Efficacy of SMA Wire in Vibration Suppression of a Stay Cable: An Experimental Investigation

Abstract

This paper analyzes the influence of three different shape memory alloy (SMA) wires (i.e., Ni-Ti, Cu-Al-Ni, and Cu-Zn wires) for mitigating model cable vibration. Shape memory alloy (SMA) wires because of their superelasticity or shape memory properties can promise to be one of the effective damping materials in suppressing stay cable oscillations. The investigation explores through experimental works, the capabilities of three Ni-Ti, Cu-Al-Ni, and Cu-Zn SMA wires by estimating the vibration amplitude suppression of the model stay cable. The effective location of the SMA wires on the model cable for vibration suppression was estimated in free as well as forced vibration cases. To impart free and forced vibration, respectively, in the cable, a transducer-based hammer and exciter were used. An accelerometer and data acquisition system was used to record and acquire vibration data. It is also very appropriate to ascertain which wire among one of the three SMA wires performs well. It is found that the Ni-Ti SMA wire has a better performance in vibration suppression as compared to Cu-Al-Ni and Cu-Zn wires. It is also established that the attachment of the SMA wire at the mid-length of the cable gives better vibration suppression for the first mode.

1 Introduction

Steel cables are commonly used in cable-stayed bridges to support the deck (Ref 1,2,3,4). Vehicular, wind, and rain-wind interactions may cause large amplitude cable vibrations because the cables show modest damping properties (Ref 5, 6). Such vibrations mainly involve vibrations in-plane and out-of-plane modes. Vibrations can result in fatigue failure of cable connection as well as damage to the corrosion protection system.

Some studies have suggested that smart materials in the form of shape memory alloys (SMAs) can be used to improve the performance of stay cable (Ref 7,8,9). The energy dissipation capacity of the SMA is closely related to the shape memory and superelasticity properties of the SMA material (Ref 10). Further studies (Ref 11,12,13,14,15,16,17,18) have shown that the hysteresis, size, length, temperature, loading range, and cyclic frequency, especially for Ni-Ti SMA wire, greatly influence the energy dissipation capability, and stay cable vibration mitigation depends on these parameters. The damping device made up of SMA wire has shown good performance in the mitigation of cable vibration (Ref 11, 19,20,21). The functional fatigue characteristics along with the maximum tensile stress, residual strains, energy dissipation, and its equivalent viscous damping have been presented for Ni-Ti shape memory alloy wire (Ref 22, 23).

Some authors through their experimental studies have attempted to establish the one-dimensional constitutive model for determining the influence of SMAs on the cable response (Ref 8, 24,25,26). A few researchers (Ref 19,20,21) have also studied experimentally about vibration dissipation in a long stay cable due to the shape memory alloy damper and they have found a satisfactory performance of SMA wire because of its good hysteresis behavior. Some authors have carried out numerical studies in the presence and absence of SMA damper to compare its performance with that of magnetorheological (MR) dampers and tuned mass dampers (TMD) (Ref 27,28,29,30,31,32,33,34,35). They have concluded that SMA wire damper performs well in comparison to the Magnetorheological damper and Tuned Mass Damper. The SMA damper can effectively diminish cable vibrations over broadband excitations. However, some researchers have concluded that SMA wires were useful only in mitigating earthquake-induced vibration of the bridge deck and rocking of piers (Ref 36, 37). Moreover, the previous studies show that SMA dampers effectively reduced cable vibration induced by wind or rain. The feasibility of theoretical and numerical studies to investigate the damping effects of a cable with an attached SMA damper was also discussed (Ref 38,39,40). Studies were further advanced to establish the location of the SMA wire along the stay cable for effective dissipation of vibration amplitude. In one study (Ref 35), it is shown that when the damping device is placed near the maximum amplitude it shows better efficiency. An analytical study on the vibration mitigation of a combined system of cable—SMA damper in single and multi-mode vibrations shows that the SMA can effectively reduce stay cable vibration in both free and forced vibration (Ref 24, 41). So, the SMA wire is substantially efficient to mitigate the cable vibration responses as described in the literatures of the present study. Therefore, the idea and necessity of using SMA wires rather than the other dampers to mitigate model cable vibration experimentally for the present study were developed.

Authors have not come across any literature which shows study on SMA wires other than Ni-Ti wire. Therefore, authors have considered three different SMA wires (i.e., Ni-Ti, Cu-Al-Ni, and Cu-Zn) to explore about the energy dissipation capability of these wires to suppress maximum vibration amplitude in the model cable using OROS-based data acquisition system under free and forced vibrations as well.

Since stay cables are more often subjected to wind-induced oscillations, they are responsible for inducing oscillations in the bridge deck and pylons causing discomfort to the bridge users and fatigue related damages. After literary review, it is felt that exploration of the type of SMA wire which would provide the greatest damping remains outstanding. Therefore, the novelty of this research work lies in the establishment of (1) the performance of the individual SMA wire out of Ni-Ti, Cu-Al-Ni, and Cu-Zn as an alternative vibration mitigation means of bridge stay cables against wind and (2) the most appropriate location of the SMA wire along the stay cable. Determination of the extent of reduction in vibration amplitude of the stay cable is also of paramount importance. Therefore, the study in this paper is presented with the following objectives.

The following are the objectives of this study:

• To assess the structural damping of the different SMA wires based on hysteresis curves obtained from loading-unloading in tension regimes.

• To obtain the maximum deformation response of the stay cable in free and forced vibration and to determine the effectiveness of the SMA wire in mitigating the oscillation amplitudes in the model stay cable with and without the SMA wire experimentally.

• To identify the critical location of the SMA wire along the model stay cable length to minimize the cable vibrations effectively.

• To identify the most effective SMA wire out of Ni-Ti, Cu-Al-Ni, and Cu-Zn alloy wires in mitigating the model cable vibration.

In the following section, the loading-unloading characteristics of the SMA wires will be experimentally determined. After this, the experimental evaluation of free and forced vibration responses of stay cable with and without the attachment of SMA wire will be evaluated. The physical and mechanical properties of the model stay cable and SMA wires which are used for the study will be experimentally determined. Comparative performances of three different SMA wires (i.e., Ni-Ti, Cu-Al-Ni, and Cu-Zn) will be investigated as a novel work for mitigating the model cable vibration using OROS-based data acquisition system in both free and forced vibrations. Based on this, the effective SMA wire among the three wires will be selected for the effective reduction of cable vibration. Moreover, the most appropriate location to attach SMA wire with model cable will be found, as well as the effective wire among the proposed wires which has not been focused in the previous studies so far.

2 Details of SMA Wires

The three SMA wires of Ni-Ti, Cu-Al-Ni, and Cu-Zn each have a 2.0 mm diameter. The chemical composition (wt.%) of the Ni-Ti SMA wire is 55.64Ni-44.36Ti, respectively. The compositions (wt.%) of the other two SMA wires are 33.85Cu-3.15Al-63Ni and 61.97Cu-39.03Zn-%.”The mass density of Ni-Ti wire is 6,500 kg/m³ as reported by the manufacturer. It is also worth noting that the mechanical properties of alloys (such as Young's modulus and damping) are reliant on temperatures, and tension. In addition, the SMA wires were first heat-treated before conducting the test using an electric furnace for a specified duration of 20 minutes. Ni-Ti wire was heat treated at 450 °C and then water-quenched. Cu-Al-Ni SMA wire underwent a two-step heat treatment process: solution treatment and aging treatment. The solution treatment involved heating the wire to a specific temperature of 750 °C followed by rapid quenching. The aging treatment involved heating the wire at a lower temperature of 350 °C for the specified duration. The heat treatment temperature for Cu-Zn SMA wire was typically taken as 700 °C. The specimens subjected to heat treatment developed superelasticity properties.

3 Details of Stay Cable

The model cable adopted in this study was a 6.70 m long strand with a diameter of 6.0 mm. The cable was made up of galvanized steel wires coaxially twisted that make a strand and these strands consisted of 7 plies. The geometrical and material properties of the steel cable are shown in Table 1.

Table 1 Geometrical and material properties of proposed steel cable

4 Tension and Loading-Unloading Tests on SMA Wires

Figure 1(a) shows the dynamic universal testing machine (UTM). The mechanical properties of all the wires were evaluated using the universal testing machine (UTM—Instron Electropuls E1000) with a dynamic loading capacity of 1 kN. Tensile loading-unloading tests were carried out using the machine's wedge-type grip. It is interesting that this grip, although best suited for testing flat specimens, is also used to test thin wires and specimens with cylindrical dog-bone shapes as per ASTM E8. This UTM is used for performing loading-unloading tests on different SMA wires as visible in Fig. 1(b). However, all the test specimens of different SMA wires are presented in Fig. 1 Each specimen SMA wire proposed for testing was 65 mm in length, with a gage length of 15 mm. All the tests were performed at a room temperature of 27 °C. The responses are recorded from the data acquisition system as shown in Fig. 1(d).

Fig. 1

Tests of Shape memory alloy wires (a) UTM machine with component details, (b) SMA wire fixed between upper and lower grip, (c) SMA wire specimens for the test, and (d) A typical plot on PC

The loading frequency was set at 0.01 Hz, and the strain rate was chosen as 2.78 × 10⁻⁴ s⁻¹. The test was performed upto 25 no. of loading cycle. The strain amplitudes were found to be 11, 2.5, and 2.75%, respectively, for Ni-Ti, Cu-Al-Ni, and Cu-Zn wires.

4.1 Tension Test Results of SMA Wires

The tensile tests of all wires were conducted at room temperature. The machine was started through a fixed strain rate of 2.78 × 10⁻⁴ s⁻¹. The tests were performed on the UTM machine for determining their tension behavior, as well as for determining the tensile load capacity of each wire. The three specimens from each wire were taken to check the accuracy of the tension test results. Moreover, the response obtained from the test using the data acquisition system is presented in Fig. 2. Each wire’s stress-strain response from the tension test is described as per the results.

Fig. 2

Result of Tension test for: (a) Ni-Ti SMA wire, (b) Cu-Al-Ni SMA wire, and (c) Cu-Zn SMA wire

The stress-strain curve for all the wires (i.e., Ni-Ti, Cu-Al-Ni, and Cu-Zn) is represented in Fig. 2(a), (b), and (c) obtained by performing an experiment. Their respective plots show the typical deformation behavior. Moreover, the mechanical properties of the SMA wires were determined from the graphs. All the specimens elastically deform under load initially. The elasticity moduli of the alloys are assessed from the initial slope of the curve. As the elastic modulus value of 31.21 GPa is recognized from the curve for Ni-Ti wire, while 60.41 and 35.81 GPa are obtained for Cu-Al-Ni & Cu-Zn wires, respectively. The ultimate tensile stress of 760 MPa is recognized corresponding to a 2.39 kN peak load for Ni-Ti wire, whereas in the case of Cu-Al-Ni wire and Cu-Zn wire, the ultimate tensile stress values, respectively, are 1010 and 660 MPa.

The stiffness of 6.54 kN/mm was found after performing an experiment for Ni-Ti wire, while the stiffness for Cu-Al-Ni and Cu-Zn wire is 12.66 kN/mm and 7.50 kN/mm, respectively. Moreover, the experimental results for all wires showed that the maximum strain value reaches in Ni-Ti wire specimen which could resist strain up to 35%, that is larger than the other wire (such as Cu-Al-Ni and Cu-Zn), they can only resist strain up to 9.45 and 9.16%, respectively, as shown in Fig. 2(b) and (c). It is noted that no significant plastic deformation is observed in the material prior to failure. The material properties of the SMA wires obtained by conducting the tension test are represented in Table 2.

Table 2 Tension test data for all the SMA wires

4.1.1 Fractography of SMA Wires

The fracture mechanism for the tensioned SMA wires (Ni-Ti, Cu-Al-Ni, and Cu-Zn) can be visually represented in a better way from the respective fractography. Numerous regimes associated with deformation show the rupture in the image obtained from the SEM test. Low to high magnification images for the crack origination regime are shown. Dominant surface defects are highlighted as can be seen in the images from Fig. 3.

Fig. 3

Fracture surface appearance of broken SMA specimens (2‐mm diameter) from SEM for: (a) Ni-Ti, (b) Cu-Al-Ni, and (c) Cu-Zn wires at (1) 5.00 KX, (2) 1.0 KX, and (3) 500 X as shown above at a different scale

SEM (scanning electron microscopy) test was conducted to obtain the typical failure surface and for observing the surface topography in the broken SMA alloys. Figure 3(a) shows the surface topography and fracture surface appearance for the Ni-Ti specimen by SEM images. Moreover, the images are shown from high to low magnification at 2 KX, 1 KX, and 500 X, as represented from the respective figure. In a similar way, the images for Cu-Al-Ni and Cu-Zn specimens are collected, which has been represented in Fig. 3(b) and (c), respectively. The fracture surfaces can be distinguished clearly in these images (Fig. 3b, c). It is found that the fracture surface and crack region with striations are smaller as compared to the fracture area categorized with a dimpled morphology as illustrated in Fig. 3(a), (b), and (c). In these images, two regions of ductile fracture and the failure effects of local surface stress are noticeable. The crack surfaces demonstrated that the material crack evolution was at an angled location. Furthermore, the fracture region retains a relatively flat surface. The zoomed images of the crack propagation region are represented by all the alloys. It is evident that the fracture is originated from an inclusion.

4.2 Loading-Unloading Test Results of SMA Wires

The test was carried out by strain (displacement) control. The machine was started with a fixed strain rate of 2.78 × 10⁻⁴ s⁻¹. Figure 4 show some hysteretic curves in terms of stress-strain of SMA wire under 25 loading cycles in the case of Ni-Ti SMA wire specimen (Fig. 4a). The same loading protocols were considered for Cu-Al-Ni as well as Cu-Zn SMA wires as shown in Fig. 4(b) and (c). The curves are significantly different for all three wires from each other as they depend on their properties. It demonstrates that the strain value was the primary factor influencing the number of loading cycles.

Fig. 4

Stress-strain response in loading-unloading cycles for: (a) Ni-Ti, (b) Cu-Al-Ni, and (c) Cu-Zn SMA wires

The maximum value of strain for Ni-Ti wire is achieved as 11.49% corresponding to the maximum stress value of 620 MPa, whereas in the case of Cu-Al-Ni wire, the maximum value of strain is found as 2.46% at 844 MPa, and for Cu-Zn wires, the maximum strain value is 2.72% corresponding to 656 MPa. It is seen from Fig. 4 that all three wires show the loading-unloading behavior as expected for the SMA-based alloys. The strain increases with each cycle. As the cycle number rises, a decrease in transformation start stress and an increase in residual strain can be seen. The enclosed portion of each loaded stress-strain loop represents the energy dissipation. Moreover, after about 10 cycles of loading in the case of Ni-Ti wire, the curves overlapped and could not be distinguished from each other shown in Fig. 4(a), while the same overlapping curves were found after 3 no. of cycles for both Cu-Al-Ni and Cu-Zn wires as indicated in Fig. 4(b) and (c). However, after imposing the same conditions for all wires during tests, the Ni-Ti wire showed a wider area enclosed by the hysteresis loop, which indicates that a higher energy dissipation capacity could be generated by Ni-Ti wire than by the other wires. The time taken for performing the loading-unloading test on the Ni-Ti wire was 40 minutes, while for Cu-Al-Ni and Cu-Zn wire, it was 23 minutes and 18 minutes, respectively. Thus, the loading-unloading behavior of the SMA wires was observed.

4.2.1 Calculating Structural Damping for the SMA Wires

“The mechanism of internal damping is structural damping or hysteretic damping. For most structures, the energy dissipated per cycle is independent of the excitation frequency, but proportional to the square of the amplitude. The energy loss during cyclic loading is obtained by measuring the area enclosed by the force-displacement curve, which takes the form of a hysteretic loop that gives its name to this damping.”

“The area of the loop denotes the energy dissipated by the damper in a cycle of motion, which is $\Delta W=\pi \omega {cX}^{2}$ and is given by (Ref 42, 43).

$\Delta W$ is the energy dissipation in a cycle of motion, $X$ is the maximum displacement amplitude from the cyclic hysteresis loop of the SMA wire. The damping coefficient c is assumed to be inversely proportional to the frequency i.e., $c=h/\omega$, here h is called the hysteresis damping constant.

It is considered that $\beta =h/k$ is a constant indicating a dimensionless measure of damping with stiffness $k$. In terms of $\beta$, the energy loss per cycle can be expressed as (substituting $h=k\beta$), then the energy dissipation becomes $\Delta W= \pi k\beta {X}^{2}$. The hysteresis logarithmic decrement can be defined as $\delta \cong 2\pi {\xi }_{\mathrm{eq}}\cong \pi \beta$, where ${\xi }_{\mathrm{eq}}$ is the equivalent damping ratio i.e., equal to half of the dimensionless measure of damping (β/2).

The enclosed hysteresis area in Fig. 4(a), (b), and (c) represent the energy dissipated per unit volume during loading-unloading cycles from the wire which is denoted by $\Delta W$. From Fig. 4(a), the maximum energy dissipation is found as 1.65 N·m according to the no. of cycles, and the hysteresis damping value is found as 177100 N/m, so finally the hysteresis damping constant becomes 0.159 in case of Ni-Ti wire.” Moreover, the energy dissipation in the case of Cu-Al-Ni and Cu-Zn wires was found as 0.42 N·m and 0.37 N·m, respectively, as from Fig. 4(b) and (c), while hysteresis damping values for both wires were 974840 N/m and 695330 N/m. It is seen that the hysteresis damping constant values for Cu-Al-Ni and Cu-Zn wires come out to be 0.137 & 0.136, respectively, as presented in Table 3.

Table 3 SMA wires Energy dissipation parameters

The energy dissipation with respect to the number of cycles is shown in Fig. 5(a), (b), and (c) for all three wires in the whole process of the repeated loading-unloading. It is shown that the dissipation energy decreases, and the smooth curve is obtained with the increasing number of cycles as by the curve fitting that could be observed in Fig. 5. It is also clear that the maximum energy dissipation takes place in the case of Ni-Ti wire rather than Cu-Al-Ni and Cu-Zn wire. However, the energy dissipation in the case of Cu-Al-Ni wire is slightly more than Cu-Zn wire. The energy dissipation with the different parameters is well represented for all the wires as mentioned in Table 3.

Fig. 5

Response between Energy dissipation vs. No. of cycles for: (a) Ni-Ti SMA wire, (b) Cu-Al-Ni SMA wire, and (c) Cu-Zn SMA wire.

5 Mechanical Properties of the Cable

A tension test of the cable was performed on the UTM at room temperature. The diameter of the steel cable was 6.0 mm. The test images have been shown in Fig. 6(a). The original gage length of the sample was taken 28.29 mm, and the initial area of the cable was 28.27 mm². The tensile test was performed at a loading rate of 0.027 to 0.030 kN/s as per the IS-1608 (Part-1). It was found that one by one its ply broke. The results obtained have been shown in Fig. 6(b) and are also well represented in the tabular form as mentioned in Table 4. The load at yield was 17.36 kN corresponding to 10.12 mm elongation. The ultimate load was 21.70 kN corresponding to 13.49 mm elongation. The final gage length was 32.57 mm. The tensile strength of the cable was 865.58 MPa and the elongation in percentage was found to be 15.13%. Figure 6(c) and (d) shows that all the strands got scattered at failure.

Fig 6

(a) Cable clamped at both ends on UTM machine for performing tension test, (b) Failure of cable wires during tension test, (c) Deformed sample after testing, and (d) The graph represents the results obtained from the UTM machine after the tension test

Table 4 Tension test data for 6 mm diameter test cable

6 Proposed Steel Cable Model Validation Using FINITE Element Modeling

A test cable of 6.70 meters in length and 6 mm diameter with material properties presented in Table 1 was taken for the experimental work. The mass per unit length of the cable is 0.133 kg/m, and the damping ratio is 0.8%. An axial tension force of 450 N is applied to the cable as a pretension.

This cable is simulated using ANSYS APDL software. To simulate cable in ANSYS, the inertia, gravity, and bending effects are neglected. The “LINK 180” element has been taken as per the suitability in the software because the link takes axial tension but does not bend as shown in Fig. 7. The finite element model has 100 no. of elements. Frequencies obtained by ANSYS as well as the theoretical formula Eq 12 are presented in Table 5.

Fig. 7

FEM modeling of a structural taut cable

Table 5 Natural frequencies determined from ANSYS(APDL), and the theoretical formula of model stay cable

7 Validation of the Proposed Steel Cable Model

The numerical simulation results and the results with the theoretical formula (appendix Eq. 12) for the frequency of a first mode show that the values are well-matched with each other. This validates the ANSYS model, which was developed by finite element based simulation.

Based on this, the idea to obtain desired natural frequency values using an experimental approach was established for the test cable. Once the natural frequency value is satisfied, thereafter an experimental study could be carried out further using combined cable-SMA wires with the help of the OROS data acquisition system without any issue.

The mode shape (pattern of deformation) corresponding to 4.34 Hz frequency obtained from ANSYS is also shown in Fig. 8.

Fig. 8

First mode deformation of the cable at a natural frequency of 4.34 Hz in ANSYS

8 Test of Model Cable Attached with SMA Wire

A special testing facility was established to examine the dynamic characteristics and vibration mitigation, as shown schematically in Fig. 9. Two saddles were connected at the ends which held the cable, and an axial tension load was applied to the cable by weight blocks. The inclination of the cable was kept at 3°. The SMA wire was fixed using fly nuts. The position of the SMA wire along the length of the cable was adjustable. The cable tension force was kept at 450 N. Each test case was conducted three times to ensure the repeatability of the experimental data. Care was taken to prevent any slippage at the anchorage during tests.

Fig. 9

Schematic diagram of the test setup

8.1 Free Vibration Analysis

A sensor-based impact hammer was used to produce the initial displacement or for exciting the model cable. An acceleration meter was fixed at the center of the cable length to pick up the vibration responses of the cable. Table 6 shows the specific details of the accelerometer and hammer, which are used in the present experimental study. The hammer and accelerometer both were connected to the OROS data acquisition system, and the data were recorded on a computer for determining the vibration responses. All the accessories and equipment details are presented in Fig. 10. The excitation in the form of initial displacement was offered by the hammer at the mid-length of the cable to observe the responses in the presence and absence of the SMA wire. The impact of the hammer was calibrated within a range of impact so that the resulting free vibration response was independent of the impact duration and magnitude. A Fast Fourier Transform analysis of the free vibration test data was used to determine the frequency of the model cable with and without the SMA wire that was automatically generated by OROS itself.

Table 6 The table represents the specifications of the accelerometer and hammer

Fig. 10

Test setup details for the free vibration test.

The test was performed first to determine the frequency of mode and structural damping. The cable-SMA wire system was tested by varying the SMA wire position from the lower anchorage to the upper anchorage side with a 0.1L interval. The test process was repeated for all three alloy wires. Table 7 shows the various lengths of SMA wire based on their position at the cable. Although, by varying the position of the SMA wire at the cable, the SMA wire length also changes because the cable was having some inclination and the bottom end of the SMA wire was kept fixed as like the real condition in the case of the bridge cable. Therefore, the different length of wire has been proposed according to position wise as represented in Table 7. The frequency band between 0 and 20 Hz was chosen for observing free vibration response using OROS. It was discovered that the outcomes of subsequent runs closely agreed.

Table 7 Shape memory alloy (SMA) wire length for model cable test in free vibration

The free vibration responses of the cable at its mid-point before and after the attachment of SMA wires are shown in Fig. 11, 12, and 13 for the three different SMA wires. It is seen that the maximum reduction in amplitude in first mode vibration occurs at the mid-span location due to any SMA wire (Fig. 11, 12, and 13). The frequencies and damping ratios of cable along with the three different SMA alloy wires by varying their positions from 0.1 to 0.9L are presented in Tables 8, 9, and 10. It clearly indicates that the tested frequency is very close to the theoretical one. Moreover, it is seen that the damping ratio of the system decreases when the SMA wire is positioned away from the mid-length of the cable. Based on the tested results, it is discovered that the Ni-Ti SMA wire effectively dissipates the acceleration response of the cable when compared to the cable without SMA wires.

Fig. 11

Free vibration response at cable midpoint without and with Ni-Ti SMA wire located at 0.5L in terms of (a) Acceleration vs. Time, and (b) Acceleration/force vs. Frequency

Fig. 12

Free vibration response at cable midpoint without and with Cu-Al-Ni SMA wire located at 0.5L in terms of (a) Acceleration vs. Time, and (b) Acceleration/force vs. Frequency

Fig. 13

Free vibration response at cable midpoint without and with Cu-Zn SMA wire located at 0.5L in terms of (a) Acceleration vs. Time, and (b) Acceleration/force vs. Frequency

Table 8 Frequency and damping ratio for combined cable and Ni-Ti wire

Table 9 Frequency and damping ratio for combined cable and Cu-Al-Ni wire

Table 10 Frequency and damping ratio for combined cable and Cu-Zn wire

The hammer-hitting location was fixed near the center. The influence of the attached SMA wires in Fig. 11, 12, and 13 clearly shows the reduction in vibration amplitudes. As compared to the unattached cases of SMA wires; the Ni-Ti wire has shown the maximum reduction in amplitude by 60-65%, whereas Cu-Al-Ni and Cu-Zn SMA wires have shown a reduction in amplitudes by 55-60, and 42-45%, respectively. Figure 11(b), 12(b), 13(b) shows the acceleration response corresponding to the dominant frequency of the cable. The reduction in acceleration amplitude is clearly visible.

Furthermore, it can be understood from Fig. 11, 12, and 13 that the maximum reduction in amplitude is found at the 0.5L location on the cable for the first mode, so this can be considered the optimum location of the SMA wire on the cable for the first mode. Figure 11, 12, and 13 shows the free vibration test records using OROS when SMA is attached at 0.5L location on the cable. However, other location results (combined cable-SMA) are presented in tabular form (Tables 8, 9, and 10).

where c & ξ stand for damping coefficient and damping ratio, respectively.

A graph in Fig. 14(a) is plotted between Frequency vs SMA wire positions to identify the effect of the location of the SMA wire on the cable. It clearly signifies that it is most suitable to set the SMA wire at mid-span to increase the first mode cable vibration frequency. The frequency corresponding to 0.5L is higher than that of other frequencies since the attachment of the SMA wire stiffens the cable. Figure 14(a) shows that the maximum frequencies are about 8 Hz for all three wires. These observations closely match and are found experimentally from the OROS data system. Moreover, Fig. 14(b) presents the first in‐plane mode vibration damping. It evidently demonstrates an optimum location of 0.5L along the cable to yield the maximum damping for the first mode. Therefore, it could be concluded that it is preferable to incorporate the SMA wire at the mid-span of the stay cable for first-mode vibration mitigation and to achieve maximum damping. The damping requirement for cable vibration mitigation with the SMA-based devices corresponds to the measured highest damping of about 0.22%.

Fig. 14

(a) Natural frequency vs. SMA wire positions, and (b) Damping ratio vs. SMA wire positions for combined cable-SMA wires

8.2 Forced Vibration Analysis

The forced vibration test was also conducted on the cable without and with the SMA wire using an exciter and amplifier as shown in Fig. 15. The exciter was used for generating the excitation in the cable and its control was done manually with the help of an oscillator. The frequency ranges in the oscillator could be set between 0 and 300 Hz. The exciter was installed at the location of 0.3L from the lower anchorage of the cable to provide excitation vertically. The frequency range and amplitude were set manually. The acceleration response of the stay cable model was measured using an accelerometer. A sweeping sinusoidal signal with a frequency band covering the interesting modal frequency (Ref 44) was induced. The frequency band for the first modal vibration was set between 0 and 100 Hz.

Fig. 15

Test setup details for forced vibration: connection of cable with exciter, oscillator, and OROS to obtain records on PC

The SMA wire length for the model cable test in forced vibration was kept at 600 mm for all the positions (such as 0.1 to 0.5L). The forced vibration response without the SMA wire was taken first, and then the responses of the cable with the three SMA wires were obtained in turn to observe the energy dissipation by the individual wires. Furthermore, the position of the SMA wires was varied from 0.1 to 0.5L for the forced vibration analysis to determine the effective position of the SMA wire for mitigating forced vibration induced by the exciter. The overall photograph schemes for the test setup are shown in Fig. 15.

In the experiment, the oscillator is associated with the amplifier, which has manual control over the frequency and amplitude. The manually applied amplitude was calibrated in terms of the electric current in Amperes (A). The oscillator in forced vibration was set to a specific frequency value with a fixed amplitude for introducing controlled forced vibration in the cable. The frequency range was specifically chosen between 20 and 40 Hz for finding out the responses, while the amplitude was fixed up to 6 mm. The displacement amplitude value of up to 6 mm was judiciously chosen for the excitation frequency range of 20-40 Hz to avoid the potential risk of damage to the whole system due to violent oscillation. All the output records were collected with the help of the OROS data acquisition system to get the responses. Figure 16, 17, and 18 shows the measured acceleration vs time and acceleration vs frequency responses.

Fig. 16

Forced vibration response with and without SMA wires at 0.5L of the cable due to excitation induced by exciter for 20 Hz excitation frequency in terms of (a) Acceleration vs. Time, and (b) Acceleration vs. Frequency

Fig. 17

Forced vibration response with and without SMA wires at 0.5L of the cable due to excitation induced by exciter for 30 Hz excitation frequency in terms of (a) Acceleration vs. Time, and (b) Acceleration vs. Frequency

Fig. 18

Forced vibration response with and without SMA wires at 0.5L of the cable due to excitation induced by exciter for 40 Hz excitation frequency in terms of (a) Acceleration vs. Time, and (b) Acceleration vs. Frequency

It is perceived that the SMA wires are good at dissipating the acceleration responses of the cable in case of forced vibration also. The cable was excited at three frequencies of 20, 30, and 40 Hz frequencies in turn. The oscillator’s amplitude was set at 6 mm for the 20 Hz excitation frequency and 4 mm for both 30 and 40 Hz excitation frequency. Figure 16(a) depicts an acceleration of 200 m/sec² at 2 seconds in the time domain, while a peak amplitude of 120 m/sec² was observed in the frequency domain without the SMA wire for the oscillator amplitude of 6 mm at 20.75 Hz. The discrepancies observed in the acceleration values in the time and frequency domain may be attributed to the mechanical errors linked with attachments as well as minor slippage and the error in conversion from time to frequency domain. However, the decrease in the acceleration amplitudes due to SMA wires are clearly visible which were determined. The SMA wire locations varied from 0.1 to 0.5L at an interval of 0.1L for all three frequencies, and responses were collected. It is seen that maximum responses were found on the 0.5L location of the cable as shown in Fig. 16, 17, and 18.

It is observed from Fig. 16, 17, and 18 that the additional damping ratio increases rapidly as the damper location approaches the cable midpoint, while the maximum response decreases sharply for the damper locations changing from 0 to 0.5L (Table 11). It implies that 0.5L is the more reasonable and feasible damper location based on forced vibration responses.

Figure 16(a), 17(a), 18(a) shows acceleration vs time response at the mid-span location of cable in the time domain, whereas Fig. 16(b), 17(b), 18(b) shows the acceleration response history at the cable mid-span due to the three oscillation frequencies before and after the attachment of SMA wires (Ni-Ti, Cu-Al-Ni, and Cu-Zn wire). The comparison between cable alone and combined cable-SMA wires for forced frequencies ranging from 20 to 40 Hz has been represented.

The results show that Ni-Ti wire dissipates the energy better than other wires at all excitation frequencies. The comparison of the cable responses with different attachment locations of SMA wire and without SMA wire attachment showed that the maximum energy dissipation in the cable occurs when the SMA wire was attached at 0.5L position (Fig. 16, 17 and 18).

Table 11 depicts the peak acceleration values of cable with varying positions of SMA wires at different frequencies. It is worth noting that the maximum acceleration values of 116 m/sec² occur at 20 Hz, 158 m/sec² occurs at 30 Hz, and 90.7 m/sec² occurs at 40 Hz on the 0.1L position.

9 Discussions of the Results

The loading-unloading responses for the SMA wires generate enclosed hysteresis loops as shown in Fig. 4. The area enclosed in the hysteresis loop shows the energy dissipation. The larger area enclosed in the hysteresis loop of the Ni-Ti wire shows a greater energy dissipation capability as compared to Cu-Al-Ni and Cu-Zn wires. This performance of Ni-Ti wire is utilized in vibration mitigation of stay cable. Figure 14(a) and (b) shows that the frequency and damping ratio of the combined cable-SMA wire depend upon the position of attachment of the SMA wire on the cable. Therefore, while selecting an SMA wire one must also give due consideration to the parameters such as size, type, location, stiffness, boundary conditions, and inclination of the cable.

In free vibration analysis and tests, the Ni-Ti wire has shown the maximum reduction in the amplitude of cable by 60-65% of the initial amplitude, whereas Cu-Al-Ni and Cu-Zn SMA wires have shown a reduction in amplitudes by 55-60, and 40-45%, respectively. From the free vibration tests, Table 8 shows that the maximum damping ratio is 0.22% corresponding to Ni-Ti wire, whereas it is 0.14 and 0.08% for Cu-Al-Ni and Cu-Zn wires as shown in Tables 9 and 10. Observation of the cable vibration test record showed that higher amplitudes occur at a lower frequency, whereas lower amplitudes take place at higher frequencies and reach nearer to the equilibrium position with time. It is also observed that the mid-point amplitude of the cable takes a long time to diminish in free vibration as compared to other points on the cable. In the forced vibration analysis, it is observed that the amplitude at 30 Hz forced frequency is larger as compared to the value of the amplitude corresponding to 40 Hz forced frequency, rather it is also more than 20 Hz excitation frequency even though a displacement amplitude of 6 mm was considered corresponding to 20 Hz frequency. It was discovered that the beat phenomenon takes place in forced vibration analysis, which is due to differences in natural and applied frequencies. In the case of the forced vibration analysis, it was observed that the response reduces by 60-65% of initial amplitude due to Ni-Ti wire as compared to cable without control. While the responses reduce by 50-55 and 45-50% the initial amplitude for Cu-Al-Ni and Cu-Zn wires corresponding to all the applied frequencies of 20, 30, and 40 Hz. The maximum values of damping and frequency of combined cable-SMA wires are obtained at 0.5L location in the first mode of vibration. So, it should be noted that 0.5L is the optimum position for vibration suppression in the first mode. However, the effective position of SMA wire may vary depending upon the modes of cable. Because several modes occur in cable vibration, so as per the dominating modes, the feasible location to attach SMA with cable should be selected where larger energy dissipation could take place. Usually, the dominating mode is considered the first mode of the cable.

10 Summary and Conclusions

The outcomes of an experimental study are presented in this paper for a taut steel cable in order to determine an efficient SMA wire from the view of vibration energy dissipated out of Ni-Ti, Cu-Al-Ni, and Cu-Zn wires, loading-unloading hysteresis curves were obtained. Free and forced vibration tests were performed on the stay cable to determine the location of the maximum amplitude of the cable. The location of the SMA wire was also determined for the effective reduction in the vibration amplitude of the cable.

The following conclusions are drawn:

• 1. Ni-Ti wire shows a wider hysteresis loop and thus a larger energy dissipation than other Cu-Al-Ni & Cu-Zn wires.

• 2. The substantial energy dissipation takes place in the cases of free as well as in forced vibrations. Ni-Ti wire shows maximum response reduction in terms of acceleration vs time or acceleration vs frequency responses than Cu-Al-Ni and Cu-Zn wires. The maximum energy dissipation is found to be 60-65% in the case of Ni-Ti wire, while Cu-Al-Ni and Cu-Zn wires show energy dissipation of 50-55 and 40-45%, respectively, for the tested model cable. So, the most effective wire is the Ni-Ti wire for suppressing the maximum cable vibration than the other wires.

• 3. The energy dissipation in the cable improved significantly due to the attachment of the SMA wire after the connection of the SMA wire. The energy dissipation in the cable is highly sensitive to the SMA wire measured attachment location along the cable.

• 4. The maximum value of frequency and damping of combined model cable-SMA wire was found at the mid-span location so it is better to attach an SMA wire at the mid-span of the stay cable for the first mode of vibration suppression.

However, full‐scale experimental work may be undertaken to get the actual outcomes.

11 Future Scope

For a more detailed examination of stay cable vibration analysis, further studies are essential to be performed by incorporating SMA dampers with the bridge cable at different inclinations. The SMA can be used in different bracing patterns and can be incorporated with innovative damping devices for mitigating large amplitude vibration in cables of cable-stayed or suspension bridges induced by wind or by the combined action of wind and rain. For cables with a significantly larger force, a larger size of SMA wire may be required. However, the damping effects of SMA wire would deteriorate when the diameter is enlarged. Therefore, an SMA strand with a bundle of thin wires should be applied instead. The investigation is still necessary to examine the effects of ambient temperature on the performance of the real cable with SMA wire based damping device.

Appendix: Schematic Diagram and Frequency Formula

In the proposed work, the stay cable is considered as a taut cable and the schematic diagram of the combined cable and SMA wire has been shown in Fig.

Fig. 19

In-plane inclined stay cable attached with SMA wire

19. In the figure, T represents the cable tension force, m is the mass per unit length, E is the elastic modulus, θ is its inclination, L is the cable length and A is the cross-‐sectional area of the model cable.

The nonlinearity of the cable due to sag can be accounted by introducing a sag parameter, ${\lambda }^{2}$ for the model cable as adopted by Zhou (Ref 20). Considering the parabolic (for simplicity) profile of the cable, the expression for which from the elementary theory is given as (Ref 10)

$$y = \frac{{{\text{mg}}L^{2} }}{2T} \frac{x}{L}\left( {1 - \frac{x}{L}} \right),$$

(1)

and, the sag d at mid-span is given as

$$d = \frac{{{\text{mg}}L^{2} }}{8T}$$

(2)

Considering a length L_e, serving as a fictitious anchor region, the length L_e for an inclined cable, as given by Zhou (Ref 20) is

$$L_{e} = L\left[ {1 + \left( {\frac{{{\text{mg}}L}}{T}\cos \theta } \right)^{2} /8} \right]$$

(3)

Let μ is the friction per unit length in the anchor region, the longitudinal equilibrium requires that μL_e = T.

The extra length of cable in the suspended portion is given as (Ref 45, 46)

$$\int\limits_{0}^{L} {\left( {{\text{d}}s - {\text{d}}x} \right) = } \int\limits_{0}^{L} {\left( {\frac{{{\text{d}}s}}{{{\text{d}}x}} - 1} \right){\text{d}}x}$$

(4)

$$\int\limits_{0}^{L} {\left( {{\text{d}}s - {\text{d}}x} \right) = \frac{1}{2}} \int\limits_{0}^{L} {\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)^{2} {\text{d}}x}$$

(5)

$$\mathop \smallint \limits_{0}^{L} \left( {{\text{d}}s - {\text{d}}x} \right) = \frac{8}{3}\left( {\frac{{\text{d}}}{L}} \right)^{2} .L$$

(6)

This extra length is supplied by the stretching of the cable in the anchor regions, namely

$$2 \times \frac{1}{2}.\frac{{{\text{TL}}_{e} }}{{{\text{EA}}}} = \frac{{T^{2} }}{{\mu {\text{EA}}}}$$

(7)

Therefore, from Eqs. (6) and (7),

$$\frac{{T^{2} }}{{\mu {\text{EA}}}} = \frac{8}{3}\left( {\frac{{\text{d}}}{L}} \right)^{2} .L$$

(8)

This can be recast, with the help of equation (2), into the following equation

$$\frac{{T^{2} }}{{\mu {\text{EA}}}} = \frac{L}{24}.\left( {\frac{{{\text{mg}}L}}{T}} \right)^{2}$$

(9)

Parameter $\lambda^{2}$ for the suspended cable is defined by [46] as

$$\lambda^{2} = \left( {\frac{{{\text{mg}}L}}{T}} \right)^{2} \left( {\frac{{{\text{LEA}}}}{{L_{e} T}}} \right)$$

(10)

From Eq 10$\lambda^{2}$ is a constant (=24). The parameter $\lambda^{2}$ depends on the weight and length parameters and dictates the nature of the response for the standard suspended cable. However, its value is obtained practically. For the cable existing at some inclination, equation (10) is taken as

$$\lambda^{2} = \left( {\frac{{{\text{mg}}L}}{T}\cos \theta } \right)^{2} \frac{{{\text{LEA}}}}{{L_{e} T}}$$

(11)

The value of λ² was designed as 0.108 in this test because sag effects were overcome by introducing high tension in the cable, which corresponded to a stay cable with a length of several hundred meters in engineering practice.

In this study, the frequency for the model cable is adopted from (Ref 20, 47) and is presented as follows:

$$f_{i} = \frac{i}{2L}\sqrt {\frac{T}{m}\left[ {1 + \frac{{2\lambda^{2} }}{{i^{4} \pi^{4} }}\left( {1 + \left( { - 1} \right)^{i + 1} } \right)^{2} } \right]}$$

(12)

Where $f_{i}$ is the modal frequency of the stay cable at i^th mode number.

In addition to damping effects, the presence of SMA wire damper also alters the vibration frequency by restraining the cable at the connection point.

The frequency of the combined system varies depending on the wire position. Moreover, the damping ratio ($\xi$ ) in the cable and combined cable-SMA wire for free vibration can be obtained by the theoretical formula given as follows (Ref 48, 49):

$${\text{The damping ratio}}\, \xi= \frac{1}{2\pi j}\ln \frac{{u_{1} }}{{u_{1 + j} }}$$

(13)

where$\frac{{u_{1} }}{{u_{1 + j} }}$ is the ratio of the first and jth peaks

$${\text{While the damping coefficient is}}\; c = \xi.c_{cr} ,{\text{ and}}\;c_{cr} = 2\sqrt {\left( \hbox{{km}} \right)}$$

(14)

Here, $k = T\pi^{2} i^{2} /2L$ at i = 1 for the first mode.

Moreover, the other symbols indicate that $c_{cr}$ is the critical damping, k and m denote the stiffness and mass of the cable, respectively.