An inverse vibration analysis of a tower subjected to wind drags on a shaking ground

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1 Applied Mathematical Modelling 26 (2002) An inverse vibration analysis of a tower subjected to wind drags on a shaking ground C.C. Kang a, C.Y. Lo b, * a Department of Mechanical Engineering, National Cheng Kung University, Tainan 701, Taiwan, ROC b Department of Aeronautical Engineering, National Huwei Institute of Technology, Huwei, Yunlin 632, Taiwan, ROC Received 14August 2000; received in revised form 14June 2001; accepted 2 July 2001 Abstract A method is presented to estimate the strength of wind drags on an elevated tower and the magnitude of the vibration of the ground on which the tower stands. The governing equations for the motions of the tower are discretized using the finite-difference method. Based on these discretized governing equations, a linear inverse model is constructed to identify the external wind drags and the ground vibration. The optimized solution of the model is determined by the linear least-square error method, which requires no numerical iteration. The uniqueness of the solution can be identified by linear algebra theory. A numerical example is given to demonstrate the feasibility of the method. The results show that the original wind drags and ground vibration may be estimated from the measured deflection at several locations along the tower. The reasonable estimations are achievable even though there exist certain measurement errors. The loading conditions are checked at different locations and the deflection information at different location may be used. The procedure is easy and effective. It may be extended to many inverse applications that the discretized governing equations were derived. Ó 2002 Elsevier Science Inc. All rights reserved. 1. Introduction Inverse problems are often found in designs and manufacturing processes whenever the direct measurement of physical quantities is either difficult or, sometimes, impossible. For an inverse problem, the major goal is to identify the unknown quantities with the limited measured data. Since Stolzd [1] did an inverse heat transfer problem, inverse problems have caught the attention of many researchers. Many works have been done in the heat conduction field [2 4]. Doyle [5 8] * Corresponding author. address: cylo@sunws.nhit.edu.tw (C.Y. Lo) X/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S X(01)

2 518 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) studied the first inverse vibration problem. In his serial papers, the inverse beam and plate vibration problems were solved according to Timoshenko beam theory and the classical plate theory. Michael and Rao [9,10] solved the inverse problem of an elastic plate subjected to an oblique force. Yen and Wu [11] used the eigen-mode expansion method to solve inverse vibration problems. Unlike most of the direct problems, inverse problems are usually ill-posed problems. Basically, an inverse problem may consist of two steps: the construction of the model for direct analysis and the optimization of the solution in inverse process. When looking for the solution of an inverse problem, most of the methods require iteration schemes to find the optimized results; however, the computational time for the iteration is costly. The strength of wind drags on a building is critical to the safety of the building; similarly, the vibration of the ground, e.g. shaking in an earthquake, may also cause sever damages to the buildings. Nevertheless, under many practical conditions, it is either difficult or impossible to measure wind drags directly. This paper presents an inverse matrix procedure to estimate the strength of wind drags on an elevated tower and the vibration of the ground on which the tower stands. The inverse model is constructed based on the discretized governing equations, which are derived by the finite different method. The optimized solution of the inverse problem is obtained by the linear least-square-error method [12,13]. The linear procedure requires no iteration and the uniqueness of the solution can be easily identified following the linear algebra theory. A numerical example is given. The results show that the procedure is capable of determining the strength of the wind drags and the magnitude of the ground vibration even when some measuring errors are present. 2. Finite-difference formula in direct analysis The tower under consideration, shown in Fig. 1, is modeled as a vertical column that has uniform distribution of mass along the longitudinal direction and an additional lumped mass Fig. 1. The model-elevated tower subjected to wind drag on shaking ground.

3 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) attached to the top. The wind exerts a horizontal drag force upon the tower when it blows across the tower. Besides the wind drags, the tower stands on the vibrating ground that may shake horizontally. The mathematical model of the problem is derived based on the Hamilton principle Z d ðt PÞdt ¼ 0; ð1þ where T is the total kinetic energy and P is the total potential energy. The governing equation derived from (1) is / o2 w ot þ o2 EI o2 w þ P o2 w ¼ Kðx; tþ: 2 ox 2 ox 2 ox ð2þ 2 Here x is the spatial coordinate along the axis of the tower, t is time, w is the transverse deflection, P is the total weight of the mass lumped on the top of the tower. Kðx; tþ is the drag force per unit length of the tower. E is the equivalent Young s modulus of the tower, I is the area moment of inertia of the cross-section of the tower, and the mass distribution function / is given by the sum of the mass per unit length k and the lumped mass in the following form: /ðxþ ¼k þ P g d ð x LÞ; ð3þ where g is the gravity acceleration, L is the height of the tower, and dðx LÞ is delta function. Sadiku and Leipholz [14] have studied the same model for direct analysis. The horizontal vibration at the root of the tower is given by a displacement function RðtÞ. Assuming that the whole tower has no deflection and velocity at the beginning ðt ¼ 0Þ and the cantilever conditions are applied to both the free end and the root of the tower; therefore, the initial conditions and boundary conditions are given by: w ¼ 0; ow ot ¼ 0 for 0 6 x 6 L; t ¼ 0; ð4þ ow ¼ 0; w ¼ RðtÞ for x ¼ 0; t P 0; ð5þ ox o 2 w ox 2 ¼ 0; EI o3 w ox þ P ow ¼ 0 for x ¼ L; t P 0: ð6þ 3 ox The main cause of the drag force acting on the tower is the velocity pressure that generates across the tower. When the wind passes the tower, the flow separates or breaks away from the rear and a pulsating wake forms behind the tower. The characteristics of the wake are dependent upon the numerical value of the dimensionless speed qud=l, called the Reynolds number Re, in which q is the density of the air, D is the diameter of the column, U is the average velocity of the air and l is the dynamic viscosity of the air. The average drag force under the different conditions can be found in many text book [15] as dimensionless drag coefficient C D vs. Reynolds number Re. The drag force per unit length of tower F D relates to q; D; U, and C D by the following equation: F D ¼ 1 2 qu 2 DC D : ð7þ

4 520 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) Due to the resistance of the ground that consumes the energy of wind, the wind velocity decreases as the altitude decreases along the axis of the tower. For engineering purposes, there are several experimental equations describing the relation between the velocity of the wind U and the altitude h from the level of the ground as the power law [16] c ; ð8þ U ¼ h U 0 L where U 0 is the velocity of the wind at the top of the tower. The index c may vary in different regions or under different constructing conditions. In the present study, index c is set to 1/7, the same value as used in the construction of Japan steel tower. The non-homogeneous term Kðx; tþ in (2) simply equals to the drag force, that is, Kðx; tþ ¼ 1 2 quðx; tþ2 DC D : The aforementioned governing Eq. (2) are then discretized based on the finite-difference method. The tower is divided into m equal parts, of dimension Dx ¼ L=m, byðm þ 1Þ nodes of which coordinates are x i ¼ idx, for i ¼ 0; 1; 2;...; m. Dt is time increment. The time at the jth step is t j ¼ jdt, for j ¼ 0; 1; 2;...; n, hence the total time span considered is t ¼ ndt. The discretized governing equations, the finite-difference formula of the original governing equations, can be expressed in the following form: / i Dt 2 þ w j 1 i þ 2/ i Dt 2 6a i Dx 4 2b i 2P Dx4 Dx 2 w j i þ / i Dt 2 w j i þ w jþ1 4a i Dx 4 i þ a i w j Dx 4 i 2 þ 4a i þ b i Dx 4 Dx þ P w j 4 Dx 2 i 1 þ b i Dx þ P w j 4 Dx 2 iþ1 þ a i w j Dx 4 iþ2 ¼ K x i;t j : ð9þ Here, the subscript i is associated with the variable at the ith node and the superscript j is associated with the jth time step. w j i is the deflect ion at x ¼ idx when t ¼ jdt. a i ¼ðEIÞ i and b i ¼ðEIÞ iþ1 2ðEIÞ i þðeiþ i 1. The corresponding boundary conditions and initial conditions are rewritten as: w 0 i ¼ 0; w 1 i ¼ w 1 i for i ¼ 1; 2; 3;...; m; ð10þ w j 1 ¼ wj 1 ; wj 0 ¼ Rðt jþ for j ¼ 0; 1; 2;...; n; ð11þ w j mþ2 ¼ 4wj m 4wj m 1 þ wj m 2 for j ¼ 1; 2; 3;...; n: 2p EI w j m w j m 1 Dx 2 ; w j mþ1 ¼ 2wj m wj m 1 After the deliberate processes of the initial conditions and boundary conditions, Eqs. (9) (12) can be used to find deflection w j i of the tower by the direct integration scheme with the appropriately chosen integrating interval. The linear system at the jth step is A j w j ¼ h j in which A j are the m m matrix consisting of the coefficients of the system equations at t ¼ jdt, w j is the m 1 column vector comprising of the deflection data of the tower at t ¼ jdt, and h j is the m 1 column vector in which all the data, e.g. the deflection at the previous time step and loading condition at the present time step, necessary for solving w j are stored. ð12þ

5 3. Solution procedure in inverse analysis C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) Eq. (9) is used to find the exact deflection data in the aforesaid direct analysis; the recursive form of the procedure can be integrated and rewritten in the following matrix form: Aw ¼ h: ð13þ Eq. (13) is a linear system in which A is a square matrix of dimension n m in which A j are at diagonal position, w and h are the column vector with n m components. w consists of all deflection data at nodes within total time span and takes the form of ðw 1 1 ; w1 2 ;...; w1 m ; w2 1 ; w2 2 ;...; w n 1 ; wn 2 ;...; wn m ÞT and h is constructed in the same manner as w is. Hence, the first m equations in (13) are associated with the solving process of the first time step in the direct analysis. The second m equations are associated with the solving process of the second time step and so on. Matrix A depends on the material properties ða; b; /; p;...þ in the physical model and the spatial and time increments ðdx; DtÞ. h is the combination of boundary conditions, initial conditions, external loading conditions and all the pre-determined variables from the previous time step. In previous direct analysis, all the procedures are well defined. There is no difficulty to construct all matrices in (13) following the appropriate numerical procedures. The inverse problem is to identify the applied unknown wind drags and the motion of the shaking ground from the deflection measurement taken at the interior points of the tower. Though the direct approach is well defined, unfortunately, the inverse problem is often ill-posed because of that the deflection measurement are only taken at finite positions in the tower instead of all nodes of the tower used in direct analysis and the continuous time history is unknown neither. Therefore, it is necessary to find the optimized solution with the limited information. Besides, drag coefficient C D cannot be determined a prior since it relates to the wind velocity that is the unknown in the inverse analysis. In the present inverse study, the unknown drag coefficient C D is replaced by the following quadratic function with three real constants C D ðtþ ¼C þ C 2 Re þ C 3 Re 2 and the non-homogeneous external loading term becomes from (7) and (8) Kðx; tþ ¼ C 1qDU0 2 x 2=7 C 2 qd 2 U 3 þ 0 x 3=7 C 3 qd 3 U 4 þ 0 x 4=7 2 L 2l L 2l L x 2=7 x 3=7 x 4=7; ¼ C I ðtþ þ CII ðtþ þ CIII ðtþ ð14þ L L L where C I ðtþ; C II ðtþ; C III ðtþ are undetermined function of time t. In the above equation, the effects of time and geometric position are separated. It is useful to decompose h into two matrices B and C. Matrix C takes the form of ðc I ðt 1 Þ; C II ðt 1 Þ; C III ðt 1 Þ; Rt ð 1 Þ; C I ðt 2 Þ; C II ðt 2 Þ; C III ðt 2 Þ; Rðt 2 Þ;...; C I ðt n Þ; C II ðt n Þ; C III ðt n Þ; Rðt n ÞÞ T ; ð15þ where C represents the time history of wind drag and shaking ground according to (9) (14), hence C is the main target to be solved by the inverse procedure in the model. B is the coefficient matrix of C and comprises components of real constants. After decomposition, Eq. (13) becomes Aw ¼ BC: ð16þ

6 522 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) As mentioned earlier, in a practical manner, the deflection measurements are taken only at several nodes. Then, some manipulations are applied to (16). The deflection column vector w are rearranged and then partitioned into ðw u.. w m Þ T, where w u is the partitioned matrix consisting of unknown deflection data at the nodes that deflection measurements are not taken, on the other hand, w m is composed of the deflection at the measuring positions. After the rearrangement and partition, the corresponding equation of (16) is further processed in a way such that the associated matrix A is reduced to the upper triangular matrix. After manipulation, Eq. (16) is partitioned into A u A um wu ¼ B u C; ð17þ 0 A m w m B m where A u ; A m are upper-triangular matrices. The linear system equation including only the deflection data at the measuring points is A m w m ¼ B m C: The estimated deflection w e is w e ¼ A 1 m B mc and the square error sum between w e and w m is e ¼ðw e w m Þ T ðw e w m Þ. The optimized solution of C derived by the least square method is ð18þ C ¼ððA 1 m B mþ T ða 1 m B mþþ 1 ða 1 m B mþ T w m : ð19þ The loading history is easy to identity from (19) if the measured deflections at some points are given. In the whole procedure, the uniqueness of the solution is readily checked based on the linear algebra theorem without difficulty. 4. Numerical example and result An example of the problem previously defined is given in this section to show the ability of the procedure in determining the history of wind drags and the motion of the shaking ground. The tower under consideration is L ¼ 20 m in height and has the diameter of D ¼ 2 m. On top of the tower is a tank of which weight is 1000 kg. The effective Young s modulus is E ¼ 2 GPa and the mass per unit length of tower is k ¼ 250 kg=m. The density of the air is q ¼ 1:23 kg=m 3 and the viscosity of the air is l ¼ 1: kg=ms. Twenty-one nodes, which divides the tower into 20 segments equal in length, are located along the axis of the tower. The nodes are ordered 0; 1; 2; 3;...; 20 from the root to the top of the tower. The finite-difference formulas were derived on the base of these nodes. Two different loading conditions are considered. In the first case, the wind velocity on the top of the tower is given by a square wave function and the motion on the shaking ground is given by the random combination of several sine functions with different amplitudes and frequencies as U 0 ðtþ ¼50½dðt 0:3Þ dðt 2:3Þþdðt 4:3Þ dðt 6:3ÞŠ; ð20þ RðtÞ ¼0:1 X5 i¼1 a i sin x i t; ð21þ

7 where d is step function, the randomly chosen amplitude of the sine functions a i are a 1 ¼ 0:84, a 2 ¼ 0:41, a 3 ¼ 0:84, a 4 ¼ 0:30, a 5 ¼ 0:41 and the randomly chosen frequency of the sine functions x 1 ¼ 5:37 s 1, x 2 ¼ 4:68 s 1, x 3 ¼ 2:87 s 1, x 4 ¼ 1:78 s 1, x 5 ¼ 1:54s 1. In the second case, the wind velocity on the top of the tower is given by a sine-wave function U 0 ðtþ ¼50 sin 2:5t ð22þ and the coefficients a i ; x i of the function of the ground motion are a 1 ¼ 0:39, a 2 ¼ 0:50, a 3 ¼ 0:15, a 4 ¼ 0:59, a 5 ¼ 0:85 and x 1 ¼ 5:90 s 1, x 2 ¼ 9:55 s 1, x 3 ¼ 5:56 s 1, x 4 ¼ 1:4 8s 1, x 5 ¼ 9:83 s 1. In both cases, the wind drags along the tower are computed from (7) and (8). The resulting wind drags are then used in the direct analysis, described in the previous section with the time interval Dt ¼ 0:1, to find the exact deflections of the tower. Random numbers simulate the inevitable measurement errors artificially. The random errors of measurement are added to these exact deflections. Thus, the measured deflections can be expressed as w measured ¼ w exact ð1 þ reþ; C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) where r is the amplitude of measurement error and e are random values between )1 and 1. It is noted that, practically, the measured deflections do not cover the whole problem domain but only a few measurement points, where the measurements are easy or feasible to conduct. The measured deflections from (23) at several nodes were then used in the aforesaid inverse procedure to determine the original wind drags and the ground motion. In the first case, the wind drag on the tower is in the shape of the square wave since the Reynolds number Re either vanishes or keeps constant; the ground vibration exhibits the behavior of combined shapes of sine wave functions. No measurement errors are included (r ¼ 0) at this step. Five nodes (x ¼ 3; 5; 9; 12; 15) are chosen as the measuring positions, and the simulated deflection data, which are calculated from the direct analysis, are adopted as the measured data of the deflection w m at these five nodes. The estimated wind drag and the exact wind drag at the node 10 are shown in Fig. 2. There is an excellent match between the estimated wind drag and the exact wind drag. The errors are quite small. As the ground motion concerns, shown in Fig. 3, the estimated ground motion is also very close to the original input functions. The results show that the proposed method is capable of determining the wind drag and the shaking ground motion when there are no measured errors. It is noted that though the deflection data are recorded only at five nodes, the data are tackled along the time span. The actual data stored may be large. In the second case, the different nodes (x ¼ 3; 6; 10; 13; 17) are used as the measuring positions and measurement errors are not included neither (r ¼ 0). The node at x ¼ 8 is used as the checking node to evaluate the results. Though the wind velocity is pure sine function, nevertheless, the wind drag is distorted from the sine curve because the drag coefficient C D varies with the wind velocity, which determines the Reynolds number. It is recalled that the exact coefficient C D determined from Reynolds number is used in the direct analysis. Their relationship may be found in the figures of many textbooks. The estimated wind drag at checking node x ¼ 8 is very close to the exact loading input as shown in Fig. 4. The estimated ground motion, shown in Fig. 5, is as good as that in the previous case. There are little errors found in the comparative curves even the frequency is approaching the sensing rate. The result indicates: (1) the method is suitable for both the square-wave loading and sine-wave loading, (2) The wind drag at different locations may be estimated by the method, and (3) the locations of measuring nodes are quite flexible. In actual ð23þ

8 524 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) Fig. 2. Comparison between the exact and estimated wind drag at x ¼ 10 in case 1, measuring points at x ¼ 3; 5; 9; 12; 15 and r ¼ 0. Fig. 3. Comparison between the exact and estimated ground movement in case 1, measuring points at x ¼ 3; 5; 9; 12; 15 and r ¼ 0. applications, the measuring positions are easy to select at the locations where the sensors are most convenient to install. Measurement errors are inevitable in actual application. To evaluate the influences of the measurement errors on the inverse solution, the artificial errors are superimposed on the exact

9 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) Fig. 4. Comparison between the exact and estimated wind drag at x ¼ 8 in case 2, measuring points at x ¼ 3; 6; 10; 13; 17 and r ¼ 0. Fig. 5. Comparison between the exact and estimated ground movement in case 2, measuring points at x ¼ 3; 6; 10; 13; 17 and r ¼ 0. deflections using Eq. (23). In Fig. 6, the measuring nodes are at x ¼ 3; 6; 10; 13; 17 and the checking point is at x ¼ 8. The amplitude of measurement errors is (r ¼ 0:2%), the results show that some deviation of the estimated wind drag from the exact wind drag has been observed. In this case the drag coefficient and Reynolds number vary over the large range that could cause difficulty to the procedure. The simple quadratic function was used to fit the relation between the drag coefficients and Reynolds number. Nevertheless, the differences between the estimated wind

10 526 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) Fig. 6. Comparison between the exact and estimated wind drag at x ¼ 8 in case 2, measuring points at x ¼ 3; 6; 10; 13; 17 and r ¼ 0:2%. drag and the exact wind drag are still within the reasonable range. Large discrepancies most possibly occur when the change of the wind acceleration is large. It may induce the high frequent errors in the solution. The how-pass filter may help reduce the errors. In this case the drag coefficient and Reynolds number vary the large range. The simple quadratic function was used to fit the relation between the drag coefficients and Reynolds numbers. Generally speaking, the inverse process for estimating the wind drag is more sensitive to the existence of the errors than that for estimating the ground motion. In Fig. 7, the measurement errors are intentionally increased to the Fig. 7. Comparison between the exact and estimated ground movement in case 2, measuring points at x ¼ 3; 6; 10; 13; 17 and r ¼ 15%.

11 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) much worser level (r ¼ 15%); the ground motion can still be identified even though there exists a couple point that deviate a lot from the exact data curves. Most of the errors are within 15% and are induced by the measurement errors not by the inverse procedure. The similar results are shown in Figs. 8 and 9 in which the measuring positions are at x ¼ 5; 8; 12; 15; 18 and the checking position at x ¼ 19. The amplitude of errors of the measured deflection is r ¼ 0:2% for estimating wind drag and r ¼ 15% for estimating ground motion. Fig. 8. Comparison between the exact and estimated wind drag at x ¼ 19 in case 2, measuring points at x ¼ 5; 8; 12; 15; 18 and r ¼ 0:2%. Fig. 9. Comparison between the exact and estimated ground movement in case 2, measuring points at x ¼ 5; 8; 12; 15; 18 and r ¼ 15%.

12 528 C.C. Kang, C.Y. Lo / Appl. Math. Modelling 26 (2002) Conclusion This study presents a matrix procedure to estimate the strength of wind drags on an elevated tower and the magnitude of the ground vibration on which the tower stands simultaneously. No pre-assumed forms of the loading are necessary. The optimized solution can be solved using the least square error method that requires no iteration. The results of the numerical example show that the procedure is simple and effective. In actual application, the procedure allows the flexibility in choosing the positions at which deflection sensors may be located. It is also easy to determine wind drag at arbitrary positions and the ground motion. The numerical examples also show that the reasonable inverse solution may be achieved even when some errors exist in measuring procedure. The procedure may be easily modified to determine the unknown external loading of certain system once the corresponding governing equations are discretized. References [1] G. Stolz Jr., Numerical solutions to an inverse problem of heat conduction for simple shapes, ASME J. Heat Transfer 82 (1960) [2] E.M. Sparrow, A. Haji-Sheikh, T.S. Lundgren, The inverse problem in transient heat conduction, ASME J. Appl. Mech. 86 (1964) [3] J.V. Beck, B. Blackwell, C.R. St. Clair, Inverse Heat Conduction Ill Posed Problem, Wiley, New York, [4] Y. Jarny, M.N. Ozisik, J.P. Bardon, A general optimization method using adjoint equation for solving multidimensional inverse heat conduction, Int. J. Heat Mass Transfer 34(1991) [5] J.F. Doyle, An experimental method for determining the dynamic contact law, Exp. Mech. 24(1) (1984) [6] J.F. Doyle, Further developments in determining the dynamic contact law, Exp. Mech. 24 (4) (1984) [7] J.F. Doyle, Determining the contact force during the transverse impact of plates, Exp. Mech. 27 (1) (1987) [8] J.F. Doyle, Experimentally determining the contact force during the transverse impact of an isotropic plate, J. Sound Vibration 118 (3) (1987) [9] J.E. Michaels, Y.H. Pao, Inverse source problem for an oblique force on an elastic plate, J. Acoust. Soc. Am. 77 (1985) [10] J.E. Michaels, Y.H. Pao, Determination of dynamic forces from wave motion measurements, ASME J. Appl. Mech. 53 (1986) [11] C.S. Yen, E. Wu, On the inverse problem of rectangular plates subjected to elastic impact, Part I: method development and numerical verification, ASME J. Appl. Mech. 62 (1995) [12] C.Y. Yang, C.K. Chen, The boundary estimation in two-dimensional inverse heat conduction problems, J. Phys. D: Appl. Phys. 29 (1996) [13] C.Y. Yang, C.K. Chen, Solution of an inverse vibration problem using a linear least-square error method, Appl. Math. Modell. 20 (1996) [14] S. Sadiku, H.H.E. Leipholz, Dynamic analysis of an elevated water tower subjected to wind gust, Trans. CSME 10 (1) (1986) [15] H.F.R. Pao, Fluid Dynamics, Charles E. Merrill Books, Columbus, OH, [16] Processes Equipment Design: 2. Tower and Vessel, edited by Society of Chemical Engineer, Maruzen Co. Ltd., 1972, pp