Nagasawa, Kaneko and Adachi, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.13, No.1 (2019)
exponential coefficient p 1 of bending resistance was almost equal to that of uni-axial tensile displacement. This means that the tensile relaxation of in-plane direction is a major factor in the bending resistance. Regarding the relationship between the release angle T 2,1 and the elapsed release time t 2ep , the linear approximation with the logarithmic term ln( t 2ep ) was introduced (Nagasawa, et al., 2016) as Eq.(3),(4). Here, the intercept b 0 was defined as T 2,1 (1) (at t 2ep =1s), b 1 was the gradient coefficient of Eq.(3), and the exponential coefficient p 2 was defined as the ratio of b 1 / b 0 . They describe the creep-recovery characteristics of folded angle.
T 2,1 = b 1 ln(t 2ep ) + b 0 T 2,1 / b 0 = ߠ ҧ ଶǡଵ = 1 p 2 ln( t 2ep ) , p 2 = b 1 / b 0
(3)
(4)
Since the transient response seems to be caused by the relaxation and creep-recovery characteristics of paperboard during the folding process, the effect of fixture’s rotational velocity Z (folding) and Z ' (unfolding) on the release angle T 2,1 ( t 2ep ) was investigated for the range of t 2ep =0~10s when J = 0.6. The synchronized condition of Z = Z ’ was mainly investigated and its value was chosen as 0.02, 0.03, 0.05, 0.1, 0.2, 0.3 and 0.4 rps (0.13, 0.19, 0.31, 0.63, 1.26, 1.88, 2.51 rad ή s ). For the sake of comparison of asynchronous condition, when the returning back velocity was chosen as a constant of Z ’=0.2 rps (1.26 rad ή s ), the folding velocity Z was chosen as 0.02, 0.03, 0.05, 0.1, 0.2, 0.3 and 0.4 rps (0.13, 0.19, 0.31, 0.63, 1.26, 1.88, 2.51 rad ή s ). 3. Results and discussion 3.1 Response of bending moment with rotational velocity In the synchronized condition of Z = Z ’= 0.2 rps (1.26 rad ή s ), Fig.8 illustrated a representative case of bending moment resistance M with the folding angle T . Since the intermediate stage (20°< T <90°) appeared to be a sort of creep response of Maxwell type two-element model in Fig.8, the values of M p1 and M 90,1 (0) were expected to increase with Z . When varying the velocity Z = Z ’), the maximum peak bending moment M p1 , and the relaxed bending moment at t 1ep = 0s, 1s with 4 = 90°: M 90,1 (0), M 90,1 (1) were measured and shown in Fig.11. Eq. (5), (6), (7) were derived as linear approximations with the logarithmic term ln( Z /0.2) from the experimental result. M p1 = 0.013 ln( Z /0.2) + 0.244 (5) M 90,1 (0) = 0.0061 ln( Z /0.2) + 0.215 (6) M 90,1 (1) = a 0 = 0.004 ln( Z /0.2) + 0.175 (7) Seeing Eq.(5),(6), the bending moment under the folding process increased with the rotational velocity Z = Z ’). This tendency matched the prediction of Maxwell type relaxation response. However, the relaxed bending moment M 90,1 (1) decreased with Z = Z ’) from Eq. (7). This means that the dissipation energy or the bending moment drop increases with the rotational velocity Z for a short duration t ep1 <1s.
0.3
Eq.(5) Eq.(6)
0.2
0.1
Eq.(7) Eq.(8)
M p1 M 90,1 (0)
M 90,1 (1) M 90,1 (20)
4 = 90
㼻 , J = 0.6
0
Z Z ǯ ) /rps
0.01
0.1
1
Rotational velocity
Figure 11 Dependency of bending moment on rotational velocity. The unfolding velocity Z ’ was equal to the folding velocity Z . As the representative quantities of bending moment resistance, the maximum peak M p1 , the relaxed three states at the tracking position M 90,1 (0), M 90,1 (1) and M 90,1 (20) were plotted as the average (with the maximum and minimum bar).
Seeing the preliminary experiment (Nagasawa, et al., 2015, Fig.14), since the exponential coefficient of relaxation p 1
© 2019 The Japan Society of Mechanical Engineers
[DOI: 10.1299/jamdsm.2019jamdsm0004]
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