Nonlinear Dynamics 18: 185–202, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands.
Experimental Identification Technique of Nonlinear Beams in Time Domain K. YASUDA and K. KAMIYA Department of Electronic-Mechanical Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8603 Japan (Received: 14 April 1997; accepted: 25 September 1998) Abstract. In previous papers, the authors proposed a new experimental identification technique applicable to elastic structures. The proposed technique is based on the principle of harmonic balance and can be classified as a frequency domain technique. The technique requires the excitation force to be periodic. This is, in some cases, a restriction. So another technique free from this restriction is of use. In this paper, as a first step for developing such techniques, a technique applicable to beams is proposed. The proposed technique can be classified as a time domain technique, two variations of which are proposed. The first method is based on the usual least-squares method. The second is based on solving a minimization problem with constraints. The latter usually yields better results. But in this method, an iteration procedure is used which requires initial values for the parameters. To obtain the initial values, the first method can be used. So both methods are useful. Finally, the applicability of the proposed technique is confirmed by numerical simulation as well as experiments. Keywords: Identification, nonlinear vibration, vibration of continuous systems.
1. Introduction Recently, techniques of experimental identification of nonlinear systems are becoming more and more important. So many techniques have been developed [1–15]. These techniques, however, are applicable to systems with relatively few degrees of freedom. When applying these techniques to many degrees of freedom systems, one should determine many parameters, which is very hard or even impossible in practice. In addition, in nonlinear distributed systems such as a beam in large defomation, the nonlinearity is dependent on global defomation of the beam. So techniques different from those in the previous papers are required. In previous papers, we proposed a new technique applicable to elastic structures [16, 17] which is based on the principle of harmonic balance and can be classified as a frequency domain technique. It requires the excitation force to be periodic. This is, in some cases, a restriction. To remove this restriction, we attempt to develop another new technique applicable to elastic structures. As a first step for developing such a technique, we propose in this paper a technique applicable to beams. The technique can be classified as a time domain one. We first examine the form of the governing equations. Then we consider methods for estimating the parameters of the governing equations. We propose two variations of the technique. The first is based on the usual least-squares method. The second is based on solving a minimization problem with constraints. The latter usually yields more accurate results than the former. But in the latter method, an iteration procedure is used which requires initial values for the parameters. To obtain these initial values, the first method can be used, so both methods are
186 K. Yasuda and K. Kamiya useful. Finally, we conduct numerical simulation and experiments to check the applicability of the technique. 2. Formulation of the Problem We consider a beam with both ends constrained in the axial direction. Along the beam in its equilibrium state, we fix the x axis whose origin O coincides with one end of the beam. We denote the deflection of the beam by w(x, t) or simply by w. We express the length of the beam by l, the density by ρ, the area of the section by A, Young’s modulus by E, and the second moment of area by I . The quantities ρ, A, E and I may be dependent on x. We suppose that the beam is subjected to viscous damping force with coefficient c. Then the equation of motion for large amplitude vibrations [18, 19] is Zl ∂ 2w ∂2 1 ∂w ∂w 2 ∂ 2 w ∂ 2w EI 2 − a u0 + = Q(x, t), (1) dx +c ρA 2 + 2 2 ∂x ∂t ∂x ∂x 2 ∂x ∂t 0
where a is a constant given by a=
Zl
−1 1 dx EA
.
(2)
0
Here, the quantity u0 is the initial stretching in the axial direction and so u0 = 0 for no stretching. The quantity Q(x, t) denotes the excitation force per unit length of the beam. Experimental identification means determination of Equation (1) by use of the data of the excitation force Q(x, t) as well as those of the responses to it. The form of Equation (1) seems difficult to determine. So we transform Equation (1) into equivalent equations of another form. For this, we introduce a set of appropriate functions φn (x) (n = 1, 2, 3, . . .), satisfying the geometrical boundary conditions of the beam. In terms of these functions, we express the deflection w in a series form as w=
∞ X
φn (x)Xn (t).
(3)
n=1
Substituting Equation (3) into Equation (1), and applying the Galerkin method, we have ∞ X mnm X¨ m (t) + cnm X˙ m (t) + knm Xm (t) + Nn m=1
= F0 φn (0) + Fl φn (l) + M0 φn0 (0) + Ml φn0 (l) + qn (t) (n = 1, 2, 3, . . .),
(4)
where the dots and primes mean differentiation with respect to t and x, respectively. The quantities mnm , cnm , knm are constants defined by Zl mnm =
ρAφn (x)φm (x) dx, 0
Experimental Identification Technique of Nonlinear Beams 187 Zl cnm =
cφn (x)φm (x) dx, 0
knm
Zl d2 φn (x) d2 φm (x) dφn (x) dφm (x) EI dx. = + au0 dx 2 dx 2 dx dx
(5)
0
These quantities satisfy mnm = mmn ,
cnm = cmn ,
knm = kmn .
(6)
The quantities Nn are the nonlinear functions of the form X βn,ij k Xi (t)Xj (t)Xk (t). Nn =
(7)
i≤j ≤k
The quantities qn (t) are the functions of time defined by Zl qn (t) =
Q(x, t)φn (x) dx.
(8)
0
Finally, the quantities F0 , Fl , M0 , Ml are defined by Zl ∂ 2 w ∂ 1 ∂w ∂w EI 2 dx − a u0 + , F0 = ∂x x=0 ∂x ∂x 2 ∂x x=0 0
∂ Fl = − ∂x
∂ 2 w EI 2 ∂x x=l
∂ w M0 = − EI 2 , ∂x x=0 ∂ 2 w Ml = EI 2 . ∂x x=l
Zl 1 ∂w ∂w dx + a u0 + , ∂x x=l 2 ∂x 0
2
(9)
These quantities with suffixes 0 and l mean the forces and moments at points x = 0 and x = l, respectively. We now examine the right-hand side of Equations (4). If the ends of the beam are free, simply-supported or fixed, the terms appearing in the right-hand side of the equations are zero except the forcing term qn (t). If the beam is supported elastically so that the forces F and the moments M at its boundaries are given by F = −kt w,
M = −kr
∂w , ∂x
(10)
then the terms appearing in the right-hand side, except the forcing term qn (t), can be transposed to the left-hand side without changing the form of the equations. In addition, the
188 K. Yasuda and K. Kamiya coefficients mnm , cnm , knm satisfy the condition of Equations (6) as before. Thus, for many boundary conditions of practical importance, the governing equations take the form ∞ X mnm X¨ m (t) + cnm X˙ m (t) + knm Xm (t) + Nn = qn (t) (n = 1, 2, 3, . . .).
(11)
m=1
These are the governing equations for our purpose. They are the ordinary differential equations with constant coefficients. We now formulate the problem of identification. From the above discussion, it follows that the problem of identification is reduced to, after defining appropriate functions φn (x), determination of Equations (11). The number of Equations (11) is infinite. However, in practical cases, not all the equations are required. In fact, by choosing appropriate φn (x), the deflection w can be approximated by a finite number of φn (x). In the following, we assume that the first M terms are enough to express w accurately. Then our problem is reduced to the determination of M equations of the form: M X mnm X¨ m (t) + cnm X˙ m (t) + knm Xm (t) + Nn = qn (t) (n = 1, 2, . . . , M).
(12)
m=1
The theoretical form of nonlinear functions Nn is given by Equation (7). In practical cases, however, they may take other forms. Thus, we put them in a general form of polynomials, as follows: X X αn,ij Xi (t)Xj (t) + βn,ij k Xi (t)Xj (t)Xk (t) + · · · , (13) Nn = i≤j ≤k
i≤j
where αn,ij , βn,ij k , . . . are unknown parameters. Now the problem of identification is reduced to determination of mnm , cnm , knm in Equations (12) as well as αn,ij , βn,ij k , . . . in Equation (13). In the following, we assume that we have experimental data of the deflection, velocity and ˙ i , tj ), w(x ¨ i , tj ), (i = 1, 2, . . . , N, j = 1, 2, 3, . . .), at N points acceleration, w(xi , tj ), w(x xi (i = 1, 2, . . . , N), at times t = tj . We also assume that we have experimental data of the excitation force Q(x, tj ), at times t = tj , at several points along the beam, which allow us to perform integration given in Equation (8). 3. The First Method of Parameter Estimation In this section, we propose the first method for estimating the parameters of the governing equations. 3.1. D ETERMINATION
OF
F UNCTIONS φn (x)
The first and important step in this method is to determine the functions φn (x). It is desirable that the functions φn (x) are such that only a few of them can express w accurately. We attempt to obtain such functions using the experimental data. As a preparation for this, we introduce vectors {w}j = [ w(x1 , tj ) w(x2 , tj ) . . . w(xN , tj ) ]T , {w} ˙ j = [ w(x ˙ 2 , tj ) . . . w(x ˙ N , tj ) ]T , ˙ 1 , tj ) w(x {w} ¨ j = [ w(x ¨ 2 , tj ) . . . w(x ¨ N , tj ) ]T , ¨ 1 , tj ) w(x
(14)
Experimental Identification Technique of Nonlinear Beams 189 the elements of which are the data of the responses at points xi , at times tj . We also introduce vectors {φ}n = [ φn (x1 ) φn (x2 ) . . .
φn (xN ) ]T ,
(15)
the elements of which are the values of unknown functions φn (x) at measuring points xi (i = 1, 2, . . . , N). From Equation (3), it follows that the experimental data {w}j , {w} ˙ j , {w} ¨ j are related to {φ}n as follows: {w}j =
N X
Xn (tj ){φ}n ,
n=1
{w} ˙ j=
N X
X˙ n (tj ){φ}n ,
n=1
{w} ¨ j=
N X
X¨ n (tj ){φ}n ,
(16)
n=1
where Xn (tj ), X˙ n (tj ), X¨ n (tj ) denote the values of Xn (t), X˙ n (t), X¨ n (t) at time t = tj . It is noted that both {φ}n and Xn (tj ), X˙ n (tj ), X¨ n (tj ) are to be determined. Now we proceed to determining vectors {φ}n . For this, we introduce a matrix ˙ 1 p2 {w} ˙ 2 . . . p3 {w} ¨ 1 p3 {w} ¨ 2 . . . , (17) [w] = p1 {w}1 p1 {w}2 . . . p2 {w} ˙ j , {w} ¨ j . Here p1 , p2 , p3 are the weighting coefficients. the elements of which are {w}j , {w} Then, we consider the eigenvalue problem defined by (18) [w][w]T {φ} = γ {φ}. We solve this, and arrange the obtained eigenvectors and eigenvalues in the descending order of magnitude of the eigenvalues. We attach subscripts n (n = 1, 2, . . . , N) to the eigenvectors and eigenvalues in this order. Finally, we normalize the eigenvectors {φ}n using the condition {φ}Tn {φ}n = 1.
(19)
To grasp the meaning of the eigenvalues γn obtained above, we substitute Equations (16) and (17) into Equation (18) and note that the eigenvectors {φ}n satisfy the orthogonality as well as Equation (19). Then we have X (20) γn = p12 Xn2 (tj ) + p22 X˙ n2 (tj ) + p32 X¨ n2 (tj ) . j
This means that if γn is large, so are Xn (t), X˙ n (t), X¨ n (t). Hence {w}j , {w} ˙ j and {w} ¨ j in Equations (16) can well be approximated using Xn (t), X˙ n (t), X¨ n (t) and {φ}n with large γn . In the following, we assume that the first M of the eigenvalues γn are large. Then it is enough to retain the first M terms in the series of Equations (16). Hence, it follows that {w}j =
M X n=1
Xn (tj ){φ}n ,
190 K. Yasuda and K. Kamiya {w} ˙ j =
M X
X˙ n (tj ){φ}n ,
n=1
{w} ¨ j =
M X
X¨ n (tj ){φ}n .
(21)
n=1
The functions φn (x) are constructed from {φ}n by interpolating them. The deflection w(x, t) at any point can be given in terms of these functions φn (x). 3.2. D ETERMINATION
OF
Xn (t), X˙ n (t), X¨ n (t) AND qn (t)
The next step in the present method is to obtain Xn (t), X˙ n (t), X¨ n (t) and qn (t) using the experimental data. To have Xn (t), we note that the first equation of Equations (21) is valid between {w}j and Xn (tj ). Using the orthogonality of the eigenvectors {φ}n in this equation as well as the normalization condition (19), we have Xn (tj ) as follows: Xn (tj ) = {φ}Tn {w}j
(n = 1, 2, . . . , M).
(22)
Similarly, from the second and the third equations of Equations (21), we have X˙ n (tj ) and X¨ n (tj ) as follows: X˙ n (tj ) = {φ}Tn {w} ˙ j,
X¨ n (tj ) = {φ}Tn {w} ¨ j.
(23)
Finally, using the experimental data of Q(x, tj ) and performing the required integration in Equation (8), we obtain qn (tj ). 3.3. E STIMATION
OF THE
PARAMETERS
OF THE
G OVERNING E QUATION
We proceed to estimating the parameters of the governing equations, using the data of Xn (tj ), X˙ n (tj ), X¨ n (tj ) and qn (tj ). Here, as mentioned above, we use the least-squares method. The errors enj contained in Equations (12), when the data of qn (tj ), Xn (tj ), X˙ n (tj ), X¨ n (tj ) are substituted, are enj = qn (tj ) −
M X mnm X¨ m (tj ) + cnm X˙ m (tj ) + knm Xm (tj ) − Nn m=1
(n = 1, 2, . . . , M, j = 1, 2, 3, . . .).
(24)
We introduce the error vectors {e}j as {e}j = [ e1j
e2j
. . . eMj ]T .
(25)
Then we can rewrite Equations (24) as {e}j = {Q}j − [A]j {S} (j = 1, 2, 3, . . .),
(26)
where the vectors {Q}j are known vectors given by {Q}j = [ q1 (tj ) q2 (tj ) . . . qM (tj ) ]T ,
(27)
Experimental Identification Technique of Nonlinear Beams 191 and where the matrices [A]j are known matrices determined by Xn (tj ), X˙ n (tj ), X¨ n (tj ) (n = 1, 2, . . . , M) and, finally, the vector {S} is an unknown vector given by T . (28) {S}j = m11 m12 . . . c11 c12 . . . k11 k12 . . . α1,11 . . . β1,111 . . . The square sum E of the errors of Equations (24) is E =
M XX j
2 enj =
X {e}Tj {e}j
n=1
j
X = ({Q}j − [A]j {S})T ({Q}j − [A]j {S}).
(29)
j
Applying the least-squares method to Equation (29), we have X X [A]Tj [A]j {S} = [A]Tj {Q}j . j
(30)
j
This gives the values of the parameters and, thus, identification is completed. 4. The Second Method of Parameter Estimation In this section, we propose another method where the unknown parameters mnm , cnm , knm and αn,ij , βn,ij k , . . . as well as the functions φn (x) are determined simultaneously. Hence, there are no separate steps for determining the functions φn (x). ˆ˙ j and {w} ˆ¨ j the vectors which are to be estimated in the form of We define by {w} ˆ j , {w} Equations (16). Then we introduce the vectors expressing errors between the measured vectors ˆ˙ j , {w} ˆ¨ j , as ˙ j , {w} ¨ j given by Equations (14) and the vectors {w} ˆ j , {w} {w}j , {w} {ew }j = {w}j − {w} ˆ j, ˆ˙ j , {ew˙ }j = {w} ˙ j − {w} ˆ¨ j . {ew¨ }j = {w} ¨ j − {w}
(31)
ˆ j the vectors which are to be estimated. Then we introduce the Similarly, we define by {Q} vectors expressing errors between the vectors {Q}j given by Equation (27) and the vectors ˆ j , as {Q} ˆ j. {e}j = {Q}j − {Q}
(32)
ˆ˙ j , {w} ˆ¨ j and Here we note that Equations (12) must be satisfied, so we impose on {w} ˆ j , {w} ˆ j the constraints {Q} ˆ j {S} = {0} (j = 1, 2, 3, . . .), ˆ j − [A] {Q}
(33)
ˆ j are the matrices defined similarly where {S} is the vector given by Equation (28), and [A] as [A]j in Equations (26) with their components determined by Xn (tj ), X˙ n (tj ) and X˙ n (tj ) of ˆ˙ j , {w} ˆ¨ j . {w} ˆ j , {w}
192 K. Yasuda and K. Kamiya To estimate the parameters of the governing equations, we minimize the square sum of Equations (31) and (32), subjected to the constraints (33). To do this, we adopt Lagrange’s method of undetermined multipliers. We define Lagrange’s function U by X 1 1 1 r1 {ew }Tj {ew }j + r2 {ew˙ }Tj {ew˙ }j + r3 {ew¨ }Tj {ew¨ }j U = 2 2 2 j 1 T T ˆ + r4 {e}j {e}j + {ζ }j {Q}j − [A]j {S} , 2
(34)
where r1 , r2 , r3 , r4 are the weighting coefficients for the errors given by Equations (31) and (32) and where {ζ }j are Lagrange’s multiplier vectors. The problem now is to find Xn (tj ), ˆ j , {φ}n , {S} and {ζ }j , so that they minimize Lagrange’s function U . The X˙ n (tj ), X¨ n (tj ), {Q} conditions for this are ∂U = 0, ∂Xn (tj ) ∂U = 0, ∂ X˙ n (tj ) ∂U = 0, ∂ X¨ n (tj ) ∂U = {0}, ˆ j ∂{Q} ∂U = {0}, ∂{φ}n ∂U = {0}, ∂{S} ∂U = {0}. ∂{ζ }j
(35)
The last four equations in Equations (35) are the set of equations obtained by setting at zero ˆ j , {φ}n, and {ζ }j . Solving Equathe derivatives of U with respect to the elements of {S}, {Q} tions (35) yields the required parameters {S} as well as other quantities and, thus, identification is completed. When needed, the functions φn (x) are obtained by interpolating {φ}n . Equations (35) are nonlinear. To solve them, an iteration method has to be used [20]. As initial values required in the iteration method, the values of Xn (tj ), X˙ n (tj ), X¨ n (tj ), {Q}j {φ}n , and {S} obtained in the first method can be used. 5. Numerical Simulation To check the applicability of the proposed technique, we conduct numerical simulation, using the data generated numerically.
w(x2,t) [mm] w(x1,t) [mm] Q(xf,t)[*10-2N]
Experimental Identification Technique of Nonlinear Beams 193 1 0 -1 1 0 -1 1 0 -1 0
2
4
6
8 10 time t [s]
12
Figure 1. Examples of data used for identification.
For this, we consider a uniform beam with both ends simply-supported. The parameters appearing in Equation (1) are taken as A = 9.5 × 10−6 m2 , ρ = 7.84 × 103 kg/m3 , u0 = 1.0 × 10−6 m,
l = 0.816 m, I = 1.93 × 10−13 m4 , E = 2.06 × 1011 Pa, c = 0.15 Ns/m2 .
To this beam we apply a concentrated fast swept sinusoidal excitation force of the form 1 2 Q(x, t) = Q0 δ(x − xf ) cos (36) λt . 2 Here Q0 , xf , λ are the magnitude, position, and angular velocity, of the excitation force. Their values are taken as Q0 = 1.0 × 10−2 N, xf = l/5, λ = 14 rad/s2 . As measuring points, we choose five points as follows: 1 x1 = l, 6
2 x2 = l, 6
3 x3 = l, 6
4 x4 = l, 6
5 x5 = l. 6
(37)
They will be denoted by points 1, 2, 3, 4, 5. 5.1. A PPLICATION
OF THE
F IRST M ETHOD
First, we show the results obtained when we applied the first method. The data used for identification were obtained as follows: we derived the modal equations from the equations of motion, using the linearized modal functions. We solved the first six modal equations numerically. Then we obtained, using these results, the deflection, velocity and acceleration at the measuring points. We considered the case in which the measuring time was 12 seconds,
φ3(x)
φ2(x)
φ1(x)
194 K. Yasuda and K. Kamiya
0
1
x /l Figure 2. Examples of functions φn (x) (n = 1, 2, 3). Table 1. Eigenvalues. γ1 γ2 γ3 γ4 γ5
1.3171 × 10−3 0.7002 × 10−3 0.2743 × 10−3 0.0004 × 10−3 0.0000 × 10−3
during which measurement was made 1200 times. Here we considered the case in which the data of the excitation contain noise. To make the data with noise, we added random numbers with the standard deviation of 5% of Q0 to the simulated data of excitation force. As examples of the data, we show in Figure 1 the excitation force and the deflection at points 1, 2. Following the procedures of the proposed method, we first determined functions φn (x). We took the weighting coefficients p1 , p2 , p3 in Equation (17) as p1 = 1, p2 = 1 × 10−2 , p3 = 1 × 10−4 so that the magnitudes of the deflection, velocity, and acceleration would be of the same order. As the interpolation function for obtaining functions φn (x) from vector {φ}n , we used the third-order B spline function. As examples of the obtained functions φn (x), we show in Figure 2 the first three of them. We show the eigenvalues in Table 1. From this, we find that the first three are far larger than the other two. So we concluded that determination of the first three of the governing equations is enough for our purpose. We assumed that the nonlinear functions in Equations (12) are derivable from the potential X X αij k Xi (t)Xj (t)Xk (t) + βij kl Xi (t)Xj (t)Xk (t)Xl (t) (38) V = i≤j ≤k
i≤j ≤k≤l
Experimental Identification Technique of Nonlinear Beams 195 Table 2. Identification results. kg m11 m12 m13 m22 m23 m33
×10−2 0.996 0.000 0.001 1.010 0.000 0.978
N/m k11 k12 k13 k22 k23 k33
×102 0.383 –0.002 –0.007 0.605 0.001 1.387
N/m2 β1111 β1112 β1113 β1122 β1123 β1133
×108 0.192 –0.003 –0.007 0.096 0.003 0.849
Ns/m c11 c12 c13 c22 c23 c33
×10−1 0.261 –0.001 –0.001 0.208 0.000 0.481
N/m2 α111 α112 α122 α123 α133 α222 α223 α233 α333
×105 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
β1222 β1223 β1233 β1333 β2222 β2223 β2233 β2333 β3333
–0.001 –0.001 –0.005 –0.016 0.012 0.001 0.212 0.008 0.955
by the formula ∂V . (39) ∂Xn (t) This reduces the number of unknown parameters and removes physical inconsistency in the form of the nonlinear functions. Following the procedures presented above, we obtained the parameters. The results are shown in Table 2. To check the accuracy of the identification results, we predicted the response of the beam to the harmonic excitation force of magnitude Q0 at point xf , and compared the response with that obtained from the original equations of motion. We show in Figure 3 the results in the form of a time history curve for deflections w(x1 , t), w(x2 , t), w(x3 , t) at points 1, 2, 3 when the excitation frequency ω is 38 and 124 rad/s. We also show in Figure 4 the results in the form of an amplitude-frequency curve for the same points. In the figures, solid lines denote the curves obtained from the identification results and • those obtained from the original equations. We clearly see that the responses predicted from the identification results agree well with those obtained from the original equations. Nn =
5.2. A PPLICATION
OF THE
S ECOND M ETHOD
Next we show the results obtained when we applied the second method. The data used for identification were obtained as follows. We first calculated the deflection, velocity, and acceleration as before. Then we added to the simulated data, random numbers with the standard deviation of 5% of root mean square of the simulated data of deflection, velocity and acceleration. Similarly, we added to the simulated data of excitation force, random numbers with the standard deviation of 5% of the magnitude Q0 of the excitation force. Before applying the second method, we applied the first method for comparison. As shown later, the results of the first method are not good in this case.
196 K. Yasuda and K. Kamiya
w(x3,t) [mm] w(x2,t) [mm] w(x1,t) [mm]
ω=38 [rad/s]
ω=124 [rad/s]
1 0 -1 1 0 -1 1 0 -1 0.0
0.4 0.0 time t [s]
0.4 time t [s]
Original 1st method
1
w2 [mm]
0 1
0
w3 [mm]
Amplitude of Deflection
w1 [mm]
Figure 3. Comparison of time history curves obtained from identification results and original equations.
1
0
50
100
150
Angular Frequency ω [rad/s] Original 1st method Figure 4. Comparison of frequency-amplitude curves obtained from identification results and original equations.
Experimental Identification Technique of Nonlinear Beams 197 w(x3,t) [mm] w(x2,t) [mm] w(x1,t) [mm]
ω=38 [rad/s]
ω=124 [rad/s]
1 0 -1 1 0 -1 1 0 -1 0.0
0.4 0.0 time t [s]
0.4 time t [s]
Original 2nd method Figure 5. Comparison of time history curves obtained from identification results and original equations.
So we applied the second method. As initial values of the parameters, we used the results obtained by the first method. As the weighting coefficients r1 , r2 , r3 , r4 in Equation (34), we took the reciprocal of the mean square of the data of deflection, velocity, acceleration and excitation force, respectively. To check the accuracy of the identification results, we predicted the response of the beam as in the previous case. We show in Figure 5 the results in the form of a time history curve for deflections w(x1 , t), w(x2 , t), w(x3 , t) at points 1, 2, 3 when the excitation frequency ω is 38 and 124 rad/s. We also show in Figure 6 the results in the form of an amplitude-frequency curve for the same points. In the figures, solid lines denote the curves obtained from the identification results and • those obtained from the original equations. In Figure 6, response predicted from the results of the first method are superimposed in dashed lines. From these figures, we see that the second method yields accurate results. 6. Experiment To check further the applicability of the proposed technique, we conducted experiments using a uniform beam made of steel.
w1 [mm]
1
0
w2 [mm]
1
w3 [mm]
Amplitude of Deflection
198 K. Yasuda and K. Kamiya
0 1
0
50
100
150
Angular Frequency ω [rad/s] Original 1st method 2nd method Figure 6. Comparison of frequency-amplitude curves obtained from identification results and original equations.
l
S1
S2
S3
M1
S4
M2 Magnet Exciting Oscillator
Sf Personal Computer
Figure 7. Experimental setup.
The experimental setup is shown in Figure 7. In the figure, B is the steel beam. Its dimensions are as follows: length : 1202 mm, width : 25.1 mm, thickness : 1.45 mm, mass : 326 g.
-1 w(x2,t) [mm] w(x1,t) [mm] Q(xf,t)[*10 N]
Experimental Identification Technique of Nonlinear Beams 199
5 0 -5 2 0 -2 2 0 -2 0
2
4
6 time t [s]
8
Figure 8. Examples of time history of excitation force and deflection.
The beam was supported at both ends through ball bearings and constrained in the axial direction. Fast swept excitation force was applied to the beam by a pair of magnetic exciters. The excitation force was given in two ways, one in which the excitation frequency was increased from 8.0 to 35.0 Hz in 8 seconds, and one in which it was decreased from 30.0 to 3.0 Hz in 8 seconds. The width of the exciter was approximately 25 mm, and its midpoint was placed at a point 7/10 times the length of the beam. We used a piezoelectric force sensor Sf to measure the excitation force. To measure the deflection, we used four optical sensors S1 , S2 , S3 , S4 placed at points 1 2 3 4 x1 = l, x2 = l, x3 = l, x4 = l 5 5 5 5 with equal intervals. The signals from the sensors were acquired by a personal computer through an A/D converter. The sampling frequency was 2048 Hz. As examples of the data, the excitation force with its frequency increasing and the deflection at points 1 and 2 induced by the excitation, are shown in Figure 8. The data of the velocity and the acceleration were obtained by differentiating the data of the deflection in the frequency domain. When carrying differentiation, we neglected components with frequencies higher than 130 Hz. First, we applied the first method. Following the procedures presented above, we determined the functions φn (x). Here, we took the weighting coefficients p1 , p2 , p3 in Equation (17) as p1 = 1, p2 = 1 × 10−2 , p3 = 1 × 10−4 so that the magnitude of deflection, velocity, and acceleration would be of the same order. As the interpolation function for obtaining functions φn (x) from vector {φ}n , we used the third-order B spline function. In Figure 9 we show the first two functions φ1 (x), φ2 (x) and in Table 3 we show the eigenvalues. From this, we find that the first two eigenvalues are far larger than the others. So we determined the parameters of the first two equations in Equation (11).
φ2(x)
φ1(x)
200 K. Yasuda and K. Kamiya
0
1
x /l Figure 9. Functions φ1 (x), φ2 (x). Table 3. Eigenvalues. γ1 γ2 γ3 γ4
1.699 × 10−2 1.093 × 10−2 0.026 × 10−2 0.009 × 10−2
Second, we determined the parameters following the second method, using the results of the first method as the initial values. Here we took the weighting parameters r1 , r2 , r3 , r4 as the reciprocals of mean squares of the deflection, velocity, acceleration and excitation force, respectively. The obtained results are shown in Table 4. Table 4. Identification results (by the second method). kg m11 m12 m22
×10−1 0.81 0.03 1.23
Ns/m
×10−1
α222
0.47
c11 c12 c22
0.10 0.52 0.19
N/m3 β1111 β1112
×106 0.70 –0.28
N/m k11 k12 k22
×103 0.32 0.00 2.23
β1122 β1222 β2222
8.57 2.00 1.49
N/m2 α111 α112 α122
×104 0.00 0.09 0.83
4 2
w2 [mm]
0 4 2 0
w3 [mm]
Amplitude of Deflection
w1 [mm]
Experimental Identification Technique of Nonlinear Beams 201
4 2 0
10
15 20 Frequency ω [Hz]
25
Experimental 1st Method 2nd Method Figure 10. Comparison of frequency-amplitude curves obtained from identification results and experiment.
To check the accuracy of the obtained results, we predicted the response of the system to harmonic excitation and compared it with that obtained experimentally. For this, we excited the beam at a point 7/10 times the length by the harmonic excitation force with magnitude 0.1N. In Figure 10 we show the results in the form of the amplitude-frequency curves at points 1, 2, 3. In the figure, solid and dashed lines denote the results obtained from the identified results and • the results of the experiment. From this, we find that the identified results agree well with the results of the experiment. 7. Conclusions We attempted to develop a new identification technique of nonlinear beams. For this, we first examined the form of the governing equations. Then we considered the methods for estimating the parameters of the governing equations. Here we proposed two methods. The first is based on the usual least-squares method. The second is based on solving a minimization problem with constraints. The latter usually yields better results. But in the latter method, an iteration technique has to be used, starting from some initial values. To have initial values, the former method can be used, so both methods are of value. We conducted numerical simulation and experiments, and showed that the proposed technique yields good identification results.
202 K. Yasuda and K. Kamiya References 1. 2. 3. 4.
5. 6.
7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
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