The model considered in this paper for improvement is the cable-nonlinear Kelvin damper system, and a derived model of the Kelvin model is adopted for the damper: A power function nonlinear spring and a power function nonlinear damper are connected in parallel. Both the nonlinearity of damping and nonlinearity of stiffness are considered for this type of dampers, and the system calculation model is shown in Figure 1. L is the length of the cable, C is the damping coefficient of the damper, K is the stiffness coefficient of the damper, m is the mass of the cable per unit length, T is the cable tension, a and a' are the distances between the installation point of the damper and the left and right anchor ends, and x and x' are the axial coordinate of the cable.

When α = 1 and β = 0, it is an ideal linear viscous damper model, and for α ≠ 1, it is a rate-dependent nonlinear model; β ≠ 0 means that the damper has stiffness, and when β = 1, it is linear stiffness, which is represented as a linear spring in the model. The relationship between damping force and velocity under different α is shown in Figure 2.

The following assumptions are adopted in derivation of the vibration equation:

1) Cable sag, bending stiffness and self-damping are ignored;

2) The tension of the cable remains unchanged during the vibration process;

3) The installation position of the damper is close to the anchor ends;

4) Damping is small, that is, the modal damping ratio ξ_n is small.

Based on the parallel tensioned string theory, the free vibration equation of a cable without dampers is [2]:

T ∂ 2 v ( x , t ) ∂ x 2 − m ∂ 2 v ( x , t ) ∂ t 2 = 0 ,     0 ≤ x ≤ l (1)

where, T is the cable tension, m is the unit mass of the cable, l is the cable length, and v(x, t) is the cable amplitude, which is a function of the time t and the distance x along the length of the cable.

The vibration equation is solved by the separation variable method, and the natural frequency of each order of the cable is:

ω n = n π l T m (2)

where, n is the vibration order.

The vibration equation of the cable-nonlinear Kelvin damper system is as follows:

T ∂ 2 v ( x , t ) ∂ x 2 − m ∂ 2 v ( x , t ) ∂ t 2 = F d δ ( x − a ) (3)

where:

F d = F k + F c = K | x | β s i g n ( x ) + C | x ˙ | α s i g n ( x ˙ ) = K | v ( x , t ) | β − 1 v ( x , t ) + C | ∂ v ( x , t ) ∂ x | α − 1 ∂ v ( x , t ) ∂ x (4)

F_k is the force caused by the spring in the nonlinear damper, F_c is the force caused by dashpot (damping) of the nonlinear damper, K and C are the stiffness coefficient and the damping coefficient respectively. The separation variable method is used to solve the problem, which assumes that:

v ( x , t ) = V ( x ) e i ω t (5)

V(x) is the lateral displacement of in-plane vibration of the cable.

The Formula (5) is substituted into Formula (3), and the derivation method [10] similar to that of the linear viscous damper is adopted to obtain:

cot ( λ a ) + cot ( λ a ′ ) = − i ω C | V a ω sin ( ω t ) | α − 1 + K | V a cos ( ω t ) | β − 1 T λ = − i η | V a ω sin ( ω t ) | α − 1 + μ λ a | V a cos ( ω t ) | β − 1 (6)

The dimensionless damping coefficient η and the dimensionless stiffness coefficient μ are respectively:

η = C T m , μ = K a T (7)

Make:

A 1 = η | V a ω sin ( ω t ) | α − 1 A 2 = μ | V a cos ( ω t ) | β − 1 (8)

Obtain:

tan ( λ L ) = ( i A 1 + A 2 λ a ) sin 2 ( λ a ) 1 + ( i A 1 + A 2 λ a ) sin ( λ a ) cos ( λ a ) (9)

The approximate solution of the vibration Equation (9) can be obtained by using the method [10] similar to that of the linear viscous damper, according to the near anchor end assumption a / L ≪ 1

λ L ≈ n π + i A 1 ( n π a L ) 2 + A 2 ( n π a L ) ( 1 + A 2 ) + i A 1 ( n π a L ) (10)

Im ( λ L ) ≈ A 1 ( n π a L ) 2 ( 1 + A 2 ) 2 + [ A 1 ( n π a L ) ] 2 (11)

From [10], when ξ_n is very small, the approximate calculation can be made by the following formula:

ξ n = Im ( λ L ) | λ L | ≈ Im ( λ L ) Re ( λ L ) = Im ( λ L ) n π (12)

Obtain:

ξ n a / L ≈ A 1 ( 1 + A 2 ) 2 + [ A 1 ( n π a L ) ] 2 = n η π a L | V a ω sin ( ω t ) | α − 1 ( 1 + μ | V a cos ( ω t ) | β − 1 ) 2 + [ n η π a L | V a ω sin ( ω t ) | α − 1 ] 2 (13)

In order to obtain the maximum damping ratio and the optimal damping coefficient [4], make:

∂ ξ n ∂ η = 0 (14)

Obtain:

Optimal damping coefficient:

η o p t = 1 + μ | V a cos ( ω t ) | β − 1 n π a L | V a ω sin ( ω t ) | α − 1 (15)

Maximum damping ratio:

ξ n , max = 1 2 1 1 + μ | V a cos ( ω t ) | β − 1 a L (16)

These are the approximate solutions of the vibration equation. As a matter of convenience, make:

η ′ = η π a L = C T m π a L (17)

Equation (9) is solved iteratively by the numerical method. The fixed point iteration is carried out, and:

λ L = n π + arctan ( i A 1 + A 2 λ a ) sin 2 ( λ a ) 1 + ( i A 1 + A 2 λ a ) sin ( λ a ) cos ( λ a ) (18)

By using this equation, the exact numerical solution can be obtained iteratively.

In order to analyze the influence of stiffness coefficient μ (see Equation (6)) on the general design curve of the damper, other parameters remain unchanged, and different μ values are taken for calculation, and the results are as shown in Figure 3 (the abscissa indicates the damping coefficient, and the ordinate indicates the damping ratio index. When the damping ratio index is maximum, the corresponding damping coefficient is the optimal damping ratio).

It can be seen that the larger the stiffness coefficient μ, the smaller the attainable damping ratio, that is, the damping effect of the damper that can be achieved under the same conditions, so the damping stiffness coefficient has an adverse effect on the damping effect of the damper.

The influence of a is discussed in the following two cases.

1) Variation of value α, n = 1

At this point, only the first-order vibration of cables is controlled. The corresponding results or different α values are shown in Figure 4.

It can be seen that the larger the α, the larger the corresponding optimal damping coefficient, that is, the larger the damping force the damper is required to provide; the smaller the α is, the smaller the damping force the damper is required to provide.

2) Combined variation of α and n

At this point, only the first-order to fifth-order vibrations of cables are controlled at the same time, and the corresponding results are shown in Figure 5.

Although the dispersion degree of the optimal damping coefficients for the first five orders is varied for different α values, however, since the curves of various orders are coincident with nη' as abscissa, the order of vibration that can be suppressed with different α values is the same (when n is the same, the attainable maximum damping ratio is the same).

It can be seen from the above figure that, to suppress the vibrations of the first five orders, the optimal damping coefficient selected should be taken as the value at the intersection of the first-order curve and the fifth-order curve marked in the figure, and the attainable damping ratios of the first and fifth orders are the same for different α values, which is about 0.37*a/L and slightly smaller than the damping ratios of the second, third and fourth orders.

For different μ values (μ = 0.2, μ = 1, μ = 1.5, μ = 2), when β changes from 0 to 2, both the optimal damping coefficient and the corresponding maximum damping ratio change accordingly, as shown in Figure 6.

As can be seen from the above figure, when β = 1, it is equivalent that the nonlinear spring in model 1 is degraded into a linear spring. When β < 1, the smaller β is, the larger the optimal damping coefficient and the smaller the

corresponding maximum damping ratio. When β > 1, the larger β is, the smaller the optimal damping coefficient and the larger the corresponding maximum damping ratio. When μ is relatively small (μ < 1), if β > 2, the curve increases very

little with the increase of β, and the longitudinal axis ξ n , max L a is basically close to the maximum value of 0.5 in the linear case. Therefore, the larger the value of β, the better, but it is not much more effective when the value of β is greater than 1.5. When μ is relatively large (μ > 1), to make the longitudinal axis ξ n , max L a close to the maximum value of 0.5 in the linear case, the required β value is larger, but its variation trend is the same as that when μ is relatively small. Therefore, the larger the value of β, the better, but it is not much more effective when the value of β is greater than a certain value.

The analysis results show that, in the design and engineering application of tension cable dampers, stiffness characteristics should be emphasized and the stiffness nonlinear coefficient should be increased as much as possible to ensure that a higher maximum damping ratio can be obtained.

1) The damper is an ideal linear damper

When the damper is an ideal linear damper (i.e. μ = 0, α = 1, β = 1), the approximate calculation results of the first five orders are compared with the iterative calculation results of the first five orders to evaluate whether the approximate calculation is accurate and reasonable. The comparison results are shown in Figure 7:

As can be seen from Figure 7, the error of the approximate calculation is smaller for the first order, and the higher the order, the larger the error. When a/L = 0.02, the error of the fifth-order approximate calculation is less than 5%; However, when a/L = 0.05, the approximate calculation error is quite large, and the error calculation is shown in Table 3:

It can be seen that the larger the damper position parameter a/L, the larger the approximate calculation error; the low-order error is smaller, and the high-order error is larger. This is consistent with the research results of Krenk [5].

2) Consider the stiffness of the damper

When the damper has stiffness (that is, μ ≠ 0, α = 1, β = 1), the approximate calculation results of the first five orders are compared with the iterative calculation results of the first five orders. The comparison results are shown in Figure 8:

As can be seen from Figure 8, when μ ≠ 0, the difference between the approximate calculation results and the iterative calculation results is basically close to that of μ = 0. Still, the larger the damper position parameter a/L, the larger the approximate calculation error; the low-order error is smaller, and the high-order error is larger. The maximum error is close to 20% when a/L = 0.05 and n = 5.

When μ ≠ 0, the larger the μ value, the smaller the maximum damping ratio and the larger the optimal damping coefficient, which is the same as the results of approximate calculation and analysis.