Passive control applied to structural optimization and dynamic analysis of a space truss considering uncertainties

Transcription

1 Passive control applied to structural optimization and dynamic analysis of a space truss considering uncertainties F. A. Pires 1, P. J. P. Gonçalves 1 1 UNESP - Bauru/SP - Department of Mechanical Engineering Avenida Engenheiro Luiz Edmundo Carrijo Coube, 14-1, Bauru, Brazil fealvespires@hotmail.com Abstract Due to the need for monitoring Earth s natural systems, fleets of satellites will be launched in the next few years. In order to minimize shuttle launch costs, space structures should have the shape of trusses because of the significant weight reduction by being assembled with elements made of light materials (e.g. aluminum). For this reason, a control method must be applied to maintain the requirements of vibration levels in these types of structures. This work applies a passive vibration control to a space lattice structure utilizing the Finite Element Method (FEM) in order to predict and compare the system s frequency response functions (FRFs). The optimization technique simulated annealing (SA) is also applied to the structure to find the coordinates of each node of the space truss that minimize the value of norm H 2 for each mode of vibration. The Monte Carlo technique is also applied so it is possible to come up with an envelope function to show whether or not the FRFs are inside it. 1 Introduction Because the increasing demand for spatial structures, it is of fundamental importance to understand how this type of system behaves. Besides, with the objective of reducing launch costs, these structures must be assembled with elements made of light materials, such as aluminum. However, this weight reduction brings flexibility to the system as a result of the components containing low structural damping [1]. Consequently, the system presents undesireable responses, known as vibrations, owing to disturbances from the environments where these structures will be exposed. In order to maintain the vibration levels in small scales, a control method should be applied. The vibrations can be controlled utilizing both passive and active control techniques. Different strategies are used to model and design the control of a flexible structure, as seen in [2] who studies a passive control technique in mechanical systems making use of dynamic vibration absorbers. [3] uses an active control of a tridimensional truss utilizing stacked piezoelectric actuators. [4], applies the norm H 2 technique to the vibration control of a truss containing piezoelectric actuators whereas [5] utilizes the norm H technique for the robust control of a MEMS (micro electro mechanical system) gyroscope. This work applies the conventional Finite Element Method (FEM) to model a space lattice structure. Such method is one of the most used computational tools to analyse structures under vibrations. [6] utilizes this technique for a vibration analysis of variable geometry trusses, and [7] makes use of this modelling method to do a fatigue and free vibration analysis of a space truss. During the past few years, the way of how different types of uncertainties are handled with this modelling technique has awaken the concern among researchers. These uncertainties might emerge because of the unknown values of physical properties (for example Young s Modulus, density and geometry). One way 4419

2 44 PROCEEDINGS OF ISMA16 INCLUDING USD16 to account for the uncertainties is to use the Monte Carlo simulations in order to treat the system with a probabilistic modelling since this approach has become practical due to advances in computational technologies. The basic idea behind Monte Carlo simulations is to carry out repeated random simulations to obtain numerical results which will represent the uncertainties in the model s response. This technique treats the parameters as random variables, rather than considering them as deterministic quantities. In the deterministic approach, the values of the parameters are constant so that there is only a single frequency response function (FRF) calculated for the structure. However, according to [8], when the values of the properties are set randomly, the behavior of the system and the FRFs can be predicted using the range of values for those parameters. Admittedly, the use of an optimization tool becomes feasiable to study the behavior of the system s FRF due to a disturbunce. The usage of the optimization technique Simulated Annealing (SA) is an alternative to this analysis. According to [9], this procedure draws an analogy between the energy of physical systems and the minimization of the objective function of structural systems. The SA s main idea is to provide a way to avoid being trapped in local minima and being able to explore globally for better solutions. [1] uses this technique to optimize the design of distinct types of lattice structures by varying the number of elements, while [11] utilizes this methodology in order to find an optimum design for trusses depeding on size, form and topology. This paper focuses on a vibration analysis by applying the optimization tool Simulated Annealing to find the best nodes coordinates to minimize the parameter norm H 2. By the finite element (FE) model, there will be a set of FRFs which were optimized in order to lower the amplitude of each resonance peak and to be able to compare them to the original FRF. This methodology has shown that the passive control method applied to the system was effective since the amplitudes of vibration have indeed diminished. The work also puts to use the Monte Carlo simulation. It represents a way to assess whether or not the found FRFs are within the range of frequency responses of the model. The approach demonstrated that the FRFs remained inside the envelope function generated by the Monte Carlo method. 2 The Structure This work studies the structure shown in figure 1. This truss contains 93 elements made of aluminum with a diameter of 7 mm, length of 15 mm and 33 nodes. The system will not be clamped which simulates the free-free boundary condition. This structure has 99 degrees of freedom (dofs) since each node can move around the 3 directions in the cartesian plane (x-y-z). Consequently, the system owns 99 natural frequencies and therefore, 99 vibration modes y z 19 x Figure 1: Space truss utilized in the finite element model.

3 USD - UNCERTAINTY IN THE AEROSPACE SECTOR Finite Element Method Applied to the Structure Consider the truss illustrated in figure 1. The system s motion is governed by the following equation of motion in a matrix form M q + K q = F(t) (1) where K and M are the structure s stiffness and mass matrices, respectively, of order 99 x 99, q the displacement vector and F(t) the external load vector, both of order 99 x 1, obtained by the finite element model using truss elements [12]. As mentioned in the previous section, the structure s boundary condition is free-free and there is no external load acting on the system at first, because in the section the pursuit is to show the natural frequencies and vibration modes of the structure. Thus, equation 1 is transformed into M q + K q = (2) Once the system s stiffness and mass matrices are computed for the truss, it is possible to analyse the truss through the eigenvalue problem. With the goal of computing the structure s natural frequencies, it is assumed a harmonic motion for the displacement in the form q=qsen(ωt). The equation that results from the eigenvalue problem is given by equation 3. (K ω 2 M) q = (3) where ω is the natural frequency and q the vector of vibration modes. These parameters are represented by the eigenvalues and engenvectors, respectively. The natural frequencies can be acquired by the square root of the diagonal of the engenvalues matrix and each vibration mode can be computed by the column of the engenvectors matrix referent to its natural frequency. For instance, if the pursuit is to analyse the seventh natural frequency, its vibration mode will be represented by the seventh column of the engenvectors matrix, in which each row denotes the displacement of each degree of freedom (dof). In this simulation, it is set the theoretical values for the material properties so that the aluminum s Young s Modulus and density are E = 7 GPa and ρ = 27 kg/m 3, respectively. The present work considers only the frequency range between 1 and 6 Hz, which embraces 6 vibration modes of the truss. It is important to point out that the paper will not consider the rigid body modes of vibration which actually are the first 6 modes of vibration so that the ones in study represent from the 7th to 12th vibration modes. These modes of vibration are the first bending (7th), second bending (8th), first torsion (9th), second torsion (1th), third bending (11th) e fourth bending (12th). Figure 2 shows the vibration modes as well as their respective natural frequencies. It s worth pointing out that they could be gotten by the finite element model assuming the principle that there is no external load acting on the system.

4 4422 PROCEEDINGS OF ISMA16 INCLUDING USD16 First Bending Hz Second Bending Hz First Torsion Hz Second Torsion - 5. Hz Third Bending Hz Fourth Bending Hz Figure 2: Modes of vibration and natural frequencies of the space lattice structure obtained by the FEM. 4 State-Space Representation A linear time-invariant (LTI) system of finite dimensions is described by the equations below ẋ = A x + B u, y = C x (4) with state space initial condition x() = x. In the equations above, x is the state vector of dimension N, x is the state space initial condition, u is the system s input vector of dimension s and y the system s output vector of dimension r. The matrices A, B and C are real and constant matrices of dimensions N x N,N x s and r x N, respectively. Moreover, the state-space representation of a system can be represented alternatively by its transfer function. The transfer function G(s) is defined in equation 5 G(s) = C(sI A) 1 B (5) 4.1 State-Space Representaion of the Truss With the objective of carrying out structural dynamic simulations and control analysis, it is convenient to represent the equations of flexible structures in state-space representaion, as seen in equation 4. By assuming that the system possesses the following equation of motion, q + M 1 D q + M 1 K q = M 1 B u y = C q q + C v q (6) It is assumed that, x = { x1 x 2 } = { } q q (7)

5 USD - UNCERTAINTY IN THE AEROSPACE SECTOR 4423 Thus, the state-space representaion of the system will be, ẋ 1 = x 2 ẋ 2 = M 1 K x 1 M 1 D x 2 + M 1 B u y = C oq x 1 + C ov x 2 (8) where M e K are the mass and stiffness matrices, respectively, represented in equation 1. Matrix D is the proportional damping matrix, which is proportional to the stiffness matrix. This study assumed that D = 1 6 K. It is known that (A,B,C) are the state-space parameters of equation 4 and by combining the above equations, it is possible to obtain state-space equations, [ ] [ ] I A = M 1 K M 1, B = D M 1, C = [ ] C B oq C ov, (9) where matrices A, B and C have dimensions 198 x 198, 198 x 1 and 99 x 198, respectively. In addition, B is the input matrix and C the output matrix of the system. This analysis assumes that there is an external load so that the parameter B is equivalent to the vector F(t) from equation 1, because in this case, the input will be a load. Also, C oq and C ov are the output displacement and output velocity matrices, respectively, and both possess dimension 99 x 99. These matrices have values only in their diagonals, with these values being either or 1. 5 Norms of a System The norms of a system are utilized to measure the intensity of a structure s response to standard excitations such as a load, white noise and unit impulse. The use of this tecnique allows the comparison between different systems. There are three types of system norms: H 2, H e Hankel. This work suggests a study that verifies whether with the minimization of norm H 2, it is possible to optimize the structure s FRF by decreasing the amplitudes of vibration. The system illustrated in figure 1 is cosidered to be discrete in the frequency domain. For structures in modal representation, theory encountered in [13], by considering (A mi, B mi, C mi ) the modal representation in state-space, each mode of vibration is independent so that the norms of the vibration modes are independent as well. 5.1 The Norm H 2 It is defined ω i as being a half-power frequency at the ith resonance, ω i = 2ζ i ω i, see [14]. Let G i (ω) = C mi (jωi A mi ) 1 B mi, be the transfer function of the ith vibration mode of the structure, where I is the identity matrix. It is known that the transfer function is in frequency domain. The norm H 2 of the ith mode of vibration is given by equation 1 G i 2 = B mi 2 C mi 2 2 ζ i ω i = B mi 2 C mi 2 2 ωi = γ 2 ωi (1) 5.2 Norm H 2 of the Structure The norm H 2 of a structure is expressed in terms of norms of each vibration modes, as shown previously. By following the theory presented by [13], the norm H 2 of the system is approximately the sum rms of the

6 4424 PROCEEDINGS OF ISMA16 INCLUDING USD16 norms of each mode of vibration, as shown in equation 11 where n is the number of vibration modes. G 2 = n G i 2 2 (11) For the simulation, the first six vibration modes of the truss will not be considered, since these are the structure s rigid body modes of vibration. In this section, the system will be considered as a forced system so that it is possible to obtain the frequency response of the structure. It is assumed that there is an external load of 1 N acting at node 1 of the truss on the positive direction of y, in other words, B (2,1) = 1, see equation 9. Furthermore, the output matrix C will be set so that the response due to the applied load at node 1 will be verified at the 3 last nodes of the trusss, at nodes 31, 32 and 33, respectively. This work just intends to get the displacement reponse, so C ov will be a matrix entirely of zeros, 99 x 99. Besides, C oq will be built up so that there will only be values in the diagonals that refer to the degrees of freedom of those 3 nodes whose response aims to be acquired. Therefore, the norm H 2 of the system, considering all of the modes of vibration but the rigid body ones, from 7 to 99, owns the value of H system 2 = As mentioned before, this work s aim is the study of the frequency range between 1 Hz and 6 Hz, which contains the 6 natural frequencies of interest. Hence, the norm H 2 related to this frequency range, that represents the vibration modes 7 to 12, possesses the value of H (7 12) 2 = It is worth mentioning that the value of the system s norm H 2 and the one related to the 6 vibration modes in study are close. It shows that the first modes of vibration are the ones that have the greatest modal values of norm H 2. This characterizes them as the most significant ones related to the system. So, it has been chosen to study the frequency range that contains these vibration modes which is from 1 Hz to 6 Hz. Thus, the analysis will be carried out by studying the modal values of norm H 2 for each of these modes. Table 1 illustrates them. i=1 Vibration Mode Norm H Table 1: Norm H 2 of each vibration mode 6 Simulated Annealing Simulated Annealing is a method used to solve optimization problems of unconstrained and bound-constrained systems. At each iteration of the simulated annealing algorithm, a new point is randomly generated. The algorithm accepts all of the new points that lower the objective, but also, with a certain probability, points that raise the objective. By accepting points that raise the objective function, the algorithm avoids being trapped in local minima and is able to globally look for better solutions. The procedure programs a way to systematically decrease the objective function as the code runs. As the objective diminishes, the algorithm reduces the extention of the search in a way to converge to a minimum value.

7 USD - UNCERTAINTY IN THE AEROSPACE SECTOR Simulated Annealing Applied to the Structure This method was used in this work so that it could be posible to obtain a minimum value of norm H 2 for the 6 vibration modes mentioned above by varying the coordinates of the 33 nodes of the truss. It is possible by assuming an initial value besides an upper and lower bounds for each nodal coordinate. It has been carried out one simulation for each mode of vibration utilizing diferent values in between the upper and lower bounds. Let X be the matrix of the original nodal coordinates of the truss illustrated in figure 1. In this simulation, it was used a standard deviation (SD) of 2% for the upper and lower bounds, in other words, LB = X.2 X and UB = X +.2 X, where.2 represents the SD. The values of the minimized norm H 2 of each vibration mode is represented in table 2. Appendix A illustrates the convergence of the algorithm for the norms H 2 shown in the table below. Vibration Mode Norm H Table 2: Norm H 2 minimized for each vibration mode 7 Uncertainty Analysis When a deterministic model is considered, the parameters of the model are constants. So, if repeated runs are computed with the same inputs the model will return the same outputs. In this case, the uncertainties are not being taken into account. In order to treat the uncertainties of a deterministic model properly, the values of the uncertain input parameters have to be considered as multivariate random variables so that the outputs of the model are also multivariate random variables. Thus, with the range of solutions obtained, statistics are made and the main result of the problem is the distribution of probability of the results. The aim of this work is also to propagate the uncertainties through the computer model to characterize the distribution. A solution to this problem is to use the Monte Carlo procedure. In this case, a vast sample is drawn from the input distribution, running the model at each sampled input configuration. The resut is a sample of outputs from each any summary of the uncertainty distribution can be estimated by using the corresponding summary statistic. As follows, Monte Carlo simulations are presented to analyse the 6 modes of vibration in consideration. The study aims to verify the uncertainty distribution of the FRF of the structure. For this analysis, the code was run 1 times keeping the material properties constant so that the nodal coordinates (inputs) of the truss were set randomly in accordance with a range by considering a SD of 2% as well and, consequently, a different FRF (output) could be computed after each iteration. As a result, it was possible to come up with an envelope function from the 1 simulations, shown in figure 3. It is well known that in an actual structure, the position of each node is uncertain and keeping in mind that FE model has the premise to pursue as close as possible the real solution, it is important to take into account the uncertainties of the model so that the likelihood of the actual value of the nodal coordinates, and consequently the actual FRF, being in the generated range is high. According to [15], the accuracy of this estimate is determined by the size of the sample. This explains why it was chosen to run the code a considerable amount of times. There are some techniques to evaluate the error on the Monte Carlo simulation s estimates [16], however, the work does not aim to perform an error evaluation.

8 4426 PROCEEDINGS OF ISMA16 INCLUDING USD16 8 Results and Analysis Figure 3: Envelope function of the FRFs. This section carries out an analysis of the FRFs by comparing the curve of the original FRF to those found for the optimized nodal coordinates when the norm H 2 is minimized for each mode of vibration considering an SD of 2%. These curves of frequency response are obtained from the finite element model. The FRFs are generated by utilizing a cost function or objective function. This cost function is based on the square of a quantity such as displacement and velocity. In this case, the objetive function is then determined as the sum of the linear squared displacements at nodes 31, 32 and 33. The values of this cost function are converted to decibels (db) and then the frequency curves are plotted. The y-axis represents the values of amplitudes in (db) and the x-axis allocates the values of frequencies in the range 1-6 Hz. In figure 4, the solid lines represent the FRFs of the non-modified truss while the dashed lines are the FRFs of the modified truss with nodal coordinates optimized and norm H 2 minimized for the 6 vibration modes in consideration. a) db Original FRF FRF for the Norm H 2 minimized Frequency [Hz] d) db Original FRF FRF for the Norm H 2 minimized Frequency [Hz] b) db Original FRF FRF for the Norm H 2 minimized Frequency [Hz] e) db Original FRF FRF for the Norm H 2 minimized Frequency [Hz] c) db f) db Original FRF FRF for the Norm H 2 minimized Frequency [Hz] Original FRF FRF for the Norm H 2 minimized Frequency [Hz] Figure 4: Comparison between the original FRF and the FRF for the minimized norm H 2 of the a) 7th vibration mode. b) 8th vibration mode. c) 9th vibration mode. d) 1th vibration mode. e) 11th vibration mode. f) 12th vibration mode. The figures above show that the original FRF s resonance peaks within the range 1-6 Hz dropped in all of the situations when compared to the FRFs plotted from the minimization of norm H 2 of each vibration mode. This confirms that the passive control method adopted of finding the optimal nodal coordinates that

9 USD - UNCERTAINTY IN THE AEROSPACE SECTOR 4427 minimize the parameter norm H 2 of each mode of vibration was effective. It is also worth mentioning that there were dramatic decreases in the amplitudes of the resonance peaks between the range -3 Hz whereas for the range 45-6 Hz the amplitudes decreased slighter. Note also that in figures 4 c) and d) for the range 45-6 Hz, there was a more significant change in the natural frequencies than for the range -3 Hz. It was chosen to analyse as well how the 3 last modes of vibration behave in the two simulations illutrated in figures 4 c) and d). These new vibration modes are represented in figures 5 and 6 a) b) c) Figure 5: New coordinates of each node considering the minimization of norm H 2 of the 9th vibration mode for the a) Second Torsion. b) Third Bending. c) Fourth Bending. a) b) c) Figure 6: New coordinates of each node considering the minimization of norm H 2 of the 1th vibration mode for the a) Second Torsion. b) Third Bending. c) Fourth Bending. Appendix B illustrates the FRFs shown in figure 4 and it points out that the FRFs are indeed inside the envelope function. It indicates that the new coordinates of the nodes as well as the FRFs are convenient. 9 Conclusion This work suggests the study of a passive control method applied to a space truss. In this scenario, the traditional finite element method has been used. The concept of norm H 2 has been introduced. This parameter serves as a measure of intensity of a structure s response to standard excitations. Together with the FEM, the optimization technique simulated annealing has been utilized to find the optimal coordinates of each node of the truss that minimize the value of norm H 2 of 6 modes of vibration within the range 1-6 Hz. Then, for every new nodal coordinates, a couple of new frequency responses could be obtained. Admittedly, it was shown that the resonance peaks of the original FRF decreased in all of the situations when compared to the FRFs obtained by the minimization of norm H 2 of each vibration mode. It shows that the passive control method used in the study was effective. Moreover, the paper applies the Monte Carlo simulation in order to take into account the uncertaties related to the the position of each node. By these means, it was possible to check if all of the FRFs remained inside the envelope function of FRFs. The analysis indicated that the new coordinates of the nodes were convenient for the study since all of the new FRFs stayed inside the envelope function.

10 4428 PROCEEDINGS OF ISMA16 INCLUDING USD16 Acknowledgements The authors acknowledge and thank CAPES (Coordenação de aperfeiçoamento de pessoal de nível superior) for the financial support during the research. References [1] P. J. P. Gonçalves, M. J. Brennan, S. Elliott, Active Vibration Control of Space Truss Structures: Power Analysis and Energy Distribution, Proceedings of ISMA-International Conference on Noise and Vibration, (6) [2] V. Steffen Jr, D. A. Rade, D. J. Inman, Using passive techniques for vibration damping in mechanical systems, Journal of the Brazilian Society of Mechanical Sciences, Vol. 22, No. 3,(), pp [3] M. Moshrefi-Torbati, A. J. Keane, S. J. Elliott, M. J. Brennan, D. K. Anthony, E. Rogers, Active vibration control (AVC) of a satellite boom structure using optimally positioned stacked piezoelectric actuators, Journal of Sound and Vibration, Vol. 292, No. 1, (6), pp [4] G. L. C. M. De Abreu, V. Lopes Jr, H2 optimal control for smart truss structure,(1). [5] Y. Fang, J. Fei, S. Wang, H-Infinity control of mems gyroscope using ts fuzzy model, IFAC-PapersOnLine, Vol. 48, No. 14, (15), pp [6] I. O. De Zarate, J. Aguirrebeitia, R. Avilés, I. Fernández, A finite element approach to the inverse dynamics and vibrations of variable geometry trusses, Finite Elements in Analysis and Design, Vol. 47, No. 3, (11), pp [7] K. Koohestani, A. Kaveh, Efficient buckling and free vibration analysis of cyclically repeated space truss structures, Finite Elements in Analysis and Design, Vol. 46, No. 1, (1), pp [8] S. W. Doebling, C. R. Farrar, Estimation of statistical distributions for modal parameters identified from averaged frequency response function data, Journal of Vibration and Control, Vol. 7, No. 4, (11), pp [9] W. A. Bennage, A. K. Dhingra, Single and multiobjective structural optimization in discrete-continuous variables using simulated annealing, International Journal for Numerical Methods in Engineering, Vol. 38, No. 16, (1995), pp [1] L. Lamberti, An efficient simulated annealing algorithm for design optimization of truss structures, Computers & Structures, Vol. 86, No. 19, (8), pp [11] O. Hasançebi, F. Erbatur, Layout optimisation of trusses using simulated annealing, Advances in Engineering Software, Vol. 33, No. 7, (2), pp [12] Y. W. Kwon, H. Bang, The Finite Element Method Using MATLAB, CRC press,(). [13] W. Gawronski, Advanced Structural Dynamics and Active Control of Structures, Springer Science & Business Media,(4). [14] R. W. Clough, J. Penzien, Dynamics of Structures, Tech. report,(1975). [15] T. E. Fricker, J. E. Oakley, N. D. Sims, K. Worden, Probabilistic uncertainty analysis of an FRF of a structure using a gaussian process emulator, Mechanical Systems and Signal Processing, Vol. 25, No. 8, (11), pp [16] J. Hammersley, Monte Carlo Methods, Springer Science & Business Media,(13).

11 USD - UNCERTAINTY IN THE AEROSPACE SECTOR 4429 Appendix A: Convergence of the Simulated Annealing a) Function value Best Function Value: 4.572e Iteration Best Function Value: e 9 b) 1 4 Function value Iteration c) 1 3 Best Function Value: e 5 d) Best Function Value: e 5 Function value 1 4 Function Value Iteration Iteration e) 1 4 Best Function Value: e 1 f) 1 4 Best Function Value: e Function Value Function Value Iteration Iteration Figure 7: Convergence of norm H 2 using Simulated Annealing for the a) 7th vibration mode. b) 8th vibration mode. c) 9th vibration mode. d) 1th vibration mode. e) 11th vibration mode. f) 12th vibration mode.

12 443 PROCEEDINGS OF ISMA16 INCLUDING USD16 Appendix B: Monte Carlo technique application Figure 8: Envelope functions with FRFs of the a) 7th vibration mode. b) 8th vibration mode. c) 9th vibration mode. d) 1th vibration mode. e) 11th vibration mode. f) 12th vibration mode.