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An efficient transient analysis method for time-varying structures based on statistical energy analysis
Mechanics Research Communications 91 (2018) 93–99
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Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
An efficient transient analysis method for time-varying structures based on statistical energy analysis Qiang Chen a,b, Qingguo Fei a,b,c,∗, Shaoqing Wu a,b, Yanbin Li a,b, Xuan Yang a,c a
Institute of Aerospace Machinery and Dynamics, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China Department of Engineering Mechanics, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China c School of Mechanical Engineering, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China b
a r t i c l e
i n f o
Article history: Received 19 November 2017 Revised 23 April 2018 Accepted 3 June 2018 Available online 15 June 2018 Keywords: Time-varying structures Statistical energy analysis Transient energy response Complex vibro-acoustic system Damping loss factor
a b s t r a c t The statistical energy analysis (SEA) is applied to predict the transient energy response of structures with time-varying parameters for the first time. With the energy governing equations derived by considering time-varying SEA parameters and energy flow item caused by time-varying damping loss factor, SEA method is firstly applied to predict transient energy response of time-varying systems. Then, numerical examples of a time-varying two-oscillator system, a time-varying L-shaped fold plate and a complex time-varying vibro-acoustic structure are investigated to demonstrate the effectiveness and accuracy of SEA method for time-varying system. The Newmark-beta method and finite element method are used to verify the accuracy of predicted transient energy response. Results show that SEA method for timevarying system is capable of predicting transient energy response of time-varying structures with sufficient accuracy and also applicable for transient analysis of complex time-varying structures with a small computational cost. © 2018 Published by Elsevier Ltd.
1. Introduction Many structures in practical engineering are time-varying structures, which are characterized by the mass, stiffness or damping properties that change with time. For instance, the rapid combustion of rocket fuel will lead to time-varying mass dynamic problems [1] and the aerodynamic heating will change the stiffness and damping characteristics of aircrafts. Moreover, moving vehicles on bridge structure will result in a time-varying vehicle-bridge interaction problem which must be well solved in order to assure the operation safety of high-speed railway [2]. Nowadays timevarying characteristics of system parameters are often neglected in dynamic analysis of engineering structures which will lead to inaccurate results in the dynamic response prediction. In order to improve the accuracy of dynamic response prediction of time-varying structures, it is increasingly important to consider the time-varying features of dynamical systems. At present, deterministic approaches are widely used to solve dynamic problems of time-varying structures. After the spatial discretization of structures to several elements, the problem is turned into an initial value problem of linear ordinary differential equa∗ Corresponding author at: Engineering Mechanics, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China. E-mail address: [email protected] (Q. Fei).
https://doi.org/10.1016/j.mechrescom.2018.06.001 0093-6413/© 2018 Published by Elsevier Ltd.
tions (ODEs) with time-varying coefficients. Then time integration methods, such as the central difference method, Newmark-beta method, can be used to solve these ODEs in time domain. For transient analysis of time-varying structures, Penny and Howard [3] developed a time finite element method (TFEM) for a time-varying single degree-of-freedom system based on Hamilton’s principle. Yu et al. [4] extended the TFEM to time-varying multiple degrees of freedom systems based on Hamilton’s law of varying action. Zhao and Yu [5] presented a transient analysis method for linear time-varying structures based on multi-level sub-structuring method which improve the computational efficiency to a certain extent. For very complex systems in practical engineering, numerous number of elements are needed in deterministic approaches to describe the vibration behavior of time-varying structures, especially in high frequency range. The computation cost will increase rapidly to predict dynamic response of complex structures. Therefore, energy based methods, such as statistical energy analysis (SEA, [6]) and extended methods of SEA, are more suitable to analyse the problem. The SEA, which is largely inspired from statistical physics [7], has been used to predict the average energy response of complex engineering systems for many years. Energy responses of subsystems can be calculated efficiently by solving energy transfer equations between subsystems in SEA. However, traditional SEA is an approach for steady state problems.
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Over the past several decades, some methods are proposed as extensions of the SEA to analyse unsteady state problem of timeinvariant structures. SEA is firstly applied to predict transient energy response of time-invariant structures by transient statistical energy analysis (TSEA, [6]). Then, Lai and Soom [8,9] proposed the concept of “time-varying coupling loss factor” for time-invariant structures in TSEA, however, it is just a concept in mathematics and hard to interpret in physics. Furthermore, Pinnington and Lednik [10,11] applied TSEA to a two-oscillator system and a couplingbeam system, results from TSEA were compared with exact analytical solutions which showed that the prediction of TSEA is sometimes far from fitting the exact reference solution. Song et al. [12] applied affine arithmetic to TSEA of a two-oscillator system, and revealed the influence of measurement errors of parameters on predicted transient energy response. Experimentally, Robinson and Hopkins [13,14] used TSEA to calculate the maximum timeweighted sound and vibration levels in built-up structures. Good agreement was achieved between measurements and predictions. The transient local energy approach (TLEA), which was proposed by Ichchou et al. [15], is another extension of SEA to predict transient energy response of structures. With consideration of timevarying item in energy flow term, TLEA has much higher precision than TSEA [16]. The current research on dynamic response prediction of time-varying structures mostly concentrates on deterministic approaches, which are prohibitly time consuming for complex structures. Thus the statistical energy based approaches are preferred. On the other hand, though SEA has been developed at 1960s, research work mainly focused on transient energy prediction of time-invariant structures and literatures on prediction of transient energy response of time-varying structures are very limited. Therefore, by deriving the energy governing equations of time-varying structures, the SEA is firstly applied to predict transient energy response of time-varying structures in this paper. The outline of this work is as follows: In Section 2, energy governing equations of time-varying structures are derived and SEA is applied to predict transient energy response of time-varying structures for the first time. In Sections 3 to 5, the effectiveness and accuracy of SEA method for time-varying system are verified by numerical examples, which include comparison with Newmark-beta method of a simply two-oscillator system under an impulse excitation, comparison with finite element method of a time-varying L-shaped folded plate with an initial energy, an application on a complex time-varying vibro-acoustic system under an impulse excitation. Finally, in Section 6, conclusions are drawn.
The local power balance for a non-loaded region can be expressed as
∂ e(s, t ) ∂ I (s, t ) + Pdiss + =0 ∂t ∂s
The dissipation power for time-varying structures is evaluated
Pdiss = η (t )ωe(s, t )
c
∂ e(s, t ) 1 ∂ I (s, t ) η (t )ωI (s, t ) + + =0 ∂s c ∂t c
∂ 2 e(s, t ) ∂ e(s, t ) ∂η (t ) ∂ 2 I (s, t ) + η (t )ω + e(s, t )ω + =0 2 ∂t ∂t ∂ s∂ t ∂t
(7)
∂ 2 I (s, t ) 2 ∂ 2 e(s, t ) ∂ I (s, t ) +c + η (t )ω =0 ∂ s∂ t ∂s ∂ s2
(8)
By subtracting Eqs. (7) and (8) leads to the expression
∂ e(s, t ) ∂ e(s, t ) +η (t )ω ∂t ∂t2 ∂η (t ) 2 ∂ 2 e(s, t ) ∂ I (s, t ) + e(s, t )ω −c −η (t )ω =0 ∂t ∂s ∂ s2 2
(9)
Substituting the expression of ∂ I(s, t)/∂ s in Eq. (4) into Eq. (9)
∂ e(s, t ) ∂ e(s, t ) ∂η (t ) +2η (t )ω + e(s, t )ω ∂t ∂t ∂t2 2 2 ∂ e (s, t ) 2 −c + (η (t )ω ) e(s, t ) = 0 ∂ s2 2
(10)
For time-varying structure consists of N subsystems, the concept of total energy rather than energy density turns the Eq. (10) into the form
∂ 2 e(s, t ) ∂ e(s, t ) ∂η (t ) + 2η (t )ω + e(s, t )ω ∂t ∂t ∂t2 2 2 ∂ e (s, t ) 2 −c + (η (t )ω ) e(s, t )] = Pi (t ) ∂ s2 [
(11)
where ∫V ei (s,t) = Ei (t), Ei (t) is time-varying energy stored in subsystem i, Pi (t) is time-varying injected power into subsystem i and N c2 2 (ηi j (t )ωEi (t )) − V (− η (t )ω ∇ e (s, t ) )dV = j=1, j=i
Some basic definitions and assumption used to derive the energy equation of time-varying structure are generally summarized as [16]
j=1, j=i
e(s, t ) = e+ (s, t ) + e− (s, t )
(1)
I (s, t ) = I+ (s, t ) + I− (s, t )
(2)
e− (s,t)
(6)
Differentiating Eq. (4) with respect to time and space, respectively, leads to the expression
N
e+ (s,t)
(5)
where η(t) is the time-varying damping loss factor (DLF), ω = 2π f is radian frequency. Substituting Eqs. (1)–(3) and (5) into Eq. (4) yields:
2. Statistical energy analysis for time-varying structures
I± (s, t ) = ±c · e± (s, t )
(4)
(η ji (t )ωE j (t )), ηji (t) is time-varying coupling loss factor
(CLF) between subsystem j to i. The power balance equation for subsystem i can be given by 1
η (t )ω
d2 Ei (t ) dt 2
+ 2 dEdi (t t ) + ηEi ((tt )) dηdi t(t ) +
i i (12) N N ηi j (t ) ωEi (t ) − ηi (t ) + η ji (t )ωE j (t ) = Pi (t ) j=1, j=i
j=1, j=i
The power balance equation for time-varying structure is driven by following second-order ODEs
(3)
where and are energy density associated with the right and left train waves, while I+ (s,t) and I− (s,t)are the incident and reflected power flows, the right train wave is considered separate from the left one. c is the energy velocity, the same as the group velocity of waves in a slight damping medium.
d2 E(t ) dE(t ) E(t ) dη (t ) +2 + + ωη (t )E(t ) = P(t ) ωη (t ) dt 2 dt η (t ) dt 1
(13)
where E(t) = [E1 (t), E2 (t),••• EN (t)]T is time-varying energy vector, η(t) is time-varying losing factor matrix of time-varying structure, P(t) = [P1 (t), P2 (t),••• PN (t)]T is time-varying input power vector, the superscript T represents matrix transpose.
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Fig. 1. Two subsystems model in SEA.
Fig. 3. Transient energy response of oscillator/subsystem two.
equation of time-varying structures can be expressed as follows
M(t )x¨ (t ) + C(t )x˙ (t ) + K(t )x(t ) = F(t )
Fig. 2. Two-oscillator system model.
For the time-varying losing factor matrix of time-varying system, elements of matrix can be given by
⎧ N ⎨ ηi (t ) + ηi j (t ), i = j η (t, i, j ) = , j =i ⎩−η (t ),j=1 i = j
(14)
ji
Specifically, for a structure consists of two subsystems shown in Fig. 1, power balance equations of subsystems can be expressed by
+ 2 dEd1t(t ) + η 1(t ) E1 (t ) dηd1t(t ) + 1 η1 (t )ωE1 (t )+η12 (t )ωE1 (t ) − η21 (t )ωE2 (t ) = P1 (t )
(15)
+ 2 dEd2t(t ) + η 1(t ) E2 (t ) dηd2t(t ) + 2 η2 (t )ωE2 (t )+η21 (t )ωE2 (t ) − η12 (t )ωE1 (t ) = P2 (t )
(16)
1
η1 (t )ω
1
η2 (t )ω
d2 E1 (t ) dt 2
d2 E2 (t ) dt 2
Under the given initial conditions and given time history of input power, by solving second-order ODEs in (13), transient energy responses of subsystems can be calculated. By considering time-varying SEA parameters η(t) and energy flow item caused by time-varying damping loss factor [(E(t)/η(t))/(dη(t)/dt)], the SEA method is applied to predict transient energy response of time-varying systems for the first time. 3. Application on a time-varying two-oscillator system As it is difficult to obtain the exact analytic solution for systems with arbitrary time-varying parameters, the SEA method for time-varying structure is verified by Newmark-beta method for a two-oscillator system in Section 3.2. Further, dynamic analysis of a two-oscillator system with different time-varying damping and different time-varying coupling stiffness are presented in Section 3.3 and Section 3.4, respectively. The transient energy results of two damped oscillators coupled by an undamped spring are investigated under an impulse excitation. In a time-varying two-oscillator system shown in Fig. 2, with mass m1 (0) = m2 (0) = 2 kg, stiffness k1 (0) = k2 (0) = 1,717,0 0 0 N/m, coupling stiffness k(0) = 280,0 0 0 N/m, damping c1 (0) = c2 (0) = 200 N•s/m. The analysis frequency is set to 10 0 0 rad/s. Consider that an initial unit impulse is applied to mass m1 , giving an initial velocity x˙ 1 (0 ) = 1 and then the initial energy is obtained as E1 (0) = 0.5m1 . 3.1. Dynamic analysis of a time-varying structures based on Newmark-beta method According to Hamilton’s law of time-varying structures, after the spatial discretization, the obtained semi-discrete dynamic
(17)
where M, C, K, F are the time-varying mass, damping, stiffness, load matrixes of time-varying structures, respectively; x,x˙ ,x¨ are the displacement, velocity, and acceleration vector of time-varying structures, dividedly. The direct time integration is a step-by-step numerical procedure. The total time T is divided into n time steps equally. Then, the Newmark-beta method [5], which is one of the most popular algorithms for numerical solutions of time-varying structural dynamic problems, is used to calculate the displacements, velocities and accelerations vector at different time steps. For a time-varying two-oscillator system shown in Fig. 2, the total energy of oscillator i can be expressed as sum of kinetic energy and potential energy
Ei (t ) =
1 1 mi (t )(x˙ i (t ))2 + (ki (t ) + k(t ))(xi (t ))2 2 2
(18)
3.2. Validation case The time-varying two-oscillator system is divided into two subsystems, namely oscillator one and oscillator two. The SEA model of system is shown in Fig. 1. The time-varying DLF of subsystem i and time-varying CLF between subsystem i to subsystem j can be expressed by
ηi (t ) =
ci (t ) 1 (χ (t ) ) ηi j (t )= mi (t )ω1 (t ) 2ηi (t ) (ωi (t ) )4 2
(19)
whereωi (t ) = (ki (t ) + k(t ))/mi (t ),χ (t ) = k(t )/ m1 (t )m2 (t ). For system under an impulse excitation, the input power, P1 (t), P2 (t) are zero, the initial energy E1 (0) = 1 J, E2 (0) = 0, and the initial energy derivatives dE1 (0)/dt = -η1 (0)ωE1 (0), dE2 (0)/dt = 0. Substituting time-varying SEA parameters in Eq. (19) to Eqs. (15) and (16). Then, by solving Eqs. (15) and (16), the transient energy responses of subsystems can be calculated. We consider a transient analysis of a two-oscillator system with time-varying damping. Assuming that the system, within 0.2 s, has linear time-varying damping c1 (t) = c2 (t )= (20 0–50 0t) N•s/m. The mass, stiffness and coupling stiffness of system are constant. The transient energy response of subsystem two is calculated by Eq. (16) and the transient energy response of oscillator two is predicted by Eq. (18). The comparison between results from SEA method for time-varying systems and Newmark-beta method is shown in Fig. 3. As shown in Fig. 3, it can be concluded that simulation results of SEA are consistent with results from Newmark-beta method. Therefore, the SEA method for time-varying system can predict transient energy response of time-varying structures with sufficient accuracy.
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Fig. 4. Transient energy response of subsystem two under different time-varying damping. Fig. 6. L-shaped folded plate.
between subsystems. And the lower CLF leads to slower energy exchange between two subsystems. Furthermore, the vibration period of oscillator two increases with the decrease of coupling stiffness. 4. Application on a time-varying L-shaped folded plate
Fig. 5. Transient energy response of subsystem two under different time-varying coupling stiffness.
3.3. Transient analysis of system with different time-varying damping A transient analysis of a two-oscillator system with timevarying damping is considered in this sub-section. Assuming that the system, within 0.2 s, has a linear time-varying damping c1 (t) = c2 (t )= (20 0–30 0t) N•s/m in case3.3.1, c1 (t) = c2 (t )= (200– 600t) N•s/m in case3.3.2 and c1 (t) = c2 (t )= (20 0–90 0t) N•s/m in case3.3.3. The mass, stiffness and coupling stiffness of system are constant. Comparison between results from different cases is shown in Fig. 4. As shown in Fig. 4, the peak energy in the same period decrease with the rise of gradient of time-varying damping. This is because the decrease of damping results in the decline of DLF of subsystems. The lower DLF leads to slower dissipation of energy. In addition, the vibration period of all three cases is equal as the variety of damping will not change the natural frequency of oscillators. 3.4. Transient analysis of system with different time-varying coupling stiffness A transient analysis of a two-oscillator system with timevarying coupling stiffness is considered in this sub-section. Assuming that the system, within 0.2 s, has a linear time-varying coupling stiffness k (t) = (280,0 0 0–20 0 0 0t) N/m in case3.4.1, k(t )= (280,0 0 0– 40 0 0 0t) N/m in case3.4.2, and k(t )= (280,0 0 0–60 0 0 0t) N/m in case3.4.3. The mass, stiffness and damping of system are constant. Comparison between results from different cases is shown in Fig. 5. As shown in Fig. 5, it can be concluded that the peak energy and rise time of each vibration period in same order decrease with the decline of gradient of time-varying coupling stiffness. This is because the decrease of coupling stiffness results in decline of CLF
In order to investigate the accuracy of SEA method for timevarying system on engineering structures, the method is verified by finite element method (FEM) for a time-varying L-shaped folded plate in Section 4.1. Further, dynamic analysis of an Lshaped folded plate with different time-varying elastic modulus and different time-varying modal damping ratio are presented in Section 4.2 and Section 4.3, respectively. The structural parameters are indicated as Fig. 6. The l-shaped folded plate is divided into two subsystems, namely plate one and plate two. The dimension of plate one and plate two are both L1 × L2 × t = 0.3 m × 0.3 m × 0.006 m. Titanium alloy with density ρ = 4420 kg/m3 , elastic modulus E(0) = 109.5 GPa, Poisson’s ratio ν = 0.33, modal damping ratio ξ (0) = 0.02 is used in calculation. The coupling edge is in free constrain condition and other edges are simply supported. 4.1. Validation case For the L-shaped folded plate, the time-varying DLF of plate one and time-varying CLF between plate one and plate two can be obtained through wave approach [17].
η1 (t )=2ξ1 (t ); η12 (t )=
2CB (t )L1 τ12 π Aω
(20)
where ξ 1 (t) is time-varying modal damping ratio of plate 1, CB (t) is time-varying bending wave speed, L1 is length of the junction of two plates, τ 12 is wave transmission coefficient defined as ratio of transmitted power to incident power, A is surface area of plate one. A unit vertical initial velocity is applied to plate one and gives plate one subsystem an initial energy E1 (0) = 1.146 J. The input power, P1 (t), P2 (t) are zero, the initial energy E1 (0) = 1.146 J, E2 (0) = 0, and the initial energy derivatives dE1 (0)/dt = -η1 (0)ωE1 (0), dE2 (0)/dt = 0. The analysis frequency is set to 20 0 0 rad/s. Substituting time-varying SEA parameters in Eq. (20) to Eqs. (15) and (16). Then, by solving Eqs. (15) and (16), transient energy responses of subsystems can be calculated. We consider a transient analysis of an L-shaped folded plate system with time-varying elastic modulus. Assuming that the system, within 0.2 s, has linear time-varying elastic modulus
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Fig. 9. Transient energy response of subsystem two under different time-varying modal damping ratio.
Fig. 7. Finite element model of L-shaped folded plate.
Fig. 10. Transient energy response of plate two subsystem under different timevarying elastic modulus. Fig. 8. Transient energy response of plate/subsystem two.
E(t )= (109.5–50t) GPa. The density and modal damping ratio of system are constant. The transient energy response of subsystem two is calculated by Eq. (16) and transient energy response of plate two is predicted by FEM software. The finite element model of structure, which is established by using four-nodded quadrilateral elements, is shown in Fig. 7. The element size is 5 mm and it is enough in each bending wave length corresponding to the analysis frequency. Simulations are performed on a desktop computer with Intel core i7 7770 processor and 8GB RAM, running on Windows 7 64bit. Comparison between results from SEA method and FEM is shown in Fig. 8. As shown in Fig. 8, it can be concluded that simulation results of SEA are consistent with results from FEM. The CPU time to solve the problem with SEA method and FEM method are 1.2 s and 416.6 s, dividedly. Therefore, the SEA method can predict the transient energy response of time-varying structures very efficiently with sufficient accuracy. 4.2. Transient analysis of system with different time-varying modal damping ratio A transient analysis of an L-shaped folded plate system with time-varying modal damping ratio is considered in this subsection. Assuming that the system, within 0.2 s, has a linear timevarying modal damping ratio ξ 1 (t) = ξ 2 (t )= (0.02 + 0.05t) N•s/m in case4.2.1, ξ 1 (t) = ξ 2 (t )= (0.02 + 0.1t) N•s/m in case4.2.2 and ξ 1 (t) = ξ 2 (t )= (0.02 + 0.15t)N•s/m in case4.2.3. The density and elastic modulus of system are constant. The comparison between results from different cases is shown in Fig. 9.
As shown in Fig. 9, the peak energy in the same period decrease with the rise of gradient of time-varying modal damping ratio. This is because the rise of damping results in increase of DLF of subsystems. The higher DLF leads to faster dissipation of energy. 4.3. Transient analysis of system with different time-varying elastic modulus A transient analysis of an L-shaped folded plate system with time-varying elastic modulus is considered in this sub-section. Assuming that the system, within 0.2 s, has a linear time-varying elastic modulus E(t )= (109.5–50t) GPa in case4.3.1, E(t )= (109.5– 100t) GPa in case4.3.2, and E(t )= (109.5–150t) GPa in case4.3.3. The density and modal damping ratio of system are constant. The comparison between results from different cases is shown in Fig. 10. As shown in Fig. 10, it can be concluded that the peak energy and the rise time of each vibration period in same order decrease with the decline of gradient of time-varying elastic modulus. This is because the decrease of elastic modulus will results in decline of CLF between subsystems. And lower CLF leads to slower energy exchange between plate one subsystem and plate two subsystem. Furthermore, the vibration period of plate two subsystem increases with the decrease of elastic modulus, which is similar to the conclusion in Section 3.4. 5. Application on a complex time-varying vibro-acoustic structure Considering the complexity of structures in engineering practice, it is necessary to validate the feasibility of the SEA method
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Fig. 12. Transient energy response of subsystems. (a): structural subsystems, (b): cavity subsystems.
Fig. 11. SEA model of a simplified launch vehicle fairing. (a): overall model, (b): shrink model.
for time-varying system on predicting transient energy response of complex time-varying systems. An application on a simplified launch vehicle fairing is investigated in this section. The SEA model of fairing is presented in Fig. 11. A transient analysis of a complex vibro-acoustic structures with exponential time-varying DLF is considered in this subsection. Assuming that the system has exponential time-varying DLF ηi (t) = 0.01e−0.5 t . The CLFs between subsystems, which are calculated by commercial software, are constant. A unit energy impulse has been applied to flat plate subsystem and focus has been placed on transient energy responses of curved plate 2 subsystem, cylinder subsystem, curved plate 1 subsystem, cavity 1 subsystem and cavity 2 subsystem. Results have been depicted in Fig. 12 for the 1/3 octave frequency bands of 10 0 0 rad/s. As shown in Fig. 12, transient energy responses of subsystems are associated with the distance between analysis subsystem and flat plate subsystem. For structural subsystems (besides flat plate subsystem), transient energy response of curved panel 2 subsystem is highest, followed by cylindrical subsystem, and energy of curved panel 1 subsystem is lowest. For cavity subsystems, transient energy response of cavity 2 subsystem is greater than cavity 1 subsystem, the peak energy of cavity subsystems show same variation trend while the rise time has opposite changing trend. 6. Conclusions In this research work, by deriving energy governing equations of time-varying structures, the SEA is applied to predict transient energy response of time-varying structures for the first time. Nu-
merical simulations are conducted, the application of SEA method to a time-varying two-oscillator system is firstly presented and the accuracy of SEA method for time-varying system is verified by Newmark-beta method; then, the application of SEA method to a time-varying L-shaped folded plate is presented and the accuracy of predicted energy response is verified by FEM; finally, to demonstrate the applicability of SEA method for time-varying system, the approach is applied to predict transient energy response of a simplified launch vehicle fairing with time-varying DLF. According to results shown in numerical simulation, it can be concluded that SEA method for time-varying system in this paper has good accuracy in the prediction of transient energy response for time-varying structures. For the two-oscillator system under an impulse excitation, the peak energy of subsystem in same period decreases with the rise of gradient of time-varying damping. The peak energy and rise time of each vibration period in same order decrease with the rise of gradient of time-varying coupling stiffness. For the L-shaped folded plate with an initial velocity, the peak energy of subsystem in the same period decreases with the rise of gradient of time-varying modal damping ratio. The peak energy and rise time of each vibration period in same order decrease with the decline of gradient of time-varying coupling stiffness. For the simplified launch vehicle fairing under an impulse excitation, transient energy responses of subsystems are associated with the distance between analysis subsystem and exited subsystem. Results show that the SEA method for time-varying system is also applicable for transient analysis of complex time-varying structures with a small computational cost. The SEA method for time-varying system has a wider application scope and can contribute to the prediction of transient vibration response of time-varying structure especially in the high frequency range.
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Acknowledgments The authors are very grateful to the supports of the National Natural Science Foundation of China (11572086, 11402052), the Fundamental Research Fundsfor the Central Universities, Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0054) and the Jiangsu Natural Science Foundation (BK20170656, BK20170022). The supports are gratefully acknowledged. References [1] A.K. Banerjee, Dynamics of a variable-mass, flexible-body system, J. Guid. Control Dynam. 23 (3) (2012) 501–508. [2] H. Ouyang, Moving-load dynamic problems: a tutorial (with a brief overview), Mech. Syst. Signal Process. 25 (6) (2011) 2039–2060. [3] J.E.T. Penny, G.F. Howard, Time-domain finite-element solutions for single degree-of-freedom systems with time-dependent parameters, J. Mech. Eng. Sci. 22 (1) (1980) 29–33. [4] K. Yu, J. Zou, Y. Zhang, G. Shi, An algorithm for structural dynamic response based on Hamilton law, J. Harbin Inst. Tech. 29 (05) (1997) 46–49 in Chinese. [5] R. Zhao, K. Yu, An efficient transient analysis method for linear time-varying structures based on multi-level substructuring method, Comput. Struct. 146 (2015) 76–90. [6] R.H. Lyon, R.G. Dejong, Theory and Application of Statistical Energy Analysis, Butterworth-Heinemann, London, 1995.
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[7] R.H. Lyon, Fluctuation theory and (very) early statistical energy analysis, J. Acoust. Soc. Am. 113 (2003) 2401–2403. [8] M.L. Lai, A. Soom, Predication of transient vibration envelopes using statistical energy analysis techniques, ASME J. Vib. Acoust. 112 (1) (1990) 127–137. [9] M.L. Lai, A. Soom, Statistical energy analysis for the time integrated transient response of vibrating systems, ASME J. Vib. Acoust. 112 (2) (1990) 206–213. [10] R.J. Pinnington, D. Lednik, Transient statistical energy analysis of an impulsively excited two oscillator system, J. Sound Vib. 189 (2) (1996) 240–264. [11] R.J. Pinnington, D. Lednik, Transient energy flow between two coupled beams, J. Sound Vib 189 (2) (1996) 265–287. [12] H. Song, K. Yu, X. Li, J. Han, Affine arithmetic applied to transient statistical energy analysis of a two-oscillator system, Mech. Res. Commun. 70 (2015) 12–16. [13] 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 Acust. Acust. 100 (1) (2014) 46–56. [14] M. Robinson, C. Hopkins, Prediction of maximum time-weighted sound and vibration levels using transient statistical energy analysis. Part 2: experimental validation, Acta Acust. Acust. 100 (1) (2014) 57–66. [15] M.N. Ichchou, A.L. Bot, Jezequel, A transient local energy approach as an alternative to transient SEA: wave and telegraph equations, J. Sound Vib. 246 (5) (2001) 829–840. [16] F.S. Sui, M.N. Ichchou, L. Jezequel, Prediction of vibroacoustics energy using a discretized transient local energy approach and comparison with TSEA, J. Sound Vib. 251 (1) (2002) 163–180. [17] A.C. Pankaj, S. Sastry, S.M. Murigendrappa, A comparison of different methods for determination of coupling factor and velocity response of coupled plates, J. Vibroeng. 15 (4) (2013) 1885–1897.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
An efficient transient analysis method for time-varying structures based on statistical energy analysis Qiang Chen a,b, Qingguo Fei a,b,c,∗, Shaoqing Wu a,b, Yanbin Li a,b, Xuan Yang a,c a
Institute of Aerospace Machinery and Dynamics, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China Department of Engineering Mechanics, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China c School of Mechanical Engineering, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China b
a r t i c l e
i n f o
Article history: Received 19 November 2017 Revised 23 April 2018 Accepted 3 June 2018 Available online 15 June 2018 Keywords: Time-varying structures Statistical energy analysis Transient energy response Complex vibro-acoustic system Damping loss factor
a b s t r a c t The statistical energy analysis (SEA) is applied to predict the transient energy response of structures with time-varying parameters for the first time. With the energy governing equations derived by considering time-varying SEA parameters and energy flow item caused by time-varying damping loss factor, SEA method is firstly applied to predict transient energy response of time-varying systems. Then, numerical examples of a time-varying two-oscillator system, a time-varying L-shaped fold plate and a complex time-varying vibro-acoustic structure are investigated to demonstrate the effectiveness and accuracy of SEA method for time-varying system. The Newmark-beta method and finite element method are used to verify the accuracy of predicted transient energy response. Results show that SEA method for timevarying system is capable of predicting transient energy response of time-varying structures with sufficient accuracy and also applicable for transient analysis of complex time-varying structures with a small computational cost. © 2018 Published by Elsevier Ltd.
1. Introduction Many structures in practical engineering are time-varying structures, which are characterized by the mass, stiffness or damping properties that change with time. For instance, the rapid combustion of rocket fuel will lead to time-varying mass dynamic problems [1] and the aerodynamic heating will change the stiffness and damping characteristics of aircrafts. Moreover, moving vehicles on bridge structure will result in a time-varying vehicle-bridge interaction problem which must be well solved in order to assure the operation safety of high-speed railway [2]. Nowadays timevarying characteristics of system parameters are often neglected in dynamic analysis of engineering structures which will lead to inaccurate results in the dynamic response prediction. In order to improve the accuracy of dynamic response prediction of time-varying structures, it is increasingly important to consider the time-varying features of dynamical systems. At present, deterministic approaches are widely used to solve dynamic problems of time-varying structures. After the spatial discretization of structures to several elements, the problem is turned into an initial value problem of linear ordinary differential equa∗ Corresponding author at: Engineering Mechanics, Southeast University, No.2 Dongnandaxue Road, Nanjing 211189, China. E-mail address: [email protected] (Q. Fei).
https://doi.org/10.1016/j.mechrescom.2018.06.001 0093-6413/© 2018 Published by Elsevier Ltd.
tions (ODEs) with time-varying coefficients. Then time integration methods, such as the central difference method, Newmark-beta method, can be used to solve these ODEs in time domain. For transient analysis of time-varying structures, Penny and Howard [3] developed a time finite element method (TFEM) for a time-varying single degree-of-freedom system based on Hamilton’s principle. Yu et al. [4] extended the TFEM to time-varying multiple degrees of freedom systems based on Hamilton’s law of varying action. Zhao and Yu [5] presented a transient analysis method for linear time-varying structures based on multi-level sub-structuring method which improve the computational efficiency to a certain extent. For very complex systems in practical engineering, numerous number of elements are needed in deterministic approaches to describe the vibration behavior of time-varying structures, especially in high frequency range. The computation cost will increase rapidly to predict dynamic response of complex structures. Therefore, energy based methods, such as statistical energy analysis (SEA, [6]) and extended methods of SEA, are more suitable to analyse the problem. The SEA, which is largely inspired from statistical physics [7], has been used to predict the average energy response of complex engineering systems for many years. Energy responses of subsystems can be calculated efficiently by solving energy transfer equations between subsystems in SEA. However, traditional SEA is an approach for steady state problems.
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Over the past several decades, some methods are proposed as extensions of the SEA to analyse unsteady state problem of timeinvariant structures. SEA is firstly applied to predict transient energy response of time-invariant structures by transient statistical energy analysis (TSEA, [6]). Then, Lai and Soom [8,9] proposed the concept of “time-varying coupling loss factor” for time-invariant structures in TSEA, however, it is just a concept in mathematics and hard to interpret in physics. Furthermore, Pinnington and Lednik [10,11] applied TSEA to a two-oscillator system and a couplingbeam system, results from TSEA were compared with exact analytical solutions which showed that the prediction of TSEA is sometimes far from fitting the exact reference solution. Song et al. [12] applied affine arithmetic to TSEA of a two-oscillator system, and revealed the influence of measurement errors of parameters on predicted transient energy response. Experimentally, Robinson and Hopkins [13,14] used TSEA to calculate the maximum timeweighted sound and vibration levels in built-up structures. Good agreement was achieved between measurements and predictions. The transient local energy approach (TLEA), which was proposed by Ichchou et al. [15], is another extension of SEA to predict transient energy response of structures. With consideration of timevarying item in energy flow term, TLEA has much higher precision than TSEA [16]. The current research on dynamic response prediction of time-varying structures mostly concentrates on deterministic approaches, which are prohibitly time consuming for complex structures. Thus the statistical energy based approaches are preferred. On the other hand, though SEA has been developed at 1960s, research work mainly focused on transient energy prediction of time-invariant structures and literatures on prediction of transient energy response of time-varying structures are very limited. Therefore, by deriving the energy governing equations of time-varying structures, the SEA is firstly applied to predict transient energy response of time-varying structures in this paper. The outline of this work is as follows: In Section 2, energy governing equations of time-varying structures are derived and SEA is applied to predict transient energy response of time-varying structures for the first time. In Sections 3 to 5, the effectiveness and accuracy of SEA method for time-varying system are verified by numerical examples, which include comparison with Newmark-beta method of a simply two-oscillator system under an impulse excitation, comparison with finite element method of a time-varying L-shaped folded plate with an initial energy, an application on a complex time-varying vibro-acoustic system under an impulse excitation. Finally, in Section 6, conclusions are drawn.
The local power balance for a non-loaded region can be expressed as
∂ e(s, t ) ∂ I (s, t ) + Pdiss + =0 ∂t ∂s
The dissipation power for time-varying structures is evaluated
Pdiss = η (t )ωe(s, t )
c
∂ e(s, t ) 1 ∂ I (s, t ) η (t )ωI (s, t ) + + =0 ∂s c ∂t c
∂ 2 e(s, t ) ∂ e(s, t ) ∂η (t ) ∂ 2 I (s, t ) + η (t )ω + e(s, t )ω + =0 2 ∂t ∂t ∂ s∂ t ∂t
(7)
∂ 2 I (s, t ) 2 ∂ 2 e(s, t ) ∂ I (s, t ) +c + η (t )ω =0 ∂ s∂ t ∂s ∂ s2
(8)
By subtracting Eqs. (7) and (8) leads to the expression
∂ e(s, t ) ∂ e(s, t ) +η (t )ω ∂t ∂t2 ∂η (t ) 2 ∂ 2 e(s, t ) ∂ I (s, t ) + e(s, t )ω −c −η (t )ω =0 ∂t ∂s ∂ s2 2
(9)
Substituting the expression of ∂ I(s, t)/∂ s in Eq. (4) into Eq. (9)
∂ e(s, t ) ∂ e(s, t ) ∂η (t ) +2η (t )ω + e(s, t )ω ∂t ∂t ∂t2 2 2 ∂ e (s, t ) 2 −c + (η (t )ω ) e(s, t ) = 0 ∂ s2 2
(10)
For time-varying structure consists of N subsystems, the concept of total energy rather than energy density turns the Eq. (10) into the form
∂ 2 e(s, t ) ∂ e(s, t ) ∂η (t ) + 2η (t )ω + e(s, t )ω ∂t ∂t ∂t2 2 2 ∂ e (s, t ) 2 −c + (η (t )ω ) e(s, t )] = Pi (t ) ∂ s2 [
(11)
where ∫V ei (s,t) = Ei (t), Ei (t) is time-varying energy stored in subsystem i, Pi (t) is time-varying injected power into subsystem i and N c2 2 (ηi j (t )ωEi (t )) − V (− η (t )ω ∇ e (s, t ) )dV = j=1, j=i
Some basic definitions and assumption used to derive the energy equation of time-varying structure are generally summarized as [16]
j=1, j=i
e(s, t ) = e+ (s, t ) + e− (s, t )
(1)
I (s, t ) = I+ (s, t ) + I− (s, t )
(2)
e− (s,t)
(6)
Differentiating Eq. (4) with respect to time and space, respectively, leads to the expression
N
e+ (s,t)
(5)
where η(t) is the time-varying damping loss factor (DLF), ω = 2π f is radian frequency. Substituting Eqs. (1)–(3) and (5) into Eq. (4) yields:
2. Statistical energy analysis for time-varying structures
I± (s, t ) = ±c · e± (s, t )
(4)
(η ji (t )ωE j (t )), ηji (t) is time-varying coupling loss factor
(CLF) between subsystem j to i. The power balance equation for subsystem i can be given by 1
η (t )ω
d2 Ei (t ) dt 2
+ 2 dEdi (t t ) + ηEi ((tt )) dηdi t(t ) +
i i (12) N N ηi j (t ) ωEi (t ) − ηi (t ) + η ji (t )ωE j (t ) = Pi (t ) j=1, j=i
j=1, j=i
The power balance equation for time-varying structure is driven by following second-order ODEs
(3)
where and are energy density associated with the right and left train waves, while I+ (s,t) and I− (s,t)are the incident and reflected power flows, the right train wave is considered separate from the left one. c is the energy velocity, the same as the group velocity of waves in a slight damping medium.
d2 E(t ) dE(t ) E(t ) dη (t ) +2 + + ωη (t )E(t ) = P(t ) ωη (t ) dt 2 dt η (t ) dt 1
(13)
where E(t) = [E1 (t), E2 (t),••• EN (t)]T is time-varying energy vector, η(t) is time-varying losing factor matrix of time-varying structure, P(t) = [P1 (t), P2 (t),••• PN (t)]T is time-varying input power vector, the superscript T represents matrix transpose.
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Fig. 1. Two subsystems model in SEA.
Fig. 3. Transient energy response of oscillator/subsystem two.
equation of time-varying structures can be expressed as follows
M(t )x¨ (t ) + C(t )x˙ (t ) + K(t )x(t ) = F(t )
Fig. 2. Two-oscillator system model.
For the time-varying losing factor matrix of time-varying system, elements of matrix can be given by
⎧ N ⎨ ηi (t ) + ηi j (t ), i = j η (t, i, j ) = , j =i ⎩−η (t ),j=1 i = j
(14)
ji
Specifically, for a structure consists of two subsystems shown in Fig. 1, power balance equations of subsystems can be expressed by
+ 2 dEd1t(t ) + η 1(t ) E1 (t ) dηd1t(t ) + 1 η1 (t )ωE1 (t )+η12 (t )ωE1 (t ) − η21 (t )ωE2 (t ) = P1 (t )
(15)
+ 2 dEd2t(t ) + η 1(t ) E2 (t ) dηd2t(t ) + 2 η2 (t )ωE2 (t )+η21 (t )ωE2 (t ) − η12 (t )ωE1 (t ) = P2 (t )
(16)
1
η1 (t )ω
1
η2 (t )ω
d2 E1 (t ) dt 2
d2 E2 (t ) dt 2
Under the given initial conditions and given time history of input power, by solving second-order ODEs in (13), transient energy responses of subsystems can be calculated. By considering time-varying SEA parameters η(t) and energy flow item caused by time-varying damping loss factor [(E(t)/η(t))/(dη(t)/dt)], the SEA method is applied to predict transient energy response of time-varying systems for the first time. 3. Application on a time-varying two-oscillator system As it is difficult to obtain the exact analytic solution for systems with arbitrary time-varying parameters, the SEA method for time-varying structure is verified by Newmark-beta method for a two-oscillator system in Section 3.2. Further, dynamic analysis of a two-oscillator system with different time-varying damping and different time-varying coupling stiffness are presented in Section 3.3 and Section 3.4, respectively. The transient energy results of two damped oscillators coupled by an undamped spring are investigated under an impulse excitation. In a time-varying two-oscillator system shown in Fig. 2, with mass m1 (0) = m2 (0) = 2 kg, stiffness k1 (0) = k2 (0) = 1,717,0 0 0 N/m, coupling stiffness k(0) = 280,0 0 0 N/m, damping c1 (0) = c2 (0) = 200 N•s/m. The analysis frequency is set to 10 0 0 rad/s. Consider that an initial unit impulse is applied to mass m1 , giving an initial velocity x˙ 1 (0 ) = 1 and then the initial energy is obtained as E1 (0) = 0.5m1 . 3.1. Dynamic analysis of a time-varying structures based on Newmark-beta method According to Hamilton’s law of time-varying structures, after the spatial discretization, the obtained semi-discrete dynamic
(17)
where M, C, K, F are the time-varying mass, damping, stiffness, load matrixes of time-varying structures, respectively; x,x˙ ,x¨ are the displacement, velocity, and acceleration vector of time-varying structures, dividedly. The direct time integration is a step-by-step numerical procedure. The total time T is divided into n time steps equally. Then, the Newmark-beta method [5], which is one of the most popular algorithms for numerical solutions of time-varying structural dynamic problems, is used to calculate the displacements, velocities and accelerations vector at different time steps. For a time-varying two-oscillator system shown in Fig. 2, the total energy of oscillator i can be expressed as sum of kinetic energy and potential energy
Ei (t ) =
1 1 mi (t )(x˙ i (t ))2 + (ki (t ) + k(t ))(xi (t ))2 2 2
(18)
3.2. Validation case The time-varying two-oscillator system is divided into two subsystems, namely oscillator one and oscillator two. The SEA model of system is shown in Fig. 1. The time-varying DLF of subsystem i and time-varying CLF between subsystem i to subsystem j can be expressed by
ηi (t ) =
ci (t ) 1 (χ (t ) ) ηi j (t )= mi (t )ω1 (t ) 2ηi (t ) (ωi (t ) )4 2
(19)
whereωi (t ) = (ki (t ) + k(t ))/mi (t ),χ (t ) = k(t )/ m1 (t )m2 (t ). For system under an impulse excitation, the input power, P1 (t), P2 (t) are zero, the initial energy E1 (0) = 1 J, E2 (0) = 0, and the initial energy derivatives dE1 (0)/dt = -η1 (0)ωE1 (0), dE2 (0)/dt = 0. Substituting time-varying SEA parameters in Eq. (19) to Eqs. (15) and (16). Then, by solving Eqs. (15) and (16), the transient energy responses of subsystems can be calculated. We consider a transient analysis of a two-oscillator system with time-varying damping. Assuming that the system, within 0.2 s, has linear time-varying damping c1 (t) = c2 (t )= (20 0–50 0t) N•s/m. The mass, stiffness and coupling stiffness of system are constant. The transient energy response of subsystem two is calculated by Eq. (16) and the transient energy response of oscillator two is predicted by Eq. (18). The comparison between results from SEA method for time-varying systems and Newmark-beta method is shown in Fig. 3. As shown in Fig. 3, it can be concluded that simulation results of SEA are consistent with results from Newmark-beta method. Therefore, the SEA method for time-varying system can predict transient energy response of time-varying structures with sufficient accuracy.
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Fig. 4. Transient energy response of subsystem two under different time-varying damping. Fig. 6. L-shaped folded plate.
between subsystems. And the lower CLF leads to slower energy exchange between two subsystems. Furthermore, the vibration period of oscillator two increases with the decrease of coupling stiffness. 4. Application on a time-varying L-shaped folded plate
Fig. 5. Transient energy response of subsystem two under different time-varying coupling stiffness.
3.3. Transient analysis of system with different time-varying damping A transient analysis of a two-oscillator system with timevarying damping is considered in this sub-section. Assuming that the system, within 0.2 s, has a linear time-varying damping c1 (t) = c2 (t )= (20 0–30 0t) N•s/m in case3.3.1, c1 (t) = c2 (t )= (200– 600t) N•s/m in case3.3.2 and c1 (t) = c2 (t )= (20 0–90 0t) N•s/m in case3.3.3. The mass, stiffness and coupling stiffness of system are constant. Comparison between results from different cases is shown in Fig. 4. As shown in Fig. 4, the peak energy in the same period decrease with the rise of gradient of time-varying damping. This is because the decrease of damping results in the decline of DLF of subsystems. The lower DLF leads to slower dissipation of energy. In addition, the vibration period of all three cases is equal as the variety of damping will not change the natural frequency of oscillators. 3.4. Transient analysis of system with different time-varying coupling stiffness A transient analysis of a two-oscillator system with timevarying coupling stiffness is considered in this sub-section. Assuming that the system, within 0.2 s, has a linear time-varying coupling stiffness k (t) = (280,0 0 0–20 0 0 0t) N/m in case3.4.1, k(t )= (280,0 0 0– 40 0 0 0t) N/m in case3.4.2, and k(t )= (280,0 0 0–60 0 0 0t) N/m in case3.4.3. The mass, stiffness and damping of system are constant. Comparison between results from different cases is shown in Fig. 5. As shown in Fig. 5, it can be concluded that the peak energy and rise time of each vibration period in same order decrease with the decline of gradient of time-varying coupling stiffness. This is because the decrease of coupling stiffness results in decline of CLF
In order to investigate the accuracy of SEA method for timevarying system on engineering structures, the method is verified by finite element method (FEM) for a time-varying L-shaped folded plate in Section 4.1. Further, dynamic analysis of an Lshaped folded plate with different time-varying elastic modulus and different time-varying modal damping ratio are presented in Section 4.2 and Section 4.3, respectively. The structural parameters are indicated as Fig. 6. The l-shaped folded plate is divided into two subsystems, namely plate one and plate two. The dimension of plate one and plate two are both L1 × L2 × t = 0.3 m × 0.3 m × 0.006 m. Titanium alloy with density ρ = 4420 kg/m3 , elastic modulus E(0) = 109.5 GPa, Poisson’s ratio ν = 0.33, modal damping ratio ξ (0) = 0.02 is used in calculation. The coupling edge is in free constrain condition and other edges are simply supported. 4.1. Validation case For the L-shaped folded plate, the time-varying DLF of plate one and time-varying CLF between plate one and plate two can be obtained through wave approach [17].
η1 (t )=2ξ1 (t ); η12 (t )=
2CB (t )L1 τ12 π Aω
(20)
where ξ 1 (t) is time-varying modal damping ratio of plate 1, CB (t) is time-varying bending wave speed, L1 is length of the junction of two plates, τ 12 is wave transmission coefficient defined as ratio of transmitted power to incident power, A is surface area of plate one. A unit vertical initial velocity is applied to plate one and gives plate one subsystem an initial energy E1 (0) = 1.146 J. The input power, P1 (t), P2 (t) are zero, the initial energy E1 (0) = 1.146 J, E2 (0) = 0, and the initial energy derivatives dE1 (0)/dt = -η1 (0)ωE1 (0), dE2 (0)/dt = 0. The analysis frequency is set to 20 0 0 rad/s. Substituting time-varying SEA parameters in Eq. (20) to Eqs. (15) and (16). Then, by solving Eqs. (15) and (16), transient energy responses of subsystems can be calculated. We consider a transient analysis of an L-shaped folded plate system with time-varying elastic modulus. Assuming that the system, within 0.2 s, has linear time-varying elastic modulus
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Fig. 9. Transient energy response of subsystem two under different time-varying modal damping ratio.
Fig. 7. Finite element model of L-shaped folded plate.
Fig. 10. Transient energy response of plate two subsystem under different timevarying elastic modulus. Fig. 8. Transient energy response of plate/subsystem two.
E(t )= (109.5–50t) GPa. The density and modal damping ratio of system are constant. The transient energy response of subsystem two is calculated by Eq. (16) and transient energy response of plate two is predicted by FEM software. The finite element model of structure, which is established by using four-nodded quadrilateral elements, is shown in Fig. 7. The element size is 5 mm and it is enough in each bending wave length corresponding to the analysis frequency. Simulations are performed on a desktop computer with Intel core i7 7770 processor and 8GB RAM, running on Windows 7 64bit. Comparison between results from SEA method and FEM is shown in Fig. 8. As shown in Fig. 8, it can be concluded that simulation results of SEA are consistent with results from FEM. The CPU time to solve the problem with SEA method and FEM method are 1.2 s and 416.6 s, dividedly. Therefore, the SEA method can predict the transient energy response of time-varying structures very efficiently with sufficient accuracy. 4.2. Transient analysis of system with different time-varying modal damping ratio A transient analysis of an L-shaped folded plate system with time-varying modal damping ratio is considered in this subsection. Assuming that the system, within 0.2 s, has a linear timevarying modal damping ratio ξ 1 (t) = ξ 2 (t )= (0.02 + 0.05t) N•s/m in case4.2.1, ξ 1 (t) = ξ 2 (t )= (0.02 + 0.1t) N•s/m in case4.2.2 and ξ 1 (t) = ξ 2 (t )= (0.02 + 0.15t)N•s/m in case4.2.3. The density and elastic modulus of system are constant. The comparison between results from different cases is shown in Fig. 9.
As shown in Fig. 9, the peak energy in the same period decrease with the rise of gradient of time-varying modal damping ratio. This is because the rise of damping results in increase of DLF of subsystems. The higher DLF leads to faster dissipation of energy. 4.3. Transient analysis of system with different time-varying elastic modulus A transient analysis of an L-shaped folded plate system with time-varying elastic modulus is considered in this sub-section. Assuming that the system, within 0.2 s, has a linear time-varying elastic modulus E(t )= (109.5–50t) GPa in case4.3.1, E(t )= (109.5– 100t) GPa in case4.3.2, and E(t )= (109.5–150t) GPa in case4.3.3. The density and modal damping ratio of system are constant. The comparison between results from different cases is shown in Fig. 10. As shown in Fig. 10, it can be concluded that the peak energy and the rise time of each vibration period in same order decrease with the decline of gradient of time-varying elastic modulus. This is because the decrease of elastic modulus will results in decline of CLF between subsystems. And lower CLF leads to slower energy exchange between plate one subsystem and plate two subsystem. Furthermore, the vibration period of plate two subsystem increases with the decrease of elastic modulus, which is similar to the conclusion in Section 3.4. 5. Application on a complex time-varying vibro-acoustic structure Considering the complexity of structures in engineering practice, it is necessary to validate the feasibility of the SEA method
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Fig. 12. Transient energy response of subsystems. (a): structural subsystems, (b): cavity subsystems.
Fig. 11. SEA model of a simplified launch vehicle fairing. (a): overall model, (b): shrink model.
for time-varying system on predicting transient energy response of complex time-varying systems. An application on a simplified launch vehicle fairing is investigated in this section. The SEA model of fairing is presented in Fig. 11. A transient analysis of a complex vibro-acoustic structures with exponential time-varying DLF is considered in this subsection. Assuming that the system has exponential time-varying DLF ηi (t) = 0.01e−0.5 t . The CLFs between subsystems, which are calculated by commercial software, are constant. A unit energy impulse has been applied to flat plate subsystem and focus has been placed on transient energy responses of curved plate 2 subsystem, cylinder subsystem, curved plate 1 subsystem, cavity 1 subsystem and cavity 2 subsystem. Results have been depicted in Fig. 12 for the 1/3 octave frequency bands of 10 0 0 rad/s. As shown in Fig. 12, transient energy responses of subsystems are associated with the distance between analysis subsystem and flat plate subsystem. For structural subsystems (besides flat plate subsystem), transient energy response of curved panel 2 subsystem is highest, followed by cylindrical subsystem, and energy of curved panel 1 subsystem is lowest. For cavity subsystems, transient energy response of cavity 2 subsystem is greater than cavity 1 subsystem, the peak energy of cavity subsystems show same variation trend while the rise time has opposite changing trend. 6. Conclusions In this research work, by deriving energy governing equations of time-varying structures, the SEA is applied to predict transient energy response of time-varying structures for the first time. Nu-
merical simulations are conducted, the application of SEA method to a time-varying two-oscillator system is firstly presented and the accuracy of SEA method for time-varying system is verified by Newmark-beta method; then, the application of SEA method to a time-varying L-shaped folded plate is presented and the accuracy of predicted energy response is verified by FEM; finally, to demonstrate the applicability of SEA method for time-varying system, the approach is applied to predict transient energy response of a simplified launch vehicle fairing with time-varying DLF. According to results shown in numerical simulation, it can be concluded that SEA method for time-varying system in this paper has good accuracy in the prediction of transient energy response for time-varying structures. For the two-oscillator system under an impulse excitation, the peak energy of subsystem in same period decreases with the rise of gradient of time-varying damping. The peak energy and rise time of each vibration period in same order decrease with the rise of gradient of time-varying coupling stiffness. For the L-shaped folded plate with an initial velocity, the peak energy of subsystem in the same period decreases with the rise of gradient of time-varying modal damping ratio. The peak energy and rise time of each vibration period in same order decrease with the decline of gradient of time-varying coupling stiffness. For the simplified launch vehicle fairing under an impulse excitation, transient energy responses of subsystems are associated with the distance between analysis subsystem and exited subsystem. Results show that the SEA method for time-varying system is also applicable for transient analysis of complex time-varying structures with a small computational cost. The SEA method for time-varying system has a wider application scope and can contribute to the prediction of transient vibration response of time-varying structure especially in the high frequency range.
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