Substructuring using Impulse Response Functions for Impact Analysis

Transcription

1 Proceedings of the IMAC-XXVIII February 1 4, 21, Jacksonville, Florida USA 21 Society for Experimental Mechanics Inc. Substructuring using Impulse Response Functions for Impact Analysis Daniel J. Rixen Delft University of Technology, Faculty of Mechanics, Maritime and Material Engineering Department of Precision and Microsystems Engineering, Engineering Dynamics Mekelweg 2, 2628 CD Delft, The Netherlands d.j.rixen@tudelft.nl ABSTRACT In the present paper we outline the basic theory of assembling substructures for which the dynamics is described as impulse response functions. The assembly procedure computes the time response of a system by evaluating per substructure the convolution product between the impulse response functions and the applied forces, including the interface forces that are computed to satisfy the interface compatibility. We call this approach the Impulse Based Substructuring method since it transposes to the time domain the Frequency Based Substructuring approach. In the Impulse Based Substructuring technique the impulse response functions of the substructures can be gathered either from experimental test using a hammer impact or from time-integration of numerical submodels. In this paper the implementation of the method is outlined for the case when the impulse responses of the substructures are computed numerically. A simple bar example is shown in order to illustrate the concept. Future work will concentrate on including in the assembly measured substructure impulse responses. The Impulse Based Substructuring allows fast evaluation of impact response of a structure when the impulse response of its components are known. It can thus be used to efficiently optimize designs of consumer products by including impact behavior at the early stage of the design. Keywords: Experimental Substructuring, assembly, impulse response functions, time integration, impact, impulse based substructuring NOMENCLATURE dof degrees of freedom FRF Frequency Response Functions IRF Impulse Response Functions FBS Frequency Based Substructuring IBS Impulse Based Substructuring u array of degrees of freedom f array of external forces H(t) matrix of Impulse Response function (s) pertaining to substructure s N s number of substructures in the system B signed Boolean matrix defining compatibility constraints λ Lagrange multipliers on interface M, K, C mass, stiffness and damping matrix of a linear(ized) system dt time-step size n pertaining to time-step n [ ] i component i of an array β, γ parameters of the Newmark time-integration scheme

2 1 INTRODUCTION Substructuring techniques allow combining the dynamics of different components. The dynamics of the components is described either using experimental data (Frequency Response Functions in the frequency domain) or with a numerical model (in terms of system matrices, computed Frequency Response Functions or component modes). See for instance 5 for a review of substructuring concepts. In the Frequency Based Substructuring approach one can assemble the Frequency Response Functions (FRFs) of substructures that were measured or that were obtained by numerical simulation. This technique can provide accurate results on simple systems 11 and gives interesting qualitative information when applied to complex engineering systems such as cars 2,8. When the substructures are described by FRFs obtained experimentally tremendous care must be taken to ensure a high degree of accuracy, for instance it must satisfy reciprocity, passivity and artifacts like additional mass effects or location/orientation errors in the sensors must be very small. In practice such errors often introduce in the assembled FRFs spurious peaks 1 and non-physical properties 1 that renders the obtained assembled model useless. To clean-up the measured FRFs of the substructures before assembly one can apply modal identification techniques in order to fit a pole-residue model to the data. When combining identified modes of the substructures with a numerical model of the measured components high quality assembled models can be obtained 7. Obtaining high quality measured FRFs is delicate since, unless slow and costly sine sweeps are used, the dynamic properties in the frequency domain are obtained through several processing steps (anti-aliasing filters, windowing, Fourier transforms) which will unavoidably alter the information contained in the measurements. Furthermore using modal identification and FRF synthesis to obtain clean FRFs is a very labour-intensive and error-prone process, and it assumes that a pole-residue model can be fitted. When high damping is present, proper identification of the poles and residues becomes difficult and if non-viscous damping (visco-elastic damping) is present the pole-residue model cannot correctly represent the frequency domain response of the components. Finally we note that if substructuring is used to simulate impact responses, working in the frequency domain requires considering a large frequency band which makes all the Frequency Based Substructuring strategies expensive and badly suited. Applying substructuring techniques to simulate impact responses of structures is a very attractive idea since it would allow to efficiently predict impact behavior at the early design stage in many fields. For instance when designing the structure of a mobile phone or a notebook, many COTS (components of the shelf) are used. If the dynamics of the COTS are characterized properly either numerically or experimentally, one could use susbtructuring techniques to rapidly optimize the housing, frame and connections (screws, rubber pads...) and thereby guarantee an improved life-time and reliability of valuable mass-produced appliances. Since Frequency Based Substructuring is not well suited for setting up a model for impact simulation (see discussion above), we propose in this paper an alternative substructuring technique. We use the same concepts as in other substructuring approaches (i.e. admittance representation of the components and dual assembly) but consider for the substructures directly the impulse response functions in the time domain measured for the input and interface degrees of freedom, instead of the modal properties or the FRFs. The method will be called Impulse Based Substructuring or IBS. This paper outlines the basic principle of the method and shows that the theory can be easily applied when the impulse response functions are computed through direct time integration of a numerical model. Future work will investigate the combination of numerical and experimental sub-models in the Impulse Based Substructuring strategy. 2 THEORY OF IMPULSE RESPONSE SUPERPOSITION FOR PARTITIONED PROBLEMS Let us call H(t) the matrix of the responses to a unit impulse at t = for a linear system that is initially at rest. In other words a coefficient [H(t)] ij of the impulse response matrix represents the response of degree of freedom (dof) i to a unit impulse on dof j. The response of the linear system to an applied force f(t) can then be evaluated by the convolution product (Duhamel s integral) between the impulse response function matrix and the applied forces: u(t) = t H(t τ)f(τ)dτ (1) This is a classical result of time analysis of linear systems, usually obtained using Laplace transforms. This convolution product can be interpreted as follows: the response at time t is an infinite sum of the responses to the infinitesimal impulses f(τ)dτ before time t (see figure 1). Each impulse at time τ gives a contribution through the impulse response

3 from τ to t, that is H(t τ). φ(τ) τ Figure 1: Forcing function as a series of impulses τ The impulse responses can be obtained either experimentally or numerically. If obtain by measurements, an impact hammer can be used: One evaluates the impulse I = t + δ(t)dt, where δ(t) is the measured force and t + the duration of the force assumed to be very short for a hammer impact. In fact the hammer impact is only an approximation of a Dirac function, but if the duration t + is much smaller then the characteristic time of the expected response u(t) to the applied force f(t) in (1) the obtained impulse response can be used in the convolution product. In other words the duration of the hammer impulse must be much shorter then the period of the frequency of the modes having a significant contribution to the response. Compute the impulse response by scaling the measured time response to the hammer impact by the impulse I. The impulse response can also be obtained from a closed-form solution of a mathematical model or by time-integration of a numerical model. In the later case, the time-step must be small enough so that the initial impulse can be considered as an impulse for the dynamic response one wants to compute with the Duhamel integral. See the next section for further discussion on how to compute the impulse response of a numerical model. Let us now assume that the problem has been decomposed into N s sub-structures. The response of each substructure can be obtained using the convolution product (1), but for the solutions to be the responses of the substructures as part of a full system the coupling forces on the interface between the substructures must be included in the forcing function. The interface forces are unknown beforehand, but we know that those interface forces coupling the interface dofs must be such that the interface is compatible in the assembled problem. Calling B (s) the signed Boolean matrices localizing the interface dofs (see for instance 3,9 ), the compatibility condition on the interface, namely the condition stating that the dofs on each side of the interface are equal, can be written as B (s) u (s) = (2) Hence the extension of Duhamel s integral (1) to a partitioned problem writes t ( ) u (s) (t) = H (s) (t τ) f (s) (τ) + B (s)t λ(τ) dτ N s B (s) u (s) (t) = N s (3) where B (s)t λ represent the interface forces, namely the reactions associated to interface compatibility constraint, λ being the Lagrange multipliers.

4 The Impulse Base Substructuring (IBS) method proposed in this paper relies on (3): if the impulse response functions are known per substructures it allows computing the impulse response (or the response to any external force) for the assembled problem. It is thus clearly a transposition to the time domain of the Frequency Based Substructuring (FBS). In practice the formulation (3) must be applied by discretizing the time integral in order to evaluate the convolution product. Indeed the impulse response H(t) is generally known only for discrete time instances, either from measurements or from numerical modelling. A way to evaluate the integral is explained in the next section. 3 IMPULSE BASED SUBSTRUCTURING WITH NUMERICAL MODELS In this section we will show how the IBS method can be applied, in particular how Duhamel s integral can be discretized in time. Here we will assume that the dynamics of the substructures are described by a numerical model. First we will explain how an impulse response is computed, then we outline how impulse response superposition can be applied for a single (non-decomposed system). We will extend the method to assemble several numerical models described by their IRFs and finally illustrate the strategy with a simple application example. 3.1 Impulse Response computation For a numerical model the impulse response is obtained by numerical time-integration. The linear dynamic equilibrium at time t n can generally be represented by the matrix equation Mü n + C u n + Ku n = f n (4) where M, C, K are the linear(ized) mass, damping and stiffness matrices, u is the set of degrees of freedom, and f are the applied forces. The subscript n indicates the time-step at which the accelerations, velocities and displacement are considered. Let us call dt the time-step size (assumed for simplicity to be constant during the time-integration) such that t = n dt = t n. Given the initial conditions u, u, the initial acceleration can be computed by ü = M (f Ku C u ) (5) To solve (4) one needs to approximate time derivatives by well chosen finite differences. In structural dynamics one classically uses the Newmark time-integration scheme 4,6 stating that u n = u n + dt u n + (.5 β)dt 2 ü n + βdt 2 ü n (6) u n = u n + (1 γ)dtü n + γdtü n (7) where β and γ are parameters used to build integration schemes with different properties. For instance when γ = 1/2, β = 1/4 one obtains an implicit and unconditionally stable scheme (equivalent to the trapezoidal integration rule). If γ = 1/2, β = the scheme is explicit but conditionally stable (equivalent to the central difference). Replacing the discretized time derivatives (6,7) in the dynamic equation (4) ( M + γdtc + βdt 2 K ) ü n = f n Kũ n C ũ n (8) where ũ n and ũ are the predictors ũ n = u n + dt u n + (.5 β)dt 2 ü n ũ n = u n + (1 γ)dtü n For an unit impulse at time t = the dynamic response can be computed in three different ways. Initial velocity step To compute the impulse response we can first compute the velocity jump at time t = due to a unit impulse. Integrating the dynamic equation in an infinitesimal interval [, + ] results in the momentum

5 equation M ( u + u ) = f(t)dt and since just before t = the system is at rest, the initial velocity resulting from a unit impulse on dof j is + M u = 1 j (9) where 1 j is a vector with a unit coefficient for dof j. Hence the impulse response can be computed by setting the applied force f(t) to zero and starting the time integration with the initial conditions then continuing the integration with (8). u = u = M 1 j (1) ü = M ( C u ) Initial applied force Another manner to compute the impulse response is to use the initial conditions and applied force u = u = (11) f = 1 j hence ü = M 1 j f n> = Note that, in the time integration scheme, this is equivalent to an impulse generated by a force being suddenly unity at time t = and decreasing linearly to at time t = dt: the IRF so obtained is in fact for an impulse equal to dt/2. The numerical impulse response is thus the time response obtained for the settings (11) divided by dt/2. In the limit where the time-step goes to zero this method is obviously converging to the impulse response as computed with the initial velocity step (see above). If the Newmark time integration scheme γ = 1/2, β = 1/4 is used the impulse response obtained by this method is actually identical to the response obtained with an initial velocity jump even for a finite time-step size. Applied force at the second time-step The two methods outlined above to compute the impulse response require factorizing the mass matrix. This is not a problem when the mass matrix is diagonal (as found for an explicit time integration), but for a consistent (non-diagonal) mass matrix the factorization cost could be significant. It that case one can avoid factorizing the mass matrix by applying a unit force on the second time-step, namely u = u = (12) ü = (13) f 1 = 1 j f n 1 = This computation represents in fact a force increasing linearly to 1 j between t and t 1, then decreasing to zero between t 1 and t 2. The related impulse value is thus dt and the unit impulse response is obtained by dividing the obtained time response by dt. Again In the limit where the time-step goes to zero this method is equivalent to the two strategies explained above. For a finite time-step the obtained impulse response is slightly different. 3.2 Impulse superposition for a single structure The impulse responses computed at time t n by one of the methods described in the previous section are stored in the Impulse Response matrix H n. The coefficient [H n ] ij is the response for dof i at time t = n dt to an unit impulse at time t = on a dof j. So H, containing the full time history of the impulse responses, can be seen as a three-dimensional matrix of dimension N p n max, calling N the number of degrees of freedom of the system (or the number of output considered), p the number of excitation (input) locations, and n max the number of time-steps

6 for which the impulse response has been computed. Note that this is similar to the Frequency Response Functions (FRFs) of a system, except that for the FRFs the third dimension is the frequency. Obviously, theoretically speaking, the FRFs of a system are the Fourier transform of the IRFs. The time response for a general applied force f(t) can then be computed by approximating the convolution integral (1) by the finite sum n u n = H n i f i dt (14) i= Note that H is not present in this series since the displacement response to an impulse is null at the instant when the impulse is applied, meaning that H =. The graphical interpretation of the discretized convolution (14) is given in figure 2. f u τ H(t-τ) t-τ t Figure 2: Discretization of Duhamel s integral In figure 2 it is seen that the applied force, in the time-integration scheme, is approximated by piece-wise linear forces between the time steps. Hence during a time step from t n to t n+1 the applied forces are coming for one half from f n and for the other half from f n+1. Let us then consider in figure 2 the response at time t 1. According to (14) the response at t 1 is solely due to the impulse created by the force at t, while in fact the force at time t 1 also produced an impulse between t and t 1. Hence one can say the the response at t 1 is due to an impulse equal to I t,t 1 = f dt/2 + f 1 dt/2 = f + f 1 dt 2

7 indicating that to evaluate the impulse the force should in fact be taken at the middle of the time step. Therefore it is more accurate to consider the following discretization of Duhamel s integral: n u n = H n i (f i + f i+1 )dt/2 (15) i= In practice however, since the time step dt is small, the difference between (15) and (14) is negligible. 3.3 Impulse superposition and assembly of substructures The matrices of the impulse responses for each substructure can be computed as indicated in the section 3.1. In order to assemble the substructures the IRFs between all interface dofs are required, in addition to the IRFs for the dofs where the external forces f (s) are applied. The convolution product and the compatibility condition of (3) can then be discretized as u (s) n N s n = i= B (s) u (s) = ( ) H (s) n i f (s) i dt + B (s)t λ i where λ i are the Lagrange multipliers related to the compatibility condition. They represent the impulse between the interface dofs needed to ensure the interface compatibility. 1 In the dual assembly formula (16) the solution at time t n is determined by the external forces and the interface impulses for t = up to t = n 1. Hence the compatibility condition at time t n determines the interface impulse at time t n. Let us rewrite (16) as where ũ (s) n u (s) n N s = ũ(s) n B (s) u (s) = + H(s) 1 B(s)T λ n is the predicted displacement when λ n =, namely n 2 ũ (s) n = i= (16) (17) ( ) H (s) n i f (s) i dt + B (s)t λ i + H (s) 1 f(s) ndt (18) From (17) the Lagrange multiplier is computed by solving the dual interface problem ( N s ) N s B (s) H (s) 1 λ B(s)T n = B (s) ũ (s) n (19) which is very similar to the dual interface problem of the Frequency Based Substructure (see e.g. 5 ). Equations (19,17) constitute the stepping algorithm for the Impulse Based Substructuring strategy (IBS) proposed in this paper. 3.4 Numerical example To illustrate the Impulse Based Substructuring technique let us consider the bar structure described in figure 3 excited by a load at its end. The structure is divided in 2 substructures of equal length, each substructure being modeled by 25 bar finite elements (the consistent mass matrices are used here). The bar is made of steel (E = Pa, ρ = 75 kg/m 3 ), has a uniform cross-section of A = m 2 and each substructure has a length of L =.5 m. In the model damping has been introduced by constructing C = K. 1 Equation (16) can also be written for the forces at half time-step as in (15). This does not modify the basics of the algorithm and provides slightly more accurate results.

8 λ f L L Figure 3: Example of a beam with two substructures Assuming the dofs of the substructures are numbered from left to right, the Boolean constraining matrices are B (1) = [ ] B (2) = [ 1 ] First we compute the Impulse Response Functions as indicated in section 3.1. Here a unit force at time t = is used. The implicit, unconditionally stable Newmark γ = 1/2, β = 1/4 scheme is used. The time-step is chosen equal to 3h crit where h cri t is the critical time step, namely the stability limit if the integration scheme would be explicit. This critical time-step is given by the CFL condition and is equal to 4 h crit = 2/ω for ω the highset eigenfrequency in the model. The obtained IRFs are plotted in figure 4 for inputs on the interface and on the end of the bar. On the right of that figure the IRFs are zoomed. H (1) LL.5 1 x x H (2) x x 1 3 H (2) L x x x 1 3 Figure 4: IRFs for the bar substructures (zoomed on the right)

9 Let us assume that one is interested in computing the dynamic response at the end of the bar when a force is applied on it. The IBS expression (17) and the dual interface problem (19) are ũ (2) n [ L] = λ n = n 2 ( ) H (2) n i [LL] f (2) i [L] dt + H (2) n i [L] λ i + H (2) 1 [LL] f (2) n [L] dt i= ũ (1) n [L] ũ (2) n [L] (2) H (1) 1 [LL] + H (2) 1 [] u (2) n [L] = ũ (2) n [L] + H (2) 1 [LL] λ n First we will apply the IBS technique to compute the response of the full bar to an impulse at its end: applying a unit force at the end of the second substructure, and dividing the obtained response by dt/2 we obtain the IRF shown in figure 5. If this impulse response is computed with a non-decomposed model, exactly the same IRF is found. H full 2L,2L 1 x x x 1 3 Figure 5: IRF for the full bar computed by IBS (zoomed on the right) Finally let us apply a step load at the end of the bar (f (2) L (t) = 1 for t ). Using again the IBS approach to compute the response based on the impulse responses of the substructures one obtains the response plotted in figure 6. Again the same response would have been found if a model of the complete bar would have been used in a direct time-integration. It is observed that the solution converges in time to the correct static solution. u (1) L u (2) L = u(2) u (1) dx 3 x x x 1 3 Figure 6: Dynamic response to a step load at the end of the bar, computed by IBS (zoomed on the right). The solutions are shown for the end of the bar, the middle point (on the interface between the substructures) and on the first node next to the fixed end. 4 CONCLUSIONS AND FUTURE WORK In this paper we propose a method that transposes the Frequency Based Substructuring technique to the time domain. It allows computing the response of a system by computing the responses of its substructures with a discretization of the Duhamel integral and enforcing the interface compatibility at every time step. Hence if the substructure impulse responses are known for all interface and input degrees of freedom, the method proposed in this paper allows predicting

10 the dynamics of the full system. The method is named Impulse Based Substructuring or IBS. The IBS (in the time domain) is the equivalent of the FBS (in the frequency domain), but we believe that when applied to the computation of impact response and shocks, the IBS can be significantly more effective since it allows using directly the impact responses of the susbstructures. The IBS approach could be used for instance when predicting the impact response of new designs. It can enable engineers to optimize their designs during the very initial design phase, if the impulse response of the components are known either from models or tests. In many cases the designer uses standard components which would then need to be characterized once, the work of the designer being then to optimize the housing and the link between the components. In this paper we outline the basic principle of the method and show that the theory can be easily applied when the impulse response functions are computed through direct time integration of numerical models. Future work will investigate the applicability of the approach when the impulse responses of the substructures are obtained from experimental tests. We will also investigate the combination of numerical and experimental sub-models in the IBS method. Finally we will perform research to combine substructures described by impulse response functions with non-linear components that need to be time-integrated simultaneously with the time-stepping in the IBS for the linear parts. REFERENCES [1] Thomas G. Carne and Clark R. Dohrmann. Improving experimental frequency response function matrices for admittance modeling. In IMAC-XXIV: International Modal Analysis Conference, St Louis, MO, Bethel, CT, February 26. Society for Experimental Mechanics. [2] Dennis de Klerk. Dynamic Response Characterization of Complex Systems through Operational Identification and Dynamic Substructuring: An application to gear noise propagation in the automotive industry. PhD thesis, Delft University of Technology, Delft, The Netherlands, March 29. [3] C. Farhat and F.-X. Roux. A method of finite tearing and interconnecting and its parallel solution algorithm. International J. Numer. Methods Engineering, 32: , [4] M. Géradin and D. Rixen. Mechanical Vibrations. Theory and Application to Structural Dynamics. Wiley & Sons, Chichester, 2d edition, [5] D. De Klerk, D. J. Rixen, and S. N. Voormeeren. General framework for dynamic substructuring: History, review and classification of techniques. AIAA Journal, 46(5): , 28. [6] N.M. Newmark. Method of computation for structural dynamics. J. Eng. Mech., 85:67 94, [7] Dana Nicgorski and Peter Avitabile. Conditioning of frf measurements for use with frequency based substructuring. Mechanical Systems and Signal Processing, (in press), 29. [8] D. Otte, J. Leuridan, H. Grangier, and R. Aquilina. Prediction of the dynamics of structural assemblies using measured frf-data: some improved data enhancement techniques. In IMAC-IX: International Modal Analysis Conference, Florence,Italy, pages , Bethel, CT, February Society for Experimental Mechanics. [9] Daniel Rixen. Encyclopedia of Vibration, chapter Parallel Computation, pages Academic Press, 22. isbn [1] Daniel J. Rixen. How measurement inaccuracies induce spurious peaks in frequency based substructuring. In IMAC-XXVII: International Modal Analysis Conference, Orlando, FL, Bethel, CT, February 28. Society for Experimental Mechanics. [11] D.J. Rixen, T. Godeby, and E. Pagnacco. Dual assembly of substructures and the fbs method: Application to the dynamic testing of a guitar. In International Conference on Noise and Vibration Engineering, ISMA, Leuven, Belgium, September KUL.