Double-ring shaped strong magnet array nonlinear dynamic vibration absorber for vibration mitigation of suspender cables and design method thereof

Abstract

A double-ring shaped strong magnet array nonlinear dynamic vibration absorber for vibration mitigation of suspender cables and design method thereof, which belongs to the field of structural vibration control. The installation positions and number are designed according to the demand of vibration mitigation, and usually one is installed at the midpoint of the suspender cable. The vibration absorber consists of the inner and outer magnet ring arrays, the additional weights, the universal wheels and a base. It feeds back the control force in the opposite direction of the motion of the suspender cable during the movement, so that the vibration energy of the suspender cable is transferred to the vibration absorber and thus less is returned to the suspender cable, and the energy dissipated through the friction between the universal wheels and the base, adding air dampers and other measures, etc.

Claims

1. A double-ring shaped strong magnet array nonlinear dynamic vibration absorber for vibration mitigation of suspender cable, comprising: an inner magnet ring array configured to be fixed at an installation position of the suspender cable, the inner magnet ring array including two inner ring subdivisions, each inner ring subdivision having a predetermined number of inner ring magnet slots, an inner ring magnet shoe being put in each of the inner ring magnet slots, inner ring steel limiters being fixed to the corresponding inner ring subdivision by bolts, non-slip gaskets being added to an inside of two inner ring subdivisions, an outer magnet ring array including two outer ring subdivisions, each outer ring subdivision having a predetermined number of outer ring magnet slots, and an outer ring magnet shoe being put in each of the outer ring magnet slots, outer ring steel limiters being fixed to the corresponding outer ring subdivisions by bolts, an additional weight installed on a periphery of the outer magnet ring array and fixed together with the outer magnet ring array, the additional weight being a square-shaped body formed by four cuboid mass blocks attached together by bolts, universal wheels attached to the outer ring subdivisions; and a base including two identical base subdivisions, non-slip gaskets being provided on an inside of the base, the two identical base subdivisions being fixed on the suspender cable by bolts to support the outer magnet ring array, the additional weight and the universal wheels, wherein the universal wheels are in direct contact with the base, the universal wheels move in any lateral direction in the plane of the base, the friction of the universal wheels with the base causes damping of the vibration absorber, wherein energy from a vibration of the suspender cable is dissipated through the damping provided by the friction, and wherein magnetic force between the inner magnet ring array and the outer magnet ring array prevents the outer magnet ring array from colliding with the inner magnet ring array during movement.

2. A design method of a double-ring shaped strong magnet array nonlinear dynamic vibration absorber for vibration mitigation of suspender cables, comprising steps of: step 1, obtaining characteristics of a suspender cable to be controlled by field survey, the characteristics including outer diameter of sheath for the suspender cable, and natural frequencies; step 2, determining design parameters of the double-ring shaped strong magnet array nonlinear dynamic vibration absorber, the design parameters including: 1) a mass ratio of an outer magnet ring array and additional weights to modal mass of the suspender cable: 1% to 5%, and within which the larger the mass ratio, the better the vibration mitigation effect, and the mass ratio determined according to demand of vibration mitigation for the suspender cable; 2) a damping ratio of the vibration absorber is equal to an optimal damping ratio of a tuned mass damper (TMD) under the same mass ratio; 3) designed values for linear and cubic stiffness of the vibration absorber: numerical model of cable-absorber system is established as shown in equation (1): $\begin{array}{cc}\{\begin{array}{c}M\stackrel{.Math.}{w}\left(t\right)+C\stackrel{.}{w}\left(t\right)+\mathrm{Kw}\left(t\right)=F\left(t\right)\\ m\stackrel{.Math.}{v}+c\left[\stackrel{.}{v}-\stackrel{.}{w}?\left(x-d\right)\right]+k_1\left[v-w?\left(x-d\right)\right]+{k_2\left[v-w?\left(x-d\right)\right]}^3=0\end{array}& \left(1\right)\end{array}$ where, w(t) denotes vector of displacements corresponding to each degree of freedom of the suspender cable, {dot over (w)}(t) and {dot over (w)}(t) denote velocity and acceleration of the suspender cable, respectively; M denotes mass matrix of the suspender cable; C denotes damping matrix of the suspender cable employing Rayleigh damping; K denotes stiffness matrix of the suspender cable; F(t) denotes sum of external loads on the suspender cable and reaction of the vibration absorber to the suspender cable; m, c, k.sub.1, k.sub.2 denote mass, damping, linear stiffness and nonlinear stiffness of the vibration absorber, respectively; ? denotes displacement of the vibration absorber, {dot over (?)} and {umlaut over (?)} denote velocity and acceleration of the vibration absorber, respectively; one end of the suspender cable is starting point of x-axis, x is x-axis coordinate of point on the suspender cable, d denotes installation location of the vibration absorber along x-axis, and ?(x?d) is Dirac function; an objective function for parameter optimization as shown in equation (2) $\begin{array}{cc}J=\min \left(\frac{{?}_{\mathrm{control}}}{?}\right)& \left(2\right)\end{array}$ where ? denotes a root-mean-square value, variance of time-history responses of displacement or velocity at a specific location along the suspender cable under an influence of white noise load, ?.sub.control denotes the root-mean-square value, variance of time-history responses of displacement or velocity at the specific location along the suspender cable under the influence of white noise load when the vibration absorber is deployed, and ? and ?.sub.control are selected or established according to the demand of vibration mitigation; dynamic responses of the suspender cable are calculated by using the numerical model of the cable-absorber system under the white noise loads at multiple points, and the optimal linear stiffness and the optimal nonlinear stiffness are obtained by automatic optimization; 4) geometric parameters of the inner magnet ring and the outer magnet ring array: with overall consideration of cost of the vibration absorber and section size of the suspender cable to be controlled, number, radius angle and inner diameter of magnet shoes in the inner magnet ring array and the outer magnet ring array are determined; a parametric electromagnetic field numerical model of the double-ring shaped strong magnet array nonlinear dynamic vibration absorber system is established, and by adjusting thickness and height of the magnet shoes of the inner magnet ring array and the outer magnet ring array, combination of the inner magnet ring array and the outer magnet ring array achieves designed linear stiffness and designed cubic stiffness; 5) geometric parameters of additional weights for the vibration absorber: total mass of the vibration absorber minus mass of the outer magnet ring array is the mass of additional weights, and then length, width and height of the additional weights are designed according to density of the additional weight; step 3, machining and fabricating the double-ring shaped strong magnet array nonlinear dynamic vibration absorber according to the design parameters obtained from step 2; and step 4, installing the manufactured double-ring shaped strong magnet array nonlinear dynamic vibration absorber on the suspender cable in the laboratory, and measuring the force-displacement relationship of the double-ring shaped strong magnet array nonlinear dynamic vibration absorber with a dynamometer and a ruler to ensure the linear and cubic stiffness of the vibration absorber to be the design values; letting the vibration absorber move freely, then measuring the displacement signal of the outer magnet ring array, and determining the damping ratio by free-vibration decay method etc., and then adjusting the damping ratio by adjusting the friction force between universal wheels and a base.

Description

DESCRIPTION OF DRAWINGS

(1) FIG. 1 is a schematic diagram of the invented double-ring shaped strong magnet array nonlinear dynamic vibration absorber for the suspender cable.

(2) FIG. 2 is a schematic diagram of an inner magnet ring array.

(3) FIG. 3 is a schematic diagram of an outer magnet ring array.

(4) FIG. 4 is a schematic diagram of an additional weight subdivision.

(5) FIG. 5 is a schematic diagram of a base subdivision.

(6) Where: 1 suspender cable; 2 inner magnet ring array; 3 outer magnet ring array; 4 additional weight; 5 universal wheel; 6 base; 7 inner ring subdivision; 8 inner ring magnet shoe; 9 inner ring steel limiter; 10 first kind of bolt hole; 11 second kind of bolt hole; 12 third kind of bolt hole; 13 fourth kind of bolt hole; 14 fifth kind of bolt hole; 15 outer ring steel limiter; 16 outer ring magnet shoe; 17 outer ring subdivision I; 18 outer ring subdivision II; 19 sixth kind of bolt hole; 20 seventh kind of bolt hole; 21 base subdivision.

DETAILED DESCRIPTION

(7) Combining the attached drawings and technical solutions, the detailed implementation method of the invention is further explained in the following parts. Taking one suspender cable of a suspension bridge as an example, the design method of the dimensional parameters and the installation process of the invented absorber are introduced in detail, and the numerical results are combined to illustrate the vibration mitigation effect of the invention for the suspender cable.

(8) A suspension bridge suspender cable is selected, which has an effective calculated length of 60.954 m and a mass of 2961 Kg, and the first order natural frequency of the suspender cable is 1.952 Hz based on the finite element modeling and analysis. According to the steps proposed by the invention, the dimensions of the double-ring shaped strong magnet array nonlinear dynamic vibration absorber are designed: the sum of the outer magnet ring array and the additional weights is 39.09 kg, the mass ratio to the suspender cable to be controlled is set to 1.32%, the optimal damping value of the vibration absorber is calculated to be 70N/(m/s), the optimal linear stiffness value is 4438 N/m, and the optimal cubic stiffness value is 1.511?10.sup.6 N/m.sup.3. Based on the finite element analysis software MAXWELL in the electromagnetic field, the parametric electromagnetic-field numerical model of the double-ring shaped strong magnet array is established, and parameters of the double-ring shaped strong magnet array nonlinear dynamic vibration absorber are calculated as follows: the number of magnet shoes for both inner and outer magnet ring arrays is 9, the thickness of such magnet shoes is 0.018 m, the height of such magnet shoes is 0.1 m, the radius angle of such magnet shoes is 30 degrees, the inner radius of magnet shoes in the inner magnet ring array is 61.5 mm, and the inner radius of magnet shoes in the outer magnet ring array is 129.5 mm; four additional weights are the same, and each has a height of 0.1 m, thickness of 0.04 m and length of 0.36 m. These parameters change as the mass ratio changes.

(9) In order to examine the vibration mitigation effect of this vibration absorber, the finite element model of the suspender cable to be controlled is established based on the SIMULINK software, the dynamic responses of the suspender cable before and after the installation of the double-ring shaped strong magnet array nonlinear dynamic vibration absorber are obtained using the transient analysis method, and the results of displacement and velocity dynamic responses before and after the installation of the damper at the midpoint of the suspender cable are shown in FIG. 6 and FIG. 7, respectively. It can be seen that under the same excitation, the displacement and velocity responses of the midpoint of the suspender cable decrease significantly after the installation of the vibration absorber.

(10) Table 1 shows the comparison results of dynamic responses at the midpoint of the suspender cable with and without a double-ring shaped strong magnet array nonlinear dynamic vibration absorber under the same band-limited white noise excitation, these results are the average values of the dynamic response calculated results under 20 sets of random white noise excitation, and it can be seen that: after the installation of the invented vibration absorber, the displacement and velocity vibration responses at the midpoint of the suspender cable are significantly reduced, and the vibration mitigation effect becomes better with the increase of the mass ratio; the double-ring shaped strong magnet array nonlinear dynamic vibration absorber has significant effect on suppressing the vibration of the suspender cable.

(11) TABLE-US-00001 TABLE 1 Reduction values of the vibration responses indices of the midpoint at a suspender cable after the installation of the invented nonlinear dynamic vibration absorber Reduction ratio of displacement Reduction ratio of velocity vibration response index vibration response index Root mean Root mean Variance square Peak Variance square Peak 1.32% mass ratio 52.75% 30.83% 20.59% 40.70% 22.45% 16.62% 2.00% mass ratio 67.69% 41.67% 33.82% 54.17% 31.78% 24.83%