Vibration Damping of a Taut Cable with the Hybrid Application of a Viscous Damper and a Tuned Mass Damper

Data Availability Statement

Acknowledgments

Notation

A_p, B_p

complex amplitude of the pth cable segment;

a

complex amplitude ratio of TMD displacement and cable displacement at the location of the TMD;

$\mathbf{C}$

matrix to displacement compatibility and force equilibrium conditions;

c

viscous damping coefficient of the VD;

c_TMD

damping coefficient of the TMD;

i

imaginary unit;

k

spring stiffness of the TMD;

L

total length of the cable;

l_p

length of the pth cable segment;

M

mass of the TMD;

m

mass per unit length of the cable;

$n,n_1,n_2,n_{\mathrm{VD},1}^{\mathrm{th}},$ $n_{\mathrm{VD},2}^{\mathrm{th}},n_{\mathrm{TMD},1}^{\mathrm{th}},n_{\mathrm{TMD},2}^{\mathrm{th}}$

cable mode number;

p

cable segment index divided by the VD and TMD;

s

iterative step index;

T

axial tension force of the cable;

t

time;

x_p

coordinate along the cable chord axis in the pth cable segment;

Y_p(x_p)

complex mode shape on the pth cable segment;

y_p(x_p, t), y_p(x_p, τ)

cable transverse displacement of the pth cable segment;

y_TMD

transverse displacement of the TMD;

α

real part of the dimensionless eigenvalue;

β

imaginary part of the dimensionless eigenvalue;

β_n

nondimensional frequency of the nth mode;

${\beta }_n^1$

nondimensional frequency of “Mode n, 1”;

${\beta }_n^2$

nondimensional frequency of “Mode n, 2”;

η

nondimensional damping constant of the VD;

${\eta }_n^{\mathrm{opt}}$

optimum nondimensional damping constant of the VD of the nth mode;

λ

complex dimensionless eigenvalue;

μ_n

mean value of the modal damping ratio of the optimized mode (nth);

${\mu }_{n_1-n_2}$

mean value of the modal damping ratios of optimized modes $(n_1^{\mathrm{th}}\text{–}{n}_2^{\mathrm{th}})$;

ξ_n

modal damping ratio of the nth mode;

${\xi }_n^1$

modal damping ratio of “Mode n, 1”;

${\xi }_n^2$

modal damping ratio of “Mode n, 2”;

ξ_TMD

damping ratio of the TMD;

${\xi }_{\mathrm{TMD},n}^{\mathrm{opt}}$

optimum damping ratio of the TMD of the nth mode;

ξ_t

target modal damping ratio;

ρ

frequency ratio of the TMD;

${\rho }_n^{\mathrm{opt}}$

optimum frequency ratio of the TMD of the nth mode;

σ_n

standard deviation of the modal damping ratio of the optimized mode (nth);

${\sigma }_{n_1-n_2}$

standard deviation of the modal damping ratios of optimized modes $(n_1^{\mathrm{th}}\text{–}{n}_2^{\mathrm{th}})$;

Γ_p

shorthand of terms related to the pth cable segment;

τ

nondimensional time;

$\mathbf{\Phi }$

vector composed of A_p and B_p;

ϕ

mass ratio of the TMD;

${\varphi }_n^{\mathrm{opt}}$

optimum mass ratio of the TMD of the nth mode;

ω

modulus of the dimensional eigenvalue;

ω_TMD

circular frequency of the TMD; and

ω_o1

fundamental circular frequency of the cable.

References