# Time-Domain Substructure Transient Vibration Transfer Path Analysis Based on Time-Varying Frequency Response Functions under Operational Excitations

Authors: Rong He; Hong Zhou
Journal: Shock and Vibration
 (2019)

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## Abstract

The time-domain substructure inverse matrix method has become a popular method to detect and diagnose problems regarding vehicle noise, vibration, and harshness, especially for those impulse excitations caused by roads. However, owning to its reliance on frequency response functions (FRFs), the approach is effective only for time-invariable linear or weak nonlinear systems. This limitation prevents this method from being applied to a typical vehicle suspension substructure, which shows different nonlinear characteristics under different wheel transient loads. In this study, operational excitation was considered as a key factor and applied to calculate dynamic time-varying FRFs to perform accurate time-domain transient vibration transfer path analysis (TPA). The core idea of this novel method is to divide whole coupled substructural relationships into two parts: one involved time-invariable components; normal FRFs could be obtained through tests directly. The other involved numerical computations of the time-domain operational loads matrix and FRFs matrix in static conditions. This method focused on determining dynamic FRFs affected by operational loads, especially the severe transient ones; these loads are difficult to be considered in other classical TPA approaches, such as operational path analysis with exogenous inputs (OPAX) and operational transfer path analysis (OTPA). Experimental results showed that this new approach could overcome the limitations of the traditional time-domain substructure TPA in terms of its strict requirements within time-invariable systems. This is because in the new method, time-varying FRFs were calculated and used, which could make the FRFs at the system level directly adapt to time-varying systems from time to time. In summary, the modified method extends TPA objects studied in time-invariable systems to time-varying systems and, thus, makes a methodology and application innovation compared to traditional the time-domain substructure TPA.


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## Body

## 1. Introduction

In the beginning of the 1980s, classical transfer path analysis (TPA) methods were developed; after decades of development, TPA has become a useful tool for load identification and contributes to investigation in the field of noise and vibration [1–3]. There are three major limitations regarding traditional TPA. First, several frequency response functions (FRFs) are required during tests, requiring considerable time and energy. Second, for estimating the coupled FRFs, it is required to remove active parts, which is not often feasible [4, 5]. The third one is that it cannot measure the exact FRFs under real operational conditions owing to time-varying systems in different circumstances [6–9].As a consequence, several new approaches, such as operational TPA (OTPA), have been developed in the last two decades, aimed at overcoming these limitations [10–20]. These methods attract some attention as they only require operational data measured at the path and reference locations under certain conditions [21]. Thus, they avoid testing FRFs and a considerable amount of testing time. Despite saving time, these approaches suffer from problems regarding the setting of appropriate paths and reference locations. In addition, matrix inversion in OTPA has errors and tolerance levels, which have to be accounted for in numerical solutions. This drawback leads to incorrect path interpretations [22, 23].Operational path analysis with exogenous inputs (OPAX), a new TPA method, has been widely used in engineering applications in recent years. The core idea of OPAX is a combination of the necessary minimal FRFs and operational data for a compromise between results accuracy and tests, consuming via parametric models for the load identification [10, 11, 20, 24]. The accuracy of this method depends on the parametric model used for load identification [25–30] and reference signals chosen for the matrix calculation. Experience and operational excitation mainly define the effectiveness of this approach.In the 2010s, a time-domain substructure transient vibration TPA method was studied by several researchers. The objectives of these studies were to expand applications of the spectral-based substructure method used under steady-state conditions to address the transient problems of a mechanical system with weak nonlinear coupling [31–33]. Time-domain convolution was used as the main theory to discretize input signals into time segments. Then, conventional substructure inverse matrix TPA methods were used to finish the computation. However, the FRFs within the coupled substructures were set as time-invariable functions during the calculation.Hence, during the 2010s, several researchers started to study substructure inverse matrix TPA (IMTPA) methods [20, 34–36]. These methods actually have some advantages over traditional TPA and OPAX [1–3, 6–8, 37, 38]. The classical TPA is more time-consuming owing to the requirement of system disassembly, and OPAX is not a suitable method for transient loads. Nevertheless, the traditional substructure inverse matrix method based on FRFs has one main limitation; the whole system should be considered as a stable unit, and its transfer characteristics are not likely to be changed by loads. Unfortunately, most current situations involve vehicles normally exhibiting different nonlinear and time-variable characteristics under severe loads [39]. In the following sections, the IMTPA, time-domain substructure IMTPA, and modified substructure IMTPA as a novel method will be discussed.In the following paragraphs, different methods will be discussed based on one simple dynamic model (seen in Figure1), which is extensively applied in the field of vehicle engineering, such as in the systems of powertrain mounts [40] and vehicle suspension systems [37, 38, 41].Figure 1
Dynamic model.This model includes several parts. Plane A represents substructure A, Plane B represents substructure B, and they are connected by three springs with various stiffnesses. This coupled system was used for path analysis comparisons in different TPA methods as a representative mechanical structure of vehicles. Figure1 presents the dynamic model of this system, where k1, k2, and k3 denote the stiffnesses of the three springs connecting active substructure B and passive substructure A and i(b) denotes the vertical excitation caused by the bench of the vibration simulator.


## 2. Methods

### 2.1. Conventional Substructure Inverse Matrix Method

The substructure inverse matrix method applied systematic FRFs between subsystems to calculate vibration transfer paths and their contribution in the coupled system as shown in Figure2.Figure 2
Coupled system with two subsystems.Generalized stiffnessKc connects substructures A and B. Fc(a) represents the force on side A, and Xc(a) represents the vibration of substructure A. In addition, basic relationships between the system-level FRFs and substructure-level FRFs can be formulated as equations (1)–(6) [1, 42–53]:(1)HA,caca=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,cbcb−Hs,cacb,(2)HB,cbcb=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,caca−Hs,cacb,(3)HA,oaca=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,cacaHs,cbcb−Hs,cacbHs,oaca,(4)HB,cbib=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,cacbHs,caca−Hs,cacbHs,caib,(5)KC=Hs,cacbHs,cacaHs,cbcb−Hs,cacbHs,cacb,(6)C=HA,caca+HB,cacb+1Kc−1,where subscripts A and B indicate the names of the substructures, subscript s represents the FRF at the system level, c represents the coupled points, o denotes the output, and i represents the force input. C in equation (6) represents an algebraic operator. The expression Hs,c(a)c(b) represents FRFs from forces Fc(b) to responses Xc(a) in the complete system. Similarly, the expression HA,o(a)c(a) represents FRFs from forces Fc(a) to output responses Xo(a) in substructure A. Clearly, equations (1)–(6) change the FRFs at the system level to those at the substructure level, and Kc is not the real spring stiffness but a generalized one. Equations (7)–(11) were deduced from equations (1)–(6) to obtain Hs,o(a)i(b):(7)HA,oaca=Hs,oaca1−CHA,caca,(8)HB,cbib=Hs,caibHA,cacaC,(9)HA,oaca=Hs,oaca1−CHA,caca,(10)HB,cbib=Hs,caibHA,cacaC.The system response function is given by(11)Hs,oaib=HA,oaCaCHB,cbib,where Hs,o(a)i(b) in equation (11) is the system response that is equal to the calculation results. Hs,c(a)c(b) and Hs,c(a)i(b) are not constant FRFs that can be easily tested in normal static conditions in the system level; they definitely change under the operational load because Kc, in many cases, is made of a nonlinear material, whose dynamic stiffness is not constant under different loads. In other words, Hs,c(a)c(b) and Hs,c(a)i(b) should be obtained using other operational methods given in equations (12) and (13):(12)Hs,cacb=acbFcbacaacb,(13)Hs,caib=aibFibacaaib.


### 2.2. Conventional Time-Domain Substructure Inverse Matrix Method

A time-domain transfer function (TTF) also generally known as the impulse response functionh(t) can be defined as the inverse Fourier transform of a frequency-domain transfer function that is often referred to as an FRF. Accordingly, the time-domain response X(t) is simply the convolution of the TTF and the time-domain excitation function f(t), where δ(t) represents the unit impulse and τ is a real-valued number. The response of a linear time-invariant system due to δ(t) is h(t). Hence, given an excitation δ(t − τ), the response is h(t − τ). Applying the theory of superposition, an excitation input of the form ∫f(t)δ(t − τ)dτ will yield the corresponding response given by ∫f(t)h(t − τ)dτ. Finally, if ∫f(t)δ(t − τ)dτ is equal to f(t), whose corresponding output is X(t) as noted above, then ∫f(t)h(t − τ)dτ is equal to X(t), which is the fundamental basis for the convolution integration applied in this study [31–33]. We considered the example discussed in the introduction as the analysis object to obtain the following equation:(14)x1tx2tx3t=h11th12th13th21th22th23th31th32th33t⊕f1tf2tf3t,where ⊕ is the convolution algorithm, and the TTF matrix with elements hij(t), i, j = 1, 2, 3 can be substituted by the following inverse Fourier transform equation, the FRF equation:(15)x1tx2tx3t=F−1H11ω⊕f1t+F−1H12ω⊕f2t+F−1H13ω⊕f3tF−1H21ω⊕f1t+F−1H22ω⊕f2t+F−1H23ω⊕f3tF−1H31ω⊕f1t+F−1H32ω⊕f2t+F−1H33ω⊕f3t,where F−1 indicates the inverse Fourier transform, Hij(ω), i, j = 1, 2, 3, and represents the system FRFs that were introduced in Section 2.1; similar to Hs,o(a)i(b), this can be computed using substructure transfer approaches described (shown in Figure 5) in Section 2.3. The primary difference here is that the TTF matrix in the substructure is derived from the FRFs calculated from the substructure theory discussed in Section 2.3.


### 2.3. Theory of a New Time-Domain Substructure IMTPA

The drawbacks of the different methods discussed above are summarized in Table1. There are two points to be considered. One is the dynamic FRFs are considered as they are absent in all the methods; the other one is that time-domain substructure IMTPA is the best way to address transient problems within several inner coupled substructures; however, accurate results cannot be obtained by ignoring dynamic FRFs during computation. Thus, the current time-domain substructure IMTPA can be considered a partial-operation-based model, and a new time-domain substructure IMTPA is necessary, which combines the time-domain substructure IMTPA and dynamic FRFs under operational excitations. The main objective is to obtain time-varying Hij(ω), (i ≠ j) mathematically and theoretically.Table 1
Comparison of TPA methods.
Current method
Load identified
Transient TPA
FRFs based
Operation based
Dynamic FRFs
OTPA
×
×
×
√
×
OPAX
√
×
√
√
×
Substructure IMTPA
√
×
√
×
×
Time-domain substructure IMTPA
√
√
√
√ (partial)
×
× indicates no function or method disadvantage; √ indicates the existence of a feature.In equation (12) given in Section 2.1, aC(b)/FC(b) represents normal constant FRFs that could be obtained under static conditions; aC(a)/aC(b) is the vibration acceleration value ratio tested under operational excitation conditions. Hs,c(a)i(b) in equation (13) is similar to Hs,c(a)c(b). When it extends to the coupled system with n connections, then the structure is shown in Figure 3. A force is applied at point 1 on the side of structure B, shown in Figure 4; the left side contains the red solid arrow. From point 1, point n, all connections in the same interface will transmit force simultaneously owing to the structural coupling of all points. It is not difficult to imagine that if we disconnect all points at Xc(b), the original Fc(b)(1) in the system can be replaced with the accumulation of Fc(b)(1)αb(1)b(1) to Fc(b)(1)αb(n)b(1) in an imaginary system shown on the right side in Figure 4. The matrix [αb(ω)] represents the dynamic forces applied with the distribution coefficient. The matrix [αb(ω)] is at the structure side in a static condition and can be obtained easily through traditional FRFs in static conditions. This coefficient will not change if the active substructure is a stiff and linear system. These two prerequisites are crucial for using this IMTPA method and influence its accuracy. There are several steps for deducing the methodology with complicated multipoint coupling interfaces. The first step is to obtain static FRFs in testing conditions without operational loads:(16)diagFcb1ωFcb2ω⋮Fcbnω−1αb1b1ωαb2b1ω⋯αb1bnωαb1b2ωαb2b2ω⋯αb2bnω⋯⋯⋯⋯αb1bnωαb2bnω⋯αbnbnωHs,cb1cb1ωHs,cb1cb2ω⋯Hs,cb1cbnωHs,cb2cb1ωHs,cb2cb2ω⋯⋯⋯⋯⋯⋯Hs,cbncb1ωHs,cbncb2ω⋯Hs,cbncbnω=diagXcb1ωXcb2ω⋮Xcbnω−1.Figure 3
Two subsystems withn-connection coupled systems.Figure 4
Original model (left side) and imaginary model (right side).The second step is to calculate under operational conditions.Fcb′(ω)αb(ω) should be identified as the imaginary excitation. Based on two sets of vibration data obtained from two sides of the coupled connection points, two formulas were developed, as shown in equations (17) and (18). Then, [Hs,c(a)c(b)(ω)] in equation (19) can be deduced using equations (16)–(18). It is easy to determine that [Hs,c(a)c(b)(ω)] is not related to [αb(ω)], which is supposed to be the virtual force distribution coefficient. For simplicity, equations (20)–(23) represent these matrix calculations. [Hs,c(a)i(b)(ω)] in equation (24) is similar to [Hs,c(a)c(b)(ω)], which is discussed in Appendix A. B is supposed to be a time-invariable and linear substructure that causes [αb(ω)] not to change under operational conditions. This prerequisite prevents the appearance of any nonlinear component in the active structure:(17)diagFcb1′ωFcb2′ω⋮Fcbn′ωTαb1b1ωαb2b1ω⋯αb1bnωαb1b2ωαb2b2ω⋯αb2bnω⋯⋯⋯⋯αb1bnωαb2bnω⋯αbnbnωHs,cb1cb1ωHs,cb1cb2ω⋯Hs,cb1cbnωHs,cb2cb1ωHs,cb2cb2ω⋯⋯⋯⋯⋯⋯Hs,cbncb1ωHs,cbncb2ω⋯Hs,cbncbnω=diagXcb1′ωXcb2′ω⋮Xcbn′ωT,(18)diagFcb1′ωFcb2′ω⋮Fcbn′ωTαb1b1ωαb2b1ω⋯αb1bnωαb1b2ωαb2b2ω⋯αb2bnω⋯⋯⋯⋯αb1bnωαb2bnω⋯αbnbnωHs,ca1cb1ωHs,ca1cb2ω⋯Hs,ca1cbnωHs,ca2cb1ωHs,ca2cb2ω⋯⋯⋯⋯⋯⋯Hs,cancb1ωHs,cancb2ω⋯Hs,cancbnω=diagXca1′ωXca2′ω⋮Xcan′ωT,(19)Hs,ca1cb1ωHs,ca1cb2ω⋯Hs,ca1cbnωHs,ca2cb1ωHs,ca2cb2ω⋯⋯⋯⋯⋯⋯Hs,cancb1ωHs,cancb2ω⋯Hs,cancbnω=Hs,cb1cb1ωHs,cb1cb2ω⋯Hs,cb1cbnωHs,cb2cb1ωHs,cb2cb2ω⋯⋯⋯⋯⋯⋯Hs,cbncb1ωHs,cbncb2ω⋯Hs,cbncbnωdiagXcb1′ωXcb2′ω⋮Xcbn′ωdiagXca1′ωXca2′ω⋮Xcan′ωT,(20)FcbωαbωHs,cbcbω=Xcbω,(21)Fcb′ωαbωHs,cbcbω=Xcb′ω,(22)Fcb′ωαbωHs,cacbω=Xca′ω,(23)Hs,cacbω=Hs,cbcbωXcb′ω−1Xca′ω,(24)Hs,caibω=Hs,cbibωXcb′ω−1Xca′ω.In the time-domain method, the entire period of outputX(t) can be divided into several time-data segments; each time segment starts from ts and ends at te, yielding the following:(25)Xt│ts⟶te=∫tste∫−∞+∞ft∣ts⟶te−τ0ht∣ts⟶te−τdτdτ0,where Xt∣ts⟶te represents the output signals between time points ts and te and ht∣ts⟶te−τ indicates the excitation from ts to te. It changes in different time-date segments; connecting every segment, X(t) can be determined continuously.For a certain time-data segment, substituting equations (20)–(24) into equation (25) and extending to n dimensions yields the following:(26)X1tnX2tnX3tn=F−1H11ωn⊕f1tn+F−1H12ωn⊕f2tn+F−1H13ωn⊕f3tnF−1H21ωn⊕f1tn+F−1H22ωn⊕f2tn+F−1H23ωn⊕f3tnF−1H31ωn⊕f1tn+F−1H32ωn⊕f2tn+F−1H33ωn⊕f3tn,where subscript n represents the nth time-data segment. In other words, before carrying out the analyses, the time-domain signal was divided into equal lengths, repeating the time-step calculation using equation (26) for each time segment. Figure 5 shows a flowchart for the calculation and test procedures.Figure 5
Calculation process chart.


## 2.1. Conventional Substructure Inverse Matrix Method

The substructure inverse matrix method applied systematic FRFs between subsystems to calculate vibration transfer paths and their contribution in the coupled system as shown in Figure2.Figure 2
Coupled system with two subsystems.Generalized stiffnessKc connects substructures A and B. Fc(a) represents the force on side A, and Xc(a) represents the vibration of substructure A. In addition, basic relationships between the system-level FRFs and substructure-level FRFs can be formulated as equations (1)–(6) [1, 42–53]:(1)HA,caca=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,cbcb−Hs,cacb,(2)HB,cbcb=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,caca−Hs,cacb,(3)HA,oaca=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,cacaHs,cbcb−Hs,cacbHs,oaca,(4)HB,cbib=Hs,cacaHs,cbcb−Hs,cacbHs,cacbHs,cacbHs,caca−Hs,cacbHs,caib,(5)KC=Hs,cacbHs,cacaHs,cbcb−Hs,cacbHs,cacb,(6)C=HA,caca+HB,cacb+1Kc−1,where subscripts A and B indicate the names of the substructures, subscript s represents the FRF at the system level, c represents the coupled points, o denotes the output, and i represents the force input. C in equation (6) represents an algebraic operator. The expression Hs,c(a)c(b) represents FRFs from forces Fc(b) to responses Xc(a) in the complete system. Similarly, the expression HA,o(a)c(a) represents FRFs from forces Fc(a) to output responses Xo(a) in substructure A. Clearly, equations (1)–(6) change the FRFs at the system level to those at the substructure level, and Kc is not the real spring stiffness but a generalized one. Equations (7)–(11) were deduced from equations (1)–(6) to obtain Hs,o(a)i(b):(7)HA,oaca=Hs,oaca1−CHA,caca,(8)HB,cbib=Hs,caibHA,cacaC,(9)HA,oaca=Hs,oaca1−CHA,caca,(10)HB,cbib=Hs,caibHA,cacaC.The system response function is given by(11)Hs,oaib=HA,oaCaCHB,cbib,where Hs,o(a)i(b) in equation (11) is the system response that is equal to the calculation results. Hs,c(a)c(b) and Hs,c(a)i(b) are not constant FRFs that can be easily tested in normal static conditions in the system level; they definitely change under the operational load because Kc, in many cases, is made of a nonlinear material, whose dynamic stiffness is not constant under different loads. In other words, Hs,c(a)c(b) and Hs,c(a)i(b) should be obtained using other operational methods given in equations (12) and (13):(12)Hs,cacb=acbFcbacaacb,(13)Hs,caib=aibFibacaaib.


## 2.2. Conventional Time-Domain Substructure Inverse Matrix Method

A time-domain transfer function (TTF) also generally known as the impulse response functionh(t) can be defined as the inverse Fourier transform of a frequency-domain transfer function that is often referred to as an FRF. Accordingly, the time-domain response X(t) is simply the convolution of the TTF and the time-domain excitation function f(t), where δ(t) represents the unit impulse and τ is a real-valued number. The response of a linear time-invariant system due to δ(t) is h(t). Hence, given an excitation δ(t − τ), the response is h(t − τ). Applying the theory of superposition, an excitation input of the form ∫f(t)δ(t − τ)dτ will yield the corresponding response given by ∫f(t)h(t − τ)dτ. Finally, if ∫f(t)δ(t − τ)dτ is equal to f(t), whose corresponding output is X(t) as noted above, then ∫f(t)h(t − τ)dτ is equal to X(t), which is the fundamental basis for the convolution integration applied in this study [31–33]. We considered the example discussed in the introduction as the analysis object to obtain the following equation:(14)x1tx2tx3t=h11th12th13th21th22th23th31th32th33t⊕f1tf2tf3t,where ⊕ is the convolution algorithm, and the TTF matrix with elements hij(t), i, j = 1, 2, 3 can be substituted by the following inverse Fourier transform equation, the FRF equation:(15)x1tx2tx3t=F−1H11ω⊕f1t+F−1H12ω⊕f2t+F−1H13ω⊕f3tF−1H21ω⊕f1t+F−1H22ω⊕f2t+F−1H23ω⊕f3tF−1H31ω⊕f1t+F−1H32ω⊕f2t+F−1H33ω⊕f3t,where F−1 indicates the inverse Fourier transform, Hij(ω), i, j = 1, 2, 3, and represents the system FRFs that were introduced in Section 2.1; similar to Hs,o(a)i(b), this can be computed using substructure transfer approaches described (shown in Figure 5) in Section 2.3. The primary difference here is that the TTF matrix in the substructure is derived from the FRFs calculated from the substructure theory discussed in Section 2.3.


## 2.3. Theory of a New Time-Domain Substructure IMTPA

The drawbacks of the different methods discussed above are summarized in Table1. There are two points to be considered. One is the dynamic FRFs are considered as they are absent in all the methods; the other one is that time-domain substructure IMTPA is the best way to address transient problems within several inner coupled substructures; however, accurate results cannot be obtained by ignoring dynamic FRFs during computation. Thus, the current time-domain substructure IMTPA can be considered a partial-operation-based model, and a new time-domain substructure IMTPA is necessary, which combines the time-domain substructure IMTPA and dynamic FRFs under operational excitations. The main objective is to obtain time-varying Hij(ω), (i ≠ j) mathematically and theoretically.Table 1
Comparison of TPA methods.
Current method
Load identified
Transient TPA
FRFs based
Operation based
Dynamic FRFs
OTPA
×
×
×
√
×
OPAX
√
×
√
√
×
Substructure IMTPA
√
×
√
×
×
Time-domain substructure IMTPA
√
√
√
√ (partial)
×
× indicates no function or method disadvantage; √ indicates the existence of a feature.In equation (12) given in Section 2.1, aC(b)/FC(b) represents normal constant FRFs that could be obtained under static conditions; aC(a)/aC(b) is the vibration acceleration value ratio tested under operational excitation conditions. Hs,c(a)i(b) in equation (13) is similar to Hs,c(a)c(b). When it extends to the coupled system with n connections, then the structure is shown in Figure 3. A force is applied at point 1 on the side of structure B, shown in Figure 4; the left side contains the red solid arrow. From point 1, point n, all connections in the same interface will transmit force simultaneously owing to the structural coupling of all points. It is not difficult to imagine that if we disconnect all points at Xc(b), the original Fc(b)(1) in the system can be replaced with the accumulation of Fc(b)(1)αb(1)b(1) to Fc(b)(1)αb(n)b(1) in an imaginary system shown on the right side in Figure 4. The matrix [αb(ω)] represents the dynamic forces applied with the distribution coefficient. The matrix [αb(ω)] is at the structure side in a static condition and can be obtained easily through traditional FRFs in static conditions. This coefficient will not change if the active substructure is a stiff and linear system. These two prerequisites are crucial for using this IMTPA method and influence its accuracy. There are several steps for deducing the methodology with complicated multipoint coupling interfaces. The first step is to obtain static FRFs in testing conditions without operational loads:(16)diagFcb1ωFcb2ω⋮Fcbnω−1αb1b1ωαb2b1ω⋯αb1bnωαb1b2ωαb2b2ω⋯αb2bnω⋯⋯⋯⋯αb1bnωαb2bnω⋯αbnbnωHs,cb1cb1ωHs,cb1cb2ω⋯Hs,cb1cbnωHs,cb2cb1ωHs,cb2cb2ω⋯⋯⋯⋯⋯⋯Hs,cbncb1ωHs,cbncb2ω⋯Hs,cbncbnω=diagXcb1ωXcb2ω⋮Xcbnω−1.Figure 3
Two subsystems withn-connection coupled systems.Figure 4
Original model (left side) and imaginary model (right side).The second step is to calculate under operational conditions.Fcb′(ω)αb(ω) should be identified as the imaginary excitation. Based on two sets of vibration data obtained from two sides of the coupled connection points, two formulas were developed, as shown in equations (17) and (18). Then, [Hs,c(a)c(b)(ω)] in equation (19) can be deduced using equations (16)–(18). It is easy to determine that [Hs,c(a)c(b)(ω)] is not related to [αb(ω)], which is supposed to be the virtual force distribution coefficient. For simplicity, equations (20)–(23) represent these matrix calculations. [Hs,c(a)i(b)(ω)] in equation (24) is similar to [Hs,c(a)c(b)(ω)], which is discussed in Appendix A. B is supposed to be a time-invariable and linear substructure that causes [αb(ω)] not to change under operational conditions. This prerequisite prevents the appearance of any nonlinear component in the active structure:(17)diagFcb1′ωFcb2′ω⋮Fcbn′ωTαb1b1ωαb2b1ω⋯αb1bnωαb1b2ωαb2b2ω⋯αb2bnω⋯⋯⋯⋯αb1bnωαb2bnω⋯αbnbnωHs,cb1cb1ωHs,cb1cb2ω⋯Hs,cb1cbnωHs,cb2cb1ωHs,cb2cb2ω⋯⋯⋯⋯⋯⋯Hs,cbncb1ωHs,cbncb2ω⋯Hs,cbncbnω=diagXcb1′ωXcb2′ω⋮Xcbn′ωT,(18)diagFcb1′ωFcb2′ω⋮Fcbn′ωTαb1b1ωαb2b1ω⋯αb1bnωαb1b2ωαb2b2ω⋯αb2bnω⋯⋯⋯⋯αb1bnωαb2bnω⋯αbnbnωHs,ca1cb1ωHs,ca1cb2ω⋯Hs,ca1cbnωHs,ca2cb1ωHs,ca2cb2ω⋯⋯⋯⋯⋯⋯Hs,cancb1ωHs,cancb2ω⋯Hs,cancbnω=diagXca1′ωXca2′ω⋮Xcan′ωT,(19)Hs,ca1cb1ωHs,ca1cb2ω⋯Hs,ca1cbnωHs,ca2cb1ωHs,ca2cb2ω⋯⋯⋯⋯⋯⋯Hs,cancb1ωHs,cancb2ω⋯Hs,cancbnω=Hs,cb1cb1ωHs,cb1cb2ω⋯Hs,cb1cbnωHs,cb2cb1ωHs,cb2cb2ω⋯⋯⋯⋯⋯⋯Hs,cbncb1ωHs,cbncb2ω⋯Hs,cbncbnωdiagXcb1′ωXcb2′ω⋮Xcbn′ωdiagXca1′ωXca2′ω⋮Xcan′ωT,(20)FcbωαbωHs,cbcbω=Xcbω,(21)Fcb′ωαbωHs,cbcbω=Xcb′ω,(22)Fcb′ωαbωHs,cacbω=Xca′ω,(23)Hs,cacbω=Hs,cbcbωXcb′ω−1Xca′ω,(24)Hs,caibω=Hs,cbibωXcb′ω−1Xca′ω.In the time-domain method, the entire period of outputX(t) can be divided into several time-data segments; each time segment starts from ts and ends at te, yielding the following:(25)Xt│ts⟶te=∫tste∫−∞+∞ft∣ts⟶te−τ0ht∣ts⟶te−τdτdτ0,where Xt∣ts⟶te represents the output signals between time points ts and te and ht∣ts⟶te−τ indicates the excitation from ts to te. It changes in different time-date segments; connecting every segment, X(t) can be determined continuously.For a certain time-data segment, substituting equations (20)–(24) into equation (25) and extending to n dimensions yields the following:(26)X1tnX2tnX3tn=F−1H11ωn⊕f1tn+F−1H12ωn⊕f2tn+F−1H13ωn⊕f3tnF−1H21ωn⊕f1tn+F−1H22ωn⊕f2tn+F−1H23ωn⊕f3tnF−1H31ωn⊕f1tn+F−1H32ωn⊕f2tn+F−1H33ωn⊕f3tn,where subscript n represents the nth time-data segment. In other words, before carrying out the analyses, the time-domain signal was divided into equal lengths, repeating the time-step calculation using equation (26) for each time segment. Figure 5 shows a flowchart for the calculation and test procedures.Figure 5
Calculation process chart.


## 3. Experimental Validation

In this section, the advantages of the new TPA method, called modified time-domain substructure IMTPA (referred to as M-2 in the remainder of the paper), are described whose methodology is explained in Section2.3. M-2 was derived from the time-domain substructure inverse matrix method (referred to as M-1 in the remainder of the paper); thus, M-2 and M-1 were compared under the same situations.


### 3.1. Experimental Model and Facility

A new experimental testing facility was designed and constructed (shown in Figure6). It includes several parts. Plane A represents substructure A, Plane B represents substructure B, and they are connected by three springs with various stiffnesses. An MTS 320 piston supplied the dynamic force required for the excitation tests [54–56]. Table 2 lists the major physical parameters of this facility. Table 3 lists the measurement conditions.Figure 6
Picture of testing facility (a) and the location of points within this facility (b).Table 2
Facility physical parameters.
Attribute
Values
Mass of plane A
6 kg
Mass of plane B
10 kg
Mass of total facility
25 kg
Size of plane A
600 × 600 mm
Size of plane B
600 × 600 mm
Statick1
15 N/mm
Statick2
25 N/mm
Statick3
45 N/mm
Load simulator
One MTS 320 piston
Acoustic environment
Semianechoic roomTable 3
Measurement details.
No.
Name
1
LMS 64 channels frontend
2
PCB ICP triaccelerators: 9 s
3
FRF force hammer: B&K force hammer
4
Excitation 1: impulsive shocking (vehicle drives over speed bumpers at 20 km/h)
5
Vibration measurement setting: B&K three-direction sensors, sampling 1024 Hz; 0.5 Hz resolution
6
FRF measurement setting: sampling 1024 Hz; 0.5 Hz resolutionThis facility has two substructures connected by three springs. The forces produced by the hydraulic piston were transmitted from pointsib1ω,ib2ω,andib3ω to points Oa1ω,Oa2ω,andOa3ω as shown in Figure 6. This model, the simplest substructure system, was chosen to study the advantages of M-2 because there are several engineering application cases in the automobile industry, like powertrain mounts and chassis suspension systems; in addition, this model better illustrates the effect of the new TPA method under transient impulsive loads, such as those in Figure 7, compared to M-1. Table 4 lists the three input forces, nine outputs, and nine paths from plate B to plate A, and Table 5 presents the FRF matrixes tested under static conditions and FRFs matrixes that need to be calculated.Figure 7
Response points (a)O(a)3(ω), (b) O(a)2(ω), and (c) O(a)1(ω) in the Z direction under same load. Red curve indicates the new time-domain substructure IMTPA M-2, green curve means conventional substructure obtained using inverse matrix method M-1, and blue curves represent the vibrations tested in this facility.
(a)
(b)
(c)Table 4
Input, output, and path points.
Position
Excitation points
Input 1
i
(b)1(ω): Z direction
Input 2
i
(b)2(ω): Z direction
Input 3
i
(b)3(ω): Z direction
Path 1 (B side)
C
(b)1(ω): X, Y, and Z directions
Path 1 (A side)
C
(a)1(ω): X, Y, and Z directions
Path 2 (B side)
C
(b)2(ω): X, Y, and Z directions
Path 2 (A side)
C
(a)2(ω): X, Y, and Z directions
Path 3 (A side)
C
(a)3(ω): X, Y, and Z directions
Path 3 (B side)
C
(b)3(ω): X, Y, and Z directions
Output 1
O
(a)1(ω): X, Y, and Z directions
Output 2
O
(a)2(ω): X, Y, and Z directions
Output 3
O
(a)3(ω): X, Y, and Z directionsTable 5
FRFs matrix in the coupled system.
Measured FRFs
Size
Calculated FRFs
Size
H
s,c(a)c(a)
9 × 9
H
s,c(a)c(b)
9 × 9
H
s,c(b)i(b)
9 × 3
H
s,c(a)i(b)
9 × 3
H
s,o(a)c(a)
9 × 9
H
s,o(a)i(b)
9 × 3In this excitation case, the load during the time domain has two sections: one from 2 s to 5 s representing stable signals and the other one from 6 s to 14 s with two vertical shocks at around 6 s and 11 s, separately, representing impulsive signals. According to the work flow shown in Figure3, all the time-domain outputs O(a)(ω) in Figure 7 can be divided into seven sections averagely and 2 s to 4 s was the second one. In other words, seven dynamic FRFs calculations would be performed along with contribution analyses using Matlab software. Then, combining those seven calculation results together yields the time-domain substructure (Figure 8).Figure 8
Time-domain vibration contribution analyses calculations using M-2 and M-1. Mea: measurements; Cal: calculation;a3: Y refers to the path of point C(a)3(ω) along with Y direction. (a) O(a)1(ω)-Z-contribution via M-2 method. (b) O(a)1(ω)-Z-contribution via M-1 method. (c) O(a)2(ω)-Z-contribution via M-2 method. (d) O(a)2(ω)-Z-contribution via M-1 method. (e) O(a)3(ω)-Z-contribution via M-2 method. (f) O(a)3(ω)-Z-contribution via M-1 method.
(a)
(b)
(c)
(d)
(e)
(f)


### 3.2. Experimental Analyses

The first two lines of the color map shown in every two parallel subfigures in Figure8 show that the response vibration amplitude obtained using M-2 was more accurate and closer to the measured values, compared to that obtained using M-1. To compare different points clearly, the value of the color bars in all subfigures was kept the same and set from a minimum of −10 to a maximum of 40.Furthermore,C(a)1(ω), C(a)2(ω), and C(a)3(ω) in the Z direction were the major paths of these shocking loads causing the color shown in the Z direction to become lighter than those in the other two directions in each subfigure, as shown in Figure 8. This was not difficult to understand because the Z direction was the major direction of the excitation forces generated and also the major direction of force transmission via springs. In addition, nine paths showing the response vibration contribution results in Figure 8 directly showed us several computation comparisons via M-1 and M-2 methods: Figures 8(b), 8(d), and 8(f) show a brighter color in each color-map line than Figures 8(a), 8(c), and 8(e); in other words, M-2 as a new approach was much better and clearer in showing the each path contribution separately, especially during transient loading at 4–6 s and 10–12 s, indicated by the red-dotted rectangle. Additionally, C(a)2(ω) and C(a)3(ω) were major paths for vibration transfer in this case owing to the stiffness of springs k2 and k3 being higher than k1.In the frequency domain, time sections 2–4 s and 10–12 s were selected as examples to investigate the frequency contribution under typical stable and transient loads, separately (seen in Figure9). Compared to the calculations in the 2–14 s section in M-1, the response vibration energy was more noticeable under impulse loads in 10–12 s in M-2; however, the response vibration energy was slightly lower under smooth loads in the 2–4 s section in M-2. This was because M-1 ignored the time-dynamic FRFs during calculation, and it averaged the FRFs during the whole testing time without considering the various FRFs at different times. Nevertheless, the two left subfigures in Figure 9 show considerable energy diversity in terms of color, which are consistent with real working conditions in that there was a big significant shock at 10–12 s and smooth random excitation at 2–4 s. It is necessary to obtain different {Hs,c(a)c(b)(ω)n} values in different time sections; the traditional time-domain TPA and substructure inverse matrix method are likely to ignore the effect of operational load owing to the method of obtaining FRFs under the static or average mode. For example, Figures 10 and 11 show {Hs,c(a)c(b)(ω)2} and {Hs,c(a)c(b)(ω)6}, considering point O(a)3(ω) in the Z direction as an example. The amplitude of {Hs,c(a)c(b)(ω)2} in M-2 was slightly smaller than static FRFs in M-1, and these curves match well in every subfigure of Figure 10; however, {Hs,c(a)c(b)(ω)6} in M-2 showed a higher energy level than that in M-1.Figure 9
Frequency-domain vibration contribution analyses results calculated using M-2 and M-1.a3: Z indicates the path of C(a)3(ω) in Z direction. Total: O(a)3(ω) vibration response calculated in the Z direction. (a) 2s-4s M-2. (b) 10s-12s M-2. (c) 2s-14s M-2.
(a)
(b)
(c)Figure 10
2–4 s time section, pointO(a)3(ω), in the Z direction. Nine paths of {Hs,c(a)c(b)(ω)2} calculated using M-2 compared to nine static paths of {Hs,c(a)c(b)(ω)} calculated using M-1; an represents C(a)n(ω) in the Z direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 11
10–12 s time section, pointO(a)3(ω), in the Z direction. Nine paths of {Hs,c(a)c(b)(ω)6} calculated using M-2 compared to nine static paths of {Hs,c(a)c(b)(ω)} calculated using M-1; an represents the path of C(a)n(ω) in the Z direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)This difference implies that {Hs,c(a)c(b)(ω)} definitely varied in line with the time sections, not the constant matrix used within other conventional TPA methods like M-1. Moreover, the higher the frequency, the higher is the difference between the two curves in every subfigure of Figure 11. This was attributed to the high-frequency dynamic stiffness of these connecting springs, much easier to increase during big compression under shocking excitation than that under small loads.In addition, this new approach overcomes the limitations of the traditional time-domain substructure TPA in terms of its strict requirements within time-invariable systems. This is because in the new method, time-varying FRFs were calculated and used, which could make the FRFs at the system level directly adapt to time-varying systems from time to time.


### 3.3. Summary

This experiment verified that the new time-domain IMTPA (M-2) had advantages in TPA carried out under unstable loads compared to M-1. Moreover, precise analysis is possible in TPA under impulse shocks in terms of the time and frequency domains. Furthermore, the operational FRFs based on substructure inverse matrix theory in M-2 could make all paths’ contribution more operational and precise than static FRFs due to the lack of consideration of operational loads. Thus, the experimental results demonstrated that M-2 is a reliable and valuable method in IMTPA than M-1 and other TPAs.The key feature of this method is optimization of the FRFs, which go over the coupled interfaces under transient loads. Computation of these FRFs was realized, as discussed in Section2.3. This method of optimizing the FRFs was determined using operational excitations and basic FRFs under static conditions. However, two requirements’ prerequisites are essential before using this method:(a)
Any substructure taken out as a single part should be a linear and time-invariable unit that is not likely to be changed by loads(b)
Coupled connections on interfaces on the active side of the substructure should be linear and time-invariable systems or structuresThis time-domain IMTPA method has substantial merits, which were verified using the experimental model described in Section3 as follows:(a)
A method for analyzing a transient transfer path under the operational load was created, which is rare in TPA. This is because OTPA and OPAX require relatively longer loading durations for error control in the matrix calculation.(b)
This time-domain IMTPA is a scalable method, enabling researchers and engineers to spend less time on measuring FRFs, operational excitation, and final matrix calculations. Besides, the FRFs at the system level instead of the part level guarantee actual measuring feasibility without dissembling the complete system.(c)
This new time-domain IMTPA was verified as yielding more accurate results in TPA calculations and path identifications than the current time-domain IMTPA. As opposed to the traditional substructure inverse matrix approach, operational excitation was introduced to improve operational FRFs, which could result in accurate calculations.


## 3.1. Experimental Model and Facility

A new experimental testing facility was designed and constructed (shown in Figure6). It includes several parts. Plane A represents substructure A, Plane B represents substructure B, and they are connected by three springs with various stiffnesses. An MTS 320 piston supplied the dynamic force required for the excitation tests [54–56]. Table 2 lists the major physical parameters of this facility. Table 3 lists the measurement conditions.Figure 6
Picture of testing facility (a) and the location of points within this facility (b).Table 2
Facility physical parameters.
Attribute
Values
Mass of plane A
6 kg
Mass of plane B
10 kg
Mass of total facility
25 kg
Size of plane A
600 × 600 mm
Size of plane B
600 × 600 mm
Statick1
15 N/mm
Statick2
25 N/mm
Statick3
45 N/mm
Load simulator
One MTS 320 piston
Acoustic environment
Semianechoic roomTable 3
Measurement details.
No.
Name
1
LMS 64 channels frontend
2
PCB ICP triaccelerators: 9 s
3
FRF force hammer: B&K force hammer
4
Excitation 1: impulsive shocking (vehicle drives over speed bumpers at 20 km/h)
5
Vibration measurement setting: B&K three-direction sensors, sampling 1024 Hz; 0.5 Hz resolution
6
FRF measurement setting: sampling 1024 Hz; 0.5 Hz resolutionThis facility has two substructures connected by three springs. The forces produced by the hydraulic piston were transmitted from pointsib1ω,ib2ω,andib3ω to points Oa1ω,Oa2ω,andOa3ω as shown in Figure 6. This model, the simplest substructure system, was chosen to study the advantages of M-2 because there are several engineering application cases in the automobile industry, like powertrain mounts and chassis suspension systems; in addition, this model better illustrates the effect of the new TPA method under transient impulsive loads, such as those in Figure 7, compared to M-1. Table 4 lists the three input forces, nine outputs, and nine paths from plate B to plate A, and Table 5 presents the FRF matrixes tested under static conditions and FRFs matrixes that need to be calculated.Figure 7
Response points (a)O(a)3(ω), (b) O(a)2(ω), and (c) O(a)1(ω) in the Z direction under same load. Red curve indicates the new time-domain substructure IMTPA M-2, green curve means conventional substructure obtained using inverse matrix method M-1, and blue curves represent the vibrations tested in this facility.
(a)
(b)
(c)Table 4
Input, output, and path points.
Position
Excitation points
Input 1
i
(b)1(ω): Z direction
Input 2
i
(b)2(ω): Z direction
Input 3
i
(b)3(ω): Z direction
Path 1 (B side)
C
(b)1(ω): X, Y, and Z directions
Path 1 (A side)
C
(a)1(ω): X, Y, and Z directions
Path 2 (B side)
C
(b)2(ω): X, Y, and Z directions
Path 2 (A side)
C
(a)2(ω): X, Y, and Z directions
Path 3 (A side)
C
(a)3(ω): X, Y, and Z directions
Path 3 (B side)
C
(b)3(ω): X, Y, and Z directions
Output 1
O
(a)1(ω): X, Y, and Z directions
Output 2
O
(a)2(ω): X, Y, and Z directions
Output 3
O
(a)3(ω): X, Y, and Z directionsTable 5
FRFs matrix in the coupled system.
Measured FRFs
Size
Calculated FRFs
Size
H
s,c(a)c(a)
9 × 9
H
s,c(a)c(b)
9 × 9
H
s,c(b)i(b)
9 × 3
H
s,c(a)i(b)
9 × 3
H
s,o(a)c(a)
9 × 9
H
s,o(a)i(b)
9 × 3In this excitation case, the load during the time domain has two sections: one from 2 s to 5 s representing stable signals and the other one from 6 s to 14 s with two vertical shocks at around 6 s and 11 s, separately, representing impulsive signals. According to the work flow shown in Figure3, all the time-domain outputs O(a)(ω) in Figure 7 can be divided into seven sections averagely and 2 s to 4 s was the second one. In other words, seven dynamic FRFs calculations would be performed along with contribution analyses using Matlab software. Then, combining those seven calculation results together yields the time-domain substructure (Figure 8).Figure 8
Time-domain vibration contribution analyses calculations using M-2 and M-1. Mea: measurements; Cal: calculation;a3: Y refers to the path of point C(a)3(ω) along with Y direction. (a) O(a)1(ω)-Z-contribution via M-2 method. (b) O(a)1(ω)-Z-contribution via M-1 method. (c) O(a)2(ω)-Z-contribution via M-2 method. (d) O(a)2(ω)-Z-contribution via M-1 method. (e) O(a)3(ω)-Z-contribution via M-2 method. (f) O(a)3(ω)-Z-contribution via M-1 method.
(a)
(b)
(c)
(d)
(e)
(f)


## 3.2. Experimental Analyses

The first two lines of the color map shown in every two parallel subfigures in Figure8 show that the response vibration amplitude obtained using M-2 was more accurate and closer to the measured values, compared to that obtained using M-1. To compare different points clearly, the value of the color bars in all subfigures was kept the same and set from a minimum of −10 to a maximum of 40.Furthermore,C(a)1(ω), C(a)2(ω), and C(a)3(ω) in the Z direction were the major paths of these shocking loads causing the color shown in the Z direction to become lighter than those in the other two directions in each subfigure, as shown in Figure 8. This was not difficult to understand because the Z direction was the major direction of the excitation forces generated and also the major direction of force transmission via springs. In addition, nine paths showing the response vibration contribution results in Figure 8 directly showed us several computation comparisons via M-1 and M-2 methods: Figures 8(b), 8(d), and 8(f) show a brighter color in each color-map line than Figures 8(a), 8(c), and 8(e); in other words, M-2 as a new approach was much better and clearer in showing the each path contribution separately, especially during transient loading at 4–6 s and 10–12 s, indicated by the red-dotted rectangle. Additionally, C(a)2(ω) and C(a)3(ω) were major paths for vibration transfer in this case owing to the stiffness of springs k2 and k3 being higher than k1.In the frequency domain, time sections 2–4 s and 10–12 s were selected as examples to investigate the frequency contribution under typical stable and transient loads, separately (seen in Figure9). Compared to the calculations in the 2–14 s section in M-1, the response vibration energy was more noticeable under impulse loads in 10–12 s in M-2; however, the response vibration energy was slightly lower under smooth loads in the 2–4 s section in M-2. This was because M-1 ignored the time-dynamic FRFs during calculation, and it averaged the FRFs during the whole testing time without considering the various FRFs at different times. Nevertheless, the two left subfigures in Figure 9 show considerable energy diversity in terms of color, which are consistent with real working conditions in that there was a big significant shock at 10–12 s and smooth random excitation at 2–4 s. It is necessary to obtain different {Hs,c(a)c(b)(ω)n} values in different time sections; the traditional time-domain TPA and substructure inverse matrix method are likely to ignore the effect of operational load owing to the method of obtaining FRFs under the static or average mode. For example, Figures 10 and 11 show {Hs,c(a)c(b)(ω)2} and {Hs,c(a)c(b)(ω)6}, considering point O(a)3(ω) in the Z direction as an example. The amplitude of {Hs,c(a)c(b)(ω)2} in M-2 was slightly smaller than static FRFs in M-1, and these curves match well in every subfigure of Figure 10; however, {Hs,c(a)c(b)(ω)6} in M-2 showed a higher energy level than that in M-1.Figure 9
Frequency-domain vibration contribution analyses results calculated using M-2 and M-1.a3: Z indicates the path of C(a)3(ω) in Z direction. Total: O(a)3(ω) vibration response calculated in the Z direction. (a) 2s-4s M-2. (b) 10s-12s M-2. (c) 2s-14s M-2.
(a)
(b)
(c)Figure 10
2–4 s time section, pointO(a)3(ω), in the Z direction. Nine paths of {Hs,c(a)c(b)(ω)2} calculated using M-2 compared to nine static paths of {Hs,c(a)c(b)(ω)} calculated using M-1; an represents C(a)n(ω) in the Z direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)Figure 11
10–12 s time section, pointO(a)3(ω), in the Z direction. Nine paths of {Hs,c(a)c(b)(ω)6} calculated using M-2 compared to nine static paths of {Hs,c(a)c(b)(ω)} calculated using M-1; an represents the path of C(a)n(ω) in the Z direction.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)This difference implies that {Hs,c(a)c(b)(ω)} definitely varied in line with the time sections, not the constant matrix used within other conventional TPA methods like M-1. Moreover, the higher the frequency, the higher is the difference between the two curves in every subfigure of Figure 11. This was attributed to the high-frequency dynamic stiffness of these connecting springs, much easier to increase during big compression under shocking excitation than that under small loads.In addition, this new approach overcomes the limitations of the traditional time-domain substructure TPA in terms of its strict requirements within time-invariable systems. This is because in the new method, time-varying FRFs were calculated and used, which could make the FRFs at the system level directly adapt to time-varying systems from time to time.


## 3.3. Summary

This experiment verified that the new time-domain IMTPA (M-2) had advantages in TPA carried out under unstable loads compared to M-1. Moreover, precise analysis is possible in TPA under impulse shocks in terms of the time and frequency domains. Furthermore, the operational FRFs based on substructure inverse matrix theory in M-2 could make all paths’ contribution more operational and precise than static FRFs due to the lack of consideration of operational loads. Thus, the experimental results demonstrated that M-2 is a reliable and valuable method in IMTPA than M-1 and other TPAs.The key feature of this method is optimization of the FRFs, which go over the coupled interfaces under transient loads. Computation of these FRFs was realized, as discussed in Section2.3. This method of optimizing the FRFs was determined using operational excitations and basic FRFs under static conditions. However, two requirements’ prerequisites are essential before using this method:(a)
Any substructure taken out as a single part should be a linear and time-invariable unit that is not likely to be changed by loads(b)
Coupled connections on interfaces on the active side of the substructure should be linear and time-invariable systems or structuresThis time-domain IMTPA method has substantial merits, which were verified using the experimental model described in Section3 as follows:(a)
A method for analyzing a transient transfer path under the operational load was created, which is rare in TPA. This is because OTPA and OPAX require relatively longer loading durations for error control in the matrix calculation.(b)
This time-domain IMTPA is a scalable method, enabling researchers and engineers to spend less time on measuring FRFs, operational excitation, and final matrix calculations. Besides, the FRFs at the system level instead of the part level guarantee actual measuring feasibility without dissembling the complete system.(c)
This new time-domain IMTPA was verified as yielding more accurate results in TPA calculations and path identifications than the current time-domain IMTPA. As opposed to the traditional substructure inverse matrix approach, operational excitation was introduced to improve operational FRFs, which could result in accurate calculations.


## 4. Conclusions

This paper introduced the development history of TPA and then briefly summarized two different TPA theories to identify their advantages and disadvantages. A new time-domain IMTPA was theoretically developed in this study by combining the substructure inverse matrix method based on FRFs with operational excitation and the time-domain TPA method. This new method can be extremely beneficial under automobile NVH impulse conditions, such as engine ignition vibration problems and chassis shocking noise when driving over speed bumps. This is because powertrain or suspension systems are connected via mounts, which show different stiffnesses under different load conditions.



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*Source: 2030326-2019-03-28.xml*