7.1.1 Characteristics of Ordinary Multibody System Dynamics Methods

With the development of modern industrial technology, many complex mechanical systems have appeared, such as weaponry, aeronautics, astronautics, vehicles, robots and precision machines, which are composed of many components with large relative motion. Under the condition of strong engineering demands, a new branch, namely multibody system dynamics, appeared. The numerical simulation of complex mechanical systems became possible with the appearance and development of the computer. Multibody system dynamics is an engineering application essential to the study of mechanical system dynamics, and systems consisting of many bodies undergoing large relative motion are its main study object. In general, the components of multibody systems include bodies and joints. Body components include rigid bodies, flexible bodies and lumped masses. A system is a multi‐rigid‐body system if the bodies involved are all rigid bodies, it is a multi‐flexible‐body system if the bodies involved are all flexible bodies and a multi‐rigid‐flexible‐body system if both rigid bodies and flexible bodies are involved. The joints are the components connecting the bodies.

The deduction of the dynamic equations of multibody systems and their numerical solutions are two main research areas of multibody system dynamics. Generally, there are two major methods for deriving dynamic equations, and their basic forms are different. The first method uses Lagrangian equations of the first kind, in which Lagrange multipliers are used to define unknown generalized constraint forces and obtain an augmented formulation. This approach leads to a mixed set of nonlinear differential and algebraic equations that have to be solved simultaneously, and it is the theoretical foundation of some commercial analysis software, such as ADAMS and DADS. It must solve a set of differential‐algebraic equations. The second method describes the motion using a set of first‐order differential equations (kinematics and dynamics) with a minimal set of unknown variables without constraint forces. From a point of view of computational efficiency, its numerical solution is more sophisticated. Differential‐algebraic equations can be transformed into a new form with a minimal number of coordinates, and a typical code is the MBOSS [246] developed by Arizona University.

With developments in engineering technology, researchers are still proposing and improving various multibody system dynamics methods, which are promoting the development of modern engineering technology. These methods have different forms, but in general they all have the two same characteristics:

1. It is necessary to build the global dynamic equations of the system, which is the most interesting and brilliant part of various methods, but is also the difficult part for engineers.
2. The orders of matrices involved in the global dynamic equations of the system are rather high (generally speaking not less than the degrees of freedom (DOFs) of the system). It has become one of the main research directions in multibody system dynamics to search for a new method to reduce the orders of relevant matrices and to avoid computational difficulty caused by high matrix order.

7.1.2 Features of the Discrete Time Transfer Matrix Method for Multibody Systems

To improve the computational efficiency of multibody system dynamics, a new method, the discrete time transfer matrix method for multibody systems (MSDTTMM) [3, 64–68], has been developed, which is different from other ordinary methods of multibody system dynamics. This method combines the advantages of the transfer matrix method and numerical integration. The dynamics equations of typical elements are derived first, followed by their discretization and linearization in the time domain based on the basic thought of numerical integration. Then the transfer matrices of these elements are obtained. The overall transfer equation and overall transfer matrix of the multibody system can be assembled. The system motion can be obtained using step‐by‐step time integration. The method is suitable for linear time‐variable, nonlinear, large motion and general multibody systems. Compared with ordinary multibody system dynamics methods, the MSDTTMM has the following features:

1. The global dynamic equations of a system are not needed, hence its application and computation are very convenient.
2. The order of involved matrices is much lower than for ordinary multibody system dynamics methods. For example, the order of relevant matrices for chain multibody systems is determined only by the highest order of differential equations of its subsystems. The computational cost is small, resulting in high computational efficiency.
3. It provides a flexible modeling instrument and the modeling procedure is highly programmable. The modeling of a multibody system looks like building blocks, the overall transfer matrix of the system can be assembled by the transfer matrices of each element directly, and its solution procedure is very simple and straightforward.
4. Complex multibody systems, such as branch, closed‐loop or network systems, can be conveniently dealt with by this method in a similar way to the transfer matrix method for linear multibody systems.
5. The study method for multi‐flexible‐body systems and the MRFS is the same as that for multi‐rigid‐body systems.
6. The transfer matrix is always a real number matrix, even when considering dampers. This simplifies the numerical algorithm.
7. The natural frequency of linear multibody systems can be obtained easily by using the MSDTTMM, and the transfer matrix method for linear multibody systems can be regarded as a special case.
8. An arbitrary appropriate numerical integration method can be introduced into the MSDTTMM, which gives the method some flexibility.

The basic principles, steps and algorithm of the MSDTTMM developed by the authors are introduced in this chapter, and the transfer matrices of typical elements are deduced.

7.3.1 Linearization of Velocity (Angular Velocity) and Acceleration (Angular Acceleration)

In dynamic equations of general elements, the unknown quantities include velocities (angular volocities), accelerations (angular accelerations), trigonometric functions and internal forces at connection points, as well as cross terms, high‐order terms and so on. For the free vibration of linear time‐invariant multibody systems, the state vectors satisfy the following relation:

where z is the physical coordinate and Z is the corresponding modal coordinate.

We can obtain

(7.26)

If the motion of a multibody system is aperiodic, Equation (7.26) does not exist. In order to develop the transfer matrix of an element, its dynamic equation must be linearized. Using numerical integral methods, velocity (angular velocity) and acceleration (angular acceleration) can be expressed as linear functions of the position coordinate (angular coordinate), that is

(7.27)

(7.28)

where , , and at time t_i are known functions with respect to time , and can be summarized as A, B_z, C and D_z.

Based on the linearization of velocities, accelerations and other variables in a small time segment during time interval , the nonlinear dynamic equation at time t_i can be expressed by linear formulas of the motion variables at time t_i. The quantities above can be determined using the following methods.

7.3.2 Step‐by‐step Time Integration Method

7.3.2.1 Forward–Euler Method

According to the Taylor expansion theorem, x(t_i) can be approximately expressed as

(7.29)

From Equation (7.30), it follows that

(7.30)

Thus, the acceleration can be expressed approximately by the first‐order difference as from Equation (7.30), that is

(7.31)

By comparing Equations (7.30) and (7.31) with Equations (7.27) and (7.28), we obtain

(7.32)

7.3.2.2 Newmark‐β Method

Expanding the variable x(t_i) at time by Taylor expansion theorem yields

(7.33)

Let

(7.34)

then

(7.35)

The velocity can be expressed by using Taylor expansion theorem as

(7.36)

Let

(7.37)

Substituting Equations (7.35) and (7.37) into Equation (7.36) yields

(7.38)

From Equations (7.35) and (7.38) we obtain

(7.39)

The Newmark‐β method originates from the trapezoid method, where only the first and second‐order derivatives of the Taylor series are adopted. The two parameters γ and β are used to compensate the contributions of the abandoned higher‐order quantities. If γ and β have different values, many formulas can be obtained. When and , this is the average acceleration method. leads to the constant acceleration method (center difference method), and yields the linear acceleration method. It can be proved that if γ ≥ 1/2, β ≥ γ/2, and the formulas are unconditionally stable. If the value of β is increased, the computational precision will reduced, and if , the computational precision is the highest, but the formula is conditionally stable. Newmark has pointed out that induces negative damping to a linear system, i.e. the vibration amplitude will be amplified in the integral computation. When , artificial damp will be introduced and finally induce reduction in the vibration amplitude. Generally, let γ ≥ 1/2. The most common choice is to let and then change β accordingly, so the method is called the Newmark‐β method. If and , this is the Forward–Euler method.

7.3.2.3 Wilson‐θ Method

The Wilson‐θ method was developed on the basis of the linear acceleration assumption. The parameter θ ≥ 1 can be introduced in the time interval , assuming acceleration is linear with respect to time, as shown in Figure 7.4. A set of equations at time , called predicted equations, is deduced first. Then the correction equations of displacement, velocity and acceleration at time are deduced. When , this is the linear acceleration method. If τ is the time variable limited by 0 ≤ τ ≤ θΔT, the acceleration during this time interval can be written as , according to the assumption of linear acceleration. Later the acceleration can be integrated in [0, τ].

(7.40)

(7.41)

If , according to Equations (7.40) and (7.41), the velocity and displacement at time can be computed by the following two formulas

(7.42)

(7.43)

Combining Equations (7.42) and (7.43) yields

(7.44)

(7.45)

According to Equations (7.44) and (7.45), we obtain

(7.46)

7.3.2.4 Houbolt Method

The basic idea of the Houbolt method is that the velocity and acceleration at can be expressed approximately by the position coordinates of four points at previous times , as shown in Figure 7.5. The relative coordinate is introduced, and , , x_n and are the position coordinates corresponding to , 1, 2 and 3, respectively. When 0 ≤ ζ ≤ 3, the position coordinate x(ζ) can be expressed approximately by the Lagrange interpolation polynomial:

(7.47)

It is not difficult to figure out the rules of this interpolation polynomial: any individual term of this polynomial is the product of a position coordinate function and three terms of the four basic functions ζ, , and . Its coefficients are obtained by the value of the function x(ζ) corresponding to each point. The first‐ and second‐order derivatives of x(ζ) are

(7.48)

(7.49)

If , then

(7.50)

Three previous time points are used in this linearization method, hence the first and second steps cannot be implemented in these formulations. The method can only be used if the information for three time points is known. This is the disadvantage of the multistep method. The two‐step and single‐step Houbolt methods are described here. Equation (7.50) yields

(7.51)

(7.52)

as

(7.53)

(7.54)

Adding Equations (7.53) and (7.54), then

(7.55)

Substituting Equation (7.55) into Equations (7.51) and (7.52), the linearization coefficients of the two‐step Houbolt method can be obtained:

(7.56)

As

(7.57)

then

(7.58)

(7.59)

Therefore, the linearization coefficients of the one‐step Houbolt method become

(7.60)

In summary, the higher‐order Houbolt method can be constructed by a higher‐order Lagrange interpolation polynomial.

For different step‐by‐step time integration methods, different formulations of A, B_z, C and D_z can be obtained. Linearization coefficient formulations for different step‐by‐step time integration methods are shown in Table 7.2.

Method	A	B z
TMM		0
Forward–Euler
Newmark‐β
Wilson‐θ
Houbolt for i ≥ 3
Houbolt for i = 2
Houbolt for i = 1
Method	C	Dz
TMM	0	0
Forward–Euler
Newmark‐β
Wilson‐θ
Houbolt for i ≥ 3
Houbolt for i = 2
Houbolt for i = 1

Let the variable z represent a column matrix, such as the position coordinates or orientation angles . At present, of the four linearization coefficients, A and C are irrespective of the state vector coefficient z, hence their expressions are invariant. B_z and D_z, however, should be rewritten as column matrices and or and , respectively. For example, in the Newmark‐β method

7.3.3 Linearization of Nonlinear Functions

Correct to ΔT², we expand the trigonometric functions at time t_i as an explicit function of previous time by the Taylor expansion theorem, that is

(7.61)

where

(7.62)

In the geometric relationship from input end to output end, which includes trigonometric functions, the trigonometric functions can be linearized by the second‐order Taylor expansion, that is

(7.63)

(7.64)

For higher‐order terms in body dynamic equations, the second‐order terms can be expressed approximately as linear functions, that is

(7.65)

The linear expression of a third‐order function is

(7.66)

The above expansion formulations neglect the effect of terms that are higher than ΔT³ order. For higher‐order terms, their linearization expression can be obtained by using the same method.

The linearization expression of a general polynomial can be written as

(7.67)

where and at time t_i are constants or functions of variables at previous times.

Substituting differential for finite difference

(7.68)

yields

(7.69)

Hence

(7.70)

Linearization formulas of the trigonometric functions sin θ(t_i), cos θ(t_i) and higher‐order functions a(t_i)b(t_i) and a(t_i)b(t_i)c(t_i) are used to derive the transfer matrices of elements. For convenience, all of these formulations are listed in Table 7.3.

Function	Linearization formulas
sin θ(ti)
cos θ(ti)
sin θ(ti)
cos θ(ti)
a(ti)b(ti)
a(ti)b(ti)c(ti)

7.3.4 Linearization of Coordinate Transformation Matrices

The angular motion of a rigid body with respect to a fixed point can be realized by three rotations along three spatial fixed axes successively. The proof can be found in Appendix A.

A motion of a rigid body B relative to reference frame A is shown in Figure 7.6, where is any vector fixed in reference system A, and is a vector fixed on rigid body B. Before B rotates relative to A, . One fixed point’s 3 DOF rotation can be described by three angles which rotate in sequence around three arbitrary non‐coplanar axes. Herein we consider the angles θ_x, θ_y, θ_z that rotate in sequence around three spatial fixed axes in inertial coordinates. Therefore, the transform matrix, from the body‐fixed coordinate system to the inertial coordinate system, can be expressed by trigonometric functions of angles θ_x, θ_y, θ_z as follows:

(7.71)

Where

(7.72)

Using multiple Taylor expansion, for transform matrix at , keeping second‐order terms, we obtain

(7.73)

According to Equation (7.71)

(7.74)

(7.75)

(7.76)

Let

(7.77)

Then

(7.78)

Similarly, it is easy to prove that

(7.79)

Combining Equations (7.71) and (7.78) yields

(7.80)

(7.81)

(7.82)

Let

Substituting Equations (7.78) to (7.82) into Equation (7.73) yields

(7.83)

where

(7.84)

7.4.1 Planar Rigid Body with One Input End and One Output End

For the planar rigid body shown in Figure 7.7 the input end is I, the output end is O and the mass center is C. The inertial coordinate system is oxy and the body‐fixed coordinate system is O₂x₂y₂ with origin I. In the coordinate system O₂x₂y₂, the position coordinates of O are (b₁, b₂) and for the mass center C they are (c_c1, c_c2). The body‐fixed coordinate system is obtained by rotating o₁x₁y₁ about the z₂ axis with angle θ. Hence,

(7.85)

(7.86)

(7.87)

(7.88)

(7.89)

where

The external forces and moments acting on the rigid body are equivalent to the external forces and moments acting on the mass center of the rigid body. Therefore, we only consider the external forces and moments acting on the mass center of rigid bodies in the following analysis. The motion equation of the mass center is

(7.90)

(7.91)

where m is the mass of the rigid body and (x_C, y_C) are the position coordinates relative to the inertial coordinate system, while q_x,I and q_y,I are internal forces acting on the input end of the rigid body, and q_x,O and q_y,O are internal forces acting on the output end. f_x,C and f_y,C are external forces acting on the mass center of the rigid body.

According to the theorem of absolute angular momentum with respect to moving point I

(7.92)

The projection of Equation (7.92) on the inertial coordinate system can be written as

(7.93)

where is the absolute moment of momentum of the rigid body with respect to origin I of the body‐fixed coordinate system, and J_I is the rotational inertia relative to point I. is the absolute angular velocity of the rigid body. r_IC is the radius vector of the mass center C of the rigid body with respect to the point I, and a_I is the absolute acceleration of point I.

Linearizing Equations (7.88) and (7.89) using Equations (7.63) and (7.64) yields

(7.94)

(7.95)

Substituting Equations (7.86) and (7.87) into Equations (7.90) and (7.91), and linearizing using Equations (7.27), (7.63) and (7.64), we obtain

(7.96)

(7.97)

Substituting Equations (7.96) and (7.97) into Equation (7.93), and linearizing using Equation (7.27), we obtain

(7.98)

Combining Equations (7.85) and (7.95) to (7.98), the transfer equation of the rigid body moving in a plane with one input and one output end can be expressed as

(7.99)

The state vector is

(7.100)

and the transfer matrix is

(7.101)

where

(7.102)

(7.103)

7.4.2 Planar Rigid Body with Multiple Input and Multiple Output Ends

In this section, we derive the transfer equations and transfer matrices of rigid bodies with multiple input and multiple output ends, rigid bodies with one input end and multiple output ends, and rigid bodies with one output end and multiple input ends. For rigid bodies with multiple input and multiple output ends, we cannot obtain the main transfer equation from one point to another point. Only the matrix relation between input and output ends can be obtained.

Figure 7.8 is the dynamic model of a rigid body moving in a plane. Assuming the input ends are and the output ends are , the mass center is C. The inertial coordinate system is oxy. The coordinate system o₁x₁y₁ is fixed on the first input end I₁, and parallels the inertial coordinate system. The coordinate system O₂x₂y₂ is the body‐fixed coordinate system and its origin is the same as o₁x₁y₁. The position coordinates of the input and output ends in the coordinate system O₂x₂y₂ are (a_1,1, a_2,1), (a_1,2, a_2,2), and (b_1,1, b_2,1), (b_1,2, b_2,2) , respectively. The position coordinates of the mass center of the rigid body in the coordinate system O₂x₂y₂ are (c_c1, c_c2).

Suppose we rotate o₁x₁y₁ by angle θ to obtain body‐fixed coordinate O₂x₂y₂. If , it is a one input end and one output end rigid body. When and , it is a rigid body with one input end and multiple output ends. When and , it is a rigid body with multiple input ends and one output end. When and , it is a rigid body with multiple input ends and multiple output ends.

Similarly to Equations (7.85) to (7.89), considering Equations (7.63) and (7.64) we obtain

(7.104)

(7.105)

(7.106)

(7.107)

(7.108)

where

(7.109)

(7.110)

(7.111)

(7.112)

(7.113)

(7.114)

(x_C, y_C) are the position coordinates of the mass center of a rigid body with respect to the inertial coordinate system. and are the position coordinates of the input end I_j and the output end O_j in the inertial coordinate system.

In the following analysis, we only consider the external force and external moment acting on the mass center of the rigid body. According to the dynamic equation of the mass center of a rigid body

(7.115)

(7.116)

where m is the mass of the rigid body, and are the internal forces acting on the input end of the rigid body, and are the internal forces acting on the output end of the rigid body, and f_x,C and f_y,C are the external forces acting on the mass center of the rigid body.

Based on the theorem of the absolute angular momentum on the moving momentum center, which is the first input end I₁, we obtain

(7.117)

The projection of Equation (7.117) on the inertial coordinate system can be written as

(7.118)

Substituting Equations (7.27), (7.106), (7.63) and (7.64) into Equations (7.115) and (7.116) yields

(7.119)

(7.120)

Substituting Equations (7.27), (7.119) and (7.120) into Equation (7.118) yields

(7.121)

Equations (7.107) and (7.104) can be written in matrix form

(7.122)

where

(7.123)

(7.124)

Equations (7.105) and (7.108) can be written in matrix form

(7.125)

where

(7.126)

Equations (7.119) to (7.121) can be written in matrix form

(7.127)

where

(7.128)

(7.129)

(7.130)

The state vector of the rigid body with multiple input ends and multiple output ends can be defined as

(7.131)

According to Equations (7.122), (7.125) and (7.127), the transfer equation of the rigid body with multiple input ends and multiple output ends is

(7.132)

The transfer matrix is

(7.133)

Similarly, the transfer equations and transfer matrices of the rigid body with multiple input ends and one output end, and the rigid body with one input end and multiple output ends can be obtained, which are provided in the following sections without the detailed deduction procedures.

7.4.2.1 Transfer Equation and Transfer Matrix of a Rigid Body moving in a Plane with Multiple Input Ends and One Output End

State vector

(7.134)

Transfer equation

(7.135)

Transfer matrix

(7.136)

All submatrices are the same as those in Equation (7.133).

For computation convenience, can be expressed by as

(7.137)

7.4.2.2 Transfer Equation and Transfer Matrix of a Rigid Body Moving in a Plane with One Input End and Multiple Output Ends

State vector

(7.138)

Transfer equation

(7.139)

Transfer matrix

(7.140)

All submatrices are the same as those in Equation (7.133).

For computation convenience, can be expressed by as

(7.141)

where

7.5.1 Spatial Rigid Body with One Input End and One Output End

The column matrix of the angular velocity of a rigid body with respect to the inertial coordinate system in the body‐fixed coordinate system of a rigid body is

(7.142)

and its skew symmetric matrix is

(7.143)

The first‐order derivative with respect to time of transformation matrix defined in Equation (7.71) is

The dynamics equation and transfer matrix are deduced using the three spatial axis angles about x–y–z, that is

(7.144)

where

It can be proved that

(7.145)

(7.146)

The first‐order derivative with respect to the time of the angular velocity ω of the rigid body in the inertial coordinate system is

(7.147)

The time derivative of Equation (7.144) is

(7.148)

where

(7.149)

For a rigid body moving in space, as shown in Figure 7.9, the column vectors of the rotation angle and positions are defined as and , respectively. For the input end and output end

(7.150)

(7.151)

where is the coordinate transformation matrix from O₂x₂y₂z₂ to o₁x₁y₁z₁.

If only the external force and moment acting on the mass center of the rigid body are considered, according to the motion equation of the mass center

(7.152)

where m is the mass of the rigid body and is the radius vector of the mass center of the rigid body with respect to the origin of the inertial coordinate system. is the internal force acting on the input end of the rigid body, is the internal force acting on the output end of rigid body and is the external force acting on the mass center of the rigid body.

The projection of Equation (7.152) on the inertial coordinate system can be written as

(7.153)

where is the coordinate column matrix of the mass center of the rigid body with respect to the inertial coordinate system:

(7.154)

According to the theorem of absolute angular momentum to the moving momentum center, we obtain

(7.155)

The projection of Equation (7.155) on the inertial coordinate system can be written as

(7.156)

(7.157)

where , and is the absolute moment of momentum of the rigid body with respect to point I. and are the internal moments acting on points I and O, respectively, is the internal forces acting on point O, is the external moment acting on the mass center of the rigid body, is the acceleration of the point I and is the inertia matrix of the rigid body with respect to point I in the body‐fixed coordinate system.

The transfer matrix is now derived. Substituting Equation (7.83) into Equation (7.151), we obtain

(7.158)

where

, and can be obtained by Equations (7.84), (7.77) and (7.20), respectively.

Equation (7.150) can be written as

(7.159)

Linearizing the left‐hand terms of Equation (7.153) yields

(7.160)

Replacing the subscript O in Equation (7.158) by C, we obtain

(7.161)

Substituting Equation (7.161) into Equation (7.160) yields

(7.162)

where

The coordinate column matrix of angular acceleration in the body‐fixed coordinate system can be obtained, that is

Let

(7.163)

then the coordinate column matrix of angular acceleration in the body‐fixed coordinate system can be rewritten as

(7.164)

Combining Equations (7.144) and (7.77) yields

(7.165)

According to Equations (7.145), (7.164) and (7.165)

(7.166)

Let

(7.167)

(7.168)

then

(7.169)

where

(7.170)

(7.171)

Substituting Equations (7.27) and (7.169) into Equation (7.156) yields

(7.172)

where

Combining Equations (7.158), (7.159), (7.162) and (7.172), the transfer equation of a rigid body moving in space with one input end and one output end can be obtained:

(7.173)

The transfer matrix is

(7.174)

7.5.2 Spatial Rigid Body with Multiple Input and Multiple Output Ends

7.5.2.1 Transfer Equation and Transfer Matrix of a Rigid Body moving in Space with Multiple Input and Multiple Output Ends

The dynamics model of a rigid body moving in space with multiple input and multiple output ends is shown in Figure 7.10. are input ends, are output ends and C is the mass center of the rigid body. The inertial coordinate system is oxyz. The origin of the coordinate system o₁x₁y₁z₁ is the first input end I₁ of the rigid body, which is parallel to the inertial coordinate system. The origin of the body‐fixed coordinate system O₂x₂y₂z₂ is the same as the coordinate system o₁x₁y₁z₁. The coordinates of the input and output ends in the body‐fixed coordinate system O₂x₂y₂z₂ are (a_1,j, a_2,j, a_3,j) and (b_1,j, b_2,j, b_3,j), respectively, and the coordinates of the mass center are (c_c1, c_c2, c_c3). Then;

(7.175)

(7.176)

(7.177)

(7.178)

where and are the coordinates column matrices of the input and output ends in the inertial coordinate system, r_C is the vector of the mass center relative to the origin of the inertial coordinate system, are the coordinates column matrix of the mass center of the rigid body in the inertial coordinate system and A is the transformation matrix from the coordinate system O₂x₂y₂z₂ to o₁x₁y₁z₁.

Linearizing Equations (7.175) and (7.176) yields

(7.179)

where

Linearizing Equations (7.175) and (7.177) yields

(7.180)

where

According to the dynamic equation of the mass center of the rigid body, we obtain

(7.181)

Projecting Equation (7.181) into the inertial coordinate system yields

(7.182)

where m is the mass of the rigid body. are the internal forces at the input ends and are the internal forces at the output ends. f_C is the external force acting on the rigid body mass center.

According to the theorem of absolute angular momentum to the moving momentum center, the dynamics equation of a rigid body with multiple input and multiple output ends can be written as

(7.183)

Projecting Equation (7.183) into the inertial coordinate system yields

(7.184)

where

Linearizing Equations (7.182) and (7.184) yields

(7.185)

where

The state vector of a rigid body with multiple input ends and multiple output ends can be defined as

(7.186)

Combining Equations (7.179), (7.180) and (7.185), the transfer equation of a rigid body with multiple input ends and multiple output ends can be obtained

(7.187)

The transfer matrix is

(7.188)

In a similar manner, the transfer equations and transfer matrices of a rigid body with multiple input ends and one output end, and a rigid body with one input end and multiple output ends can be found. The results are given in the following sections.

7.5.2.2 Transfer Matrix of a Rigid Body with Multiple Input Ends and One Output End

State vector

(7.189)

Transfer equation

(7.190)

Transfer matrix

(7.191)

All submatrices are the same as those in Equation (7.188).

For ease of calculation, can be expressed by , hence

(7.192)

7.5.2.3 Transfer Matrix of a Rigid Body with One Input End and Multiple Output Ends

State vector

(7.193)

Transfer equation

(7.194)

Transfer matrix

(7.195)

All submatrices are the same as those in Equation (7.188).

For ease of calculation, can be expressed by , hence

(7.196)

where

7.6.1 Elastic Hinges

Here, the elastic hinge consists of linear (or nonlinear) spring hinges and linear (or nonlinear) rotary spring hinges. Figures 7.11 and 7.12 illustrate the models for linear spring and nonlinear spring forces, respectively.

Ignore the mass and the initial length of the spring and assume

(7.197)

where is the change in length of a spring in the xy plane. O denotes the output end and I denotes the input end. is the spring force, and K₁ and K₂ are the stiffness coefficients of a nonlinear spring.

Projection of Equation (7.197) into the ox and oy directions results in

(7.198)

Substituting Equation (7.66) into Equation (7.198) yields

(7.199)

where

According to Equation (7.199), we obtain

(7.200)

(7.201)

where

The moment of a nonlinear rotary spring is

(7.202)

where and are the stiffness coefficients.

Similarly, linearizing Equation (7.202), we obtain

(7.203)

For linear spring and rotary spring hinges whose masses are neglected, we obtain

(7.204)

(7.205)

(7.206)

Rewriting Equations (7.200), (7.201) and (7.203) to (7.206) in terms of matrices, the transfer equation of a spring moving in a plane is

(7.207)

The state vector is

and the transfer matrix is

(7.208)

where

For a linear spring, that is, , the transfer matrix can be derived from Equation (7.208) as

(7.209)

7.6.2 Damper Hinges

For a viscous damper, the viscous damped forces and damped moment are

(7.210)

where C_d and are the damping coefficients of translational dampers and rotary dampers, respectively.

Linearizing Equation (7.210), we obtain

(7.211)

(7.212)

(7.213)

From Equations (7.211) to (7.213), the transfer equation of a viscous damper hinge moving in a plane is

(7.214)

The state vector is

and the transfer matrix is

(7.215)

where

7.6.3 Smooth Hinges

For the smooth hinge shown in Figure 7.13, at the two ends the position coordinates and internal forces are equal and the internal torques are constant zero:

(7.216)

The numbers of inboard and outboard bodies of the smooth hinge j are and , respectively. In the following, the rotation angle θ_O of the outboard body of a smooth hinge is discussed for two cases.

7.6.3.1 Smooth Hinge whose Outboard is a Rigid Body and with Internal Torques at the Output End of Zero

If another point of the outboard rigid body is also connected to a smooth hinge or a free boundary, then we can obtain the transfer equation of the outboard rigid body of the smooth hinge

(7.217)

where u₄₁, u₄₂, u₄₃, u₄₅, u₄₆ and u₄₇ are elements of the transfer matrix of the outboard rigid body.

Substituting Equation (7.216) into Equation (7.217) yields

(7.218)

Combining Equations (7.216) and (7.218), and referring to the state vectors defined by Equation (7.21), we obtain the transfer equation of the smooth hinge whose outboard is a rigid body and whose internal torques of the output end are zero, namely

The transfer matrix is

(7.219)

In Equation (7.219), the element u_4,4 could also be written as “1”. However, if we set , the computational precision will be improved. According to the above derivation, if the internal moments of the outboard body at the two ends of the smooth hinge are equal to zero, the above transfer equation and transfer matrix are valid.

7.6.3.2 Smooth Hinge whose Outboard is a Rigid Body and with Nonzero Internal Torques at the Output End

The output end of the outboard rigid body of a smooth hinge is connected to an elastic hinge or a damper hinge, as shown in Figure 7.14. Among the outboard elements of the smooth hinge, if there is such an element whose internal moment is always zero or with free boundaries, then the transfer equation of this smooth hinge can be deduced in the following way. The transfer equation of the subsystem from the smooth hinge to the hinge whose internal moment is zero can be derived as

The transfer matrix is

(7.220)

where are the transfer matrices of the outboard elements of the smooth hinge.

Because the internal moments in the state vectors of the output end of the smooth hinge and the state vector of the output end of the subsystem are all equal to zero, we can obtain the transfer equation and transfer matrix of the smooth hinge which are similar to the form of Equation (7.219), where, are elements of the transfer matrix .

7.7.1 Elastic and Damper Hinges

The elastic damper hinge is composed of a linear spring hinge with a parallel connected damper, combining a rotary spring with a parallel connected damper. As shown in Figure 7.15, element is the elastic joint and damper, whose inboard body and outboard body are rigid bodies j and , respectively. According to the force equilibrium at points O and I, we obtain

(7.221)

where

K_x, K_y and K_z are the stiffness coefficients of the spring along the three axes of the inertial coordinate system, respectively, and C_d,x, C_d,y and C_d,z are the damper coefficients of the translation damper in the three axes of the inertial coordinate system, respectively.

Linearizing the velocity item yields

(7.222)

where

and is defined as in Equation (7.20).

For the rotational spring and damper we obtain

(7.223)

where

The definitions of and can be seen in Equations (7.71) and (7.144). , and are the stiffness coefficients of the rotational spring along the three axes of the inertial system, respectively, and , and are the damper coefficients of the rotational damper about the three axes of the inertial system, respectively.

Linearizing Equation (7.223), we obtain

(7.224)

where

The transfer equation of the spring joint and damper moving in space can be written as

The transfer matrix is

(7.225)

7.7.2 Smooth Ball‐and‐socket Hinge whose Outboard is a Rigid Body and with Internal Torques at the Output End of Zero

For a smooth ball‐and‐socket hinge, using a similar method as for the smooth hinge moving in a plane, its mass and size can be neglected. The position coordinates and internal forces are equal and the internal moments are constant zero at its two ends, that is

(7.226)

The and denote the numbers of the inboard body and outboard body of the smooth ball‐and‐socket hinge j. For a smooth ball‐and‐socket hinge, if its outboard is a rigid body and the internal moment of the output end of this rigid body is zero, then by using the transfer equation of the outboard body of the smooth ball‐and‐socket joint, we obtain:

(7.227)

where , , and are submatrices of the transfer matrix of the smooth ball‐and‐socket joint’s outboard body. is defined in Equation (7.20).

Substituting Equation (7.226) into Equation (7.227) yields

(7.228)

Combining Equations (7.226) and (7.228), the state vectors are derived by Equation (7.20), and the transfer equation of the smooth ball‐and‐socket joint can be derived as

(7.229)

The transfer matrix is

(7.230)

7.7.3 Smooth Ball‐and‐socket Hinge whose Outboard is a Rigid Body and with Nonzero Internal Torques at the Output End

If the internal moment of the output end of the outboard rigid body of the smooth ball‐and‐socket joint is nonzero, the transfer matrix and transfer equation of smooth ball‐and‐socket joint can be obtained by using the same method as in section 7.6.3. The form is the same as in Equation (7.220).

7.9.1 Chain Multibody System Dynamics

For the chain multibody system shown in Figure 7.16, the transfer equations of hinge element j and body element are

(7.231)

where are the transfer matrices of hinge j and body .

The overall transfer equation of the system can be assembled by only using the transfer matrices of these elements, that is

(7.232)

where the overall transfer matrix is

(7.233)

For a chain multibody system, the overall transfer matrix of the system can be obtained by successive multiplication of the transfer matrices of the elements. The order of the overall transfer matrix of the system is equal to the highest order of the transfer matrices of the elements, and it does not change as the number of DOFs in the system increases. For a chain multi‐rigid‐body system moving in a plane, the highest order of the transfer matrix is (7 × 7), while for a chain multi‐rigid‐body system moving in space, the highest order is (13 × 13).

Substituting the boundary conditions into the overall transfer equation of the system, the unknown quantities in the state vectors of the system boundaries can be computed. The state vectors and the motion quantities of each element at time t_i can be computed by repeatedly using the corresponding transfer equations of elements. The quantities of velocity, angular velocity, acceleration and angular acceleration at time t_i can be obtained. Repeating the entire procedure can allow the system motion to be obtained.

7.9.2 Numerical Example of a Chain Multi‐rigid‐body System Moving in a Plane

7.9.3 Numerical Example of a Chain Multi‐rigid‐body System Moving in Space