In this work a machine cam with five different profiles was used to investigate the linear and nonlinear dynamical behavior of asymmetric Stockbridge damper with excitation frequencies in the range of
The Stockbridge damper is presently the most common type of transmission line damper. In general, the absorber consists of two weights attached to the end of stranded cables, which are known as
A detailed mathematical description of conductor motion is difficult due to the stranded construction of a conductor. An example of this problem is the study carried out by Nawrocki and Labrosse [
In Stockbridge dampers of transmission line, mechanical energy is dissipated in wire cables “damper or messenger cables”. The damping mechanism is due to statical hysteresis resulting from Coulomb (dry) friction between the individual wires of the cable undergoing bending deformation. In order to test this dynamical model of Stockbridge damper, the typical experimental impedance curves are compared with numeric results. This procedure was used by [
Zhu and Meguid [
López and Venegas [
In this work, the physical parameters are adjusted through the comparison between numerical and experimental results. The numerical models are obtained using the Finite Element Method. The experimental results are obtained using a cam machine with five different eccentricities. This way, the motion amplitude of the excitation is maintained constant. The experimental results are compared with linear and nonlinear numerical results. The nonlinear system contains nonlinear stiffness and damping elements. The dynamic responses of the linear and nonlinear systems are also compared with the experimental results (impedance curves) obtained through conventional testing using an electromechanical shaker to excite the system.
The mathematical models of the Stockbridge damper system are shown in this section.
The messenger wire is modeled by the EulerBernoulli beam finite element. In this element, the transversal displacement is interpolated using the wellknown
In order to take into consideration the cable
The suspended masses of the Stockbridge damper are modeled with a rigid body plane motion hypothesis and the admissible displacements are shown in Figure
References and admissible displacements.
After assembling all the elements of the messenger wire, each weight of the Stockbridge damper contributes to two terms of the dynamical equilibrium. The first contribution is in the mass matrix (inertia force)
These two terms are obtained using the first variation of the kinetic energy (Hamilton Principles) and the rigid body plane motion hypothesis for modeling the suspended Stockbridge damper weight. With the convention defined in Figure
Taking into consideration the hypotheses of plane motion of rigid body,
The variation of
The Stockbridge damper dynamical equilibrium equation is obtained after assembling all finite elements and it can be conventionally written as
Considering the base excitation as harmonic,
The amplitude of the displacement vector is calculated solving Equation (
The parameter estimation of the nonlinear system can be made by approximating the numeric and experimental Frequency Response Function (FRF) curves. The analysis can be exemplified considering the motion equation of a simple oscillator subjected to a harmonic excitation [
It is possible to find a linearized coefficient
In order to find the nonlinear coefficient
The nonlinear coefficient
The mathematical model of a cubic stiffness element can be expressed as [
The nonlinear friction damping can be obtained using a similar approach to the cubic stiffness development. The nonlinear restoring force becomes
The linearized coefficient
Expanding the idea of the simple oscillator introduced in (
For additive nonlinearities, it is possible to expand the nonlinear vector into individual nonlinear restoring forces, as follows:
Introducing the newly redefined nonlinear coefficients into (
The motion equation of a general nonlinear system subjected to harmonic excitation can be described by the following nonlinear ordinary differential equation:
The nonlinear component of the system is represented by the nonlinear vector
Considering a harmonic response
The linear receptance can be defined as
The solution of (
Figure
Schematic cam machine.
Figure
Cam machine with Stockbridge damper coupled.
Five different disk cams with eccentricities of 0.25, 0.5, 0.75, 1.25, and 1.5 mm were used. The tests were carried out varying the excitation frequency between 5 and 17 Hz with increments of 0.25 Hz. This lower frequency range was used due to the mechanical limitation of the excitation system. Figures
Acceleration ratio curves of accelerometer 1.
Acceleration ratio curves of accelerometer 2.
Curves of natural frequencies and damping ratio versus base displacement.
The linear and nonlinear parameters used in the numerical models were adjusted considering five different cam profiles. The results are shown in Table
Parameters adjusted for the linear and nonlinear systems
Eccentricity (mm)  Absolute error 








0.25  linear  0.589  1.90  9.82  
nonlinear  0.600  1.98  11.46  −7.9e9  −0.0036  −0.0264  −0.0502  
0.5  linear  0.454  1.74  6.21  
nonlinear  0.480  1.69  7.68  −9.59e8  −0.0308  −0.0416  −0.0964  
0.75  linear  0.496  1.65  4.40  
nonlinear  0.464  1.67  6.57  −4.59e8  −0.0077  −0.0445  −0.0084  
1.25  linear  0.568  1.52  2.94  
nonlinear  0.569  1.54  3.72  −1.67e8  −0.0182  −0.0490  −0.0537  
1.5  linear  0.828  1.43  2.56  
nonlinear  0.797  1.08  11.24  −5.20e8  −0.1601  −0.0147  −0.0996 
To simulate the hysteretic damping of the system, it was considered
Figure
Base displacement and acceleration.
Figures
Curve of the flexural stiffness and loss factor versus base displacement.
Figure
Cable acceleration.
Figure
Impedance curves.
The results showed that the natural frequency of the system varies with the amplitude of the excitation motion, that is, the higher the motion amplitude is the lower the natural frequency gets. The same behavior was verified for the damping ratio of the system, that is, the higher the motion amplitude is the lower the damping ratio gets.
It was verified that the real and imaginary parts of the complex Young modulus decrease with the motion amplitude increase when the parameters are fitted for a linear system.
For the frequencies range
As the test was done with a full Stockbridge damper, it is possible that the smallest cable and the smallest mass have influenced the results. The mathematical model was based only on the Stockbridge damper half model containing the largest weight.
When the tests are performed with constant base motion amplitude, it is possible to fit a single value of the parameters according to the motion amplitude.
When the tests are carried out using an electromechanical shaker, the motion amplitude is varied according to the excitation frequency. In this case, the fitted parameters are frequency dependent.
The typical numerical and experimental impedance curves present good agreement.
The authors gratefully acknowledge the support for this work provided by the Brazilian Science Foundation CNPq.