An Augmented State Formulation for Modeling and Analysis of Multibody Distributed Dynamic Systems

Noh, K.; Yang, B.

doi:10.1115/1.4026124

Abstract

Multibody distributed dynamic systems are seen in many engineering applications. Developed in this investigation is a new analytical method for a class of branched multibody distributed systems, which is called the augmented distributed transfer function (DTFM). This method adopts an augmented state formulation to describe the interactions among multiple distributed and lumped bodies, which resolves the problems with conventional transfer function methods in modeling and analysis of multibody distributed systems. As can be seen, the augmented DTFM, without the need for orthogonal system eigenfunctions, produces exact and closed-form solutions of various dynamic problems, in both frequency and time domains.

Introduction

Multibody distributed dynamic systems are seen in many engineering applications, such as buildings, bridges, railways, automobile suspension systems, light-weight deployable space structures, complex rotor systems, piping systems, and multilayer heat-conduction systems. The dynamics of multibody distributed systems thus have been extensively studied, and are of continued research interest today. For better understanding of the physical behaviors of these systems and for their optimal design and operation, both analytical and numerical tools for modeling and analysis have been in constant development. Among the available methods are the transfer matrix method [1], Lagrange multiplier method [2,3], wave propagation method [4,5], modal analysis [6], the spectral element method [7,8], the Rayleigh–Ritz method [9], the distributed transfer function method [10,11], and of course the finite element method (FEM).

Although analytical and semianalytical methods can produce accurate results and provide deep physical insight into the problem under investigation, most of them are limited to single-body distributed systems with simple configurations. For branched multibody distributed systems with complicate configurations, numerical tools such as the FEM often have to be used. The distributed transfer function method (DTFM), which combines the closed form of distributed transfer functions and the concept of finite element analysis, is capable of modeling certain types of multibody distributed systems [12,13]. This method, however, has been limited to frequency–domain solutions, mainly because of the singularities of the global dynamic equilibrium equation that is assembled from the transfer functions of distributed subsystems. Although the DTFM-based analysis has been recently extended to obtain exact transient solutions for stepped distributed systems [14,15], it is not directly applicable to general branched multibody systems.

The objective of the current investigation is to develop a new analytical method for branched multibody distributed dynamic systems that can systematically deliver closed-form solutions in both frequency and time domains. The branched multibody system in consideration is composed of one-dimensional distributed parameter subsystems, lumped parameter subsystems, and feedback controllers. These subsystems are interconnected at certain points that are called nodes. Each distributed subsystem is bounded by two nodes. Shown in Fig. 1 is an example of multibody systems (a flexible structure in vibration) from Ref. [12], which has 19 nodes, 14 distributed subsystems, two lumped subsystems (one oscillator at node 3, and one rigid-body mounted at nodes 10 and 11), and one feedback controller with a sensor at node 6 and an actuator at node 5. Nodes 1, 13, 14, 15, and 19 form the boundary of the structure. Additionally, a spring and a damper are placed at node 2, a distributed constraint (say elastic foundation) is imposed on the subsystem with nodes 7 and 8, and a viscoelastic bounding is reinforced between the subsystem with nodes 10 and 11 and the subsystem with nodes 16 and 17.

Fig. 1

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Schematic of a branched multibody distributed system [12]

Mathematically, the dynamic responses of branched multibody systems like the one shown in Fig. 1 are governed by a mixture of partial differential equations (for distributed subsystems) and ordinary differential equations (for lumped subsystems), which are coupled through multibody interactions (“force balance” and “displacement compatibility”) at interconnecting nodes. Because of the branched configuration, in which more than two distributed subsystems are interconnected at one node, single-body solutions by conventional analytical methods (such as eigenfunction expansion, the Rayleigh–Ritz method and transfer matrix method) cannot be applied herein. Moreover, a feedback controller with noncollocated sensors and actuators, which are also described by ordinary differential equations, can arbitrarily link several distributed subsystems at multiple nodes, rendering multibody interactions more complicated. In addition, the partial differential equations for distributed subsystems in general are nonself-joint due to damping, gyroscopic and circulatory forces, and other effects, yielding nonorthogonal eigenfunctions. These issues make the determination of analytical solutions for a branched multibody distributed system extremely difficult, and as a result, numerical methods almost always have to be used. In the literature, no analytical method is available for systematic determination of closed-form solutions of branched multibody distributed systems in both frequency and time domains.

The new modeling and analysis method proposed in this work is called the augmented DTFM. The novelty of this method lies in its utility of an augmented state formulation for description of multibody interactions, by which the global distributed transfer functions of the multibody distributed system are obtained. This approach does not need to assemble a global dynamic equilibrium equation from subsystem transfer functions, and as such totally eliminates the singularities with the previous DTFM [12]. Another advantage of the proposed method is that the augmented state formulation is flexible in describing the combination of distributed subsystems, lumped subsystems, and feedback controllers, which, as discussed previously, involves a set of coupled partial and ordinary differential equations. With the augmented DTFM, eigensolutions, frequency response, and transient time response of a wide class of branched multibody distributed systems can be systematically obtained in exact and closed form.

The remainder of the paper is organized as follows. A brief review of the previous DTFM in Sec. 2 reveals two major issues that have limited the utility of the method in dynamic analysis of multibody distributed systems. Presented in Sec. 3 is an augmented s-domain state formulation, which allows systematic description of the interactions of distributed and lumped bodies. Also presented in the section is a comparison study that explains why the issues with the previous DTFM can be resolved by the proposed method. Outlined in Sec. 4 are the solution formulas for dynamic problems of multibody distributed systems. The treatment of certain specific connections of distributed and lumped parameter bodies is given in Sec. 5. The proposed method is illustrated in Sec. 6, where the eigensolutions of two beam structures are computed.

Issues With the Previous DTFM

To understand the significance of this work, two issues with the previous distributed transfer function method (DTFM) are discussed in this section. To this end, some results in Refs. [11] and [12] are briefly reviewed.

Let the dynamic response w(x, t) of a uniformly distributed subsystem be governed by the nth order, linear partial differential equation

\begin{matrix} \sum_{k = 0}^{n} (a_{k} \frac{\partial^{2}}{\partial t^{2}} + b_{k} \frac{\partial}{\partial t} + c_{k}) \frac{\partial^{k}}{\partial x^{k}} w (x, t) = f (x, t), for 0 < x < 1 and t > 0 \end{matrix}

(1)

where a_k, b_k, c_k are constants representing the physical properties of the subsystem, such as inertia, damping, stiffness, gyroscopic effects, and circulatory forces; f(x, t) is an external disturbance; t is a temporal parameter; and x is a nondimensional, local spatial coordinate with x = 0 and 1 designating the two ends (nodes) of the subsystem. In a heat conduction problem, all coefficients a_k that are related to the temporal operator ∂²/∂t² are zeros. If the dynamic response of the subsystem involves multiple variables, such as the displacement components of a three-dimensional beam in vibration, Eq. (1) becomes a set of partial differential equations of similar format, with a_k, b_k, c_k being matrices and w(x, t), f(x, t) being vectors.

Taking Laplace transform of Eq. with respect to t and casting the resulting equation into a spatial state form, yields [11]

\frac{\partial}{\partial x} η (x, s) = F (s) η (x, s) + q (s), 0 < x < 1

(2)

where s is the complex Laplace transform parameter, η(x, s) is the state vector of the form

η (x, s) = {\begin{matrix} \bar{w} (x, s) & \frac{\partial \bar{w} (x, s)}{\partial x} & \dots & \frac{\partial^{n - 1} \bar{w} (x, s)}{\partial x^{n - 1}} \end{matrix}}^{T}

(3)

with

\bar{w} (x, s)

being the Laplace transform of w(x, t), F(s) is an n-by-n matrix consisting of coefficients a_k, b_k, c_k, and vector q(s) contains the Laplace transform of f(x, t) and the initial disturbances of the subsystem. The boundary and matching conditions of the subsystem at its two nodes can be expressed as

M_{b} (s) η (0, s) + N_{b} (s) η (1, s) = γ_{b} (s)

(4)

where M_b(s), N_b(s) are n-by-n matrices, and γ_b(s) is a vector consisting of the unknown nodal displacements of the subsystem and/or boundary disturbances.

The s-domain response of the distributed subsystem is obtained by solving the state equation with the boundary condition and it takes the form [11]

η (x, s) = \int_{0}^{1} G (x, ξ, s) q (ξ) d ξ + H (x, s) γ_{b} (s), 0 < x < 1

(5)

Here matrices G(x, ξ, s) and H(x, s) are the distributed transfer functions of the subsystem, and they are in the exact and closed form

\begin{matrix} G (x, ξ, s) = {\begin{matrix} H (x, s) M_{b} (s) e^{- F (s) ξ}, ξ \leq x \\ - H (x, s) N_{b} (s) e^{F (s) (1 - ξ)}, ξ > x \end{matrix} \\ H (x, s) = e^{F (s) x} {[M_{b} (s) + N_{b} (s) e^{F (s)}]}^{- 1} \end{matrix}

(6)

With the transfer functions of the distributed subsystems given in Eq. , imposing “force balance” and “displacement compatibility” at the nodes yields a global dynamic equilibrium equation of the multibody system in s domain [12]

K (s) U (s) = Q (s)

(7)

where K(s) is the global dynamic stiffness matrix that is assembled from the elements of the subsystem transfer functions, and U(s) and Q(s) are the global nodal displacement and nodal force vectors, respectively. In the formation of Eq. (7), which is called the distributed transfer function synthesis [12,13], no discretization or series truncation has been made.

Equation (7) can be applied to obtain solutions for various static and dynamic problems of multibody distributed systems. For instance, the eigenequation of a multibody distributed system is K(s) U(s) = 0, by which the system eigenvalues are the roots of the transcendental equation $det K (λ_{k}) = 0$ ⁠, $k = 1, 2, 3, . . .$ ⁠. The frequency response of the system subject to an excitation of frequency ω can be estimated by $u (t) = K^{- 1} (J ω) q_{0} e^{J ω t}$ ⁠, where $J = \sqrt{- 1}$ ⁠, u(t) is the vector of nodal displacements, and q₀ is a vector of force amplitudes. The solution of a static problem is found to be $U (0) = K^{- 1} (0) Q (0)$ ⁠, where s = 0 indicating that the temporal operators have been removed. In the above static and dynamic problems, substitution of the nodal displacements into Eq. (5) eventually gives the response of each distributed subsystem. Furthermore, with the transfer function formulation provided by Eqs. (5) and (7), stability analysis and design of distributed dynamic systems with feedback controllers can be performed [16].

Although the DTFM has found success in modeling, analysis, and control of distributed dynamic systems, it has two mathematical issues in dealing with multibody systems. The first issue is that the DTFM is inconvenient in transient analysis of a multibody system. According to Eq. (7), to estimate the system transient response, one needs to determine the nodal displacements via inverse Laplace transform of $U (s) = K^{- 1} (s) Q (s)$ ⁠. Because K(s) is assembled from the elements of G(x, ξ, s) and H(x, s) of the subsystems, the analytical expressions of inversed matrix $K^{- 1} (s)$ in general are not available, which renders the inverse Laplace transform of $K^{- 1} (s) Q (s)$ extremely difficult. Without the knowledge of the nodal displacements in time domain, determination of the transient response of the distributed subsystems by Eq. (5) is impossible. One may think about numerical evaluation of inverse Laplace transform, which, however, is impractical due to the second issue with the DTFM as follows.

The second issue with the DTFM is the singularities of G(x, ξ, s) and H(x, s), which are characterized by the poles of the transfer functions or the roots of the transcendental equation $det [M_{b} (s) + N_{b} (s) e^{F (s)}] = 0$ ⁠; see Eq. (6). These roots are the eigenvalues of a “fully confined” subsystem, which in general are not the eigenvalues of the multibody system. The singularities of the transfer functions of distributed subsystems can cause the following three problems: it makes root finding for the characteristic equation $det K (λ_{k}) = 0$ difficult and inefficient, and in some cases even misses roots; it may yield unbounded frequency response at a nonresonant excitation frequency, which is physically incorrect; and it renders numerical inverse Laplace transform of $K^{- 1} (s) Q (s)$ erroneous and unreliable.

The above-mentioned issues have limited the utility of the DTFM for branched multibody distributed systems, especially in transient response analysis. It is upon this understanding that the current investigation introduces an augmented state formulation that can resolve these issues, as shown in the subsequent sections.

Augmented State Formulation

Assume that the multibody system in consideration has N uniformly distributed subsystems. The s-domain response of the subsystems is governed by the state equation

\frac{\partial}{\partial x} {\hat{η}}_{i} (x, s) = F_{i} (s) {\hat{η}}_{i} (x, s) + {\overset{\land}{p}}_{i} (x, s), 0 < x < 1, i = 1, 2, \dots, N

(8)

where

{\hat{η}}_{i}, F_{i} (s), {\overset{\land}{p}}_{i} (x, s)

are similarly defined as

η (x, s), F (s), q (s)

in Eq. . In the proposed method, the state equations for the N subsystems are cast into an augmented form

\frac{\partial}{\partial x} \hat{η} (x, s) = F (s) \hat{η} (x, s) + \overset{\land}{p} (x, s), 0 < x < 1

(9)

where

\begin{matrix} \hat{η} (x, s) & = (\begin{matrix} {\hat{η}}_{1} (x, s) \\ {\hat{η}}_{2} (x, s) \\ : \\ {\hat{η}}_{N} (x, s) \end{matrix}), \\ \overset{\land}{p} (x, s) & = (\begin{matrix} {\overset{\land}{p}}_{1} (x, s) \\ {\overset{\land}{p}}_{2} (x, s) \\ : \\ {\overset{\land}{p}}_{N} (x, s) \end{matrix}), F (s) = \underset{0 \leq k \leq N}{diag} {F_{k} (s)} \end{matrix}

(10)

The dimension of the state vector $\hat{η} (x, s)$ is given by $n_{1} + n_{2} + \dots + n_{N}$ ⁠, where n_i is the dimension of ${\hat{η}}_{i} (x, s)$ for the ith distributed subsystem.

Now consider the boundary and matching conditions of the multibody system at its nodes. Denote a node of the ith subsystem by

ξ_{i}^{*}

⁠, the value of which can be either 0 or 1, depending on the chosen coordinate system. If

ξ_{i}^{*}

is a boundary node of the multibody system, like node 19 in Fig. 1, the boundary conditions at the node are of the form

Γ_{i} (s) {\hat{η}}_{i} (ξ_{i}^{*}, s) = {\hat{γ}}_{i} (s)

(11)

where

Γ_{i} (s)

is a matrix of proper dimensions, and

{\hat{γ}}_{i} (s)

is a vector of boundary disturbances. If

ξ_{i}^{*}

is a point where subsystems

i, j, \dots, k

are interconnected, like node 12 in Fig. 1, the matching conditions at the node can be written as

C_{ii} (s) {\hat{η}}_{i} (ξ_{i}^{*}, s) + C_{ij} (s) {\hat{η}}_{j} (ξ_{j}^{*}, s) + \dots + C_{ik} (s) {\hat{η}}_{k} (ξ_{k}^{*}, s) = {\hat{γ}}_{i} (s)

(12)

where

ξ_{i}^{*}, ξ_{j}^{*}, ξ_{k}^{*}

may take different values (either 0 or 1) but they all refer to the same node; matrices

C_{ii} (s), C_{ij} (s), \dots, C_{ik} (s)

describe the force balance and displacement compatibility at the node; and

{\hat{γ}}_{i} (s)

is a vector of external disturbances at the node. When a feedback controller with an actuator at node

ξ_{i}^{*}

of the ith subsystem and a sensor at node

ξ_{l}^{*}

of the lth subsystem is implemented, like nodes 5 and 6 in Fig. 1, the actuation action at the

ξ_{i}^{*}

is described through modification of Eq. by

C_{ii} (s) {\hat{η}}_{i} (ξ_{i}^{*}, s) + C_{ij} (s) {\hat{η}}_{j} (ξ_{j}^{*}, s) + \dots + C_{ik} (s) {\hat{η}}_{k} (ξ_{k}^{*}, s) + G_{il} (s) {\hat{η}}_{l} (ξ_{l}^{*}, s) = {\hat{γ}}_{i} (s)

(13)

where the added matrix

G_{il} (s)

contains the transfer functions of the sensor, actuator, and controller. Here

ξ_{i}^{*}

and

ξ_{l}^{*}

may refer to two different nodes (for a noncollocated control system) or the same node (for a collocated control system). A feedback control system with multiple sensors and actuators can be treated similarly. Assembly of the boundary and matching conditions for all the subsystems yields the boundary conditions for the state equation

M_{G} (s) \hat{η} (0, s) + N_{G} (s) \hat{η} (1, s) = {\hat{γ}}_{G} (s)

(14)

where matrices $M_{G} (s), N_{G} (s)$ are composed of those matrices in Eqs. (11) to (13), and vector ${\hat{γ}}_{G} (s)$ consists of the disturbances at the related nodes. Examples on boundary and matching conditions via this augmented form are given in Sec. 6.

It is easy to see that the augmented state equation and boundary condition (14) for a multibody distributed system have the same form as Eqs. and . Consequently, the s-domain response of the multibody distributed system is given by

\overset{\land}{η} (x, s) = \int_{0}^{1} \overset{\land}{G} (x, ξ, s) \overset{\land}{p} (ξ, s) d ξ + \overset{\land}{H} (s) {\hat{γ}}_{G} (s), 0 < x < 1

(15)

where

\overset{\land}{G}

and

\overset{\land}{H}

⁠, the global distributed transfer functions of the multibody system, are given by

\begin{matrix} \overset{\land}{G} (x, ξ, s) = {\begin{matrix} \overset{\land}{H} (x, s) M_{G} (s) e^{- F (s) ξ}, ξ \leq x \\ - \overset{\land}{H} (x, s) N_{G} (s) e^{F (s) (1 - ξ)}, ξ > x \end{matrix} \\ \overset{\land}{H} (x, s) = e^{F (s) x} {[M_{G} (s) + N_{G} (s) e^{F (s)}]}^{- 1} \end{matrix}

(16)

The state equation (9) subject to the boundary condition (14) and the solution (15) are the core of the augmented DTFM, which lays out a new foundation for modeling and analysis of branched multibody distributed dynamic systems. The augmented DTFM does not make any approximation or discretization. Neither does it rely on the orthogonality of system eigenfunctions. A comparison of the augmented DTFM with the previous DTFM [12] is given in Table 1. Different from the previous DTFM, the augmented DTFM does not need to form a global dynamic equilibrium equation like Eq. (7). Because of this, the singularities with the previous DTFM are completely avoided and inverse Laplace transform of Eq. (15) can be systematically and accurately performed without any difficulty [14].

Table 1

Comparison of the previous DTFM and the augmented DTFM

	Previous DTFM [12]	Proposed method (augmented DTFM)
Global (system-level) equation	Dynamic equilibrium equation $K (s) U (s) = Q (s)$	Augmented state equation (9) with boundary condition (14)
Singularities in global equation	Yes, due to the singularities of K(s) that come from subsystem transfer functions	No
Coordinates used to describe multibody interactions	Global coordinates	Local coordinate (x) of distributed subsystems
Transfer functions of distributed subsystems	Yes, given in Eq. (6)	No
Transfer functions for the entire multibody system	No	Yes, given in Eq. (15)
Characteristic equation for system eigenvalues	$det K (s) = 0$ with singularities	$det [M_{G} (s) + N_{G} (s) e^{2 F (s)}] = 0$ continuous and smooth, without singularities
Root search for eigenvalues	Less efficient and possible to miss roots	Highly efficient and accurate
Transient analysis via inverse Laplace transform	Difficult and erroneous due to the singularities in K(s)	Yes, it can be performed systematically and accurately

	Previous DTFM [12]	Proposed method (augmented DTFM)
Global (system-level) equation	Dynamic equilibrium equation $K (s) U (s) = Q (s)$	Augmented state equation (9) with boundary condition (14)
Singularities in global equation	Yes, due to the singularities of K(s) that come from subsystem transfer functions	No
Coordinates used to describe multibody interactions	Global coordinates	Local coordinate (x) of distributed subsystems
Transfer functions of distributed subsystems	Yes, given in Eq. (6)	No
Transfer functions for the entire multibody system	No	Yes, given in Eq. (15)
Characteristic equation for system eigenvalues	$det K (s) = 0$ with singularities	$det [M_{G} (s) + N_{G} (s) e^{2 F (s)}] = 0$ continuous and smooth, without singularities
Root search for eigenvalues	Less efficient and possible to miss roots	Highly efficient and accurate
Transient analysis via inverse Laplace transform	Difficult and erroneous due to the singularities in K(s)	Yes, it can be performed systematically and accurately

Solution of Dynamic Problems Via the Augmented DTFM

The augmented DTFM presented in Sec. 3 is applicable to various problems of branched multibody distributed systems. As pointed out previously, with similar formats, the results in DTFM analysis of single-body systems can be conveniently borrowed by the augmented DTFM for multibody systems.

Eigenvalue Problem.

The eigenvalue problem of a multibody distributed system by the augmented state formulation is described by the homogeneous equation [14]

\frac{\partial}{\partial x} ψ (x) = F (s) ψ (x), 0 < x < 1

(17)

subject to the boundary condition

M_{G} (s) ψ (0) + N_{G} (s) ψ (1) = 0

(18)

where s is an eigenvalue and ψ(x) is the associate eigenfunction. The system eigenvalues are the roots of the transcendental characteristic equation

det [M_{G} (s) + N_{G} (s) e^{F (s)}] = 0

(19)

The associate eigenfunctions have the form

ψ (x) = e^{F (s) x} a

(20)

where a is a nonzero constant vector that is a solution of the homogeneous equation

[M_{G} (s) + N_{G} (s) e^{F (s)}] a = 0

(21)

Unlike $det K (λ_{k}) = 0$ in the previous DTFM, the characteristic function in Eq. (19) is smooth and well-behaved, and does not contain any singularity at all. This shall also be demonstrated in a numerical example in Sec. 6.

Frequency Response.

For a multibody distributed system subject to a harmonic excitation

p_{0} (x) e^{J ω t}

⁠, where

J = \sqrt{- 1}

⁠, ω is the excitation frequency, and vector

p_{0} (x)

gives a spatial distribution of the excitation, its steady-state response by Eq. is

η_{ss} (x, t) = \int_{0}^{1} \overset{\land}{G} (x, ξ, J ω) p_{0} (ξ) d ξ e^{J ω t}

(22)

If a boundary excitation

γ_{0} e^{J ω t}

is applied, the steady-state response, again by Eq. , can be written as

η_{ss} (x, t) = \overset{\land}{H} (x, J ω) γ_{0} e^{J ω t}

(23)

The $\overset{\land}{G} (x, ξ, J ω)$ and $\overset{\land}{H} (x, J ω)$ are the frequency response functions of the multibody system.

Transient Response.

The transient response of a branched multibody distributed dynamic system can be obtained via inverse Laplace transform of Eq. . This yields the following convolution integral:

\begin{matrix} η (x, t) = \int_{0}^{t} {\int_{0}^{1} G (x, ξ, t - τ) p (ξ, τ) d ξ + H (x, t - τ) γ_{G} (τ)} d τ, 0 < x < 1 \end{matrix}

(24)

where the time-domain quantities

η (x, t), p (ξ, t), γ_{G} (t)

are the inverse Laplace transforms of

\overset{\land}{η} (x, s), \overset{\land}{p} (ξ, s), {\hat{γ}}_{G} (s)

⁠, respectively;

G (x, ξ, t), H (x, t)

are the Green's functions of the multibody distributed system, and they are the inverse Laplace transforms of

\overset{\land}{G} (x, ξ, s), \overset{\land}{H} (x, s)

⁠. Following Ref. [14], the Green's functions can be expressed as

\begin{matrix} G (x, ξ, t) = \sum_{k = 1}^{\infty} e^{F (s_{k}) x} R_{k} D (x, ξ, s_{k}) e^{s_{k} t} H (x, t) = \sum_{k = 1}^{\infty} e^{F (s_{k}) x} R_{k} e^{s_{k} t} \end{matrix}

(25)

where s_k is the kth eigenvalue of the multibody system that is a root of the characteristic equation ; R_k is the residue of

{[M_{G} (s) + N_{G} (s) e^{F (s)}]}^{- 1}

at s_k; and

D (x, ξ, s) = {\begin{matrix} M_{b} e^{- F (s_{k}) ξ}, & ξ \leq x \\ - N_{b} e^{F (s_{k}) (1 - ξ)}, & ξ > x \end{matrix}

(26)

According to Ref. [14], an analytical formula for determination of the transfer function residues R_k is obtainable. Limited by space, the derivation of this formula is neglected. Note that the transient solution given herein avoids dealing with the inverse Laplace transform of $K^{- 1} (s) Q (s)$ as in the previous DTFM (see Sec. 2), which is difficult to compute, and erroneous due to the singularities of K(s).

Treatment of Certain Types of Connection of Distributed Subsystems

Section 3 gives the general guidelines of the augmented DTFM. This section presents the treatment of three types of connection of distributed subsystems, which requires proper modification of the state equation (9) or the boundary condition (14).

Serially Connected Distributed Systems.

Consider an N-body serially connected system in Fig. 2, where $X_{0}, X_{N}$ are the boundary nodes, and $X_{1}, X_{2}, \dots, X_{N - 1}$ are the interior nodes with constraints represented by $C_{1}, C_{2}, \dots, C_{N - 1}$ ⁠. Distributed systems like this are also referred as stepped systems. The state equation of this type of systems is the same as Eq. (9). The boundary conditions at $X_{0}, X_{N}$ are described by

Fig. 2

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Distributed subsystems serially connected

A_{0} {\hat{η}}_{1} (0, s) + A_{N} {\hat{η}}_{N} (1, s) = {\hat{γ}}_{b} (s)

(27)

and the matching conditions at

X_{1}, X_{2}, \dots, X_{N - 1}

can be written as

A_{i} {\hat{η}}_{i} (0, s) + B_{i - 1} {\hat{η}}_{i - 1} (1, s) = {\overset{\land}{ν}}_{i} (s), i = 2, 3, \dots, N

(28)

where

A_{i}

and

B_{i}

are matrices describing the subsystem interactions and constraints, and

{\hat{γ}}_{b} (s), {\overset{\land}{ν}}_{1} (s), {\overset{\land}{ν}}_{1} (s), \dots, {\overset{\land}{ν}}_{N} (s)

are the vectors of disturbances at the nodes. Equations and can be cast into the state form of Eq. , with

\begin{matrix} M_{G} & = [\begin{matrix} A_{0} & 0 & \dots & 0 \\ 0 & A_{1} & : \\ : & 0 \\ 0 & \dots & 0 & A_{N - 1} \end{matrix}], \\ N_{G} & = [\begin{matrix} 0 & \dots & 0 & A_{N} \\ B_{1} & 0 & \dots & 0 \\ 0 & : \\ 0 & 0 & B_{N - 1} & 0 \end{matrix}], {\overset{\land}{γ}}_{G} = (\begin{matrix} {\hat{γ}}_{b} (s) \\ {\overset{\land}{ν}}_{1} (s) \\ : \\ {\overset{\land}{ν}}_{N - 1} (s) \end{matrix}) \end{matrix}

(29)

Pointwise Connection of Distributed Subsystems.

In Fig. 3, two distributed subsystems are connected at their nodes by some mechanism C (e.g., the spring connection between nodes 12 and 18 in Fig. 1). The state equation of this type of systems is the same as Eq. (9). The boundary and matching conditions of the subsystems are of the form

Fig. 3

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Pointwise connection of two distributed subsystems

\begin{matrix} M_{b} (s) {\hat{η}}_{1} (0, s) + C_{11} (s) {\hat{η}}_{1} (1, s) + C_{12} (s) {\hat{η}}_{2} (0, s) = γ_{b 1} (s) C_{21} (s) {\hat{η}}_{1} (1, s) + C_{22} (s) {\hat{η}}_{2} (0, s) + N_{b} (s) {\hat{η}}_{2} (1, s) = γ_{b 2} (s) \end{matrix}

(30)

where $M_{b} (s) and N_{b} (s)$ specify the boundary conditions at nodes $X_{0} and X_{3}$ ⁠, $C_{ij} (s)$ describe the connection mechanism C, and $γ_{b 1} (s)$ and $γ_{b 2} (s)$ represent prescribed boundary disturbances. Because Eq. (30) is a special case of Eq. (12), it can be cast in the form of boundary condition (14).

Distributed Connection of Two Distributed Subsystems.

In Fig. 4(a) two distributed subsystems (I) and (II) are connected throughout their domains. One example of such connection is seen in Fig. 1, where a viscoelastic bounding is reinforced between the subsystem of nodes 10 and 11 and the subsystem of nodes 16 and 17. This type of connection can also be used to model the vibration of carbon nanotubes [9]. Because the connection is imposed at the interior points of the subsystems, the state equation (9) has to be modified. (The boundary condition (14) remains the same.) Following a setting in Ref. [12], the state equations of the subsystems can be written as

Fig. 4

View large Download slide

Distributed connection of two distributed systems

\frac{\partial}{\partial x} {\hat{η}}_{i} (x, s) = F_{i} (x, s) {\hat{η}}_{i} (x, s) + {\overset{\land}{p}}_{i} (x, s) + Q_{ci} (x, s), 0 < x < 1, i = 1, 2

(31)

where

i = 1

and 2 refers to subsystems (I) and (II), respectively;

Q_{c 1} (x, s) and Q_{c 2} (x, s)

denote the constraint forces applied at the subsystems, in order to maintain the distributed connection, as shown in Fig. 4(b). Assume that the constraint forces are given by

(\begin{matrix} Q_{c 1} (x_{1}, s) \\ Q_{c 2} (x_{2}, s) \end{matrix}) = [\begin{matrix} C_{11} (s) & C_{12} (s) \\ C_{21} (s) & C_{22} (s) \end{matrix}] (\begin{matrix} {\hat{η}}_{1} (x_{1}, s) \\ {\hat{η}}_{2} (x_{2}, s) \end{matrix})

(32)

with matrices

C_{ij} (s)

describing the connection. Substitution of Eq. into Eq. yields a coupled state equation

\frac{\partial}{\partial x} {\hat{η}}_{c} (x, s) = F_{c} (s) {\hat{η}}_{c} (x, s) + {\overset{\land}{p}}_{c} (x, s)

(33)

where

\begin{matrix} F_{c} (s) & = [\begin{matrix} F_{1} (s) + C_{11} (s) & C_{12} (s) \\ C_{21} (s) & F_{2} (s) + C_{22} (s) \end{matrix}] \\ {\hat{η}}_{c} (x, s) & = (\begin{matrix} {\hat{η}}_{1} (x, s) \\ {\hat{η}}_{2} (x, s) \end{matrix}), {\overset{\land}{p}}_{c} (x, s) = (\begin{matrix} {\overset{\land}{p}}_{1} (x, s) \\ {\overset{\land}{p}}_{2} (x, s) \end{matrix}) \end{matrix}

(34)

Thus, to implement the distributed connection, $F_{c} (s)$ is used to modify matrix $F (s)$ in the state equation (9). Note that $F (s)$ after the modification may not be a block-diagonal matrix.

Numerical Examples

The augmented DTFM is applied to compute the eigensolutions of two flexible structures in vibration: a coupled beam structure and a frame structure.

Example 1: A Coupled Beam Structure.

In Fig. 5, two cantilever Euler–Bernoulli beams of unity length are coupled by a spring of coefficient k at their tips (nodes X₁ and X₂). The governing equations of the beams are given by

Fig. 5

View large Download slide

A coupled beam structure

ρ_{i} \frac{\partial^{2}}{\partial t^{2}} w_{i} (x, t) + {EI}_{i} \frac{\partial^{4}}{\partial x^{4}} w_{i} (x, t) = f_{i} (x, t), 0 < x < 1

(35)

where

i = 1 and 2, {EI}_{i}

and

ρ_{i}

are the bending stiffness and linear density of the beams, respectively, and the local coordinate

x = x_{i}

has been used. The boundary and matching conditions of the system are

\begin{matrix} w_{1} (0, t) = 0, \frac{\partial}{\partial x} w_{1} (0, t) = 0, w_{2} (1, t) = 0, \frac{\partial}{\partial x} w_{2} (1, t) = 0, \\ {EI}_{1} \frac{\partial^{2}}{\partial x^{2}} w_{1} (1, t) = 0 - {EI}_{1} \frac{\partial^{3}}{\partial x^{3}} w_{1} (1, t) = {kw}_{2} (0, t) - {kw}_{1} (1, t), \\ - {EI}_{2} \frac{\partial^{2}}{\partial x^{2}} w_{2} (0, t) = 0 {EI}_{2} \frac{\partial^{3}}{\partial x^{3}} w_{2} (0, t) = {kw}_{1} (1, t) - {kw}_{2} (0, t) \end{matrix}

(36)

Define the state vectors of the beam elements as

\begin{matrix} {\hat{η}}_{i} (x, s) = {\begin{matrix} {\bar{w}}_{i} (x, s) & \frac{\partial {\bar{w}}_{i} (x, s)}{\partial x} & \frac{\partial^{2} {\bar{w}}_{i} (x, s)}{\partial x^{2}} & \frac{\partial^{3} {\bar{w}}_{i} (x, s)}{\partial x^{3}} \end{matrix}}^{T} \end{matrix}

(37)

where

{\bar{w}}_{i} (x, s)

is the Laplace transform of

w_{i} (x, t)

with respect to time t. Equations and are cast into the augmented state equations and , in which

\begin{matrix} F (s) = [\begin{matrix} F_{1} (s) & 0 \\ 0 & F_{2} (s) \end{matrix}], M_{G} = [\begin{matrix} A_{0} & 0 \\ 0 & A_{1} \end{matrix}], N_{G} = [\begin{matrix} 0 & A_{2} \\ B_{1} & 0 \end{matrix}] \end{matrix}

(38)

with

\begin{matrix} F_{i} (s) & = [\begin{matrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \frac{- ρ_{i} s^{2}}{{EI}_{i}} & 0 & 0 & 0 \end{matrix}] for i = 1, 2 \\ A_{0} & = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}], A_{1} = [\begin{matrix} 0 & 0 & 0 & 0 \\ - k & 0 & 0 & 0 \\ 0 & 0 & - {EI}_{2} & 0 \\ k & 0 & 0 & {EI}_{2} \end{matrix}] \\ A_{2} & = [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{matrix}], B_{1} = [\begin{matrix} 0 & 0 & {EI}_{1} & 0 \\ k & 0 & 0 & - {EI}_{1} \\ 0 & 0 & 0 & 0 \\ - k & 0 & 0 & 0 \end{matrix}] \end{matrix}

(39)

By Eq. , the natural frequencies of the beam coupled structure are the roots of the characteristic equation

Δ (ω) \equiv det (M_{G} + N_{G} e^{F (J ω)}) = 0, ω \geq 0, J = \sqrt{- 1}

(40)

With the nondimensional values of the system parameters given in Fig. 5, the first 10 natural frequencies of the coupled beam structure are computed by the proposed method and the finite element method. The results obtained are listed in Table 2. As can be seen, the results obtained by the augmented DTFM are the limits of the finite element result as the number N of elements increases.

Table 2

The natural frequencies of the coupled beams (rad/s)

		Finite element method
k	Augmented DTFM	N = 8	N = 16	N = 20
1	27.8999	27.9011	27.9000	27.8999
2	49.4029	49.4074	49.4032	49.4030
3	142.3247	142.4970	142.3365	142.3296
4	222.2395	222.5032	222.2576	222.2470
5	391.2485	394.2897	391.4877	391.3486
6	617.6240	622.4137	618.0005	617.7816
7	765.1783	776.2695	766.8945	765.9085
8	1209.3491	1226.8996	1212.0596	1210.5022
9	1264.3418	1443.1774	1271.6601	1267.5303
10	1888.4428	2281.5682	1910.8978	1898.6379

		Finite element method
k	Augmented DTFM	N = 8	N = 16	N = 20
1	27.8999	27.9011	27.9000	27.8999
2	49.4029	49.4074	49.4032	49.4030
3	142.3247	142.4970	142.3365	142.3296
4	222.2395	222.5032	222.2576	222.2470
5	391.2485	394.2897	391.4877	391.3486
6	617.6240	622.4137	618.0005	617.7816
7	765.1783	776.2695	766.8945	765.9085
8	1209.3491	1226.8996	1212.0596	1210.5022
9	1264.3418	1443.1774	1271.6601	1267.5303
10	1888.4428	2281.5682	1910.8978	1898.6379

For comparison purposes, the characteristic equation $det K (J ω) = 0$ in the previous DTFM [12] is also considered, where K(s) is given in Eq. (7). In Fig. 6, the characteristic functions $det K (J ω)$ and $Δ (ω)$ are plotted against ω, where the zeros of the functions are the natural frequencies of the structure. As can be seen, $det K (J ω)$ in Fig. 6(a) is piecewise continuous, and becomes unbounded at the certain frequencies. These frequencies are not the natural frequencies of the beam structure, but those of the beam elements that are “fully constrained,” that is, the beam elements with clamped-clamped boundaries. The singularities of $det K (J ω) = 0$ can cause problems in root finding, as discussed in Sec. 2. On the other hand, $Δ (ω)$ in Fig. 6(b) is continuous and smooth in the entire frequency region (⁠ $0 \leq ω < + \infty$ ⁠). A well-behaved $Δ (ω)$ renders root search more efficient, and avoids the possibility of missing roots as with the previous DTFM. Moreover, due to its smoothness, $Δ (ω)$ can be scaled to maintain relatively small values in computation. For instance, one may use $Δ (ω) / (1 + ω^{μ})$ ⁠, with μ being a properly selected positive number.

Fig. 6

View large Download slide

The characteristic functions of the coupled beam structure: (a) $\det K (J ω)$ in Ref. [12]; and (b) $Δ (ω)$ in Eq. (40)

Example 2: A Frame Structure.

A frame of three beam elements is shown in Fig. 7, where the numbers in parentheses are the element numbers, $X_{0}, X_{1}, X_{2}, X_{3}$ are the nodes of the frame, and x is the nondimensional local coordinate of the elements. Let the $ρ_{i}, {EA}_{i}, {EI}_{i}, L_{i}$ (i = 1, 2, 3) be the linear density, longitudinal stiffness, bending stiffness, and length of the beam elements, respectively. The vibrations of the beam elements are governed by the partial differential equations

Fig. 7

View large Download slide

A frame structure

\begin{matrix} ρ_{i} \frac{\partial^{2}}{\partial t^{2}} u_{i} (x, t) - \frac{{EA}_{i}}{L_{i}^{2}} \frac{\partial^{2}}{\partial x^{2}} u_{i} (x, t) = f_{A, i} (x, t) ρ_{i} \frac{\partial^{2}}{\partial t^{2}} w_{i} (x, t) + \frac{{EI}_{i}}{L_{i}^{4}} \frac{\partial^{4}}{\partial x^{4}} w_{i} (x, t) = f_{T, i} (x, t) \end{matrix}

(41)

for

0 < x < 1

⁠, where

u_{i} (x, t)

and

w_{i} (x, t)

are the longitudinal and transverse displacements of the ith beam, respectively,

f_{A, i} (x, t) and f_{T, i} (x, t)

are external forces, and the

1 / L_{i}^{4}

is associated with the spatial derivatives because the local coordinate x is nondimensional. Nine boundary conditions of the frame structure are prescribed at nodes

X_{0}, X_{2} and X_{3}

\begin{matrix} u_{1} (0, t) = 0, w_{1} (0, t) = 0, \frac{1}{L_{1}} \frac{\partial}{\partial x} w_{1} (0, t) = 0 u_{2} (1, t) = 0, w_{2} (1, t) = 0, \frac{1}{L_{2}} \frac{\partial}{\partial x} w_{2} (1, t) = 0 u_{3} (1, t) = 0, w_{3} (1, t) = 0, \frac{1}{L_{3}} \frac{\partial}{\partial x} w_{3} (1, t) = 0 \end{matrix}

(42a)

With the assumption of rigid connection of the beam elements, there are nine matching conditions imposed at $X_{1}$ ⁠:

(i)
Six conditions of displacement compatibility
$\begin{matrix} u_{1} (1, t) = u_{2} (0, t) \cos α + w_{2} (0, t) \sin α \\ w_{1} (1, t) = w_{2} (0, t) \cos α - u_{2} (0, t) \sin α \\ u_{1} (1, t) = w_{3} (0, t), w_{1} (1, t) = - u_{3} (0, t) \\ \frac{1}{L_{1}} \frac{\partial w_{1} (1, t)}{\partial x} = \frac{1}{L_{2}} \frac{\partial w_{2} (0, t)}{\partial x} = \frac{1}{L_{2}} \frac{\partial w_{3} (0, t)}{\partial x} \end{matrix}$
(42b)
(ii)
Three conditions of force balance
$\begin{matrix} - \frac{{EI}_{1}}{L_{1}^{2}} \frac{\partial^{2} w_{1} (1, t)}{\partial x^{2}} + \frac{{EI}_{2}}{L_{2}^{2}} \frac{\partial^{2} w_{2} (0, t)}{\partial x^{2}} + \frac{{EI}_{3}}{L_{3}^{2}} \frac{\partial^{2} w_{3} (0, t)}{\partial x^{2}} = 0 - \frac{{EA}_{1}}{L_{1}} \frac{\partial u_{1} (1, t)}{\partial x} + \frac{{EA}_{2}}{L_{2}} \frac{\partial u_{2} (0, t)}{\partial x} \cos α \\ - \frac{{EI}_{2}}{L_{2}^{3}} \frac{\partial^{3} w_{2} (0, t)}{\partial x^{3}} \sin α - \frac{{EI}_{3}}{L_{3}^{3}} \frac{\partial^{3} w_{3} (0, t)}{\partial x^{3}} = 0 \frac{{EI}_{1}}{L_{1}^{3}} \frac{\partial^{3} w_{1} (1, t)}{\partial x_{1}^{3}} - \frac{{EA}_{2}}{L_{2}} \frac{\partial u_{2} (0, t)}{\partial x} \sin α \\ + \frac{{EI}_{2}}{L_{2}^{3}} \frac{\partial^{3} w_{2} (0, t)}{\partial x^{3}} \cos α - \frac{{EA}_{3}}{L_{1}} \frac{\partial u_{3} (0, t)}{\partial x} = 0 \end{matrix}$
(42c)

Again, factors $1 / L_{i}, 1 / L_{i}^{2}, 1 / L_{i}^{3}$ in the previous equations are related to the utility of the nondimensional coordinate x.

Define the state vectors of the beam elements as

\begin{matrix} {\hat{η}}_{i} (x, s) \\ = {\begin{matrix} {\bar{u}}_{i} (x) & \frac{\partial {\bar{u}}_{i} (x)}{\partial x} & {\bar{w}}_{i} (x, s) & \frac{\partial {\bar{w}}_{i} (x, s)}{\partial x} & \frac{\partial^{2} {\bar{w}}_{i} (x, s)}{\partial x^{2}} & \frac{\partial^{3} {\bar{w}}_{i} (x, s)}{\partial x^{3}} \end{matrix}}^{T} \end{matrix}

(43)

for

i = 1, 2, 3

⁠. With Eqs. to , the state equation and the boundary condition can be easily formed. By the augmented state formulation, the eigensolutions of the frame structure can be determined through use of the formulas in Sec. 4.1. In numerical simulation, the following nondimensional values of the frame parameters are assigned:

\begin{matrix} Element (1) : ρ_{1} = 78, {EA}_{1} = 2 \times 10^{7}, {EI}_{1} = 2000, L_{1} = 1 \\ Element (2) : ρ_{2} = 27, {EA}_{2} = 7 \times 10^{6}, {EI}_{2} = 700, L_{2} = 1, α = π / 4 \\ Element (3) : ρ_{3} = 27, {EA}_{3} = 7 \times 10^{6}, {EI}_{3} = 700, L_{3} = \sqrt{2} / 2 \end{matrix}

(44)

The first eight natural frequencies of the frame structure are computed by the augmented DTFM, and tabulated in Table 3. For comparison, the results computed by the finite element method are also included in the table. It is seen that as the number of elements increases, the finite element predictions converge to the results given by the proposed method. Additionally, the first four mode shapes of the frame structure are computed by Eqs. (20) and (21), and they are plotted in Fig. 8.

Fig. 8

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The first four mode shapes of the frame structure: (a) mode 1: $ω_{1} = 83.524$ ⁠, (b) mode 2: $ω_{2} = 113.05$ ⁠, (c) mode 3: $ω_{3} = 206.44$ ⁠, and (d) mode 4: $ω_{4} = 271.71$

Table 3

The natural frequencies of the frame structure (rad/s)

		Finite element method
k	The proposed method	N = 9	N = 18	N = 27
1	83.5236	83.7067	83.5354	83.5259
2	113.0478	113.5050	113.0774	113.0537
3	206.4408	207.4903	206.5098	206.4546
4	271.7135	276.2557	272.0697	271.7852
5	307.5947	313.7979	308.1738	307.7115
6	519.4175	562.7119	521.7580	519.9003
7	570.5345	628.6682	573.3994	571.1386
8	620.1257	700.5947	622.6839	620.6544

		Finite element method
k	The proposed method	N = 9	N = 18	N = 27
1	83.5236	83.7067	83.5354	83.5259
2	113.0478	113.5050	113.0774	113.0537
3	206.4408	207.4903	206.5098	206.4546
4	271.7135	276.2557	272.0697	271.7852
5	307.5947	313.7979	308.1738	307.7115
6	519.4175	562.7119	521.7580	519.9003
7	570.5345	628.6682	573.3994	571.1386
8	620.1257	700.5947	622.6839	620.6544

Conclusions

This paper presents a new analytical method, namely the augmented distributed transfer function method, for modeling and analysis of one-dimensional multibody distributed dynamic systems. The proposed method has the following special features.

(a)
By using an augmented state formulation, Eqs. (9), (14), and (15), the proposed method is flexible and convenient in description of interactions of multiple distributed and lumped bodies that are governed by a mixture of partial and ordinary differential equations. With this setting, various dynamic problems of branched multibody systems can be systematically solved.
(b)
The augmented DTFM does not use dynamic equilibrium equations that are assembled from the transfer functions of distributed subsystems. Because of this, the singularity problems with the previous DTFM and the transfer matrix method are completely eliminated. Indeed, the characteristic function in Eq. (19) is smooth and well-behaved for any value of s, which renders root finding efficient, and avoids the problem of missing roots in computation. Due to this advantage of the augmented DTFM over conventional transfer function methods, inverse Laplace transform for time–domain solutions can be accurately and easily performed.
(c)
In modeling and analysis, the augmented DTFM does not make any approximation or discretization; neither does it require orthogonal system eigenfunctions. To the authors' knowledge, the augmented DTFM is the first solution technique that delivers exact and closed form solutions for a wide class of branched multibody distributed systems, in both frequency and time domains.
(d)
In the proposed method, modeling of different systems with different boundary and matching conditions just needs simple formation of the state matrix F(s) in Eq. (9) and the boundary matrices $M_{G} (s) and N_{G} (s)$ in Eq. (14); solution procedures and algorithms are essentially same. This feature of symbolic manipulation renders the proposed method easy in use and convenient for computer coding, as has been shown in the numerical examples.

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An Augmented State Formulation for Modeling and Analysis of Multibody Distributed Dynamic Systems

Abstract

Introduction

Issues With the Previous DTFM

Augmented State Formulation

Solution of Dynamic Problems Via the Augmented DTFM

Eigenvalue Problem.

Frequency Response.

Transient Response.

Treatment of Certain Types of Connection of Distributed Subsystems

Serially Connected Distributed Systems.

Pointwise Connection of Distributed Subsystems.

Distributed Connection of Two Distributed Subsystems.

Numerical Examples

Example 1: A Coupled Beam Structure.

Example 2: A Frame Structure.

Conclusions

References

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An Augmented State Formulation for Modeling and Analysis of Multibody Distributed Dynamic Systems

Abstract

Introduction

Issues With the Previous DTFM

Augmented State Formulation

Solution of Dynamic Problems Via the Augmented DTFM

Eigenvalue Problem.

Frequency Response.

Transient Response.

Treatment of Certain Types of Connection of Distributed Subsystems

Serially Connected Distributed Systems.

Pointwise Connection of Distributed Subsystems.

Distributed Connection of Two Distributed Subsystems.

Numerical Examples

Example 1: A Coupled Beam Structure.

Example 2: A Frame Structure.

Conclusions

References

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