1 Introduction

The transient mass and structural characteristics of a typical multistage spacecraft vehicle coupled with boosters, as shown in Fig. 1 for example, require that the natural vibration characteristics of the vehicle are known. These natural vibration characteristics are important during the ignition and burnout time of each stage of flight, and frequently needed in other cases such as those at Mach number of one, maximum dynamic pressure, and minimum lift. As the slender ratio increases, i.e., structures become more flexible, the flexural rigidity of the rocket and the natural frequencies of transverse vibrations are reduced.

The vibration theory of single-beam systems is well-developed and studied in detail in hundreds of contributions. The vibration of systems composed of a uniform double-beam coupled by translational springs or elastic layers has been studied extensively in the literature [1–7]. The transverse free vibrations of a system where two beams are coupled by a spring have been studied in Ref. [8]. In Ref. [9], the free vibration of an undamped triple-beam system on elastic foundation was analyzed. The system consists of three Euler–Bernoulli beams of the same length that are arranged in parallel and continuously connected by elastic layers. However, there are only a few contributions dealing with the vibration of triple-beam systems. That is probably because the general vibration analyses of an elastically connected multi-beam system are complicated and laborious in view of a large variety of possible combinations of boundary conditions, and, thus, the solution of the governing coupled partial differential equations is difficult.

Applications of the transfer matrix method of linear multibody systems (MSTMM) range from vibration analysis, modeling of composite structures and multibody systems (MS) for computing vibration characteristics, static deformations, and control and dynamical response, to damage identification [10]. MSTMM and its specific extension to MSs is a rather exceptional approach in the MS community. It has proven its strength, especially for linear systems involving both discrete elements like rigid bodies, springs or dampers, and continuous elements like Euler–Bernoulli or Timoshenko beams, where it is even able to find exact solutions [11]. This is achieved by transforming ordinary and partial differential equations into algebraic relations between input and output states consisting of displacements (both translational and rotational) and forces as well as moments. On an element level, these relations are captured by transfer matrices; on the system level, the transfer matrices have to be just multiplied in a proper sequence, depending on the topology of the system [12]. According to the natural attribute of bodies, a complicated MS can be represented by various bodies (e.g., rigid/elastic bodies, lumped masses, etc.) interconnected by hinges (e.g., spherical/sliding/cylindrical joints, dampers, springs, etc.). In MSTMM, there are different topologies of dynamics models such as chain, tree, and closed-loop systems as illustrated in Fig. 2.

Fig. 1
figure 1

Typical multistage spacecraft vehicle coupled with boosters

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Primarily, based on MSTMM with chain topology, the natural vibration characteristics of a flexible rocket/satellite launch vehicle were explored theoretically in Ref. [13]. The free-free non-uniform beam was subjected to variable axial compressive force and modeled by both Euler–Bernoulli and Timoshenko theory. The latter solution included the influences of rotary inertia and shear deformation. For forward applications, the beam was divided into piecewise constant property segments. For the free vibration, the exact solution was obtained using the beam functions. Further, an approach based on MSTMM with implicit force scenarios was developed to analyze the free vibration of a multi-level beam, coupled by spring/dashpot systems attached to them in-span. The Euler–Bernoulli model was used for the transverse vibration of the beams, and the spring/dashpot system represented a simplified model of a viscoelastic material [14].

Fig. 2
figure 2

Examples of general multibody systems with a Chair. b Tree. c Closed closed-loop topology

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The present paper is a continuation of the work described in Refs. [13, 14]. Therefore, the full theory of MSTMM, including sign convention, MS topologies, state vector (SV), state variables, transfer direction, transfer equation, transfer matrix, overall system transfer matrix, overall system state vector, and Eigenfrequency equation of the whole MS are presented. The reader may download these two articles from the Internet for more details.

This paper presents a new scenario called virtual cutting in the context of MSTMM closed-loop topology for computing the natural vibration characteristics of elastically coupled flexible launch vehicle boosters. This issue is very important for aeroelasticity, flight mechanics, and control. In the present approach, the coupled system is idealized as a triple-beam system-like structure coupled by four linear translational springs, where a non-uniform free-free Euler–Bernoulli beam is used. A large thrust-to-weight ratio leads to large axial accelerations that result in an axial inertia load distribution from nose to tail. Consequently, it causes the development of significant compressive forces along the length of the launch vehicle. Therefore, it is important to take into account this effect in the transverse vibration model.

2 Mathematical formulations

2.1 Problem statement

The complex structure configuration shown in Fig. 1 may be modeled as an MS. As illustrated in Fig. 3, the elastically coupled system is vibrating in the xy-plane and composed of continuous elements (beams) and discrete elements (springs). Each beam is divided into n-segments. Beam 2 has the following properties: length \(L_2\), bending stiffness and mass per unit length distributions along the beam length are \(EI_2 (x)\) and \(\bar{{m}}_2 (x)=\rho _2 (x)A_2 (x)\), respectively, where \(E,I,\rho \), and A are the modulus of elasticity, area moment of inertia, material density and cross section area, respectively. Beams 1 and 3 are identical, i.e., \(L_1 \equiv L_3 \), \(EI_1 (x)\equiv EI_3 (x)\) and \(\bar{{m}}_1 (x)\equiv \bar{{m}}_3 (x)\). The three free-free beams are loaded by three compressive forces (thrusts) at the left ends, which lead to axial forces in each beam section, namely, beam 2 with \(T_2 (x)\rightarrow S_2 (x)\) and beams 1 and 3 with \(T_1 (x)\equiv T_3 (x)\rightarrow S_1 (x)\equiv S_3 (x)\). The launch vehicle with boosters is assumed to move with constant instantaneous acceleration at all points of the trajectory. This results in reduction of the distributed compressive force as the total vehicle mass depletes [15]. Within each segment, bending stiffness, axial compressive force and mass per unit length distributions can be approximated as constants. Such an approach has the advantage that it is capable of taking into account fairly complex configurations just by increasing the number of segments [13, 15]. There are four linear translational springs with stiffness \(K_{y_1}, K_{y_2}, K_{y_3}\), and \(K_{y_4}\); each two are located at \(x_1\) and \(x_2\), respectively, from the beams rear ends. However, these springs are identical. It should be noted that there is no inherent difficulty in extending the current method to a system consisting of any number of uniform or non-uniform beams, any kind of springs, damper, and rigid bodies.

Fig. 3
figure 3

Elastically coupled system

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2.2 Components partitioning

The strategy of MSTMM for treating complex linear MS is to partition the structure into simple components whose features are easily expressed in matrix form. The matrix of each component can be regarded as a building block. Each component is characterized by its transfer matrix, which may be saved in the MSTMM library. The overall transfer matrix and overall SV of the entire system are deduced by assembling these blocks following the MS topology in action. The mechanics characteristic of such as an Eigen problem of the global system is then obtained. It is natural to divide the system shown in Fig. 3 into components like beam segments and linear springs. However, in order to handle the connections between beam segments and springs more clearly and simply, it is also necessary to introduce the infinitesimal rigid massless body (dummy body). Then the whole system can be regarded as collections of the components shown in Fig. 4

Fig. 4
figure 4

System collected with beam segments, springs and dummy bodies components

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2.3 Transfer equations of the components

In the following, transfer equations and transfer matrices of the beam and linear translational spring components vibrating in a plane are briefly presented.

Fig. 5
figure 5

Components in MS-TMM library. a Beam segment under axial load. b Linear translational spring

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The governing differential equation of motion of an Euler–Bernoulli beam segment of length l under axial load S(x) (positive in tension) and small rotation due to small deformation as shown in Fig. 5a can be written as

$$\begin{aligned}&EI(x)\frac{\partial ^{4}y}{\partial x^{4}}-S(x)\frac{\partial ^{2}y}{\partial x^{2}}+\bar{{m}}(x)\frac{\partial ^{2}y}{\partial t^{2}}=0 \,\,\,\underrightarrow{y(x,t) = Y(x)\;{{\hbox {e}}^{\mathrm{i}\omega t}}} \nonumber \\&\frac{\mathrm{d}^{4}Y(x)}{\mathrm{d}x^{4}}-\frac{S(x)}{EI(x)}\frac{\mathrm{d}^{2}Y(x)}{\mathrm{d}x^{2}}-\frac{\bar{{m}}(x) \omega ^{2}}{EI(x)}Y(x)=0, \end{aligned}$$
(1)

where \(\omega \) is the angular frequency. The general solution of Eq. (1) is \(Y(x)=A_1 \cosh \beta _1 x+A_2 \sinh \beta _1 x+A_3 \cos \beta _2 x+A_4 \sin \beta _2 x\), where Y is the displacement in the modal coordinate system, \(A_1 ,A_2,\ldots ,A_4\) are arbitrary constants depending on the boundary conditions, \(\beta _1^2 =(\sqrt{a^{2}+4\lambda ^4}+a)/2\) and \(\beta _2^2 =(\sqrt{a^{2}+4\lambda ^4}-a)/2\), where \(a=S(x)L^{2}/(EI(x))\) and \(\lambda ^4=[\bar{{m}}(x) \omega ^{2}L^{4}/(EI(x))]\) are the dimensionless axial force and dimensionless frequency parameter [13]. For the Euler–Bernoulli beam, the linearized relations \(\varTheta _z ={Y}^{\prime }\), \(M_z =EI{Y}^{\prime \prime }\) and \(Q_y ={M}^{\prime }_z\) (where \(\varTheta _z ,Q_y\), and \(M_z\) are the rotation, shear force, and moment in the modal coordinate system, respectively) may be added to end up with the transfer relation

$$\begin{aligned}&{{\mathbf {Z}}}(x)={{\mathbf {B}}}(x)\;\;{{\mathbf {a}}} \nonumber \\&\left[ {{\begin{array}{l} Y \\ {\varTheta _z} \\ {M_z} \\ {Q_y} \\ \end{array}}} \right] _x = \left[ {{\begin{array}{l@{\quad }l@{\quad }l@{\quad }l} {\hbox {cosh}\beta _1 x}&{} {\hbox {sinh}\beta _1 x}&{} {\cos \beta _2 x}&{} {\sin \beta _2 x} \\ {\beta _1 \hbox {sinh}\beta _1 x}&{} {\beta _1 \hbox {cosh}\beta _1 x}&{} {-\beta _2 \sin \beta _2 x}&{} {\beta _2 \cos \beta _2 x} \\ {EI\beta _1^2 \hbox {cosh}\beta _1 x}&{} {EI\beta _1^2 \hbox {sinh}\beta _1 x}&{} {-EI\beta _2^2 \cos \beta _2 x}&{} {-EI\beta _2^2 \sin \beta _2 x} \\ {EI\beta _{1}^3 \hbox {sinh}\beta _1 x}&{} {EI\beta _1^3 \hbox {cosh}\beta _1 x}&{} {EI\beta _2^3 \sin \beta _2 x}&{} {-EI\beta _2^3 \cos \beta _2 x} \\ \end{array}}} \right] \left[ {{\begin{array}{l} {A_1} \\ {A_2} \\ {A_3} \\ {A_4} \\ \end{array}}} \right] , \end{aligned}$$
(2)

where \({\mathbf {Z}}\) is the SV in the modal coordinate system, and \({{\mathbf {a}}}=[A_1 ,A_2 ,A_3 ,A_4 ]^{\mathrm{T}}\) is the coefficient vector that summarizes the unknown constants to be adapted to boundary conditions. At the input end \({\mathbf {Z}}_I(x=0)\), one gets \({\mathbf {Z}}_I =\left[ {{\mathbf {B}}(0)} \right] {{\mathbf {a}}}\). For the beam output end at \(x=l\), Eq. (2) and \({\mathbf {a}}=\left[ {{\mathbf {B}}}(0) \right] ^{-1} {\mathbf {Z}}_O \) give

$$\begin{aligned} {\mathbf {Z}}_O= & {} \left[ {{\mathbf {B}}(l)} \right] {\mathbf {a}} \nonumber \\= & {} \left[ {{\mathbf {B}}(l)} \right] \left[ {{\mathbf {B}}(0)} \right] ^{-1} {\mathbf {Z}}_I\nonumber \\= & {} {\mathbf {U}}^\mathrm{B}{\mathbf {Z}}_I, \end{aligned}$$
(3)

where \({{\mathbf {U}}}^\mathrm{B}\) is the transfer matrix of the Euler–Bernoulli beam component [13].

The massless linear translational spring with stiffness \(K_y\) may vibrate transversely as shown in Fig. 5b. The SV in the physical coordinate system of this one-dimensional vibration at the input and output ends is given as \({{\mathbf {z}}}_I =[y_I ,q_I ]^{\mathrm{T}}\) and \({{\mathbf {z}}}_O =[y_O ,q_O ]^{\mathrm{T}}\), respectively. By considering the equilibrium condition of forces acting on the linear spring, we find

$$\begin{aligned} \left\{ {\begin{array}{l} q_{yO} =q_{yI} \\ -q_{yO} -K_y (y_O -y_I )=0 \\ \end{array}} \right. \quad \hbox {or}\quad \left\{ {\begin{array}{l} y_O =y_I -{q_{yI}}/{K_y} \\ q_{yO} =q_{yI} \\ \end{array}} \right. .\nonumber \\ \end{aligned}$$
(4)

For simple harmonic vibrations, we may apply the transformation \(y=Y\hbox {e}^{\mathrm{i}\omega t},\;q_y =Q_y \hbox {e}^{\mathrm{i}\omega t}\), then substitute it into Eq. (4), resulting in the following transfer equation

$$\begin{aligned}&\left[ {{\begin{array}{l} Y \\ {Q_y} \\ \end{array}}} \right] _O =\left[ {{\begin{array}{ll} 1&{} {-1/{K_y}} \\ 0&{} 1 \\ \end{array}}} \right] \left[ {{\begin{array}{l} Y \\ {Q_y} \\ \end{array}}} \right] _I \quad \hbox {or} \quad {{\mathbf {Z}}}_O ={{\mathbf {U}}}^{\mathrm{LS}} {{\mathbf {Z}}}_I \; \nonumber \\&{{\mathbf {U}}}^{\mathrm{LS}} =\left[ {{\begin{array}{ll} 1&{} {-1/{K_y}} \\ 0&{} 1 \\ \end{array}}} \right] , \end{aligned}$$
(5)

where \({{\mathbf {U}}}^{\mathrm{LS}}\) is the transfer matrix of a linear translational spring component that vibrates transversely [10].

As for the transfer equation of the infinitesimal rigid massless body, it can be deduced as a rigid body with multiple input ends and a single output end. In the viewpoint of MSTMM, for each component with more than two ends, only one of the ends is considered as the output end, and all the other ends are input ends, i.e., \({\mathbf {Z}}_{j,O} ={\mathbf {U}}_{j,I_1} {\mathbf {Z}}_{j,I_1} +{\mathbf {U}}_{j,I_2} {\mathbf {Z}}_{j,I_2} +\cdots +{\mathbf {U}}_{j,I_L } {\mathbf {Z}}_{j,I_L}\) [12], where \({\mathbf {Z}}_{j,O}\) and \({{\mathbf {Z}}}_{j,I_k} (k=2,3,4,\ldots , L)\) are the SVs of the output end and the k-th input end of the component, respectively. L is the number of input ends. \({\mathbf {U}}_{j,I_k}\) is the corresponding transfer matrix from \({\mathbf {Z}}_{j,I_k}\) to \({\mathbf {Z}}_{j,O}\) [12]. The subscript j is the sequence number of the component. O and \(I_1 ,I_2 ,I_3, \ldots , I_L\) denote the output end and input ends, respectively, where the first input end \(I_1\) is considered as the dominant input end. Since the deduction is quite easy, the results can be directly found in the Appendix. To study the dynamics problems of a multi-rigid-flexible body system in the sense of tree topology, it is needed to build-up the main transfer equations (similar to the chain topology) and the geometrical equations according to the transfer directions and geometrical relationship between its first input end and output end, which are fixed on the same element body. In some engineering applications, these equations are not enough to represent the full dynamics equations of motion, and therefore, they needs supplementary equations as we can see in the Appendix.

Fig. 6
figure 6

MSTMM closed loop methodology: state vectors and transfer directions of multistage vehicle coupled with boosters

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2.4 Overall transfer equation of the system

In the context of MSTMM, the SVs and transfer directions of the complex structure shown in Fig. 3 are demonstrated in Fig. 6. It should be pointed out that Fig. 3 is a closed-loop system according to the MS topologies shown in Fig. 2c. The best way to deal with this closed-loop system is to open the loop(s) by a virtual cutting at any one junction point between two different components in a closed-loop and to carry out the dynamic analysis by applying the same producers used for chain or tree topology (open loops). Moreover, only one boundary is considered as the root of the entire system, all the other boundaries are tip(s). For simplicity, the SV at the root is denoted as \({{\mathbf {Z}}}_{1,0}\), while that at the tip(s) is denoted as \({{\mathbf {Z}}}_{l,0}\). Herein, subscript 0 denotes the boundary and \(l=2,3,\ldots ,n_\mathrm{b} -1\), where \(n_\mathrm{b}\) is the total number of the entire system boundaries. The transfer directions of a system are always from the tip(s) to the root. However, caution should be taken that the transfer equation of an element makes sense only if the conversion of the transfer direction for element (from its input end to its output end) and the orientation of the inertial reference frame coincides with those used in the deduction process exhibited in Fig. 4. The dotted arrows in Fig. 6 show the transfer directions for elements whose transfer direction does not match with the orientation of the global reference frame xy. Thus, by introducing the local inertia reference frame \(\bar{{x}}\bar{{y}}\), the transfer equation for the element stands in \(\bar{{x}}\bar{{y}}\). Therefore, these components need to be transformed from \(\bar{{x}}\bar{{y}}\rightarrow xy\) system coordinates (See Appendix, Eq. (21)).

In Fig. 6, the vibrating system is comprised of 19 components. Dummy rigid body (massless rigid body) components 2, 5, 9 and 11 are branches. It has two input ends and one output end. Components 1, 4, 7, 8, 10, 12, 15, 17 and 19 are beam segments (and each segment can be divided into any number of segments). Spring components are 3, 6, 13 and 14. Before virtual cutting of the closed loops, the entire system has totally six boundaries (\(n_\mathrm{b} =6\)), namely, \({{\mathbf {Z}}}_{1,0} \) (root), \({{\mathbf {Z}}}_{12,0} \) (tip 1), \({{\mathbf {Z}}}_{8,0} \) (tip 2), \({{\mathbf {Z}}}_{7,0} \) (tip 3), \({{\mathbf {Z}}}_{19,0} \) (tip 4) and \({{\mathbf {Z}}}_{15,0} \) (tip 5). After virtually cutting the junction between the dummy body 9 and the spring 3, a couple of “new boundaries” denoted as \({{\mathbf {Z}}}_{9,0} \) and \({{\mathbf {Z}}}_{9,3} \), which are identical, will emerge at the “cutting point”. Similarly, another two new boundaries; namely, \({{\mathbf {Z}}}_{2,0} \) and \({{\mathbf {Z}}}_{2,13} \) will be added after cutting the junction between the dummy body 2 and the spring 13. Then, as shown in Fig. 6, the closed-loop system becomes a tree system with four “new boundaries” SVs: \({{\mathbf {Z}}}_{9,0} \) (tip 6), \({{\mathbf {Z}}}_{9,3} \) (tip 7), \({{\mathbf {Z}}}_{2,0} \) (tip 8) and \({{\mathbf {Z}}}_{2,13} \) (tip 9) at the “cutting points”. Therefore, \(n_\mathrm{b} =10\).

The compressive force \(S_{i,m} (x)\) acting on the \(m{\text{- }}\mathrm{th}\) segment of beam i may be calculated at the centre of gravity of the segment by summing up the inertias of all the segments preceding the current one and taking the average inertia force for the current segment [13, 15]. For the constant thrust trajectory, it can be presented as

$$\begin{aligned} S_{i,m} (x)=T_i /M_{o_i} \left[ M_{o_i} -\sum _{k=1}^{m-1} {\rho A(x)_{i,k} l_{i,k} -0.5\rho A(x)_{i,m} l_{i,m}} \right] .\nonumber \\ \end{aligned}$$
(6)

Here in, \(M_o\) and l are the total mass of the beam i and length of the beam segment, respectively.

Now, we are in the position to write the main transfer equation (TE) of the system. That is

$$\begin{aligned} {{\mathbf {Z}}}_{1,0}= & {} {{\mathbf {U}}}_1 {{\mathbf {Z}}}_{1,2} ={{\mathbf {U}}}_1 ({{\mathbf {U}}}_{2,I_1} {{\mathbf {Z}}}_{12,6} +{{\mathbf {U}}}_{2,I_2} {{\mathbf {Z}}}_{4,2} +{{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} ) \\= & {} {{\mathbf {U}}}_1 ({{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {Z}}}_{4,5} +{{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} ) \\= & {} {{\mathbf {U}}}_1 [{{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 ({{\mathbf {U}}}_{5,I_1} {{\mathbf {Z}}}_{5,6} +{{\mathbf {U}}}_{5,I_2} {{\mathbf {Z}}}_{7,5} \\&+\,{{\mathbf {U}}}_{5,I_3} {{\mathbf {Z}}}_{5,14} )+{{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} ] \\= & {} {{\mathbf {U}}}_1 [{{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 ({{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {Z}}}_{11,6} \\&+\,{{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0}+{{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {Z}}}_{18,14} )+{{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} ] \\= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {Z}}}_{11,6} \\&+\,{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0}\\&+\, {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {Z}}}_{18,14} +{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} \\= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 ({{\mathbf {U}}}_{11,I_1} {{\mathbf {Z}}}_{12,11}\\&+\,{{\mathbf {U}}}_{11,I_2} {{\mathbf {Z}}}_{10,11} )+ {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0}\\&+\, {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} ({{\mathbf {U}}}_{18,I_1} {{\mathbf {Z}}}_{19,18} +{{\mathbf {U}}}_{18,I_2} {{\mathbf {Z}}}_{17,18} )\\&+\, {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} \\= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} \\&{{\mathbf {U}}}_6 ({{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0}+{{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {Z}}}_{10,9} ) \\&+\, {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} + {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} \\&({{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19} {{\mathbf {Z}}}_{19,0}+{{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {Z}}}_{17,16} )+ {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} \\ \end{aligned}$$
$$\begin{aligned}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6[{{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12} \\&{{\mathbf {Z}}}_{12,0}+{{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} ({{\mathbf {U}}}_{9,I_1} {{\mathbf {Z}}}_{8,9} +{{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} )]\\&+\, {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} + {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14}\\&[{{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19} {{\mathbf {Z}}}_{19,0}+{{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} ({{\mathbf {U}}}_{16,I_1} {{\mathbf {Z}}}_{15,16}\\&+\, {{\mathbf {U}}}_{16,I_2} {{\mathbf {Z}}}_{16,13} )]+{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} \\= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6\\&[{{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} +{{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} ({{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8 {{\mathbf {Z}}}_{8,0}\\&+\,{{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} )] + {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} \\&+\, {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} [{{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19} {{\mathbf {Z}}}_{19,0}\\&+\,{{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} ({{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15} {{\mathbf {Z}}}_{15,0} \\&+\, {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13} {{\mathbf {Z}}}_{2,13})]+{{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_3} {{\mathbf {Z}}}_{2,0}. \end{aligned}$$

Or,

$$\begin{aligned} {{\mathbf {Z}}}_{1,0}= & {} {{\mathbf {T}}}_{12-1} {{\mathbf {Z}}}_{12,0} +{{\mathbf {T}}}_{8-1} {{\mathbf {Z}}}_{8,0} +{{\mathbf {T}}}_{7-1} {{\mathbf {Z}}}_{7,0} +{{\mathbf {T}}}_{19-1} {{\mathbf {Z}}}_{19,0} \nonumber \\&+\,{{\mathbf {T}}}_{15-1} {{\mathbf {Z}}}_{15,0}+{{\mathbf {T}}}_{9-1} {{\mathbf {Z}}}_{9,0} +{{\mathbf {T}}}_{3-1} {{\mathbf {Z}}}_{9,3}\nonumber \\&+\,{{\mathbf {T}}}_{2-1} {{\mathbf {Z}}}_{2,0} +{{\mathbf {T}}}_{13-1} {{\mathbf {Z}}}_{2,13}, \nonumber \\ \hbox {where}&\nonumber \\ {{\mathbf {T}}}_{12-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12}, \nonumber \\ {{\mathbf {T}}}_{8-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8, \nonumber \\ {{\mathbf {T}}}_{7-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7, \nonumber \\ {{\mathbf {T}}}_{19-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19}, \nonumber \\ {{\mathbf {T}}}_{15-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15}, \nonumber \\ {{\mathbf {T}}}_{9-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_2}, \nonumber \\ {{\mathbf {T}}}_{3-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_1} {{\mathbf {U}}}_3, \nonumber \\ {{\mathbf {T}}}_{2-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_3}, \nonumber \\ {{\mathbf {T}}}_{13-1}= & {} {{\mathbf {U}}}_1 {{\mathbf {U}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13}. \end{aligned}$$
(7)

In Eq. (7), the subscript \(j-1\) denotes the transfer direction from the component j to the root component 1. \({{\mathbf {U}}}_1 ,\;{{\mathbf {U}}}_4 ,\;{{\mathbf {U}}}_7 ,\;{{\mathbf {U}}}_8 ,\;{{\mathbf {U}}}_{10} ,\;{{\mathbf {U}}}_{12} ,\;{{\mathbf {U}}}_{15} ,\;{{\mathbf {U}}}_{17} \) and \({{\mathbf {U}}}_{19} \) are transfer matrices of beam component \({{\mathbf {U}}}^\mathrm{B}\) (Eq. (3)). \({{\mathbf {U}}}_3 ,\;{{\mathbf {U}}}_6 ,\;{{\mathbf {U}}}_{13} \) and \({{\mathbf {U}}}_{14} \) are transfer matrices of linear translational spring component \({{\mathbf {U}}}^{\mathrm{LS}}\) (Eq. (5)). \({{\mathbf {U}}}_{2,I_1} ={{\mathbf {U}}}_{5,I_1} ,\;{{\mathbf {U}}}_{2,I_2 } ={{\mathbf {U}}}_{5,I_2} ,\;{{\mathbf {U}}}_{2,I_3} ={{\mathbf {U}}}_{5,I_3} \) are transfer matrices of a massless rigid body with three input ends and single output end (See Appendix, Eq. (18)). \({{\mathbf {U}}}_{9,I_1} ={{\mathbf {U}}}_{16,I_1} ,\;{{\mathbf {U}}}_{9,I_2} ={{\mathbf {U}}}_{16,I_2} \) and \({{\mathbf {U}}}_{11,I_1} ={{\mathbf {U}}}_{18,I_1} ,\;{{\mathbf {U}}}_{11,I_2} ={{\mathbf {U}}}_{18,I_2} \) are transfer matrices of a massless rigid body with two input ends and a single output end (See Appendix, Eqs. (14) and (16), respectively).

Equation (7) illustrates the kinematics and kinetics’ relationships. However, it does not describe the geometrical relationship between different input ends. Also, the number of unknown variables is larger than that of algebraic equations. Therefore, another set of equations named geometrical equations (GEs) should be introduced, which describes the geometrical relationship between the first input end and r-th \((r=2,3,4,\ldots ,L)\) input end of the component. GE can be written in the following form \({{\mathbf {H}}}_{j,I_1} {{\mathbf {Z}}}_{j,I_1} ={{\mathbf {H}}}_{j,I_r} {{\mathbf {Z}}}_{j,I_r} \). As an example, component 2 has three input ends and one output end. Therefore, it has two GEs. The first one is

$$\begin{aligned}&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {Z}}}_{2,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {Z}}}_{4,2} =\mathbf 0 , \\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {Z}}}_{4,5} =\mathbf 0 ,\\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 ({{\mathbf {U}}}_{5,I_1} {{\mathbf {Z}}}_{5,6} +{{\mathbf {U}}}_{5,I_2} {{\mathbf {Z}}}_{7,5}\\&\quad +\,{{\mathbf {U}}}_{5,I_3} {{\mathbf {Z}}}_{5,14} )=\mathbf 0 ,\\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 ({{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {Z}}}_{11,6}\\&\quad +\,{{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} +{{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {Z}}}_{18,14})=\mathbf 0 , \\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {Z}}}_{11,6}\\&\quad +\,{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {Z}}}_{18,14} =\mathbf 0 , \\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 ({{\mathbf {U}}}_{11,I_1} {{\mathbf {Z}}}_{12,11} \\&\quad +\,{{\mathbf {U}}}_{11,I_2} {{\mathbf {Z}}}_{10,11} )+{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} +\ldots \\&+{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} ({{\mathbf {U}}}_{18,I_1} {{\mathbf {Z}}}_{19,18} +{{\mathbf {U}}}_{18,I_2} {{\mathbf {Z}}}_{17,18} )=\mathbf 0 , \\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 ({{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} \\&\quad +\,{{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {Z}}}_{10,9})+\ldots \\&+{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} ({{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19} {{\mathbf {Z}}}_{19,0}\\&\quad +\,{{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {Z}}}_{17,16})=\mathbf 0 \\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 [{{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} \\&\quad +\,{{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} ({{\mathbf {U}}}_{9,I_1} {{\mathbf {Z}}}_{8,9} +{{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} )]\\&\quad +\,{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} [{{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19}{{\mathbf {Z}}}_{19,0} \\&+\ldots +{{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} ({{\mathbf {U}}}_{16,I_1} {{\mathbf {Z}}}_{15,16} +{{\mathbf {U}}}_{16,I_2} {{\mathbf {Z}}}_{16,13} )]=\mathbf 0 , \\&-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 [{{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0}\\&\quad +\,{{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} ({{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8 {{\mathbf {Z}}}_{8,0} +{{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} )]+\ldots \\&+{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7 {{\mathbf {Z}}}_{7,0} +{{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} [{{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19} {{\mathbf {Z}}}_{19,0}\\&\quad +\,{{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} ({{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15} {{\mathbf {Z}}}_{15,0} +\ldots + {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13} {{\mathbf {Z}}}_{2,13} )]=\mathbf 0 . \end{aligned}$$

Or,

$$\begin{aligned}&{{\mathbf {G}}}_{12-2} {{\mathbf {Z}}}_{12,0} +{{\mathbf {G}}}_{8-2} {{\mathbf {Z}}}_{8,0} +{{\mathbf {G}}}_{7-2} {{\mathbf {Z}}}_{7,0} +{{\mathbf {G}}}_{19-2} {{\mathbf {Z}}}_{19,0} \nonumber \\&\quad +\,{{\mathbf {G}}}_{15-2} {{\mathbf {Z}}}_{15,0} +{{\mathbf {G}}}_{9-2} {{\mathbf {Z}}}_{9,0}+ {{\mathbf {G}}}_{3-2} {{\mathbf {Z}}}_{9,3} +{{\mathbf {G}}}_{13-2} {{\mathbf {Z}}}_{2,13} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{12-2} ={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1 } {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12}, \nonumber \\&{{\mathbf {G}}}_{8-2} \;\;={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8, \nonumber \\&{{\mathbf {G}}}_{7-2} \;={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_2} {{\mathbf {U}}}_7, \nonumber \\&{{\mathbf {G}}}_{19-2} ={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19}, \nonumber \\&{{\mathbf {G}}}_{15-2} ={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15}, \nonumber \\&{{\mathbf {G}}}_{9-2} \;\;={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_2}, \nonumber \\&{{\mathbf {G}}}_{3-2} =-{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3, \nonumber \\&{{\mathbf {G}}}_{13-2} ={{\mathbf {H}}}_{2,I_2} {{\mathbf {U}}}_4 {{\mathbf {U}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13}. \end{aligned}$$
(8a)

And the second GE of component 2 is

$$\begin{aligned} -{{\mathbf {H}}}_{2,I_1} {{\mathbf {U}}}_3 {{\mathbf {Z}}}_{9,3} +{{\mathbf {H}}}_{2,I_3} {{\mathbf {Z}}}_{2,0} =\mathbf 0 . \end{aligned}$$

Or,

$$\begin{aligned}&{{\mathbf {G}}}_{3-2} {{\mathbf {Z}}}_{9,3} +{{\mathbf {G}}}_{2-2} {{\mathbf {Z}}}_{2,0} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{2-2} ={{\mathbf {H}}}_{2,I_3}. \end{aligned}$$
(8b)

Similarly, the GEs corresponding to components 5, 9, 11, 16, and 18 can be deduced as follows

$$\begin{aligned}&{{\mathbf {G}}}_{12-5} {{\mathbf {Z}}}_{12,0} +{{\mathbf {G}}}_{8-5} {{\mathbf {Z}}}_{8,0} +{{\mathbf {G}}}_{7-5} {{\mathbf {Z}}}_{7,0} +{{\mathbf {G}}}_{9-5} {{\mathbf {Z}}}_{9,0} =\mathbf 0 , \nonumber \\&{{\mathbf {G}}}_{12-5} {{\mathbf {Z}}}_{12,0} +{{\mathbf {G}}}_{8-5} {{\mathbf {Z}}}_{8,0} +{{\mathbf {G}}}_{19-5} {{\mathbf {Z}}}_{19,0} +{{\mathbf {G}}}_{15-5} {{\mathbf {Z}}}_{15,0},\nonumber \\&\quad +\,{{\mathbf {G}}}_{9-5} {{\mathbf {Z}}}_{9,0} +{{\mathbf {G}}}_{13-5} {{\mathbf {Z}}}_{2,13} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{12-5} =-{{\mathbf {H}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_1} {{\mathbf {U}}}_{12}, \nonumber \\&{{\mathbf {G}}}_{8-5} =-{{\mathbf {H}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8, \nonumber \\&{{\mathbf {G}}}_{7-5} ={{\mathbf {H}}}_{5,I_2} {{\mathbf {U}}}_7, \nonumber \\&{{\mathbf {G}}}_{19-5} ={{\mathbf {H}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_1} {{\mathbf {U}}}_{19}, \nonumber \\&{{\mathbf {G}}}_{15-5} ={{\mathbf {H}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15}, \nonumber \\&{{\mathbf {G}}}_{9-5} =-{{\mathbf {H}}}_{5,I_1} {{\mathbf {U}}}_6 {{\mathbf {U}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_2}, \nonumber \\&{{\mathbf {G}}}_{13-5} \;={{\mathbf {H}}}_{5,I_3} {{\mathbf {U}}}_{14} {{\mathbf {U}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13},\end{aligned}$$
(8c)
$$\begin{aligned}&{{\mathbf {G}}}_{8-9} {{\mathbf {Z}}}_{8,0} +{{\mathbf {G}}}_{9-9} {{\mathbf {Z}}}_{9,0} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{8-9} =-{{\mathbf {H}}}_{9,I_1} {{\mathbf {U}}}_8, \nonumber \\&{{\mathbf {G}}}_{9-9} \;={{\mathbf {H}}}_{9,I_2}, \end{aligned}$$
(8d)
$$\begin{aligned}&{{\mathbf {G}}}_{12-11} {{\mathbf {Z}}}_{12,0} +{{\mathbf {G}}}_{8-11} {{\mathbf {Z}}}_{8,0} +{{\mathbf {G}}}_{9-11} {{\mathbf {Z}}}_{9,0} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{12-11} =-{{\mathbf {H}}}_{11,I_1} {{\mathbf {U}}}_{12}, \nonumber \\&{{\mathbf {G}}}_{8-11} \; ={{\mathbf {H}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8, \nonumber \\&{{\mathbf {G}}}_{9-11} \; ={{\mathbf {H}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_2}, \end{aligned}$$
(8e)
$$\begin{aligned}&{{\mathbf {G}}}_{15-16} {{\mathbf {Z}}}_{15,0} +{{\mathbf {G}}}_{13-16} {{\mathbf {Z}}}_{2,13} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{15-16} =-{{\mathbf {H}}}_{16,I_1} {{\mathbf {U}}}_{15}, \nonumber \\&{{\mathbf {G}}}_{13-16} \;={{\mathbf {H}}}_{16,I_2} {{\mathbf {U}}}_{13}, \end{aligned}$$
(8f)
$$\begin{aligned}&{{\mathbf {G}}}_{19-18} {{\mathbf {Z}}}_{19,0} +{{\mathbf {G}}}_{15-18} {{\mathbf {Z}}}_{15,0} +{{\mathbf {G}}}_{13-18} {{\mathbf {Z}}}_{2,13} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {G}}}_{19-18} =-{{\mathbf {H}}}_{18,I_1} {{\mathbf {U}}}_{19}, \nonumber \\&{{\mathbf {G}}}_{15-18} \;={{\mathbf {H}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15}, \nonumber \\&{{\mathbf {G}}}_{13-18} \;={{\mathbf {H}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13}. \end{aligned}$$
(8g)

In Eq. (8), \({{\mathbf {H}}}_{2,I_1} ={{\mathbf {H}}}_{5,I_1} ,{{\mathbf {H}}}_{2,I_2} ={{\mathbf {H}}}_{5,I_2} \), \({{\mathbf {H}}}_{2,I_3} ={{\mathbf {H}}}_{5,I_3} \) and \({{\mathbf {H}}}_{11,I_1} ={{\mathbf {H}}}_{18,I_1} ,\;{{\mathbf {H}}}_{11,I_2} ={{\mathbf {H}}}_{18,I_2} \) are geometric matrices at the massless rigid body with multi-input/single output (See Appendix, Eq. (19)). \({{\mathbf {H}}}_{9,I_1} ={{\mathbf {H}}}_{16,I_1} ,\;{{\mathbf {H}}}_{9,I_2} ={{\mathbf {H}}}_{16,I_2} \) and \({{\mathbf {H}}}_{11,I_1} ={{\mathbf {H}}}_{18,I_1} ,\;{{\mathbf {H}}}_{11,I_2} ={{\mathbf {H}}}_{18,I_2} \) are geometric matrices at the massless rigid body with two input ends and a single output end (See Appendix, Eqs. (15) and (17), respectively).

Up to now, the equilibrium condition of the moments acting on the dummy bodies 11 and 18 are still missing, that is: \(m_{11,I_1} +m_{11,I_2} =0\) and \(m_{18,I_1} +m_{18,I_2} =0\), respectively. Consequently, the so-called supplementary equation (SE) can be obtained for body 11 as an example

$$\begin{aligned}&{{\mathbf {E}}}_{11,I_1} {{\mathbf {Z}}}_{12,11} ={{\mathbf {E}}}_{11,I_2} {{\mathbf {Z}}}_{10,11}, \nonumber \\&-{{\mathbf {E}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} +{{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {Z}}}_{10,9} =\mathbf 0 , \nonumber \\&-{{\mathbf {E}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} +{{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} ({{\mathbf {U}}}_{9,I_1} {{\mathbf {Z}}}_{8,9} +{{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} )=\mathbf 0 , \nonumber \\&-{{\mathbf {E}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} +{{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} ({{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8 {{\mathbf {Z}}}_{8,0}\nonumber \\&\quad +\,{{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} )=\mathbf 0 , \nonumber \\&-{{\mathbf {E}}}_{11,I_1} {{\mathbf {U}}}_{12} {{\mathbf {Z}}}_{12,0} +{{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8 {{\mathbf {Z}}}_{8,0}\nonumber \\&\quad +\,{{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_2} {{\mathbf {Z}}}_{9,0} =\mathbf 0 . \end{aligned}$$

Or,

$$\begin{aligned}&{{\mathbf {S}}}_{12-11} {{\mathbf {Z}}}_{12,0} +{{\mathbf {S}}}_{8-11} {{\mathbf {Z}}}_{8,0} +{{\mathbf {S}}}_{9-11} {{\mathbf {Z}}}_{9,0} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {S}}}_{12-11} =-{{\mathbf {E}}}_{11,I_1} {{\mathbf {U}}}_{12}, \nonumber \\&{{\mathbf {S}}}_{8-11} ={{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_1} {{\mathbf {U}}}_8, \nonumber \\&{{\mathbf {S}}}_{9-11} ={{\mathbf {E}}}_{11,I_2} {{\mathbf {U}}}_{10} {{\mathbf {U}}}_{9,I_2}. \end{aligned}$$
(9a)

Similarly, the SE of body 18 is

$$\begin{aligned}&{{\mathbf {S}}}_{19-18} {{\mathbf {Z}}}_{19,0} +{{\mathbf {S}}}_{15-18} {{\mathbf {Z}}}_{15,0} +{{\mathbf {S}}}_{13-18} {{\mathbf {Z}}}_{2,3} =\mathbf 0 , \nonumber \\&\hbox {where} \nonumber \\&{{\mathbf {S}}}_{19-18} =-{{\mathbf {E}}}_{18,I_1} {{\mathbf {U}}}_{19}, \nonumber \\&{{\mathbf {S}}}_{15-18} ={{\mathbf {E}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_1} {{\mathbf {U}}}_{15}, \nonumber \\&{{\mathbf {S}}}_{13-18} ={{\mathbf {E}}}_{18,I_2} {{\mathbf {U}}}_{17} {{\mathbf {U}}}_{16,I_2} {{\mathbf {U}}}_{13}. \end{aligned}$$
(9b)

In Eq. (9), \({{\mathbf {E}}}_{11,I_1} ={{\mathbf {E}}}_{18,I_1} \) and \({{\mathbf {E}}}_{11,I_2} ={{\mathbf {E}}}_{18,I_2} \) are the supplementary matrices at the massless rigid body with two input ends and a single output end (See Appendix, Eq. (7)).

To this end, by combining Eqs. (7), [SubEquationDirect](8a), and (9), the overall transfer equation of the entire system shown in Fig. 6 can be obtained as

$$\begin{aligned} \left. {{{\mathbf {U}}}_{\mathrm{all}}} \right| _{16\times 32} \left. {{{\mathbf {Z}}}_{\mathrm{all}}} \right| _{32\times 1} =\mathbf 0 , \end{aligned}$$
(10)

where,

$$\begin{aligned} {{\mathbf {U}}}_{\mathrm{all}}= & {} \left[ {{\begin{array}{cccccccccc} {{{\mathbf {T}}}_{12-1}}&{} {{{\mathbf {T}}}_{8-1}}&{} {{{\mathbf {T}}}_{7-1}}&{} {{{\mathbf {T}}}_{19-1}}&{} {{{\mathbf {T}}}_{15-1}}&{} {{{\mathbf {T}}}_{9-1}}&{} {{{\mathbf {T}}}_{3-1}}&{} {{{\mathbf {T}}}_{2-1}}&{} {{{\mathbf {T}}}_{13-1}}&{} {-{{\mathbf {I}}}} \\ {{{\mathbf {G}}}_{12-2}}&{} {{{\mathbf {G}}}_{18-2}}&{} {{{\mathbf {G}}}_{17-2}}&{} {{{\mathbf {G}}}_{19-2}}&{} {{{\mathbf {G}}}_{15-2}}&{} {{{\mathbf {G}}}_{9-2}}&{} {{{\mathbf {G}}}_{3-2}}&{} \mathbf 0 &{} {{{\mathbf {G}}}_{13-2}}&{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{3-2}}&{} {{{\mathbf {G}}}_{2-2}}&{} \mathbf 0 &{} \mathbf 0 \\ {{{\mathbf {G}}}_{12-5}}&{} {{{\mathbf {G}}}_{8-5}}&{} {{{\mathbf {G}}}_{7-5}}&{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{9-5}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 \\ {{{\mathbf {G}}}_{12-5}}&{} {{{\mathbf {G}}}_{8-5}}&{} \mathbf 0 &{} {{{\mathbf {G}}}_{19-5}}&{} {{{\mathbf {G}}}_{15-5}}&{} {{{\mathbf {G}}}_{9-5}}&{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{13-5}}&{} \mathbf 0 \\ \mathbf 0 &{} {{{\mathbf {G}}}_{18-9}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{9-9}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 \\ {{{\mathbf {G}}}_{12-11}}&{} {{{\mathbf {G}}}_{8-11}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{9-11}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{15-16}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{13-16}}&{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{19-18}}&{} {{{\mathbf {G}}}_{15-18}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{13-18}}&{} \mathbf 0 \\ {{{\mathbf {S}}}_{12-11}}&{} {{{\mathbf {S}}}_{8-11}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {S}}}_{9-11}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {S}}}_{19-18}}&{} {{{\mathbf {S}}}_{15-18}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {S}}}_{13-18}}&{} \mathbf 0 \\ \end{array}}}\right] ,\\ {{\mathbf {Z}}}_{\mathrm{all}}^{\mathrm{T}}= & {} \left[ {{\begin{array}{cccccccccc} {{{\mathbf {Z}}}_{12,0}}&{} {{{\mathbf {Z}}}_{8,0}}&{} {{{\mathbf {Z}}}_{7,0}}&{} {{{\mathbf {Z}}}_{19,0}}&{} {{{\mathbf {Z}}}_{15,0}}&{} {{{\mathbf {Z}}}_{9,0}}&{} {{{\mathbf {Z}}}_{9,3}}&{} {{{\mathbf {Z}}}_{2,0}}&{} {{{\mathbf {Z}}}_{2,3}}&{} {{{\mathbf {Z}}}_{1,0}} \\ \end{array}}} \right] ^{\mathrm{T}}. \end{aligned}$$

The internal force in \({{\mathbf {Z}}}_{9,0} \) (tip 6) and \({{\mathbf {Z}}}_{9,3} \) (tip 7) and similarly in \({{\mathbf {Z}}}_{2,0} \) (tip 8) and \({{\mathbf {Z}}}_{2,13} \) (tip 9) are of the same quantity but with opposite direction due to the sign conventions, that is

$$\begin{aligned} {{\mathbf {Z}}}_{9,0} ={{\mathbf {C}}}\;{{\mathbf {Z}}}_{9,3};\quad {{\mathbf {Z}}}_{2,0} ={{\mathbf {C}}}\;{{\mathbf {Z}}}_{2,13}, \end{aligned}$$
(11a)

where

$$\begin{aligned} {{\mathbf {C}}}=\left[ {{\begin{array}{cc} 1&{} 0 \\ 0&{} {-1} \\ \end{array}}}\right] . \end{aligned}$$
(11b)

Substituting Eq. (11) into Eq. (10), the overall transfer equation can be written in the following form with less order

$$\begin{aligned} \left. {{{\mathbf {U}}}_{\mathrm{all}}} \right| _{16\times 28} \left. {{{\mathbf {Z}}}_{\mathrm{all}}} \right| _{28\times 1} =\mathbf 0 , \end{aligned}$$
(12)

where,

$$\begin{aligned} {{\mathbf {U}}}_{\mathrm{all}}= & {} \left[ {{\begin{array}{cccccccccc} {{{\mathbf {T}}}_{12-1}}&{} {{{\mathbf {T}}}_{8-1}}&{} {{{\mathbf {T}}}_{7-1}}&{} {{{\mathbf {T}}}_{19-1}}&{} {{{\mathbf {T}}}_{15-1}}&{} {{{\mathbf {T}}}_{9-1} {{\mathbf {G}}}+{{\mathbf {T}}}_{3-1}}&{} {{{\mathbf {T}}}_{2-1} {{\mathbf {G}}}+{{\mathbf {T}}}_{13-1}}&{} {-{\mathbf {I}}} \\ {{{\mathbf {G}}}_{12-2}}&{} {{{\mathbf {G}}}_{18-2}}&{} {{{\mathbf {G}}}_{17-2}}&{} {{{\mathbf {G}}}_{19-2}}&{} {{{\mathbf {G}}}_{15-2}}&{} {{{\mathbf {G}}}_{9-2} {{\mathbf {G}}}+{{\mathbf {G}}}_{3-2}}&{} {{{\mathbf {G}}}_{13-2}}&{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{3-2}}&{} {{{\mathbf {G}}}_{2-2} {{\mathbf {G}}}}&{} \mathbf 0 \\ {{{\mathbf {G}}}_{12-5}}&{} {{{\mathbf {G}}}_{8-5}}&{} {{{\mathbf {G}}}_{7-5}}&{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{9-5{{\mathbf {G}}}}}&{} \mathbf 0 &{} \mathbf 0 \\ {{{\mathbf {G}}}_{12-5}}&{} {{{\mathbf {G}}}_{8-5}}&{} \mathbf 0 &{} {{{\mathbf {G}}}_{19-5}}&{} &{} {{{\mathbf {G}}}_{9-5} {{\mathbf {G}}}}&{} {{{\mathbf {G}}}_{13-5}}&{} \mathbf 0 \\ \mathbf 0 &{} {{{\mathbf {G}}}_{18-9}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{9-9} {{\mathbf {G}}}}&{} \mathbf 0 &{} \mathbf 0 \\ {{{\mathbf {G}}}_{12-11}}&{} {{{\mathbf {G}}}_{8-11}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{9-11} {{\mathbf {G}}}}&{} \mathbf 0 &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{15-16}}&{} \mathbf 0 &{} {{{\mathbf {G}}}_{13-16}}&{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {G}}}_{19-18}}&{} {{{\mathbf {G}}}_{15-18}}&{} \mathbf 0 &{} {{{\mathbf {G}}}_{13-18}}&{} \mathbf 0 \\ {{{\mathbf {S}}}_{12-11}}&{} {{{\mathbf {S}}}_{8-11}}&{} \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {S}}}_{9-11} {{\mathbf {G}}}}&{} \mathbf 0 &{} \mathbf 0 \\ \mathbf 0 &{} \mathbf 0 &{} \mathbf 0 &{} {{{\mathbf {S}}}_{19-18}}&{} {{{\mathbf {S}}}_{15-18}}&{} \mathbf 0 &{} {{{\mathbf {S}}}_{13-18}}&{} \mathbf 0 \\ \end{array}}} \right] ,\\ {{\mathbf {Z}}}_{\mathrm{all}}^{\mathrm{T}}= & {} \left[ {{\begin{array}{cccccccccc} {{{\mathbf {Z}}}_{12,0}}&{} {{{\mathbf {Z}}}_{8,0}}&{} {{{\mathbf {Z}}}_{7,0}}&{} {{{\mathbf {Z}}}_{19,0}}&{} {{{\mathbf {Z}}}_{15,0}}&{} {{{\mathbf {Z}}}_{9,3}}&{} {{{\mathbf {Z}}}_{2,3}}&{} {{{\mathbf {Z}}}_{1,0}} \\ \end{array}}} \right] ^{\mathrm{T}}. \end{aligned}$$
Table 1 Common boundary conditions for a beam vibrating in a plane
Full size table
Fig. 7
figure 7

a System consisting of unloaded free-free and loaded pinned-pinned beams connected with three translational springs. b State vectors and transfer directions

Full size image

The overall transfer equation (Eq. (12)) involves only the boundary SVs, and the SVs at all other connection points do not appear. For common boundary conditions of the beam listed in Table 1, half of state variables of \({{\mathbf {Z}}}_{\mathrm{all}} \) are zeros due to known constraints, and it should be noted that the displacements and forces in the SVs \({{\mathbf {Z}}}_{9,3} \) and \({{\mathbf {Z}}}_{2,13} \) are unknown too. Thus, Eq. (12) is reduced to \({\bar{{\mathbf {U}}}}_{\mathrm{all}} {\bar{{\mathbf {Z}}}}_{\mathrm{all}} \mathbf{=0}\) where \({\bar{{\mathbf {Z}}}}_{\mathrm{all}} \) is composed of the unknown state variables, and \({\bar{{\mathbf {U}}}}_{\mathrm{all}} \) is a square matrix resulting from elimination of all columns of \({{\mathbf {U}}}_{\mathrm{all}} \) associated zeros in \({{\mathbf {Z}}}_{\mathrm{all}} \). Finally, \({\bar{{\mathbf {U}}}}_{\mathrm{all}} \) is only a function of the unknown eigenvalues of the system. For non-trivial solutions, the characteristic equation \(\varDelta (\omega )=\det {\bar{{\mathbf {U}}}}_{\mathrm{all}} \mathop {=}\limits ^{!} 0\) has to be fulfilled.

A novel algorithm is implemented to determine the natural frequencies. This algorithm is based on the methodology to reduce the zero-search of the reduced overall transfer matrix’s determinate to a minimization problem (refer to Ref. [11] for more details).

3 Numerical results

To validate the accuracy and efficiency of MSTMM with a virtual cutting scenario, a numerical example is given in the following. A system consisting of two beams connected with three symmetrically distributed linear translational springs \(K_{y_1} =K_{y_2} =K_{y_3 } =K_y \) located respectively at positions \(\xi _1, \xi _2 =0.5\) and \(\xi _3 =1-\xi _1 \), where \(\xi _i =x_i /L\), is shown in Fig. 7b. The beams are supposed to have the same material and geometrical parameters:

\(L_1 =L_2 =L=\hbox {1\,m}, \quad EI_1 =EI_2 =EI=166.67\, \hbox {Nm}^{2}, \quad \bar{{m}}_1 =\bar{{m}}_2 =\bar{{m}}=0.78~\hbox {kg/m}\) and \(K_{y_1} =K_{y_2} =K_{y_3} =K_y =100EI/L^{3}\) [2]. The first beam is unloaded free-free (\(S_1 =0\rightarrow a_1 =0)\) and the second beam is loaded pinned-pinned with an axial tensile force (\(S_2 =666.68\hbox { N}\rightarrow a_2 =4\,a_2 =4.0)\) all through this beam.

Following the previous derivations in Sect. 2 and the transfer directions shown in Fig. 7b, the overall transfer equation of the system can be summarized as follows

(13)
Fig. 8
figure 8

System consisting of unloaded free-free and loaded pinned-pinned beams connected with three translational springs. a fMin1D function determinant \((k_{y}=100EI/L^{3} \hbox { and } \xi _{1}=0.2)\). b Dimensionless frequency parameter \(\bar{\omega }\) values as a function of the spring locations \((\xi _{1}=0\rightarrow 0.5)\)

Full size image

Applying the boundary conditions shown in Table 1 to Eq. (13) for the beam ends and noting that \({{\mathbf {Z}}}_{12,3} \)(tip 5) and \({{\mathbf {Z}}}_{14,6} \) (tip 7) are unknown too. Thus, the overall transfer equation is reduced to \(\left. {{\bar{{\mathbf {U}}}}_{\mathrm{all}}} \right| _{12\times 12} \left. {{\bar{{\mathbf {Z}}}}_{\mathrm{all}}} \right| _{12\times 1} \mathbf{=0}\) that is ready for the eigenproblem.

Applying the fMin1D algorithm \([(1, 3700), N_{x0} =1000,\varepsilon =10^{-6}]\) [11], the absolute determinant of the non-dimensional natural frequencies (\(\bar{{\omega }}=\root 4 \of {{\bar{{m}} \omega _i^2 L^{4}}/{(EI)}})\) are presented in Fig. 8a for springs located at \((\xi _1 =0.2)\). The system shows symmetric and anti-symmetric vibrations as illustrated in Fig. 8b. To summarize, (1) MSTMM results are in good agreement with Fig. 3a, b of Ref. [2] (page 133, where the Green function method is used for solving analytically the problem of free vibration of beams with elastic supports) and (2) MSTMM can elegantly handle this kind of problem. However, the reader may download the reference from the library or from the Internet because the authors of the present paper cannot include these figures because of the Journal’s copyright.

4 Conclusions

The vibration problem of beam-type structures is of particular urgent importance in many branches of modern aerospace, mechanical, and civil engineering. Natural vibration frequencies and modes are one of the most important dynamic characteristics of these kinds of systems. The virtual cutting scenario based on the transfer matrix method of linear multibody systems with closed-loop topology is developed to analyze the free vibration of elastically coupled flexible launch vehicle boosters. It is modeled as a triple-beam under axial load, coupled by springs. The Euler–Bernoulli model is used for the transverse vibration of the beams. MSTMM reduces the dynamics problem, precisely the Eigenvalue problem to an overall transfer equation, which only involves boundary state vectors. The state vectors at the boundaries are composed of displacements, rotation angles, bending moments, and shear forces, which are partly known and partly unknown, and end up with a reduced overall transfer matrix. A non-trivial solution requires the coefficient matrix to be singular to yield the required natural frequencies. In the present paper, a novel algorithm is implemented based on the methodology to reduce the zero search of the reduced overall transfer matrix’s determinate to a minimization problem. The proposal scenario is easy to formulate, systematic to apply, and simple to code.