Damping of Cable with HDR Damper Accounting for Restraint Boundary Conditions

Data Availability Statement

Acknowledgments

Notation

a

damper location from the left end of cable;

C_i

constants of mode shape of cable;

EI

bending stiffness of cable;

F_a

damper force;

i

imaginary number, a part of complex number;

$K,\overline{K}$

spring stiffness of HDR damper and its nondimensional parameter;

${\overline{K}}_{\mathrm{str}}^{\mathrm{opt}},{\overline{K}}_f^{\mathrm{opt}},{\overline{K}}_h^{\mathrm{opt}},{\overline{K}}_r^{\mathrm{opt}}$

optimal coefficient of HDR damper for taut string cable, fixed end cable, hinged end cable and rotational restraint end cable, respectively;

${\overline{K}}_{rd}^{\mathrm{opt}}$

optimal coefficient of HDR damper for fixed end cable under rotational restraint consideration between cable and damper;

K_r

elastic stiffness of rotational restraint;

${\overline{k}}_r$

nondimensional parameter of K_r;

l

cable length;

m

mass per unit length of cable;

q

intermediate parameter of cable bending stiffness;

R_f

reduction factor of damping due to bending stiffness for fixed end cable;

R_h

increase factor of damping due to bending stiffness for hinged end cable;

R_r

modification factor of damping due to bending stiffness for rotational restraint end cable;

R_rd

reduction factor of damping due to rotational restraint at damper location;

r

nondimensional parameter of damper location;

T

chord tension of cable;

t

time coordinate;

v(y, t)

cable transverse displacement;

y

coordinate;

${\tilde{v}}_a$

amplitude of mode shape at y = a;

$\tilde{v}(y)$

mode shape of cable;

φ

loss factor of rubber material;

η_f, η_h, η_r

modification factor of $\overline{K}$ due to cable bending for fixed end cable, hinged end cable and rotational restraint end cable, respectively;

η_rd, η_rd1, η_rd2

modification factor of $\overline{K}$ due to rotational restraint at damper location;

β_n

nth wave number of a cable with HDR damper;

β_tn

nth wave number of a taut string without damper;

δ_1,2

parameter of cable mode shape;

η

nondimensional parameter of cable bending stiffness;

ξ_n

damping ratio corresponding to nth mode of cable; and

ω_n

nth complex natural frequency of cable.

References