurrence
q, qn , q Distributed load, load vector at cross-sectional level, n = y, z or θ
R, R External load, reaction force, external load vector at system level
R, R Modal load, Modal load vector
r, ri , r Displacement or rotation, displacement vector, i = 1, 2, . . .
rel (x), r el Element cross-sectional displacement, displacement vector
St Strouhal number
S, S, S j Auto or cross-spectral density, cross-spectral density matrix
s Cross-sectional surface coordinate, relative time variable
TM , Tm Motion energy of system body masses
t, T Time, total length of time window
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Matrices and Vectors
Matrices are in general bold upper-case Latin or Greek letters, e.g., K or Φ.
Vectors are in general bold lower-case Latin or Greek letters, e.g., q or ϕ.
diag[·] is a diagonal matrix whose content is written within the brackets.
det(·) is the determinant of the matrix within the brackets.
tr (·) is the trace of a matrix. √
x is the norm of vector x, i.e., x = x T x.
Imaginary Quantities
√
i is the imaginary unit (i.e., i = −1). Re(·) and Im(·) are the real and imaginary
parts of the variable within the brackets, respectively.
Superscripts and Bars Above Symbols
Super-script T indicates the transposed of a vector or a matrix.
Super-script * indicates the complex conjugate of a quantity.
Dots above symbols (e.g., ṙ, r̈) indicates the time derivatives, i.e., d/dt, d 2 /dt 2 .
Prime on a variable (e.g., C L or φ ) indicates the first derivative, e.g., φ = dφ/d x,
two primes are the second derivative (e.g., φ = d 2 φ/d x 2 ), and so on.
−
Bar (−) above a variable (e.g., r ) indicates its time-invariant average value.
Tilde (~) above a symbol (e.g., M̃n ) indicates a modal quantity.
Hat (∧) above a symbol (e.g., H η ) indicates a normalised quantity.
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,xii Notation
The Use of Indices and Superscript
Index x, y or z refers to structural axis. i and j are general indices on variables.
n and m are mode shapes or element numbers. p and k are node numbers.
Abbreviations
CC and SC are short for the centre of cross-sectional neutral axis and the shear centre.
tot is short for total. max, min are short for maximum and minimum.
L or A means integration over the entire length or the area of the system.
Latin Letters
A, A j Area, cross-sectional area, coefficient associated with variable j
A∗1 − A∗6 Aerodynamic derivatives associated with the motion in torsion
A, Am , An Connectivity matrix (associated with element m or n)
a, ai Distance, coefficient, amplitude
a x , a y , az Vector components associated with motion in x, y, z directions
B Cross-sectional width
b, bi Coefficient, bandwidth parameter
bc Distance between cable planes is a suspension bridge
bq Buffeting dynamic load coefficient matrix at cross-sectional level
C, C Damping coefficient, matrix containing damping coefficients
Cae , C ae Aerodynamic damping, aerodynamic damping matrix
c, ci Coefficient, damping coefficient at cross-sectional level
c, cae Damping matrix at element level, aerodynamic damping matrix
Co, C o Co-spectral density, co-spectral density matrix
C ov j Covariance matrix associated with variable j
D, d Cross-sectional depth, coefficient
d, di , dk Element displacement vector, element end displacement component
E, E i Modulus of elasticity, impedance function (i = 1, 2, . . . )
E0 Matrix containing cross-sectional elastic constants
e, ec Exponential number (≈ 2.718281828), cable sag
F, F, F n Force, force vector, element force vector
f, f n Frequency [Hz], eigenfrequency associated with mode n
f (·) Function of variable within brackets
G Modulus of elasticity in shear
G0 Matrix containing cross-sectional time invariant stress resultants
g(·), g Function of variable within brackets, gravity constant
H (t), H̄ Horizontal cable force component, its mean value
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, Notation xiii
H1∗ − H6∗ Aerodynamic derivatives associated with the across-wind motion
Hn , H r Frequency response function, frequency response matrix
Hη , H η Modal frequency response functions, matrix containing Hηn
hc, hm Length of suspension bridge hangers, hanger length at mid-span
hr Vertical distance between shear centre and hanger attachment
h0 Height (above girder) of suspension bridge tower
It , Iw St Venant torsion and warping constants
I y , Iz Moment of inertia with respect to bending about y or z axis
Ij Turbulence intensity of flow components j = u, v or w
I Identity matrix √
i The imaginary unit (i.e., i = −1)
J, J Joint acceptance function, joint acceptance matrix
j Index variable
K, K Stiffness, stiffness matrix
K ae , K ae Aerodynamic stiffness, aerodynamic stiffness matrix
k Index variable, node, or sample number
kp Peak factor
k, kae Stiffness matrix at element level, aerodynamic stiffness matrix
L , L(t, r, ṙ) Length (of structural system), Lagrange function
m
Ln Integral length scales (m = x, y or z, n = u, v or w)
e Effective length
M, Mg , M g Mass, concentrated mass, mass matrix containing Mg
Mn Cross-sectional bending moment (n = y or z)
M x , Mθ Cross-sectional torsion moment, external torsion moment
m Index variable
mx , m y, mz Mass per unit length associated with motion in x, y, z directions
M Mass matrix
mn Modally equivalent and evenly distributed mass (n = x, y or z)
m0 , m Mass matrix at a cross-sectional level, mass matrix at element level
N , Nr Number, number of elements, number of degrees of freedom
N , Nx , N y Normal force, normal force in x or y directions
n Index variable
P, PF , Pq External load energy
P1∗ − P6∗ Aerodynamic derivatives associated with the along-wind motion
p, p() Index variable, node or sample number, probability of occurrence
q, qn , q Distributed load, load vector at cross-sectional level, n = y, z or θ
R, R External load, reaction force, external load vector at system level
R, R Modal load, Modal load vector
r, ri , r Displacement or rotation, displacement vector, i = 1, 2, . . .
rel (x), r el Element cross-sectional displacement, displacement vector
St Strouhal number
S, S, S j Auto or cross-spectral density, cross-spectral density matrix
s Cross-sectional surface coordinate, relative time variable
TM , Tm Motion energy of system body masses
t, T Time, total length of time window
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