Effect of mass eccentricity on vibration of a stockbridge damper

Abstract

Natural frequency is a system’s most important dynamic attribute. Resonance occurs when a time-varying force is applied to a system with a frequency that is equal to the system's natural frequency. This will result in the system's maximum amplitude, which could lead to system failure. A stockbridge damperis typically used worldwide as well as Bangladesh for damping vibration of overhead transmission lines. This work aims to investigate the change in vibration characteristics when the mass eccentricity of a stockbridge damper triggers a torsional mode of vibration in addition to the existingbending mode. We considered typical data of a stockbridge damper used in Bangladesh and modeled this stockbridge damper as a double cantilever beam having long and short sides. Each side was divided into two DOFS, three DOFS, and four DOFS.The natural frequencies are calculated on both sides by keeping the lumped masses at equal and unequal distances.

This study is based on two methods–The Myklestad’s method used for uncoupled bending analysis and coupled flexure-torsion vibration method used for bending-twisting analysis.In this respect, a computer program is developed.This developed computational program has a graphical representation that showsstockbridge damper’s characteristics in terms of the displacement (y) versus natural frequency (ɷ) curves.

First,it is analyzed how mass eccentricity affectsvibration. It is found that the natural frequency of coupled flexure-torsion vibration is greater than that for the uncoupled bending vibration.Natural frequency is increasedwhenmass eccentricity is gradually increasedfor 2DOFS. The natural frequency is further increased for 3DOFS and 4DOFS. Brieflyit can be concluded that if the value of mass eccentricity increases, the natural frequency will also increase, and vice versa forboth sides of the stockbridge damper.

Second, the impact of lumped mass distance on vibration is analyzed.Lumped mass distance has shown a great impact on the natural frequency of the stockbridge damper. The natural frequency was calculated first keeping lumped masses at equal distances and keeping the same lumped masses at unequal distances.For the long side of 2DOFS,ω_1for mode shape 1 is increased by 20.1% andω_2for mode shape 2 is decreased by 45.3% at unequal distance compared to equal distance.Where ω_1 and ω_2represent the first and second natural frequencies of the system respectively. However, on the short side, natural frequency is increased in both ω_1 and ω_2 by 30.2% and 63.4%, respectively. On the long side of the stockbridge damper and unequal distancesof lumped masses for 3DOFS, the natural frequencyω_1is increased for mode shape 1 by 15%,ω_2for mode shape 2 by 22.2%, while ω_3is decreased for mode shape 3 by 39% compare to the case when lumped masses were kept at equal distance.But on the short side of unequal distance, the natural frequencies are increasedby 43.7%, 83.4%,and 81.1% respectively.Now for long side unequal distance of 4DOFS, the natural frequencyω_1for mode shape 1 is increased by 8.5%, ω_2 for mode shape 2 by 17.7%, but ω_3 is decreased for mode shape 3 by 17.8% and ω_4 for mode shape 4 by 26.5%, where ω_3 and ω_4 represent the third and fourth natural frequency of the system respectively. However, in short side, the natural frequency ω_1 for mode shape 1is increased by 9.2%,ω_2 for mode shape 2is increased by 22.6%, and ω_3 for mode shape 3 is increased by 25.6% but ω_4 is changed randomly.

Third, the effect of DOF on vibration is analyzed.The natural frequency is changed as DOF increases. Between 2 and 3 DOFS, the natural frequency is increased in 3DOFS by 50.3% for mode shape 1(ω_1) and by 44.9% for mode shape 2 (ω_2) on the long sideequal distance and also increased48.2 % and 45.6 % for long sideunequal distance.On the short side and an equal distances, the natural frequencies in mode shapes 1(ω_1) and 2(ω_2) are increased by 51.2% and 15.1%, respectively. Regarding the unequal distance, the natural frequencies in mode shapes 1(ω_1) and shape 2(ω_2) have likewise increased by 50% and 73.6%, respectively.

In comparison to 3 DOFS with 4 DOFS, the natural frequency is increased in 4DOFS by 22.4% for mode shape 1(ω_1), 27% for mode shape 2(ω_2), and 17.1% for mode shape 3(ω_3) in long side equal distance and 17.2%, 25.3% and 26.9% for long side unequal distance.In short side and equal distance, natural frequency in mode shape 1(ω_1), mode shape 2(ω_2) and mode shape 3(ω_3) is decreased by 54.4%, 28.8% and 75.9% respectively and also decreased 69%, 52.9% and 76.1% for unequal distancerespectively.Thus we can conclude that mass eccentricity, the separation between lumped masses, and DOF have a significant influence on vibration in both coupled flexure torsion vibration and uncoupled bending vibration of stockbridge damper.

The loss of energy from the oscillatory system results in the decay of amplitude of vibration.Generally, in forced vibration with damping, energy dissipation(W_d) depends on many factors such as temperature, frequency or amplitude. We considered the simplest case of energy dissipation with viscous damping. The energy dissipated per cycle becomes,W_d = πcωX^2. Where, c= damping co-efficient,ω = natural frequency,X = amplitude of vibration.The preceding equation at resonance becomes, W_d= 2ζπkX^2 where,ζ= damping ratio,k= spring constant. In present analysis ζ is very small and therefore was not considered in mathematical modelling. Natural frequency is a system’s most important dynamic attribute. Resonance occurs when a time-varying force is applied to a system with a frequency that is equal to the system's natural frequency. This will result in the system's maximum amplitude, which could lead to system failure. A stockbridge damperis typically used worldwide as well as Bangladesh for damping vibration of overhead transmission lines. This work aims to investigate the change in vibration characteristics when the mass eccentricity of a stockbridge damper triggers a torsional mode of vibration in addition to the existingbending mode. We considered typical data of a stockbridge damper used in Bangladesh and modeled this stockbridge damper as a double cantilever beam having long and short sides. Each side was divided into two DOFS, three DOFS, and four DOFS.The natural frequencies are calculated on both sides by keeping the lumped masses at equal and unequal distances.

This study is based on two methods–The Myklestad’s method used for uncoupled bending analysis and coupled flexure-torsion vibration method used for bending-twisting analysis.In this respect, a computer program is developed.This developed computational program has a graphical representation that showsstockbridge damper’s characteristics in terms of the displacement (y) versus natural frequency (ɷ) curves.

First,it is analyzed how mass eccentricity affectsvibration. It is found that the natural frequency of coupled flexure-torsion vibration is greater than that for the uncoupled bending vibration.Natural frequency is increasedwhenmass eccentricity is gradually increasedfor 2DOFS. The natural frequency is further increased for 3DOFS and 4DOFS. Brieflyit can be concluded that if the value of mass eccentricity increases, the natural frequency will also increase, and vice versa forboth sides of the stockbridge damper.

Second, the impact of lumped mass distance on vibration is analyzed.Lumped mass distance has shown a great impact on the natural frequency of the stockbridge damper. The natural frequency was calculated first keeping lumped masses at equal distances and keeping the same lumped masses at unequal distances.For the long side of 2DOFS,ω_1for mode shape 1 is increased by 20.1% andω_2for mode shape 2 is decreased by 45.3% at unequal distance compared to equal distance.Where ω_1 and ω_2represent the first and second natural frequencies of the system respectively. However, on the short side, natural frequency is increased in both ω_1 and ω_2 by 30.2% and 63.4%, respectively. On the long side of the stockbridge damper and unequal distancesof lumped masses for 3DOFS, the natural frequencyω_1is increased for mode shape 1 by 15%,ω_2for mode shape 2 by 22.2%, while ω_3is decreased for mode shape 3 by 39% compare to the case when lumped masses were kept at equal distance.But on the short side of unequal distance, the natural frequencies are increasedby 43.7%, 83.4%,and 81.1% respectively.Now for long side unequal distance of 4DOFS, the natural frequencyω_1for mode shape 1 is increased by 8.5%, ω_2 for mode shape 2 by 17.7%, but ω_3 is decreased for mode shape 3 by 17.8% and ω_4 for mode shape 4 by 26.5%, where ω_3 and ω_4 represent the third and fourth natural frequency of the system respectively. However, in short side, the natural frequency ω_1 for mode shape 1is increased by 9.2%,ω_2 for mode shape 2is increased by 22.6%, and ω_3 for mode shape 3 is increased by 25.6% but ω_4 is changed randomly.

Third, the effect of DOF on vibration is analyzed.The natural frequency is changed as DOF increases. Between 2 and 3 DOFS, the natural frequency is increased in 3DOFS by 50.3% for mode shape 1(ω_1) and by 44.9% for mode shape 2 (ω_2) on the long sideequal distance and also increased48.2 % and 45.6 % for long sideunequal distance.On the short side and an equal distances, the natural frequencies in mode shapes 1(ω_1) and 2(ω_2) are increased by 51.2% and 15.1%, respectively. Regarding the unequal distance, the natural frequencies in mode shapes 1(ω_1) and shape 2(ω_2) have likewise increased by 50% and 73.6%, respectively.

In comparison to 3 DOFS with 4 DOFS, the natural frequency is increased in 4DOFS by 22.4% for mode shape 1(ω_1), 27% for mode shape 2(ω_2), and 17.1% for mode shape 3(ω_3) in long side equal distance and 17.2%, 25.3% and 26.9% for long side unequal distance.In short side and equal distance, natural frequency in mode shape 1(ω_1), mode shape 2(ω_2) and mode shape 3(ω_3) is decreased by 54.4%, 28.8% and 75.9% respectively and also decreased 69%, 52.9% and 76.1% for unequal distancerespectively.Thus we can conclude that mass eccentricity, the separation between lumped masses, and DOF have a significant influence on vibration in both coupled flexure torsion vibration and uncoupled bending vibration of stockbridge damper.

The loss of energy from the oscillatory system results in the decay of amplitude of vibration.Generally, in forced vibration with damping, energy dissipation(W_d) depends on many factors such as temperature, frequency or amplitude. We considered the simplest case of energy dissipation with viscous damping. The energy dissipated per cycle becomes,W_d = πcωX^2. Where, c= damping co-efficient,ω = natural frequency,X = amplitude of vibration.The preceding equation at resonance becomes, W_d= 2ζπkX^2 where,ζ= damping ratio,k= spring constant. In present analysis ζ is very small and therefore was not considered in mathematical modelling.