Prediction of high-frequency responses in the time domain by a transient scaling approach

Prediction of high-frequency responses in the time domain by a transient scaling approach X.. Li 1 1 Beijing Municipal Institute of Labor Protection, Beijing Key Laboratory of Environmental Noise and Vibration, Taoranting Road 55, Beijing 100054, China e-mail: lixianh@vip.sina.com Abstract Many engineering structures are subject to excitations with large amplitudes over a short time duration. These impulsive excitations can cause structural failure or unwanted noise especially in the high-frequency band. This paper presents a transient scaling approach to predict the high-frequency vibration of impulsively excited structures. General scaling laws are derived from the transient statistical energy analysis (TSEA). The similitude between the scaled model and the original system is demonstrated by the equivalence of the TSEA governing equations. Specific forms of the general scaling laws are formulated for a two-oscillator system and a coupled beam system. Computational speedup is achieved from both the reduced order of the original system and the accelerated energy evolution in the scaled model. Numerical validation demonstrates that time domain responses can be obtained efficiently by the transient scaling approach. 1 Introduction Many engineering structures may be subject to excitations with large amplitudes over a short time duration. These impulsive excitations can cause structural failure or unwanted noise especially in the high-frequency band. To better describe their dynamics, transient responses from an energetic point of view are often used. Although many methods can predict high-frequency responses in the steady state, only a few deal with the prediction of the high-frequency responses in the transient state. An earlier work on the subject was published by Manning and Lee [1] in the context of TSEA. They proposed a method based on a steady state power balance equation to deal with the mechanical shock transmission. Reasonable agreement between measured and predicted vibration responses was shown for a beam-plate system with a transient structureborne excitation. However, the application of steady state coupling loss factors (CLFs) in the transient condition requires further justification. Lai and Soom [2] proposed using time-varying CLFs to describe the modal interactions in the case of impulsive excitation. It is shown that the time-varying CLFs increase from low values to almost time-independent values and approach the steady-state CLFs when increasing the system damping. More recently, Pinnington and Lednik [3, 4] investigated the transient responses of a twooscillator system and a coupled-beam system. It is shown that TSEA gives good estimates of peak energy transmitted for modal overlaps less than unity. As an alternative method to TSEA, thermal analogy is used to obtain the energy distribution in subsystems. Nefske and Sung [5] generalized the thermal analogy in the time domain, but only steady state results have been shown. Ichchou et al. [6] pointed out that the analogy between the vibrational energy flow and the the thermal flow is no longer valid for the transient state. They proposed a transient local energy approach (TLEA) where the governing equation is a telegraph-type equation for one-dimensional damped structures. Sui et al. [7] compared time domain results for TLEA and TSEA to the exact solution of the two-oscillator system which shows better agreement for peak level and peak location for TLEA than TSEA. 1927

1928 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 Although TSEA and TLEA have been validated on some simple structures, their extension in the time domain lacks a rigorous theoretical foundation. Recently, Savin et al. [8, 9] proposed a transport model for high-frequency vibrational power flows in coupled heterogeneous structures. It is shown that the energy densities associated to high-frequency elastic waves in each single structure satisfy transport equations when the correlation lengths of the heterogeneous materials comparable with the wavelength. The energy densities are coupled at the junctions where the reflection and transmission phenomena of the energy fluxes are described by power flow reflection and transmission operators. After long times and/or propagation distances, the transport model evolves into a diffusion model when the energy rays are fully mixed due to scattering on the heterogeneities or multiple reflections on the boundaries [10]. This work provides a rational theoretical framework for the validation and generalization of heuristics such as SEA and thermal analogy and their extension in the time domain. The above methods can be categorized as energetic approaches in which the primary variables are the quadratic terms related the subsystem energy or energy density. The merits of the energetic approaches lie in the reduced computational cost and automatically taking into account the unavoidable uncertainties in the high-frequency band, however, it is very challenging to compute the transient high-frequency responses of a complex structure due to lack of a mature software to implement these approaches. On the other hand, finite element (FE) methods are well established and transient solvers are readily available in most FE softwares. Although FE methods are limited to low frequencies due to intensive computation, efforts have been made in the past decades to extend the their applications to high frequencies. It is an interesting observation that two different systems may yield similar responses in the high-frequency band. If a scaled model is available which is dynamically similar to the original system and has a much lower modal density, one can build a finite element model with much fewer degrees-of-freedom (DOFs) and solve it with relatively low computational cost. The responses of the scaled model can be used to estimate those of the original system in the high-frequency band. There are mainly two ways to construct the scaled model. De Rosa et al. [11, 12] proposed asymptotic scaled modal analysis (ASMA) to obtain an acceptable approximation of high-frequency vibrations using a reduced modal base. The scaled model in ASMA is determined using an energy distribution approach [13], resulting in a reduction in the dimensions not involved in energy propagation and accordingly an increase of the original damping loss factors. Li [14, 15] proposed to use statistical energy analysis (SEA) similitude to construct the scaled model. In this paper, the approach is extended to deal with the transient problem. Besides the structural parameters, the time variable is also scaled to achieve the similitude in the time domain. The scaled model usually has a larger damping and a faster time variable, making the energy dissipate in subsystems and transmit to adjacent subsystems at a faster rate than in the original system. Therefore, one can obtain the transient responses from the scaled model in a shorter simulation time, which is then scaled back to the original time to obtain the final responses. Preliminary results of the transient scaling approach are presented. In Sec. 2, the general scaling laws are derived from TSEA governing equations. Then specific forms of the scaling laws are derived in Sec. 3 for a two-oscillator system and a coupled-beam system. Numerical validation is presented in Sec. 4 and some conclusions are given in Sec. 5 about the application of the transient scaling approach in the high-frequency response prediction in the time domain. 2 General scaling laws TSEA is based on the instantaneous power flow balance among coupled subsystems. Although the justification of TSEA is out of the scope of this paper, one could list some basic prerequisites for the successive application of TSEA. One group of prerequisites comes from SEA, which requires that the system can be partitioned into a set of weakly coupled subsystems and each subsystem carries a diffusive field. The latter requirement is likely to be satisfied when the subsystems are lightly damped and have many modes. The other group of prerequisites comes from the fact that TSEA describes a diffusive type of high-frequency vibrations, which requires multiple scatterings on the heterogeneities and reflections on the boundaries to justify

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1929 a diffusive vibrational energy field. This requirement is usually met after the very beginning of vibrations. Hereafter the addressed problem is assumed to be in the diffusion regime and all the above prerequisites are met. Furthermore, we assume that steady state CLFs can be used in TSEA with acceptable accuracy. The TSEA governing equation for the jth subsystem at a band center frequency ω is de j /dt + ωη j E j + k ωη jk n j (E j /n j E k /n k ) = Π in,j, (1) where η j is the loss factor of subsystem j, E j is the instantaneous subsystem energy, n j is the modal density of the subsystem, Π in,j is the power input to the subsystem from external excitations, and η jk is the CLF from subsystem j to subsystem k. In practice, the TSEA equations can be solved by a time difference method. At each time step, the discrete form of Eq. (1) is essentially a steady state SEA power balance equation and time-averaged subsystem energy is solved for that time interval. The discretization time interval should be long enough to allow the time-averaged net power flow to occur and short enough to make the stationary representation of the subsystem energy accurate [16]. Equation (1) can be rewritten in a matrix form as N de/dt + Se = Π, (2) where N is a diagonal matrix with the jth diagonal element as n j, S is a matrix with the (j, k) element as (µ j + l µ jl)δ jk µ kj, in which µ j = ωη j n j, µ jk = µ kj = ωη jk n j, and δ jk is the Kronecker delta, e is a vector with the jth component as the modal energy E j /n j, and Π is a vector with the jth component as Π in,j. Similarly, the TSEA governing equations for the scaled model are ˆN dê/dˆt + Ŝê = ˆΠ, (3) where the caret stands for the scaled model. Denote γ as the scaling factor and its subscript refers to the corresponding scaled parameter. For example, the power input vector is scaled as ˆΠ(ˆt) = ˆΠ(γ t t) = γ Π Π(t). (4) It can be derived from Eqs. (2) and (3) that the TSEA governing equations are scaling invariant when all subsystems follow the scaling laws given by and γ η γ t = 1, (5) γ µ = γ η γ n = γ Π, (6) where no scaling in the frequency parameter ω is assumed. The subscript denoting a term associated with a subsystem is dropped in Eqs. (4) to (6) because the same scaling laws are applied to all subsystems, or, γ n is chosen to be a constant across all the subsystems. The scaling law (5) is unique for the transient analysis. It implies that the time variable should be scaled inverse-proportionally to the (coupling) loss factor. Since the scaled model usually has a larger damping than that of the original system, the subsystem energy will evolve faster in the scaled model. The scaling law (6) implies that the loss factor η j and the coupling loss factor η jk scale by the same factor. For a complex system consisting of N coupled statistical subsystems, it means each loss factor η j needs to scale by the same factor as (N 1) CLFs η jk,(k j). This requirement distinguishes the present scaling approach from those derived for an isolated system. At the first sight, it seems impossible to fulfill the requirement for a complex coupled structure. The key to resolve the problem is to isolate each SEA subsystem from the rest of the system with an adapter structure. A procedure to implement the scaling law (6) was developed in [17]. The auxiliary requirements of the general scaling laws derived for the steady state analysis also apply to the transient case. The scaling factor γ n has a lower limit to guarantee that each subsystem contain a number of resonant modes over the analysis band of interest and the modal overlap factor is large enough. One should choose an appropriate value for γ n to balance the computational efficiency and the accuracy of the scaled model.

1930 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 3 Specific scaling forms In this section, a two-oscillator system and a coupled beam system are considered, both having been thoroughly investigated in the literature [3, 4, 7]. These systems are simple enough and suitable for demonstrating the transient scaling approach. The general scaling laws can be significantly simplified for the problems addressed in this section. 3.1 Two-oscillator system Consider a two-oscillator system shown in Fig. 1. The TSEA equations are Figure 1: Two-oscillator system de 1 /dt + ωη 1 E 1 + ωη 12 E 1 ωη 21 E 2 = Π in,1, (7) de 2 /dt + ωη 2 E 2 + ωη 21 E 2 ωη 12 E 1 = Π in,2, (8) where all parameters follow the definition in Refs [3, 7]. The TSEA equations have the solutions E 1 (t) = E 1 (0)e aωt [ (D 1 /ω + η b ) e bωt (D 2 /ω + η b ) e bωt] /2b, (9) E 2 (t) = E 1 (0)η 12 e aωt [ e bωt e bωt] /2b, (10) where a = (η a +η b )/2, b = (η a η b ) 2 + 4η 12 η 21 /2, η a = η 1 +η 12, η b = η 2 +η 21, and D 1,2 = ω(a b). The CLF between the two oscillators is [3] η 12 = k 2 2η 1 (k + k 1 ) 2 m 1 m 2 k2 2η 1 k 2 1 where the weak coupling assumption k k 1 is used to obtain the last approximation. m 1 m 2, (11) One procedure is scaling the coupling stiffness k and the damping coefficient c 1,2 by the same factor γ whereas no scaling in k 1,2 and m 1,2. Then according to Eq. (11), the CLF η 12 also scales by the factor γ, so do the parameters a and b. Suppose the system is excited by an impulse on the left mass, the power input can be written in the form as Π in,1 = Aδ(t). By integrating the input power over an infinitesimal interval around t = 0 yields E 1 (0) = A. Similarly, the power input to the scaled model and its initial energy are ˆΠ in,1 = Âδ(ˆt) and Ê1(0) = Â, respectively. According to Eq. (4) and the property of the delta function, one obtains  = γ tγ π A, which yields  = A according to the scaling laws (5) and (6). Substituting all above quantities into Eqs. (9) and (10), one finally obtains Ê1(ˆt) = E 1 (t) and Ê2(ˆt) = E 2 (t), which verifies that the scaled model has the same transient responses as the original system in terms of subsystem energies.

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1931 Figure 2: The coupled-beam system 3.2 Coupled-beam system A coupled-beam system shown in Fig. 2 is investigated in Ref. [4]. It can also be modeled by Eqs. (7) and (8). The solutions are of the same form as Eqs. (9) and (10), except that the subscripts are replaced by s and r standing for the source and receiver. The CLF between the two beams is defined as η sr = τ ω(1 τ/2)n s, (12) where the transmission efficiency τ = (M s /M r ) T 2, in which M s,r are the characteristic impedance moduli of the beams and T is the transmission coefficient [4]. One procedure is scaling the loss factor of both beams by γ and their length by 1/γ. Since only the damping and the length are the parameters being scaled, it can be verified that the η sr scales by γ because all terms are invariant in the current scaling procedure except for the modal density n s, which scales by 1/γ. Following the same line as in the last subsection, one obtains Ê s0 = γ t γ Π E s0 = γ n E s0, (13) where the last equivalence is derived using the scaling laws (5) and (6). Substituting all above quantities into Eqs. (9) and (10) yields Ês,r(ˆt) = γ n E s,r (t). Due to the relationship of e s,r = E s,r /n s,r, the modal energies of the scaled model are the same as those of the original system. The dynamic similitude of the scaled model and the original system can be explained physically as following. The impulsive power feeds in the source beam with an initial energy 1/γ times of that in the original system. The energy is traveling along the source beam until it hits the junction. This process takes exactly 1/γ of the traveling time in the original system. Meanwhile it dissipates the same portion of the total energy as in the original system. After that, the energy is partially transmitted to the receiver beam and partially reflected back to the source beam. From the point of view of the incident wave, the source and receiver beams are equivalent to two semi-infinite beams. Hence there is no difference in the energy transmission and reflection process before and after scaling. In summary, the energy in the scaled model is 1/γ times of that in the original system and evolves γ times faster than it does in the original system. After converting back to the original time, the scaled responses is the same as the original responses. Since the above argument does not depend on any specific coupling form, it can be generalized to beam assemblies with junctions different to those considered in Ref. [4]. 4 Numerical validation Firstly, the application of the scaling procedure is demonstrated by the transient response prediction of the two-oscillator system. Since the TSEA solutions are proved to be equal in Section 3.1, only the exact solutions of the scaled model and the original system are compared. The parameters of the two-oscillator

1932 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 0.18 0.16 0.14 0.12 Energy (J) 0.1 0.08 0.06 0.04 0.02 0 0 0.02 0.04 0.06 0.08 0.1 (a) 0.012 0.01 0.008 Energy (J) 0.006 0.004 0.002 0 0 0.02 0.04 0.06 0.08 0.1 (b) Figure 3: Energy results of the two-oscillator system: (a) Input energy; (b) transmitted energy. Solid line: original system; Dotted line: scaled model with γ = 3; Dash-dotted line: scaled model with γ = 5. system are taken from Test B given in Table 1 in Ref. [7], where the coupling stiffness is small enough to meet the weak coupling assumption. The scaling procedure is presented in Section 3.1, where the coupling stiffness k and the dashpot c 1,2 are scaled by the same factor γ. The stiffness k 1,2 are adjusted to keep the block frequencies unchanged. Note that the time in the scaled model is γ times faster than that in the original system. Hence the total simulation time for the scaled model is only 1/γ that of the original system. When the responses of the scaled model are available, they need to be scaled back to the original time to obtain the final results. Figure 3 gives the comparison of the exact energy results of the original system and the scaled model. Good agreement is observed over all the simulation time for the scaled model with γ = 3. When a larger scaling factor γ = 5 is used, the scaled model can predict the decay rate in the source oscillator and the rise time in the receiver oscillator, however, it cannot correctly predict the peak level in the latter for the reason that the weak coupling assumption is no longer satisfied. The second example involves two steel beams connected by a hinge. The parameters of the system are taken from Test E given in Table 1 in Ref. [4]. The loss factors of both beams are changed to 0.01, or equivalently the modal overlaps around 1, in order to meet the requirement of SEA. The scaling procedure is presented in Section 3.2, where the lengths of the beams are scaled by 1/γ and the internal loss factors are scaled by γ. Figure 4 gives the comparison of the averaged energy density in the source and receiver beams obtained from the original system and the scaled model. Good agreement is observed over all the simulation time for the scaled model with γ = 3. The decay rate in the source beam and the rise time and the peak level in the

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1933 10 2 10 1 Averaged enrgy density (N) 10 0 10 1 10 2 10 3 0 0.01 0.02 0.03 0.04 0.05 (a) 10 1 10 0 Averaged enrgy density (N) 10 1 10 2 10 3 10 4 0 0.01 0.02 0.03 0.04 0.05 (b) Figure 4: Averaged energy density of the flexurally excited beams: (a) source beam; (b) receiver beam. Solid line: original system; Dotted line: scaled model with γ = 3. receiver beam can be correctly estimated from the scaled model. The last example involves an aluminum beam simply supported at both ends and the mid-point. Each half of the beam is of 3 m long, 0.01 m wide, and 0.002 m high. A concentrated force is applied at a point z 0 = 0.45 m away from the left end. The load is generated from a Gaussian white noise with an envelop of 1 exp( t/t 0 ), where the time constant T 0 = 0.06 s. One implementation of the load is shown in Fig. 5. The scaling procedure is presented in Section 3.2, where the lengths of the beams are scaled by 1/γ and the internal loss factors are scaled by γ. Besides, the load position z 0 and the time constant T 0 are scaled by 1/γ. By so doing, the load case in the original system is exactly reproduced in the scaled model when evaluated with the new time and dimension scales. Figure 6 gives the comparison of the averaged energy density in the left and right beams obtained from the original system and the scaled model. It is observed that the scaled model provides a good estimate of the impulsive responses of the original system, especially over the short time period immediately after the load is applied. When the system is loaded for a long time, it enters the steady state whose responses fluctuate around a constant value. 5 Conclusions The paper presents a transient scaling approach to predict the high-frequency vibration of impulsively excited structures. The relationship of instantaneous power flow balance among subsystems is employed to derive

1934 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 4 3 2 1 Force (N) 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 Figure 5: Time history of the concentrated force. 10 2 10 1 Averaged energy density (N) 10 0 10 1 10 2 10 3 10 4 0 0.1 0.2 0.3 0.4 0.5 (a) 10 2 10 1 Averaged energy density (N) 10 0 10 1 10 2 10 3 10 4 0 0.1 0.2 0.3 0.4 0.5 (b) Figure 6: Averaged energy density of the flexurally excited beams: (a) source beam; (b) receiver beam. Solid line: original system; Dotted line: scaled model with γ = 3.

MEDIUM AND HIGH FREQUENCY TECHNIQUES 1935 the general scaling laws, which guarantee the dynamic similitude between the scaled model and the original system. Specific forms of the scaling laws are formulated for a two-oscillator system and a coupled-beam system. Scalings are taken in the structure parameters and the time variable as well. Numerical validation demonstrates that a good response prediction in the time domain can be obtained from the scaled model. Since the damping is usually increased in the scaled model, the energy dissipates and transmits faster in the scaled model than in the original system. This property implies that one can simulate a short time period to get a broad picture of the evolution of the subsystem energies over a long time period. Together with the reduced model size, a small size finite element model can be used to efficiently solve for the transient responses. In this sense, the present approach offers a way to extend the FE methods to the transient highfrequency vibration analysis. Acknowledgements This work is financially supported by the National Science Foundation of China (No. 11174041). References [1] J. E. Manning, K. Lee, Predicting mechanical shock transmission, Shock and Vibration Bulletin vol. 37, (1968), pp. 65 70. [2] M. L. Lai, A. Soom, Prediction of transient vibration envelopes using statistical energy analysis techniques, Journal of Vibration and Acoustics, vol. 112, (1990), pp. 127 137. [3] R. J. Pinnington, D. Lednik, Transient statistical energy analysis of an impulsively excited two oscillator system, Journal of Sound and Vibration, vol. 189, (1996), pp. 249 264. [4] R. J. Pinnington, D. Lednik, Transient energy flow between two coupled beams, Journal of Sound and Vibration, vol. 189, (1996), pp. 265 287. [5] D. J. Nefske, S. H. Sung, Power flow finite element analysis of dynamic systems: basic theory and application to beams, NCA Publication, vol. 3, (1987), pp. 47 54. [6] M. N. Ichchou, A. Lebot, L. Jezequel, A transient local energy as an alternative to transient SEA: wave and telegraph equations, Journal of Sound and Vibration, vol. 246, (2001), pp. 829 840. [7] F. S. Sui, M. N. Ichchou, L. Jezequel, Prediction of vibroacoustics energy using a discretized transient local energy approach and comparison with TSEA, Journal of Sound and Vibration, vol. 251, (2002), pp. 163 180. [8] E. Savin, A transport model for high-frequency vibrational power flows in coupled heterogeneous structures, Interaction and Multiscale Mechanics, vol. 1, (2007), pp. 53 81. [9] Y. L. Guennec, E. Savin, A transport model and numerical simulation of the high-frequency dynamics of three-dimensional beam trusses, Journal of the Acoustical Society of America, vol. 130, (2011), pp. 3706 3722. [10] E. Savin, Diffusion regime for high-frequency vibrations of randomly heterogeneous structures, Journal of the Acoustical Society of America, vol. 124, (2008), pp. 3507 3520. [11] S. De Rosa, F. Franco, A scaling procedure for the response of an isolated system with high modal overlap factor, Mechanical Systems and Signal Processing, vol. 22, (2008), pp. 1549 1565.

1936 PROCEEDINGS OF ISMA2016 INCLUDING USD2016 [12] S. De Rosa, F. Franco, On the use of the asymptotic scaled modal analysis for time-harmonic structural analysis and for the prediction of coupling loss factors for similar systems, Mechanical Systems and Signal Processing, vol. 24, (2010), pp. 455 480. [13] B. R. Mace, Statistical energy analysis, energy distribution models and system modes, Journal of Sound and Vibration, vol. 264, (2003), pp. 391 409. [14] X. Li, A scaling approach for the prediction of high-frequency mean responses of vibrating systems, Journal of the Acoustical Society of America, vol. 127, (2010), pp. EL209 EL214. [15] X. Li, A scaling approach for high-frequency vibration analysis of line-coupled plates, Journal of Sound and Vibration, vol. 332, (2013), pp. 4054 4058. [16] M. Robinson, C. Hopkins, Prediction of maximum time-weighted sound and vibration levels using transient statistical energy analysis. Part 1: theory and numerical implementation, Acta Acustica United with Acustica, vol. 100, (2014), pp. 46-56. [17] X. Li, A scaling approach for the vibro-acoustic analysis of a complex system in the mid-frequency region, in Proceedings of NOVEM 2015, Dubrovnik, Croatia, 2015, April 13-15, Dubrovnik (2015), pp. 48858-1-11.