Analysis of mechanical properties in bending processes and optimal design of simple tape spring

3.1 Numerical simulations of tape spring

The geometric model of tape spring is set up and analyzed through finite element software ABAQUS/Standard. The finite element model of tape spring is shown in Figure 2. The material of tape spring is spring steel 65Mn with mass density 7.85 g/cm³, Young’s modulus 200GPa, Poisson’s ratio 0.3. The boundary condition is shown in Figure 3. In order to simulate the folding process of tape spring, two reference nodes, A and B, at either end are connected to the edges of the end cross sections through kinematic constraints. All the degrees of node A are restrained except the rotation about the global z-axis. Node B is only allowed to rotate about the z-axis and translate along the global y-axis. The specified opposite and equal angular displacements with value of 90° are applied on reference nodes defined in section centroid to achieve pure bending. The complicated behavior of tape spring under pure bending is analyzed by using an improved arc-length scheme which is introduced into the nonlinear finite element procedure. The numerical results are obtained in Table 1.

Figure 2

FE model of tape spring.

Figure 3

Boundaries for tape spring in the bending process.

Table 1

Comparison of experiment and simulation results in the bending process.

Rotation(°)	The equal sense bending process		The opposite sense bending process
	Experiment	Simulation	Experiment	Simulation
1
10
20
30
40
50
60
70
80
90

3.2 Experiments of tape spring

In order to verify the validation of the numerical models, the properties of tape spring during folding process are experimentally investigated with a specially designed test rig, shown in Figure 4.

Figure 4

Test apparatus for tape spring ① Angle sensors of driving shaft ② Angle sensors of driven shaft ③ Tube holders ④ Drive motor ⑤ Tape spring ⑥ Guide rail ⑦ Computer.

The test rig is equipped with a driven shaft and a driving shaft that being fixed to a rigid base. The driven shaft is the one passive driven shaft, and the driving shaft is the other active driving shaft. All the freedoms of driving shaft are limited except the rotation around itself. The driven shaft is allowed to move along the horizontal direction, which is mounted on the guide rail. Two angle sensors are connected to the computer on driving and driven shafts, which are used to monitor the rotational speed and rotation angles. As the rotation angle rises, the driving shaft rotates slowly, which driven by the motor, and the driven shaft rotates and moves along the slide way. The behavior of simple tape spring during the folding process is obtained, which is shown in Table 1.

3.3 Comparison of experiments and simulations

The properties of tape spring with initial design during folding process are investigated by means of experiment and numerical simulation. The process of equal and opposite folding is studied and the results of experiment and numerical simulation are compared in Table 1. The deformation and stress contours show that the tape spring has a large displacements and large rotations under the moment at each end. Both of the equal and opposite folding indicates that the longitudinal curvature changes from 0 to 1/R and the transverse curvature varies from 1/R to 0. During the equal process, the tape spring showed a smooth folding as the raise of rotation angle, however, in the case of opposite folding, a sudden snap through buckling occurred when the rotation angle is increasing to the critical value. The experimental results agree well with the numerical results.

Figure 5 shows that the maximum stress varies with the bending angle of folding process. The maximum stress increases to 997.1MPa during opposite folding process when the rotation angle is around 20°, and then slightly slops down due to the snap through buckling. The equal folding also has high value of maximum stress, which is 892.9MPa. The results obtained in these two cases show that the property of tape spring is including the geometric nonlinearity and material nonlinearity. Therefore, the linear buckling analysis is already unsuitable for predicting the buckling behavior of tape springs. The improved arc-length scheme is adapted to solve the nonlinear problem.

Figure 5

Variation curve of maximum stress with the bending angle.

The moment rotation profiles obtained by means of simulation and experiment are drawn in Figure 6. The experiments for tape spring include two main steps, the first step is to fold the tape spring to a fully folded configuration; the second step is deployment of tape spring, starting from the completely folded state. In the picture, the starts for both deployment simulation and experiment are corresponding to the rotation of90∘. For equal sense deployment process, the steady moments of simulation and experiment agree well with each other when the rotation changes from 90∘to20∘, in which the steady moments from simulation and experiment are 0.02723N.m and 0.0219N.m, respectively; while in the range of rotation from 20∘to2.5∘, the moment variation in simulation is more smooth than the experiment; the simulation captures the tape spring’s snap back at the rotation of 2.5∘, the corresponding peak moment is 0.43365N.m. Regarding the opposite sense, both simulation and experiment have the similar moment variation as rotation decreases from 90∘to20∘, in which the steady moments for simulation and experiment are respectively are 0.0465N.m and 0.0521N.m; in the range of rotation from 20∘ to2.7∘, the moment in experiment has a relatively bigger slop; when the rotation at2.7∘, the moment in simulation has a maximum which corresponds to the snap back of tape spring, and the maximum is 0.78956N.m.The results obtained have a good agreement with the estimates in Seffen et al. [3].

Figure 6

Comparison of moment-rotation profiles during deployment.

4.1 Thickness effects

The effects of tape spring thickness on the steady moment and maximum strain energy are investigated; meanwhile the maximum strain energy is obtained in the fully folded configuration as shown in Figures 7 and 8. It can clearly be seen that the steady moment and maximum strain energy grow rapidly as thickness rises, which indicates that the driving capability of simple tape spring get enhanced remarkably. The variations of maximum strain energy and steady moment with thickness are shown in Figure 7 and Figure 8.

Figure 7

Variation in steady moment with thickness.

Figure 8

Variation in maximum strain energy with thickness.

When the thickness of tape spring varies from 0.05 mm to 0.3 mm while keeping other parameters using initial design, as shown in Table 2 the steady moment has a high growth rate which is more than 67.3 % during equal folding process, while the opposite gives a value of 68.2 %, see Figure 7. The values of maximum strain energy stored for deployment of tape spring increase by more than 68.8 % and 68.1 %, corresponding to equal and opposite folding, see Figure 8.

4.2 Subtended angle effects

The variations of maximum strain energy and steady moment with the subtended angle of tape spring are obtained, see Figures 9 and 10. When the subtended angle varies from 25°to 120°with other parameters using initial design, as listed in Table 3 it is obvious that the driving capability of simple tape spring has an vast improvement as the growing of subtended angle. The steady moment has a growth percentage of 40.5 % for equal folding and the value of opposite folding is 44.1 %. For the maximum strain energy in equal and opposite bending of the tape spring, it will be a significant improvement in growth rate 48.4 % and 53.7 % as the rising of subtended angle respectively.

4.3 Length effects

For the end of tape spring is held by two rigid clamps, the actual working length is L₁. Each of the rigid clamps has initial length of 25 mm, the initial design of parameters is adopted apart from L₁ changing from 80 mm to 180 mm.The variation tendencies of steady moment and maximum strain energy with the length of tape spring are investigated, shown in Figures 11 and 12. Whether equal or opposite bending of the tape spring, a similar variation tendency will be obtained for the steady moment and strain energy. The two indices drop down when L₁ varies from 80 mm to 160 mm with 10 mm increments, and an approximately stable state has been achieved when L₁ arrived at 160 mm. It should be noted that the improvement of length is not beneficial for the driving capability of tape spring, which also can be found in Table 4.

4.4 Cross section radius effects

The variation tendencies of driving capability with respect to cross section radius are studied and the other parameters use initial design. When the radius varies from 8 mm to 22 mm with 2 mm increments, the energy and steady moment responses are listed in Table 5, it can be seen that both of the maximum strain energy and the steady moments have a slight falling, once the radius increases to more than 12 mm, they begin to rise. The relationships between cross section radius and steady moment are obtained in Figure 1 and the variation of maximum strain energy is given in Figure 14.

As discussed above, the steady moment and maximum strain energy are treated as two crucial indices to measure mechanical properties of simple tape spring, and both of them have close relationships with thickness, subtended angle, length and cross section radius of tape spring. From Figure 8–Figure 14, it can be found that the thickness and subtended angle of tape spring have a huge influence on the driving capability; however the length and cross section radius weaken the driving capability in a certain degree. So it is necessary to find an optimal geometry for the best driving performance.

Figure 9

Variation in steady moment with subtended angle.

Figure 10

Variation in maximum strain energy with subtended angle.

Figure 11

Variation in steady moment with length.

Figure 12

Variation in maximum strain energy with length.

Figure 13

Variation in steady moment with transverse radius.

Figure 14

Variation in maximum strain energy with transverse radius.

5.1 Establishment of optimal model

During the folding process, the stress concentration occurs at the local deformation area, and the maximum stress has already exceeded the allowable stress, as investigated in previous part. All of the four key geometric parameters have effects on the driving capability to some different degrees. The optimal design will be studied to find an optimal geometry to get the best driving capability. Therefore, it aims at maximum strain energy during the tape spring hinge folding, and subjected allowable stress of tape spring. The optimal model is established as follows:

Where x=[x1,……,xi]T(i=1,2,3,4)denotes design variables, U(x) presents objective function. σmax(x)and σsare the maximal stress and allowable stress of tape spring. L1=140mm and x_i，xˉi are the lower and upper limit design variables.

5.2 Explicit objective function and constraint function

RSM is a collection of mathematical and statistical techniques, which used for obtaining the relationship between the input and output of complex system. In this paper, RSM is introduced to solve the problems of optimal design. The basic idea is to construct a polynomial with explicit expression to represent implicit function about design variables, so as to find out the optimal response value of the input variables.

The material of tape spring is spring steel 65Mn with mass density 7.85 g/cm³, Young’s modulus 200GPa, Poisson’s ratio 0.3. x₁ is the thickness of the tape spring ranging from 0.05 mm to 0.25 mm, x₂ represents the subtended angle of section varying from 70° to 90°, x₃ is the cross section radius changing from 14 mm to 22 mm, x₄ represents the total length of tape spring ranging from 120 mm to 200 mm.

According to the design area, the central principle of expansion is used to obtain the initial center point [x₁,x₂,x₃,x₄]^T=[0.15,90,18,140]^T and fitting radius δ⁰=[0.05,10,2,10]^T. Center symmetric method is adopted to design test sample points. In the range of ±μδ⁰(where μ = 1,2), the center symmetric design points are selected to obtain 17 test sample points, initial center point included. And then, the responses of the maximum strain energy and stress are obtained by means of simulation with finite element software ABAQUS. The results of 17 design points are listed in Table 6.

The fitting of response surface function with maximum difference minimization method is introduced to construct corresponding response surface model, and the complex nonlinear relationships between the objective function and design variables are approximatively expressed [16]. The objective function can be written in terms of a series basic function as follows:

Where α1, …, αi+1is undetermined coefficient.

Thus the quadratic response polynomial function expression of objective function obtained as follows:

Where U(x₁,x₂,x₃,x₄) presents the maximum strain energy.

Then quadratic response polynomial function of constraint function is obtained as follows:

Where σ(x₁,x₂,x₃,x₄) is the maximum stress that should be less than the allowable stress σ_s= 1000MPa.

5.3 Precision test of response surface fitting

In order to ensure the accuracy and applicability of the response surface model, the precision of response surface fitting for objective function and constraint function should be tested. If the fitting precision could not satisfy the required one, it is necessary to add the design points and adjust the initial center point. Then setting μ=0.5, testing the boundary points of the effective range firstly, and adding the boundary points of the initial range for special point test. The added 8 sample points are shown in Table 6, and the relative error between the FEA results and RS results for the 8 added sample points are listed in Table 6. According to the definition of response surface, fitting precision is given as follows:

Where n_e is the sample size for testing, y_i represents the real response value, ŷ_i is the response value obtained by observation model,yˉrepresents the mean value of real response value.

RMSE(root mean square error) represents the difference between response surface fitting value and test value. The smaller the RSME, the higher accuracy of response surface fitting. The relative errors of maximum strain energy and maximum stress between the FEA results and RS results are shown in Table 7. The fitting error of objective function is 0.6281 %, and the fitting error of constraint function is 0.8082 %, both of them are less than 1 %, which can meet the precision demand.

5.4 Optimal results and discussion

In order to solve the optimal model conveniently, the general nonlinear programming model is introduced as follows:

Where H,cdenote parameter matrixes in the standard form of objective function.

The optimal model is translated into a nonlinear programming and solved by the interior point method based on MATLAB software platform. The number of the variable is reduced and the model’s scale is minified. In this paper, the convergence criterion is chosen as follows:

Where x^k is the design variable of the k iteration, r_k represents obstacle factor, r_k>0，r₁>r₂> … >r_k> … >0，ε is the convergence precision, ε = 0.001.

If r_k+1<r_k (r_k+1=r_k/c, c≥2), k=k+1, it will be back to the previous step iteration cycle, and does not jump out of the loop until convergence criteria is satisfied. Otherwise, it will directly output the final optimal results.

After 33 iterations, the optimal solution is obtained and the iteration history of objective function is shown in Figure 15.

Figure 15

Iteration history of the objective function.

The comparisons between original and optimized results are carried out, as depicted in Table 8. The maximum stress after optimization is 999.1MPa, which is reduced by 32.81 % before optimization. The maximum strain energy after optimization is 0.25425J, which is reduced by 1.52 % before optimization. It is obvious that the best driving capability of simple tape spring is achieved and the optimal geometry size is obtained under the condition of meeting the allowable stress.

Figures 16 and 17 are the stress distributions of simple tape spring when fully folded before and after optimization obtained with ABAQUS. Figure 16 shows that the maximum stress mainly locates at the edge of the central part before optimization, the maximum stress after optimization is 993.2MPa, the maximum strain energy after optimization is 0.2550J, which are almost equal with the values obtained with RSM. Whereas the optimal design has a homogeneous stress distribution, shown in Figure 17. The stress concentration of fully folded tape spring is improved remarkably and the maximum stress of tape spring is controlled within the range of allowable stress. The optimal program flowchart of simple tape spring is shown in Figure 18.

Figure 16

The stress distribution before optimization.

Figure 17

The stress distribution after optimization.

Figure 18

Optimal program flowchart of simple tape spring.