Luận án tiến sĩ: Three dimensional strain measurement by a Bragg grating sensor subjected to axial and transverse load simultaneously

Abstract The objective of this thesis is to establish mathematical models for predicting strains in the axial and other two orthogonal transverse directions of a multi-parameter Bragg gr

SK data ccc 4

SK Axial loading f†€S ôch ki 106

Figure 4.6 illustrates the strain prediction results from the axial loading test, comparing the strains computed by the 3-by-3 linear model (ecmp) for i = 1, 2, 3 with the strains obtained through Finite Element Analysis (e(FEM)) This comparison aims to evaluate the accuracy of the K matrix derived from a simplified approach in predicting strains for tests that were used to determine this matrix The results demonstrate a strong correlation, as anticipated for the axial data.

Figure 4.6: SK axial loading test

Error at each data point was calculated by the following equation

Table 4.6 presents the average errors across all data points, revealing that the prediction for € is highly accurate While the predictions for € and € are marginally less precise, they remain within acceptable limits This observation supports the validity of the linear assumption of the AA vs P curve in axial loading tests.

CHAPTER 4: 3-BY-3 LINEAR MODEL FOR STRAIN ANALYSIS

Table 4.6 — Errors in strain analysis: SK axial loading test

SK Transverse loading test se 107

Figures 4.7 through 4.14 show a representative sample of the results of strain comparison in the transverse loading tests for different values of angle 8 Figures at only some angles

This chapter omits figures for all angles to maintain brevity, presenting only key comparisons In the figures, e,(cmp) represents strain calculated using a 3-by-3 linear model, while e(FEM) denotes strain derived from Finite Element Analysis (FEA) The K matrix, established from tests at angles of 0° and 135°, shows excellent comparison in Figures 4.7 and 4.8 However, discrepancies arise at 27°, with unacceptable differences at 45° as illustrated in Figures 4.9 and 4.10 Figures 4.11 and 4.12 also reveal poor comparisons, including for €, Notably, the comparison improves for angles 135° and 162°, as shown in Figures 4.13 and 4.14.

SK Transverse Loading Test (0 deg)

Figure 4.7: SK transverse loading test (0 = 0°)

SK Transverse Loading Test (9 deg) 0.0008 +-

Figure 4.8: SK transverse loading test (@ = 9°)

0.0001 se + 5 Đ si [T8 "Sim oe 8 TA 05 1 1⁄8 2 2S | A s8(emp) 1 H c2Gmp) me 4 A a a ———e! (FEM)

+0 | [ter e2 GEM) on on K— e Dez (emp)

Table 4.7 presents the average errors calculated for all data points using Eq (4.31) at each 8 The errors in euros are excluded from the table since the value of e,(FEM) is zero; consequently, the error approaches infinity in Eq (4.31) if e,(cmp) is not zero.

Table 4.7 — Errors in strain analysis: SK transverse loading test

The analysis reveals significant errors in most cases, particularly at specific angles of 0° and 135°, where the K matrix was derived However, a notable error occurs at 9 = 135° The primary reason for these large discrepancies is the non-linearity observed in the experimental data Figures 3.18 to 3.28 illustrate that the AA vs Q curves in transverse loading tests exhibit considerable non-linearity, undermining the assumption of linearity This non-linearity leads to substantial errors in determining the slope of the AA vs Q curves in equations (4.4) to (4.7), complicating the strain analysis using the 3-by-3 method.

SK Combined loading tesfs cà coi 112

The analysis of combined loading test data was conducted through three distinct approaches In the first analysis, the actual combined loading test data were omitted; instead, axial and transverse loading test data were superimposed and treated as combined loading data The second analysis focused on a different methodology, which will be detailed further.

CHAPTER 4: 3-BY-3 LINEAR MODEL FOR STRAIN ANALYSIS actual combined loading test data which underwent the data shift were used In Analysis Ill, the raw combined loading test data were used The idea in Analysis I was to check the validity of superposition Theoretically, the superposed data in Analysis I should be exactly the same as the actual experimental data But in reality, they are different due to experimental errors, or problems with the assumption of superposition itself By comparing the results of Analysis I and other two, those problems can be assessed and discussed later The idea in Analysis II was to make the analysis consistent Since the axial and transverse loading test data underwent the data shift, the combined loading test data were also subject to the data shift The idea in analysis III was to see how the results were changed when data shift was not applied

4.3.1.3.1 SK Combined loading test — Analysis I

The combined loading test data were derived from superimposing axial and transverse loading test results, utilizing a 3-by-3 K matrix to compute strains, with findings compared to Finite Element Analysis (FEA) results Figures 4.15 through 4.24 illustrate a representative sample of these results, while including all angles would have made the chapter excessively lengthy The combined loading tests were conducted at two axial load values, P = 0.427 and 0.684 N, revealing that the combined loading results align more closely than those from transverse loading alone, as shown in Figures 4.7 through 4.14 Notably, the strain e,(cmp) closely matches e,(FEM), highlighting the dominant influence of axial loading due to its larger values, which diminishes the impact of errors from transverse loading The K matrix was established using data from these tests.

6 = 0 and 135°, the comparison is very good in Figures 4.15, 4.18, 4.20 and 4.23

However, similarly to the transverse loading test, the comparison starts to deteriorate especially for €, and ¢, when 8 becomes greater as seen in Figures 4.16, 4.17, 4.21 and 4.22

Figure 4.15: SK combined loading test — Analysis I (6 = 0°, P = 0.427N)

H e2(œmmp› § ° a A , | A es (cmp) ð a os a 1,4 4&5 2 25 re foe

Figure 4.16: SK combined loading test — Analysis I (@ = 45°, P = 0.427N)

= ì ứ lrị “ s8 (cmp) ọ 0 BS fro =—=—— i joe? (FEM) et FEM) ee BR an Oe OE 0.8 1 1.2 14 16 1/@ |—:—::e8 ŒEM)

Figure 4.17: SK combined loading test ~ Analysis I (6 = 108°, P= 0.427N)

TH A & e3 (omp) wee Ta ˆ i e1 EM)

Figure 4.19: SK combined loading test — Analysis I (6 = 162°, P = 0.427N)

: © el (emp) c 00002 | © e2 (omp) ứ AÁ e8(cmp) a ° deere TT a - J—! Fem ae “ ie Poe? ŒEM) pen ee 1 1.5 2 25 |_ e8 (FEM)

0.0004 © el (cmp) ce H e2(mp) ° ứ A e8 (emp)

Figure 4.21: SK combined loading test —- Analysis I (@ = 45°, P = 0.684N)

Figure 4.22: SK combined loading test ~ Analysis I (6 = 108°, P = 0.684N)

0.0008 © 41 (emp) c 0.0004 1 e2 (emp) fg A e8(emp) ð e1 (FEM)

Figure 4.23: SK combined loading test— Analysis I (6 = 135°, P = 0.684N)

SK Combined Analysis | (162 deg, P=0.684N) © ei (emp) ủ 82 (cmp)

Figure 4.24: SK combined loading test ~ Analysis I (@ = 162°, P = 0.684N)

4.3.1.3.2 SK Combined loading test - Analysis IT

The experimental data from the combined loading test, which experienced a data shift, were analyzed using a 3-by-3 K matrix to compute strains The results were compared with those obtained from Finite Element Analysis (FEA), as illustrated in Figures 4.25 to 4.34 Similar to Analysis I, the findings showed better agreement than the transverse loading data due to the dominance of axial loading Notably, the results were strong at an angle of @ = 0°, as depicted in Figures 4.25 and 4.30 However, discrepancies arose for angles ¢ and € as @ increased, particularly evident in Figures 4.26, 4.27, 4.31, and 4.32 The comparison improved again at angles of @ = 135° and 162°, as shown in Figures 4.28, 4.29, 4.33, and 4.34.

SK Cambined Analysis Il (O deg, P=0.427N)

5 02 04 very 0.6 08 1 12 1⁄4 1.6 18 | ——-sz @EM) 1, [=——et (FEM)

Figure 4.25: SK combined loading test —- Analysis II (9 = 0°, P = 0.427N)

SK Combined Analysis ll (45 deg, P=0.427N)

Figure 4.26: SK combined loading test — Analysis IT (@ = 45°, P = 0.427N)

Bf -.< aA `” a ——+! ŒEM) ° OTs ren 1 2 (FEM)

Figure 4.27: SK combined loading test —- Analysis II (9 = 108°, P = 0.427N)

Figure 4.28: SK combined loading test— Analysis II (9 = 135°, P= 0.427N)

Figure 4.29: SK combined loading test — Analysis II (6 = 162°, P = 0.427N)

SK Combined Analysis ll (135 deg, P=0.427N)

SK Combined Analysis lf (162 deg, P=0.427N)

SK Cambined Analysis II (0 deg, P=0.684N)

BR nb em em A e1 (FEM) ° 02 04 Of AT 88 1 cl 1.2 14 , 1.6 r 1P Í_ R’* as follows

Also, define the following quantity for A(*) in Eq (5.7.13) ứ(Ä,(*))=[@(4()) (AP) > o( 4 (*)) | (5.730)

The coordinate transformation in the space E(D) is defined in a similar way §.1.7.5.3 BGS space and properties of data type

In section 5.1.7.5.1, A(D) is identified as a subset of a four-dimensional Euclidean space R* We refer to the space where the orthogonal bases (5.7.27) are applied to A(D) as the "Bragg Grating Sensor space" (BGS space), denoted as BGS(A(v), {oy}) Additionally, we define another BGS space, BGS(E(0), {el}) It is important to highlight that each of these spaces possesses distinct bases The relationships among these spaces and the "applied load space," which includes the x1, x2, and x3 axes (referred to as the (x1, x2, x3)—space), are illustrated in Figure 5.3.

CHAPTER 5: NON-LINEAR MODEL FOR STRAIN ANALYSIS: FQI METHOD

BGS global coordinate A system space on A(D)

BGS global coordinate ơ ve t system space on E(D) e,“? Ve = ( _XxY V4) ị @

Figure 5.3: Bragg Grating Sensor space in the global coordinate system

[I] Global representation of data in BGS space

Write Eq (5.7.28) for an arbitrary v € A(D) as shown below v=v,ef) + ve 4,29 + y,e% (5.7.31);

On the other hand, from Eq (5.7.20), this v can be written as follows: v= XyAL0(#) + X,Al(#) + XpAP(*) +X, (#4 X,AM(*) (73

Applying the transformation @ in Eq (5.7.29) to these equations, we obtain, tị

= X, “ +X, +X, “ +X, “ +X, * (5.732); Đ P› P; pH P› ph po p? pÿ p ph P (k} where |", = (AS (*)) (k=0,0,8,7.6) (5.7.32),

[II] Properties of data in BGS space

In this case, the following holds

In this case, the following relation holds vy, =0 or X,=0 (5.7.34)

In this case, Egs (5.7.33) and (5.7.34) do not necessarily hold

By analyzing an experimental data point represented as vector v, it becomes feasible to predict the loading type—whether axial, transverse, or combined—by mapping it within the BGS space (¥;, V2, V3, V4).

5.1.7.6 Local coordinate system and its structure

In this section, as the second preparation to construct the inverse matrices {A, @} and

{B,(*)}Ƒ, a local coordinate system will be introduced to each block in Eq (5.7.19) and

Figure 5.2 i : @1 §.1.7.6.1 Local coordinate system in BGS A(D),{e } k=1

Substituting Eqs (5.7.21), through (5.7.21), into (5.7.31), the following relation is obtained:

= X,A0(*)+ XA) + XpAlP (+ XAM (+ KAMA) G5739 where

This relation can be transformed as follows: v= KyA0(2)+ HE (25) HE (ry) AC) + (5) (25) AP)

= X,AI)(*)+ HỆ (x,)C“®(x,)+ HỆ, (x,)Cữ®(x,) (5.737) where

[oes] HE Gar Co 15) AP() cự? (x,)= HW” (x, ) Ay” (*)+ Ho ( 3) Ay ” (*) n

Note that C)(x,) and C\%)(x,) depend only on x,, and these vectors are not zero- vectors Then Eq (5.7.37) can be translated as follows:

1) The three vectors AS (*), C(x,) and C%*) (x,) are coordinate axes in the block [§„;6„ |x[n,n,„„ ], and generally oblique coordinate axes These three vectors can be defined in each block in Figure 5.2

2) An arbitrary vector v < A(D) can be represented with respect to the coordinate axes

AO (*), C9 (x,) and C%) (x,), as shown in Eq (5.7.37) Then the following hold i) A) (*) is the axis which represents the axial loading in each block

CHAPTER 5: NON-LINEAR MODEL FOR STRAIN ANALYSIS: FQI METHOD ii) C(x.) and C* (x,) are the axes which represent the transverse loading, and determined by x, = cos®, depending on the value of @ iii) In Eq (5.7.37), © x, A (*) corresponds to the axial loading test, and X, is equal to P © H® (x,)c@") (x,)+ H®), (x,)C%” (x,) corresponds to the transverse loading test

3) The following mathematical interpretation can be made i) The condition of the axial loading can be represented by one point taken on the straight line {aa (*):a¢ Rh In other words, if a data point is on the A) (*)-axis, that data point was taken in the axial loading test ii) The condition of the transverse loading can be represented by the local coordinate system {o,,c-® (x,),Cữ®(z, )} (O, is the local origin;

X,A (*) if the axial load exists, and the axis origin if there is no axial load.) determined in each block iii) | The condition of general combined loading can be represented as follows: _ (a) It has the local origin O, on the line in i) that depends on the condition of the axial loading, and

(b) It is represented in the local coordinate system in ii) that depends upon the condition of the transverse loading

The position of point O in the local coordinate system is influenced by the axial load P, while the rotation of the two coordinate axes is determined by the value of 6 This combined loading condition can be effectively represented by the local coordinate system {o,,c@" (x,),C%) (x, )}, which is contingent upon these variables.

Figure 5.4 describes the idea in the above (a) through (c) In the figure, C, and C, are Cc) (x,) and C%)(x,) at © = , respectively C,, C,®, C,® and C, are defined in a similar way

[The local origin O, moves depending on the fluctuation of P, and the axes of the local coordinate system C and C\) rotate depending on the fluctuation of 9.]

The position of vector v within the local coordinate system is influenced by the fluctuations of Q, as illustrated by its placement along the dashed line in the accompanying figure.

Figure 5.4: Representation of combined loading in local coordinate system

5.1.7.6.2 Local eoordinate system in BGS E(Ð),{e bà

The above discussion in section 5.1.7.6.1 is valid in BGS [E(o).{2 là too Figure 5.5 describes the relations between the local coordinate system spaces and the applied load space

BGS local coordinate system space on A(D) e,” ca) e e9

BGS local coordinate system space on E(D)

Figure 5.5: Bragg Grating Sensor space in the global coordinate system

The third step in constructing the inverse matrices {A, (*)} and {B, (*)b involves noise processing, as outlined in our model based on Eqs (5.4.6) However, these equations may not hold true due to experimental errors, necessitating a careful examination of the experimental data To ensure accuracy, the data graphs must be "zero-shifted" to pass through the origin This shifting was initially performed intuitively using a 3-by-3 linear model in chapter 4 Given that such shifting is not feasible in the general space for the non-linear FQI model, we introduce a new, more precise method for data shifting It is essential to consider specific factors in this process.

1) The value given by A(x) in Eq (5.4.6), is a point in R’ in the global coordinate system

2) However, in the local coordinate system, that value is three-dimensional as shown in Eq (5.7.37)

3) Therefore, the noise that may be included in the data of each A(x) can be categorized in the following two types:

(i) Four-dimensional noise, which does not belong to the three- dimensional space with the bases {A® (*),C@® (x,),C% (x, )} We call this “Noise Type I.”

(ii) | Three-dimensional noise which belongs to the space described in above (i) We call this “Noise Type II.” §.1.7.7.1 Erasing Noise Type I

The three-dimensional space defined by the bases {ae (#), C@ (x,), c%) (x, )} is referred to as Span {ao (*), c@) (x,), C(x, )} This represents one of the hyper-planes within the four-dimensional Euclidean space R’ We denote this hyper-plane as z, The normal vector of the plane z, is represented by n,, indicating that z, is orthogonal to the vectors A (*), C@)(x,), and C%*) (x,).

CHAPTER 5: NON-LINEAR MODEL FOR STRAIN ANALYSIS: FQI METHOD orthogonal projection v, of an arbitrary experimental data v € A(D) onto the hyper-plane

7, is given by the following equation (see Appendix B.11): vp =v- (v.",) n (5.738) n

The data point v should ideally lie on the hyper-plane Z„; however, if it does not, it signifies the presence of four-dimensional noise known as Noise Type I To address this, the computation of v in Equation (5.7.38) effectively shifts v onto the hyper-plane, thereby eliminating Noise Type I This process serves to "erase" the unwanted noise, ensuring that v is accurately represented on Z„.

C2 (afA Án LH HH HH TH TT Tà nọ TH TH ng 1e 294

Figure 5.45 illustrates the findings from the axial loading test analysis, where e,(cmp) (i = 1,2,3) represents the strain calculated using the non-linear (FQI) model, while e,(FEM) denotes the strain derived from finite element analysis (FEA) Notably, in contrast to the SK data, the C2 axial loading test only provided two available data points.

PM Figure 5.45: C2 axial loading test

Table 5.4 presents the average errors of all data points calculated using Eq (4.31), revealing excellent results; however, this is primarily due to the limited number of two data points in the axial loading test To enhance the accuracy of strain prediction, additional data points are essential for a more comprehensive analysis.

Table 5.4 — Errors in strain analysis: C2 axial loading test

Figures 5.46 to 5.53 present a representative sample of results from the transverse loading tests, showcasing only select angles to maintain brevity In these figures, e(cmp) represents the strain calculated by the non-linear (FQI) model, while e(FEM) denotes the strain derived from finite element analysis (FEA) The results demonstrate that the non-linear (FQI) model accurately predicts all three strains across the various angles tested.

C2 Transverse Loding Test (105 deg) © el (cmp)

~1,00E-04 i ủ e2 (comp) sa & 3 Comp) sa e1 ŒEM)

CHAPTER 5; NON-LINEAR MODEL FOR STRAIN ANALYSIS: FQI METHOD

Table 5.5 presents the average errors calculated for all data points using Equation (4.31) at each 9 Notably, errors in € have been excluded due to the fact that e,(FEM) equals zero; consequently, if e,(cmp) is non-zero, the error approaches infinity in Equation (4.31).

Table 5.5 — Errors in strain analysis: C2 transverse loading test

The non-linear FQI model demonstrated strong predictive capabilities for strains during transverse loading tests across all angles In contrast, the 3-by-3 linear model failed to produce satisfactory results, as indicated in Table 4.14, highlighting the superior accuracy of the non-linear FQI model.

Similarly to the SK data, the C2 combined loading test data were analyzed in three ways §.2.2.3.1 C2 Combined loading test —- Analysis I

The combined loading test data were derived by superposing axial and transverse loading test data Using the non-linear (FQI) model, the K matrix was applied to compute strains, with results compared to those from finite element analysis (FEA) Figures 5.54 to 5.63 present a representative sample of these results, although including figures for all angles would have made the chapter excessively lengthy The combined loading tests were conducted at two axial load values, P = 0.736 N and P = 1.472 N, and the results demonstrated a high level of accuracy across all angles.

: © el (emp) c 2.00604 ————— H e2(mp) ứ eo I A e8 (emp)

Figure 5.54: C2 combined loading test — Analysis I (@ = 90°, P = 0.736N)

Figure 5.55: C2 combined loading test - Analysis I (@ = 105°, P = 0.736N)

1.00E-Q8 © el (cmp) c 5.00E-04 ủ s2(œnp) ứ A e3 (cmp) a : e1 (FEM)

9.00E+0O0 os 1 Ts SEs se olen

Figure 5.60: C2 combined loading test — Analysis [ (9 = 105°, P = 1.472N)

Figure 5.61: C2 combined loading test — Analysis I (0 = 135°, P = 1.472N)

0.5 Dae nage RS ere 2.5 $ |—-— e8 (FEM) tr TT wb Kg :

Figure 5.63: C2 combined loading test — Analysis I (6 = 180°, P= 1.472N) §.2.2.3.2 C2 Combined loading test — Analysis II

The experimental data from the combined loading test, which experienced a data shift, were utilized for analysis Using the K matrix, strains were calculated and compared to results obtained through Finite Element Analysis (FEA) The findings, illustrated in Figures 5.64 to 5.73, demonstrate excellent agreement across all angles.

Figure 5.64: C2 combined loading test - Analysis IT (8 = 90°, P = 0.736N)

Figure 5.65: C2 combined loading test - Analysis II (9 = 105°, P = 0.736N)

Figure 5.66: C2 combined loading test — Analysis IT (@ = 135°, P = 0.736N)

Figure 5.67: C2 combined loading test — Analysis II (@ = 165°, P = 0.736N)

Figure 5.68: C2 combined loading test — Analysis II (8 = 180°, P= 0.736N)

OS BS Lee eon T7 2 25 $ |c:7-:e9 ŒEM)

Figure 5.69: C2 combined loading test - Analysis II (8 = 90°, P= 1.472N)

Figure 5.70: C2 combined loading test — Analysis II (0 = 105°, P = 1.472N)

Figure 5.71: C2 combined loading test — Analysis II (9 = 135”, P = 1.472N)

Figure 5.72: C2 combined loading test - Analysis IT (6 = 165°, P = 1.472N)

OS ee! = ưan oy 25A $ |—-— *2ŒEMĐ

Figure 5.73: C2 combined loading test ~ Analysis II (6 = 180°, P = 1.472N)

CHAPTER 5: NON-LINEAR MODEL FOR STRAIN ANALYSIS: FQl METHOD ã.2.2.3.3 C2 Combined loading test — Analysis III

The raw experimental combined loading test data without data shift were used Figures 5.74 through 5.83 show the results The non-linear (FQI) model predicted the strains quite accurately

Figure 5.74: C2 combined loading test — Analysis III (0 = 90°, P = 0.736N)

@ el (emp) c 200504 1 e2(mp) Ỷ wee A e8 (cmp) ọ eo 21 (FEM)

Figure 5.75: C2 combined loading test — Analysis III (@ = 105°, P = 0.736N)

Figure 5.78: C2 combined loading test — Analysis HI (@ = 180°, P= 0.736N)

Figure 5.80: C2 combined loading test — Analysis III (9 = 105°, P= 1.472N)

TT CS TRE 1=||A|- |A Í: l6ứJ è bị bị

This indicates that in solving Ax = b with large cond(A), a small error in b can make the solution x quite inaccurate

[Example 2] Case in which the ‘=’ sign approximately holds in Eq (B.0.12)

Therefore, fol 10 v1 °-—~=cond(A l4] fe 1 io al

This indicates that in solving Ax = b with large cond(A), a small error in Acan make the solution x quite inaccurate

(1) Figure B.1.1 illustrates a parallelotope in R', R’ and RẺ

(A parallelotope in R* cannot be illustrated.)

Figure B.1.1: Parallelotope in several dimensions

(2) If the mappings AE" and EA” can be represented as matrices with constant elements, AE" is so-called the K matrix, and EA’ is its inverse

(3) The “T” or “x,” in sections 5.1.1.1 and 5.1.1.2 is the temperature change from the room temperature 23°C If the temperature is 23°C during the experiment, T = x, = 0

This article examines two key mathematical assumptions relevant to the conditions outlined in section 5.1.2: the separation of variables and the principle of superposition We will verify the validity of these assumptions in this discussion.

(1) Two possibilities of separation of variables

There are the following two possibilities of separation of variables:

We assume that section 5.1.2.3 (ii) and (iii) can be represented by the form of (b)

If —,(x) and @,(x) are solutions of a certain linear equation, then, c@,(x)+c,ỉ,(x) — (c; and c; are any constants) is also a solution of that equation

The ideas in section 5.1.2.3 (ii) and (iii) are not this type of superposition

(1) Taylor’s formula for a vector function with vector variables

If a vector function fis a p times continuously differentiable mapping, then,

APPENDIX B: MATHEMATICAL BACKGROUND f(eta)= fat as (a) x4 f(a): x9 bon

(p-1) i(1-g)?? œ) () li” ? (a+£x)đ€ cư, where x™ means (x, x, ô++, X) (K times) (Dieudonne, 1969)

Differential f’ of vector function fis expressed as follows:

Let xy f(x) x4 #@) xel il, f(x)=| : x, f,, (x) where each f,(x) is a scalar function

Then, ly Or, (x) a, (x) ey Bey] ar, (x) yy 35⁄0) 3

The right hand side of the above equation is called a Jacobian matrix

(2) Mean value theorem for a function with one variable

Equation (BR-A) in section 5.1.3.1 can be obtained by denoting G, = Š

As shown in Figure B.4.1, the behavior of AA, (k = 1,2,3,4) in transverse loading tests is generally non-linear about the transverse load Q (= X,)

The non-linear curve depicted in Figure B.4.1 can be segmented into multiple sections using dashed lines, allowing each section to be approximated by a straight line Consequently, the entire curve can be represented as a piecewise linear function In this context, the slope of the curve, denoted as 0AA,/Ax, = 04., is analyzed within each segment, as illustrated in Figure B.4.2.

Since Figure B.4.1 is piecewise linear, Figure B.4.2 becomes piecewise constant This is [P] (i) in section 5.1.4.2

Figures B.4.1 and B.4.2 can be considered at each value of @ Now the value of a, in the first section of Figure B.4.2 is picked up for all 9, and graphed as Figure B.4.3 below

Figure B.4.3 can be considered for the second, third - sections in Figure B.4,1 in the similar way Since Figure B.4.2 is piecewise constant, Figure B.4.3 becomes piecewise linear This is [P] (ii) in section 5.1.4.2

A hat function y = H(x) is illustrated in Figure B.5.1

The x-axis is decomposed into many sections by t’s The set of these t’s is called the

“knot sequence” for the x-axis As shown in Figure B.5.1, the value of H(x) is 1 atx = %;, and zero if x < T, or X > t; Therefore, art m

Tiêu đề	Three Dimensional Strain Measurement by a Bragg Grating Sensor Subjected to Axial and Transverse Load Simultaneously
Tác giả	Tadamichi Mawatari
Người hướng dẫn	Drew V. Nelson, Kosuke Ishii, Sheri D. Sheppard
Trường học	Stanford University
Chuyên ngành	Mechanical Engineering
Thể loại	Dissertation
Năm xuất bản	2006
Thành phố	Stanford

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Số trang	499
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