# Characteristics Analysis and Testing of SMA Spring Actuator

Authors: Jianzuo Ma; Haolei Huang; Jin Huang
Journal: Advances in Materials Science and Engineering
 (2013)

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## Abstract

The biasing form two-way shape memory alloy (SMA) actuator composed of SMA spring and steel spring is analyzed. Based on the force equilibrium equation, the relationship between load capacity of SMA spring and geometric parameters is established. In order to obtain the characteristics of SMA spring actuator, the output force and output displacement of SMA spring under different temperatures are analyzed by the theoretical model and the experimental method. Based on the shape memory effect of SMA, the relationship of the SMA spring actuator's output displacement with the temperature, the stress and strain, the material parameters, and the size parameters is established. The results indicate that the trend of theoretical results is basically consistent with the experimental data. The output displacement of SMA spring actuator is increased with the increasing temperature.


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## Body

## 1. Introduction

Shape memory alloy (SMA) is known as a kind of new intelligent material. SMA may undergo mechanical shape changes at relatively low temperatures, retain them until heated, and then come back to the initial shape [1, 2]. The outstanding quality characteristics of SMA are shape memory effect (SME) and super elasticity (SE) [3]. The shape memory effect, which allows the deformed material to recover a memorized shape when heated above the transformation temperature, can be exploited effectively in microrobots, automobile, automatic adjustment devices, aerospace, home appliances and daily necessities, [4–8] and so on.An actuator based on these materials is made up of an SMA element that works against a contrasting element (a weight or other constant force, a conventional spring, or a second SMA element). At low temperature, the contrasting element overcomes the resistance of the easily deformable SMA element. The actuator is activated by heating the SMA element above the transformation temperature. The resulting increase in stiffness enables the SMA element to overcome the resistance of the contrast, thus generating useful displacements and producing mechanical work [3, 9–11].In this paper, we present the biasing form SMA actuator, which is able to generate displacement and force. Based on the force equilibrium equation, the output force and output displacement of SMA spring under different temperatures are analyzed by the theoretical model and the experimental method. Based on the shape memory effect of SMA, the relationship of the SMA spring actuator’s output displacement with the temperature, the stress and strain, the material parameters, and the size parameters is established. The output displacement of SMA spring actuator is increased with the increasing temperature.


## 2. Properties of SMA

The most commonly used SMA elements for actuators are helical springs, which for this form produce a large displacement. The force that a spring of any material produces at a given deflection depends linearly on the shear modulus of the material. SMAs exhibit a large temperature dependence on the material shear modulus. The relationship between shear modulus and temperature for SMAs is given by(1)G={GMwhenT<MfandT<As,G(γ,T)whenMf≤T≤Af,GAwhenT>AfandT>Ms,
where G is the shear modulus of SMAs. T is temperature and Ms, Mf, As, and Af are the start and finish transformation temperatures of martensite and austenite, respectively, as shown in Figure 1. GM and GA are the shear moduli of martensite and austenite, respectively. When Mf≤T≤Af, in absence of stress, shear modulus of SMAs can be expressed approximately as
(2)G(T)=GM+GA-GM2[1+sinϕ(T-Tm)].Figure 1
Transformation temperatures of martensite and austenite.In the process of heating,Tm=(As+Af)/2, ϕ=π/(Af-As); in the process of cooling, Tm=(Ms+Mf)/2, ϕ=π/(Ms-Mf).When the SMA wire is heated or cooled, the heat balance equation is(3)ρ1cVdTdt=-hA(T-Tf),
where ρ1 is the mass density of SMA, c is the specific heat, V is the volume of SMA exposed in air, t is the time, h is the heat exchange coefficient, A is the superficial area of SMA, and Tf is the temperature of airflow.IfT=T0, when t=0, the temperature variation of SMA wire with time is
(4)T=(T0-Tf)e-t/φ+Tf,
where T0 is the initial temperature and φ is the time constant of SMA wire, φ=ρ1cV/hA.If the material and structural parameters of SMA have been determined, the time constant is inversely proportional to the heat exchange coefficient. Under three different heat exchange coefficients, the temperature variation of SMA wire with the around airflow temperature is shown in Figure2. As shown in Figure 2, in a different heat exchange coefficient, the temperature of SMA wire changes faster when the time constant is smaller and the lag of time is shorter. When the time constant is less than 2.5, the lag time is less than 2 seconds.Figure 2
The temperature variation of SMA wire with the around airflow temperature under three different heat exchange coefficients.


## 3. Operational Principle of SMA Actuator

The SMA drive element uses the properties of low yield stress at martensitic state and returns to the high yield stress at austenite phase state when heated. Thus, the action form of a single SMA part is one-way. To obtain two-way characteristics of SMA elements, the structures of differential form and biasing from are used commonly. The differential form uses two or more SMA elements to obtain the two-way characteristics. The biasing form combines the one-way SMA with other parts to obtain two-way characteristics, shown in Figure3, with the SMA helical spring working against a conventional steel spring (referred here as the “biasing” spring). At low temperatures, the steel spring is able to completely deflect the SMA spring to its compressed length. When increasing the temperature of the SMA spring, it expands, compressing the steel spring and moving the push rod.Figure 3
The operational principle of the SMA actuator.


## 4. Property Analysis of SMA Spring

Relative to the free length of the spring, the SMA spring provides a large action stroke, shown in Figure4.Figure 4
A compression helical SMA spring.The expression for shear stress in an SMA spring is described as(5)τ=κ8FDπd3=κ8FCπd2,
where the axial load is F, D is the average diameter of the spring, d represents the wire diameter, C is the spring index, C=D/d, and κ is known as the Wahl correction factor applied:
(6)κ=4C-14C-4+0.615C.Shear stressτ has a relationship with shear strain γ which is
(7)τ=G·γ.The stretch of spring under the loadF is
(8)δ=8FD3nd4G,
where n is the number of turns in the spring.The relationship between compressed lengthδ and shear strain γ for SMA spring is given by
(9)δ=nπD2dγ.The wire diameter for the actuator can be obtained from (5) for acceptable values of C ranging from 3 to 12:
(10)d=κ8FCπτ.The number of turns in the spring can be obtained from (9):
(11)n=ΔδdπΔγD2,
where Δδ represents the stroke of the actuator and Δγ is the strain difference at high and low temperatures:
(12)Δγ=γL-γH.


### 4.1. The Output Force of SMA Spring under Different Temperatures

The experimental system for the output force of SMA spring versus temperature under the constraint of displacement is shown in Figure5 and the experimental device is shown in Figure 6.Figure 5
The experimental system for output force.Figure 6
The experimental device for output force.As shown in (8), when Mf≤T≤Af, the axial load at temperature T can be expressed as
(13)F(T)=δ(T)G(T)δLGLFL.The axial load at low temperature is expressed as(14)FL=d4GL8D3nδL.When the axial displacement of SMA spring is restricted, the compressed length of SMA spring is kept as(15)δ(T)=δL.WhenMf≤T≤Af, the output force at temperature T can be obtained from (13), (14), and (15) as
(16)F(T)=G(T)d48D3nδL.In this study, Ti-49.8at.%Ni SMA spring is used, shown in Figure7; its start and finish temperatures of the martensitic and austenitic phase transformation are Ms=78°C, Mf=50°C, As=74°C, and Af=95°C, respectively. The shear moduli of martensite and austenite are GM=7.5GPa and GA=25GPa, respectively. The wire diameter of SMA spring is d=1mm, the angle of inclination is α=6°, the diameter of SMA spring is D=8.6mm, and the number of turns is n=7. When δ=15mm, the theoretical and the experimental results of the relationship between the output force and temperatures of SMA spring are shown in Figure 8. The trend of theoretical results is basically consistent with the experimental data. The output force is increased with the rising of temperature.SMA spring sample.
(a)
(b)Figure 8
Output force versus temperature under the constraint of displacement.


### 4.2. The Output Displacement of SMA Spring under Different Temperatures

The experimental system for output displacement of SMA spring under different temperatures is shown in Figure9 and the experimental device is shown in Figure 10.Figure 9
The experimental system for output displacement.Figure 10
The experimental device for output displacement.As shown in (9), the compressed length can be expressed as
(17)δ=πD2nγd.
When Mf≤T≤Af, shear strain γ is
(18)γ=GLγmaxG.The output displacement can be obtained from (17) and (18) as
(19)Δδ=πD2nd(1-GLG)γmax.The typical SMA spring sample is shown in Figure7; when the maximum shear strain is γmax=2%, the theoretical and the experimental results of the relationship between the output displacement and temperatures of SMA spring are shown in Figure 11. The trend of theoretical results is basically consistent with the experimental data. The output displacement is increased with the rise of temperature.Figure 11
The output displacement of SMA spring versus temperature under constant load.


## 4.1. The Output Force of SMA Spring under Different Temperatures

The experimental system for the output force of SMA spring versus temperature under the constraint of displacement is shown in Figure5 and the experimental device is shown in Figure 6.Figure 5
The experimental system for output force.Figure 6
The experimental device for output force.As shown in (8), when Mf≤T≤Af, the axial load at temperature T can be expressed as
(13)F(T)=δ(T)G(T)δLGLFL.The axial load at low temperature is expressed as(14)FL=d4GL8D3nδL.When the axial displacement of SMA spring is restricted, the compressed length of SMA spring is kept as(15)δ(T)=δL.WhenMf≤T≤Af, the output force at temperature T can be obtained from (13), (14), and (15) as
(16)F(T)=G(T)d48D3nδL.In this study, Ti-49.8at.%Ni SMA spring is used, shown in Figure7; its start and finish temperatures of the martensitic and austenitic phase transformation are Ms=78°C, Mf=50°C, As=74°C, and Af=95°C, respectively. The shear moduli of martensite and austenite are GM=7.5GPa and GA=25GPa, respectively. The wire diameter of SMA spring is d=1mm, the angle of inclination is α=6°, the diameter of SMA spring is D=8.6mm, and the number of turns is n=7. When δ=15mm, the theoretical and the experimental results of the relationship between the output force and temperatures of SMA spring are shown in Figure 8. The trend of theoretical results is basically consistent with the experimental data. The output force is increased with the rising of temperature.SMA spring sample.
(a)
(b)Figure 8
Output force versus temperature under the constraint of displacement.


## 4.2. The Output Displacement of SMA Spring under Different Temperatures

The experimental system for output displacement of SMA spring under different temperatures is shown in Figure9 and the experimental device is shown in Figure 10.Figure 9
The experimental system for output displacement.Figure 10
The experimental device for output displacement.As shown in (9), the compressed length can be expressed as
(17)δ=πD2nγd.
When Mf≤T≤Af, shear strain γ is
(18)γ=GLγmaxG.The output displacement can be obtained from (17) and (18) as
(19)Δδ=πD2nd(1-GLG)γmax.The typical SMA spring sample is shown in Figure7; when the maximum shear strain is γmax=2%, the theoretical and the experimental results of the relationship between the output displacement and temperatures of SMA spring are shown in Figure 11. The trend of theoretical results is basically consistent with the experimental data. The output displacement is increased with the rise of temperature.Figure 11
The output displacement of SMA spring versus temperature under constant load.


## 5. Analysis of SMA Actuator

The scheme of the proposed actuator with an SMA spring and conventional steel against spring is illustrated in Figure3, where at low temperature the SMA spring will be compressed and when heated will extend with a pushing actuation. For the SMA actuator in Figure 3, the axial load of SMA spring F has the relationship with the compressed length of SMA spring δ as follows:
(20)F(T)δ(T)G(T)=FLδLGL,F(T)=FL+FH-FLΔδS(T),
where F(T), δ(T), and G(T) are the axial load, compressed length, and shear modulus of SMA spring at temperature T, respectively; FL, δL, and GL are the axial load, compressed length, and shear modulus of SMA spring at low temperature, respectively; FH is the axial load at high temperature; and S(T) is the output displacement of SMA spring actuator:
(21)S(T)=δL-δ(T).The output displacement of SMA spring actuator can be obtained from (1), (2), (9), (20), and (21)
(22)S(T)=(G(T)-GL)γL(d/nπD2)G(T)+((FH-FL)/ΔδFL)G(T)γL.The experimental system for output displacement of SMA actuator under different temperatures is shown in Figure12. The SMA helical spring works against a conventional steel spring to obtain the two-way SMA actuator. The typical SMA spring sample is shown in Figure 7. The stroke of the actuator is Δδ=10mm. The effect of temperature on the output displacement of SMA spring actuator is analyzed by the theoretical model and the experimental method, shown in Figure 13. The axial loads of SMA spring at low and high temperatures are FL=10 N and FH=30 N, respectively. The low temperature shear strain is γL=1.5%. As shown in Figure 13, the output displacement of SMA spring actuator is increased with the increasing temperature.Figure 12
The experimental system for output displacement of SMA actuator versus temperature.Figure 13
The output displacement versus temperature.


## 6. Conclusions

The characteristics and test method of SMA spring and SMA actuator are analyzed in this paper. The output force and output displacement equations of SMA spring are derived. The output force and output displacement are increased with the rise of temperature. The relationship of the SMA spring actuator’s output displacement with the temperature is investigated theoretically and experimentally. With the increase of the temperature acting on SMA actuator, the output displacement of SMA spring actuator is increased proportionally.



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*Source: 823594-2013-10-23.xml*