Vibration Analysis of Uniform and Tapered Composite Beams with Embedded Shape Memory Alloy

In this study, laminated composite materials were hybridized with E-glass fiber and Nitinol (Nickel-Titanium) wires. Hand lay-up technique was used to prepare the samples, epoxy resin type (Sikadur 52 N) was used as matrix reinforced by one fiber from E-glass fiber woven roving with embedded nitinol wires with a diameter 0.5 mm for samples and number of wires such as 0, 1, 3, 5 and 9 to find the effect of the number of wires on the natural frequency. The samples were fixed as a cantilever beam. The effects of increasing the number of nitinol wires, the diameter of nitinol wires, the length of the cantilever beam and the thickness of beam on the natural frequencies of the beam were studied. Also, the effects of the tapered in width side and thickness side on the natural frequencies of cantilever beam were studied. The results showed that the increasing in the number of nitinol wires and the diameter of nitinol wires lead to decrease the natural frequency in martensite phase and increase the natural frequency in austenite phase. Also, the increasing in thickness of beam and width ratio of the beam lead to increase the natural frequency. As well as, the increasing in the thickness ratio leads to increase the first natural frequency and decrease the second and third ones. In addition, the increasing in the length of the beam decreases the natural frequency.


Symbol
The ratio of the total cross sectional area of wires to the cross sectional area of the beam.

𝜌 𝑛
The density of the nitinol Kg/m 3

𝜌 𝑏
The density of the specimen Kg/m 3

𝜔 𝑛
The natural frequency rad/sec

Width ratio
The ratio width of root side of cantilever beam to width of free side of cantilever beam (bin/bout) Thickness ratio The ratio thickness of root side of cantilever beam to thickness of free side of cantilever beam (hin/hout)

Introduction
Recently, Shape Memory Alloys (SMAs) have been on the front of research due to that the SMAs are unique alloys having ability to remember an original shape after being deformed.The natural frequencies have a big influence on the design of system that exposed to vibrations because when the system worked at a frequency near to the natural frequencies, the acceleration will be at the highest value, which lead to the failure of the system.Lau et al. [1] estimated the natural frequency of glass fiber composite beams with embedded shape memory alloy (SMA) wires.The results showed the natural frequencies of all the beams decrease with increasing number of SMA wires in martensitic phase.Dezfuli et al. [2] used the nitinol wires as reinforcement, into plate made up Aluminum as a matrix.The results showed the phase transformation starting from martensite to austenite phase and that led to increasing in the stiffness of the plate, which results in increasing in the natural frequency.M. Yuvaraja and M. Senthilkumar [3] compared between utilizing piezoelectric (PZT) based and SMA wires composites on the vibration characteristics.The smart composite cantilever beam contains a glass fiber reinforced polymer (GFRP) with attached SMAs externally and with surface bonded PZT.The results demonstrated using SMA wires is more effective than using PZT.Cem and Mustafa [4] investigated the free vibration analysis on a blade of Air X 4140W horizontal axis wind turbine.4140 steel, shape memory alloys (Ni-Ti, Cu-Zn-Al and Cu-Al-Ni) in blade root connection was used.The results showed that the maximum total deformation was observed in the fifth mode of the natural frequency for the blade root connection from Cu-Zn-Al alloy.Gupta et al. [5] studied the effect of shape memory alloys to increase the damping of glass fiber reinforced plastic (GFRP) composites.The results exhibited the damping ratio of SMA hybrid composite beam was found to be higher as compared to the pristine and steel hybrid GFRP composite beam.
The .main target .ofthis work .is to study the effect of the number and diameter of nitinol wires and the length, thickness, width ratio and thickness ratio of cantilever beams on the natural frequency.

Used Materials
The materials used to prepare the samples consist of epoxy resin as a matrix (type Sikadur 52 N), Table (4.1)lists the properties of the epoxy used in this work [6].And two reinforced composite materials, which are shape memory alloy wires (Nitinol), a near ideal 50/50 combination of nickel and titanium.The diameters used of nitinol wires are 0.5 mm.Table (4.2) contains the physical properties of Nitinol [7] and E-glass fiber woven roving.Table (4.3)lists the properties of the fibers type E-glass [8].

Samples
The technique used to form the samples is hand lay-up.The uniform samples have dimensions (220 mm x 50 mm x 4 mm), as shown in figure (1), while the tapered samples are designed as follows: the width of root side is constant 50 mm and the width of free side is variable (20 mm, 30 mm and 40 mm), the thickness is 4 mm and the length is 220 mm, in all samples the length decreases 20 mm for fixation.

Figure (1): Samples of composite materials with embedded nitinol wires 2.3 Vibration Test
The vibration system was used to find the natural frequency, see figure (2).

Figure (2): Vibration system
The experimental frequency-amplitude curve was found from Fast Fourier Transformation (FFT) screen by using LABVIEW program in vibration system test, see figure (3).

Rule of Mixture
This rule was used to estimate the density and modulus of elasticity.

Exact solution of natural frequency of cantilever beam
When the beam is uniform and assumed cantilever beam as Euler-Bernoulli beam as shown in figure (4).

Figure (4): Cantilever beam
The natural frequencies can be found from the equation [10]:

Finite Element Analysis (FEA)
ANSYS Workbench software, version 17, is for the finite element analysis to investigate the natural frequencies and its vibration mode shapes, see figure (5).

The Density and Modulus of Elasticity
The rule of mixtures was used to calculate the density and the modulus of elasticity for uniform beam.
In figures (6 and 7), the samples contain epoxy reinforced by E-glass fiber layer with embedded nitinol wires.In figure (6) the density increased by 7.74% if the number of embedded nitinol wires rose from 1 to 20 wires.And in figure (7) the modulus of elasticity increased by 5.02% if the number of embedded nitinol wires rose from 1 to 20 in the phase of martensite, also the modulus of elasticity increased by 13.914% the number of embedded nitinol wires raised from 1 to 20 in the phase of austenite.

The natural frequency of uniform cantilever beam
Table (4) reveals the verification of the results, which include the theoretical results of the density, the theoretical results of the modulus, and the theoretical, numerical and experimental results of the vibration natural frequency modes.Theoretical results of natural frequency were found from the equation (6).

The effect of the numbers of nitinol wires
The samples of the figures (8), ( 9) and (10) are containing epoxy and one E-glass fiber layer with changing the numbers of wires (diameter is 0.5 mm) that embedded in the samples.The results of these figures were calculated theoretically by equation (6).Figures (8), ( 9) and (10) illustrate the behavior of changing the beam natural frequencies which embedded with SMA wires in two phases, the first is nitinol wires totally martensite, and the second is nitinol wires totally austenite, for martensite case, the initial 3 natural frequencies decreased by 1.44% if embedded nitinol wires raised from 1 to 20 wires.And in austenite phase, the initial 3 natural frequencies raised by 3.39% if embedded nitinol wires raised from 1 to 20.

The effect of the diameter of nitinol wires
Figures (11), ( 12) and ( 13) elucidate the behavior of changing the natural frequencies of epoxy reinforced by one E-glass fiber layer and embedded by three nitinol wires with increased diameters of these three wires at martensite and austenite phases.The results of these figures were calculated theoretically by equation ( 6).
for martensite case, the initial 3 natural frequencies decreased by 6.79% if diameter of the embedded nitinol wires was raised from 0.5 mm to 3.3 mm.And in austenite phase, the initial 3 natural frequencies raised by 13.20% if diameter of the embedded nitinol wires was increased from 0.5 mm to 3.3 mm.

The effect of the length of the beam
Figure ( 14) displays the trend of changing the cantilever natural frequencies of composite of epoxy reinforced via one E-glass fiber layer with increased length of beam.The results of this figure were calculated theoretically by equation (6).In this figure, if the beam length is increased, the natural frequencies will decrease in all modes.The initial 3 natural frequencies reduced by 75% if increasing length of beam from 200 mm to 400 mm.Table (5) show of the theoretical results of the density, the theoretical results of the modulus of elasticity, the verification from the experimental natural frequency, and with numerical methods by using ANSYS program.And, in all these tables, the cantilever beam is containing epoxy reinforced by E-glass fiber layer.

Conclusions
The major goals of this work are to study the effect of the geometrical and mechanical properties on the natural frequency of uniform and tapered cantilever beam: 1. Increasing the number and the diameter of nitinol wires which embedded in the composite beam leads to decrease the natural frequency in martensite phase and leads to increase the natural frequency in austenite phase.2. Increasing the thickness of the cantilever composite beam leads to increase the natural frequencies.
Also, increasing the length of the beam leads to decrease the natural frequencies.3. Increasing width ratio of the tapered cantilever beam embedded with shape memory alloy leads to increase the natural frequencies.

Figure ( 8
Figure (8): The variation of first natural frequencies with numbers of embedded nitinol wires

Figure ( 10
Figure (10): The variation of third natural frequencies with numbers of embedded nitinol wires

Figure ( 14 4 . 2 . 4 Figure ( 15
Figure (14): The variation of the first three natural frequencies with length of beam 4.2.4The effect of the thickness of the beam Figure (15) clarifies the relation between the thickness of beam containing epoxy and one Eglass fiber layer with the first three natural frequencies.The results of this figure were calculated theoretically by equation (6).In this figure the increasing in thickness leads to increasing in the natural frequencies.The first three natural frequencies rose by 50% if the thickness was raised from (2-4) mm.

Figure ( 16 ) 2
Figure (16)  shows the relation between first three natural frequencies with width ratio of cantilever beam contain epoxy and E-glass fiber layer by ANSYS program.In this figure the first three natural frequencies increased by 22.9%, 7.9% and 2.7%, respectively if the ratio width was raised from (1-2.5) mm.

Figure ( 17 )
Figure (17): The variation of first three natural frequencies with thickness ratio (hin/hout)