15.2.1 One‐dimensional Longitudinal Vibration

The state vector is

The transfer matrix is

(15.1)

where K_x is the stiffness of the spring.

15.2.2 Longitudinal Vibration in Plane

The state vector is

The transfer matrix is

(15.2)

where K_x and K_y are the stiffnesses of the springs in the x and y directions, respectively.

15.2.3 Longitudinal Vibration in Space

The state vector is

The transfer matrix is

(15.3)

where K_x, K_y and K_z are the stiffnesses of the springs in the x, y and z directions, respectively.

15.3.1 One‐dimensional Torsional Vibration

The state vector is

The transfer matrix is

(15.4)

where is the torsional stiffness of the rotary spring.

15.3.2 Torsional Vibration in Plane

The state vector is

The transfer matrix is

(15.5)

where are the torsional stiffnesses of the rotary springs about the x and z directions, respectively.

15.3.3 Torsional Vibration in Space

The state vector is

The transfer matrix is

(15.6)

where are the torsional stiffnesses of the rotary springs about the x, y and z axes, respectively.

15.4.1 The Planar Elastic Hinge

The state vector is

The transfer matrix is

(15.7)

where K_x and K_y are the stiffness of the springs in the x and y directions, respectively, and is the torsional stiffness of the rotary spring in the z direction.

15.4.2 The Spatial Elastic Hinge

The state vector is

The transfer matrix is

(15.8)

where

where K_x, K_y and K_z are the stiffnesses of the springs in the x, y and z directions, respectively, and are the torsional stiffness of the rotary springs about the x, y and z axes, respectively.

15.5.1 Lumped Mass with Longitudinal Vibration

The state vector is

The transfer matrix is

(15.9)

where m is the mass of the lumped mass and ω is the eigenfrequency.

15.5.2 Lumped Mass with Longitudinal Vibration in a Plane

The state vector is

The transfer matrix is

(15.10)

15.5.3 Lumped Mass with Longitudinal Vibration in Space

The state vector is

The transfer matrix is

(15.11)

15.6.1 Planar Vibration of a Rigid Body with One Input End and One Output End

The state vector is

The transfer matrix is

(15.12)

where ω is the eigenfrequency, m is the mass of the rigid body and J_I is the moment of inertia with respect to point I. (b₁, b₂) are the position coordinates of the output end O and (c_c1, c_c2) are the position coordinates of the mass center C.

If only the transverse displacement and torsional vibration of the rigid body are considered, the longitudinal displacement can be neglected and the state vector is defined as

The transfer matrix is

(15.13)

15.6.2 Spatial Vibration of a Rigid Body with One Input End and One Output End

The state vector is

The transfer matrix is

(15.14)

where

which is decomposed in a body‐fixed coordinate system with origin the input end I, is the inertia matrix with respect to point I, (b₁, b₂, b₃) are the position coordinates of the output end O and (c_c1, c_c2, c_c3) are the position coordinates of the mass center C.

The state vector is defined as

The corresponding transfer matrix is

(15.15)

where

15.6.3 Spatial Vibration of a Rigid Body with N Input Ends and L Output Ends

The state vectors are

The transfer equation is

(15.16)

The transfer matrices are

(15.17)

The related structural parameters are described in the body‐fixed coordinate system whose origin is the first input end I₁.

15.6.4 Spatial Vibration of a Rigid Body with N Input Ends and One Output End

The state vectors are

The transfer equation is

The transfer matrix is

(15.18)

where is the nth input end. The related structural parameters are described in the body‐fixed coordinate system whose origin is the first input end I₁ and is the inertia matrix with respect to I₁.

15.6.5 Spatial Vibration of a Rigid Body with One Input End and L Output Ends

The state vectors are

The transfer matrix is

(15.19)

where is the lth output end. The related structural parameters are described in the body‐fixed coordinate system whose origin is located at input end I. It should be noted that the transfer equation in this case is .

15.7.1 Euler–Bernoulli Beam with Transverse Vibration

The transfer matrix is

(15.20)

where S, V, U and T are the Кpылoв functions:

where l is the length of the beam, , EI is the bending stiffness of the beam and is the line mass density of the beam.

15.7.2 Timoshenko Beam with Transverse Vibration

The transfer matrix is

(15.21)

where

where l is the length of the beam, EI is the bending stiffness of the beam and is the line mass density of the beam. A is the cross‐section area, I_z is the moment of inertia of the cross‐section with respect to the neutral line and ρ_z is the gyration radius of the cross‐section with respect to the neutral line. is the shear stiffness and κ_s is the shape factor determined by the shape of the cross‐section.

15.7.3 Massless Euler–Bernoulli Beam with Transverse Vibration

The transfer matrix is

(15.22)

Considering the influence of shear deformation, the transfer matrix of the massless Euler–Bernoulli beam with transverse vibration is

(15.23)

15.7.4 Massless Euler–Bernoulli Beam with Axial Load and Shear Deformation

The transfer matrix is

(15.24)

where for the axial compressive force P, and . For the axial tensile force P, and .

15.7.5 Massless Euler–Bernoulli Beam with Elastic Foundation

The transfer matrix is

(15.25)

where

where z , k* is the Winckler (elastic) modulus of the foundation and .

15.7.6 Massless Transverse Vibrational Euler–Bernoulli Beam with Elastic Foundation

The massless transverse vibrational Euler–Bernoulli beam with elastic foundation is shown in Figure 15.4.

The transfer matrix is

(15.26)

where K* is the Winckler (elastic) modulus of the foundation, K′* is the rotational (elastic) modulus of the foundation and r is the gyration radius of the cross‐section about the z axis.

15.7.7 Flexible Point Support

Different kinds of flexible point supports are shown in Figure 15.5.

The transfer matrices of the flexible points supported by spring, rotary spring and elastic hinges are

(15.27)

where K is the translational stiffness of the support and K′ is the torsional stiffness.

If there is a lumped mass at the support point, the rotary inertia of the lumped mass is considered. Then the corresponding transfer matrix is

(15.28)

where m is the mass of the lumped mass and J is the moment of inertia of the lumped mass.

15.7.8 Nonuniform Cross‐section Euler–Bernoulli Beam with Axial Compressive Force

A nonuniform cross‐section Euler–Bernoulli beam with axial compressive force is shown in Figure 15.6.

The transfer matrix is

(15.29)

where

15.8.1 Torsional Vibration of an Elastic Shaft with Uniform Cross‐section

The transfer matrix is

(15.30)

where l is the length of the shaft, and ρ is the mass density of the shaft. J_p is the polar inertia of the cross‐section of the shaft and GJ_p is the torsional stiffness of the shaft.

15.8.2 Torsional Vibration of a Massless Elastic Shaft with Uniform Cross‐section

The transfer matrix is

(15.31)

15.8.3 Massless Torsional Vibrational Elastic Shaft with Elastic Foundation and Uniform Cross‐section

The transfer matrix is

(15.32)

where and K_t is the elastic modulus of the foundation.

15.8.4 Torsional Vibration of a Rigid Shaft with Elastic Foundation

The transfer matrix is

(15.33)

15.8.5 Torsional Vibration of an Elastic Shaft with Elastic Foundation and Uniform Cross‐section

The transfer matrix is

(15.34)

where . If , and Equation (15.32) are used.

15.8.6 Flexible Point Support of a Lumped Mass on a Torsional Vibrational Shaft

The transfer matrix is

(15.35)

where is the torsional stiffness of the flexible point support and J is the moment of inertia of the lumped mass.

15.9.1 Longitudinal Vibration of an Elastic Rod with Uniform Cross‐section

The transfer matrix is

(15.36)

where l is the length of the rod, EA is the tensile stiffness, and is the mass per unit length.

15.9.2 Longitudinal Vibration of a Massless Elastic Rod with Uniform Cross‐section

The transfer matrix is

(15.37)

15.9.3 Flexible Point Support of a Lumped Mass on a Translational Vibrational Rod

The transfer matrix is

(15.38)

where K_b is the elastic coefficient of the flexible point support and m is the mass of the lumped mass.

15.10.1 Euler–Bernoulli Beam Vibrating Longitudinally in the x Direction and Transversely in the y Direction

The state vector is

The transfer matrix is

(15.39)

where 0 ≤ x₁ ≤ l, and .

15.10.2 Euler–Bernoulli Beam Vibrating Torsionally about the x Axis and Transversely in the z Direction

The state vector is

The transfer matrix is

(15.40)

where 0 ≤ x₁ ≤ l, and .

15.10.3 Euler–Bernoulli Beam Vibrating Torsionally about the x Axis, Longitudinally in the x Direction and Transversely in the y and z Directions

The state vector is

The transfer matrix is

(15.41)

where

If the longitudinal motion in the x direction and the rotation about x axis are regarded as rigid motion, then

(15.42)

15.11.1 Dynamic Equations of an Anisotropic Rectangle Plate

For an orthogonal anisotropic rectangle plate, as shown in Figure 15.7, the basic equations of the bending motion [38] are

(15.43)

where

(15.44)

(15.45)

(15.46)

P_x and P_y are the planar pressure, w is the transverse deflection, D_x, D_y and D_xy are the flexural stiffness, and μ_x and μ_y are the Poisson’s ratios with respect to the x and y axes, respectively. ρ is the mass density, K is the elastic modulus of the foundation and p is the distributed transversely loading intensity. E_x and E_y are the material elastic moduli with respect to the x and y axes, respectively. G is the shear elastic modulus of the material, h is the thickness of the plate and ΔT is the temperature variation. α_x and α_y are the heat expansion coefficients with respect to the x and y axes, respectively. M_x and M_y are the bending moments, M_yx and M_xy are torsional moments, and Q_x and Q_y are the shear forces.

Equation (15.43) can be written as

(15.47)

where .

If the pressures P_x and P_y are replaced by and , then Equation (15.43) and Equation (15.47) are valid for the case with a tensile force.

If the proposed plate is simply supported on both sides of and , the variables w, θ, M_x and Q_x can be expanded in Fourier series. Deleting the variable y in Equation (15.46) yields

(15.48)

The mechanical load and heat load can be expanded as follows

(15.49)

therefore

(15.50)

Substituting Equations (15.49) and (15.50) into Equations (15.43) and (15.52) gives

(15.51)

The state vector is defined as

(15.52)

The corresponding transfer matrix can be obtained as follows.

15.11.2 Massless Isotropic Rectangle Plate

The transfer matrix is

(15.53)

where , and , and

15.11.3 Isotropic Rectangle Plate with Axial Force

The transfer matrix is

(15.54)

where , , and .

For the plate with free vibration

For the plate under the planar pressure and , we get

15.11.4 General Rectangle Plate

The transfer matrix is

(15.55)

where

P_x and P_y are the pressures, if the planar force is tensile, P_x and P_y should be replaced by , and e₀, e₁, e₂, e₃ and e₄ are shown in Table 15.1.

e0
e1		A
e2		B
e3					is fit for
					is fit for
					is fit for
					is fit for

A, B, C, D, g, a and b in Table 15.1 are shown in Table 15.2.

For the orthogonal anisotropic plate

For the isotropic plate

15.12.1 Dynamic Equations of a Disk

As shown in Figure 15.8, the disk acts on a symmetrical load. Its basic dynamic equations of flexural motion are

(15.56)

where

(15.57)

(15.58)

(15.59)

P_r and P_ϕ are the planar pressures, w is the transverse deflection and D_r, D_ϕ and D_rϕ are the bending stiffnesses. μ_r and μ_ϕ are the Poisson’s ratios with respect to the r and ϕ directions, respectively. ρ is the mass density, and i_r and i_ϕ are the rotational gyration radius around the axial and tangent directions, respectively. K is the elastic modulus of the foundation, p is the distributed transverse loading intensity, and E_r and E_ϕ are the material elastic modulus with respect to the direction of r and ϕ. G is the material shear elastic modulus, h is the thickness of the plate and ΔT is the temperature variation. α_r and α_ϕ are the heat expansion coefficients with respect to the direction of r and ϕ. M_r and M_ϕ are the bending moments, and M_rϕ and M_ϕr are the torsional moments.

If an isotropic disk only vibrates transversely, Equation (15.56) is usually written as the fourth‐order basic equation:

(15.60)

where

If the pressures P_x and P_y are replaced by , then Equation (15.56) or Equation (15.60) is also valid for tensile forces.

The variables w, θ, M_r and Q_r can be expanded in Fourier series as follows

(15.61)

(15.62)

For a symmetrical motion (), Equation (15.61) can be further simplified:

(15.62)

The mechanical load and heat load are expanded as follows:

(15.63)

Substituting Equations (15.61) and (15.63) into Equation (15.56) yields

(15.64)

where or .

The state vector is defined as

(15.65)

The corresponding transfer matrix can be obtained as follows.

15.12.2 Massless Isotropic Disk

The transfer matrix is

1. 
   (15.66a)
2. 
   (15.66b)
3. m ≥ 2
   (15.66c)

15.12.3 Massless Rigid Disk

The transfer matrix is

1. 
   (15.67a)
2. 
   (15.67b)
3. m ≥ 2
   (15.67c)

15.12.4 Massless Disk with a Rigid Support at its Center

The transfer matrix is

1. 
   (15.68)
2. 

   The transfer matrix can be obtained from Equation (15.66b) or Equation (15.66c).

15.12.5 Massless Disk with Planar Symmetrical Pressure P_r

The transfer matrix is

(15.69)

where J₀(αr) and J₁(αr) are the Bessel functions and .

15.12.6 Symmetrical Disk

The transfer matrix is

(15.70)

where

15.13.1 Massless Beam Strip

The beam strip i in a two‐dimensional plate, as shown in Figure 15.9, comprises m massless elastic beams. The state vectors of its input and output ends are defined as

(15.71)

(15.72)

where

(15.73)

(15.74)

The sequence number of the row of beam element (i, j) is denoted by index j, .

The transfer equation of the beam strip i is

(15.75)

The transfer matrix is

(15.76)

where

(15.77)

is the transfer matrix of single beam element (i, j), and EI_i,j, l_i,j and (GJ_p)_i,j are the bending stiffness, length and torsional stiffness of the beam element (i, j), respectively.

15.13.2 Lumped Mass Strip

The lumped mass strip i in a two‐dimensional plate, as shown in Figure 15.10, comprises m lumped masses and massless beams. The state vectors of its input and output ends are defined as

(15.78)

(15.79)

where

(15.80)

(15.81)

The sequence number of the row of lumped mass element (i, j) is denoted by index j.

The transfer equation of the lumped mass strip i is

(15.82)

The transfer matrix is

(15.83)

where

(15.84)

(15.85)

(15.86)

(15.87)

If the beam element (i, j) has a simply supported boundary, when and , we obtain

(15.88)

(15.89)

If the beam element (i, j) has a fixed boundary, when and , in Equation (15.86) can be determined by Equation (15.87).

If the beam element (i, j) has a free boundary, when and ,

(15.90)

(15.91)

15.14.1 Dynamic Equations of a Thick‐walled Cylinder

As shown in Figure 15.11, the basic equations [38] for the radial motion of a thick‐walled cylinder are

(15.92)

where

(15.93)

r and ϕ are the radial coordinate and the circumferential coordinate. u is the radial displacement, σ_r is the radial stress, σ_ϕ is the shear stress, p_r is the distributed radial loading intensity, ρ is the mass density, α is the heat expansion coefficient, ΔT is the temperature variation, G is the material shear elastic modulus, λ is the Reynolds coefficient, E is the material elastic modulus and μ is the material Poisson’s ratio.

The state vector is defined as

(15.94)

The corresponding transfer matrix can be obtained as follows.

15.14.2 Massless Isotropic Cylinder

The transfer matrix is

(15.95)

where a_k is the radial coordinate of the inner surface of the cylinder.

15.14.3 Massless Isotropic Cylinder without a Center Hole

The transfer matrix is

(15.96)

15.14.4 Isotropic Cylinder

The transfer matrix is

(15.97)

where

J_γ(βr) and y_γ(βr) are the first type and second type of Bessel functions with λ, respectively.

15.14.5 Isotropic Cylinder without a Center Hole

The transfer matrix is

(15.98)

where and J₁(βr), J₂(βr) are the first kind Bessel functions.

15.15.1 Dynamic Equations of a Thin‐walled Cylinder

As shown in Figure 15.12, the basic equations [38] for the radial motion and flexural displacement of a thin‐walled cylinder are

(15.99)

where x and ϕ are the axial coordinate and the circumferential coordinate, v is the radial displacement, θ is the rotational angle of the displacement course, R is the radius of the cylinder, M is the axial moment per unit circumferential length, Q is the shear force per unit circumferential length, D_Q is the shear stiffness, K_x and K_ϕ are the extention stiffnesses, D_x and D_ϕ are the flexural stiffnesses, p and c are the distributed loading intensities, ρ is the mass density, μ_x and μ_ϕ are the Poisson ratios of the material, r_y is the gyration radius of the cross‐section around the y axis, P_x is the axial pressure, and P_Tx and P_Tϕ are the temperature effect forces.

For an isotropic homogeneous material

where α is the heat expansion coefficient, ΔT is the temperature variation and E is the material elastic modulus.

The state vector is defined as

(15.100)

The corresponding transfer matrix can be obtained as follows.

15.15.2 Massless Thin‐walled Cylinder with Shear Deformation

The transfer matrix is

(15.101)

15.15.3 General Thin‐walled Cylinder

The transfer matrix is

(15.102)

where

e₀, e₁, e₂, e₃ and e₄ are shown in Table 15.3.

e0		0
e1		1	A
e2		x	B
e3
e4

A, B, C, D, g, a and b in Table 15.3 are shown in Table 15.4.

			,
			,

15.17.1 Linearization

1. Linearization of velocity (angular velocity) and acceleration (angular acceleration) leads to
   

   According to the different numerical integration methods, the choices of , , and are shown in Table 7.2.

2. Linearization of nonlinear functions

   See Table 7.3.

3. Linearization of the coordinate transformation matrix

   The direction cosine matrix may be given as rotations about the spatially fixed three axes . Thus the angular velocity vector in the body‐fixed reference system is

   

The linearization of the direction cosine matrix is

(15.109)

where

(15.110)

15.17.2 State Vectors

1. Planar rigid body
   (15.111)
2. Spatial rigid body
   (15.112)
3. Planar beam
   (15.113)
4. Spatial beam
   (15.114)

15.18.1 Spring Hinge whose Inboard and Outboard Bodies are Planar Rigid Bodies

Neglecting the mass and the initial length of the spring, the constitutive relationships of the nonlinear spring hinge are

where is a variable quantity of the length of the spring and is the elastic force. K₁ and K₂ are the longitudinal stiffness coefficients of the nonlinear translational spring, and and are the torsional stiffness coefficients of the nonlinear rotational spring.

The transfer matrix is

(15.115)

where

a₁, a₂, a₃, b₁, b₂ and b₃ are the linearization coefficients, as shown in Equation (8.199).

15.18.2 Damper Hinge whose Inboard and Outboard Bodies are Planar Rigid Bodies

For a viscous damper hinge whose inboard body and outboard body are planar rigid bodies, the damper force and damper torques are

where C_d and are the translational and rotary damper coefficients of dampers.

The transfer matrix is

(15.116)

where

C, , , , , and are the linearlization coefficients.

15.18.3 Spring and Damper Hinge whose Inboard and Outboard Bodies are Spatial Rigid Bodies

The transfer matrix is

(15.117)

where

15.19.1 Smooth Pin Hinge whose Inboard and Outboard Bodies are Planar Rigid Bodies

The transfer matrix is

(15.118)

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

15.19.2 Smooth Ball‐and‐socket Hinge whose Inboard and Outboard Bodies are Spatial Rigid Bodies

The transfer matrix is

(15.119)

where are elements of the transfer matrix of the outboard rigid body.

15.20.1 Rigid Body with One Input End and One Output End Moving in a Plane

The transfer matrix is

(15.120)

where

15.20.2 Planar Rigid Body with Multiple Input and Multiple Output Ends

The state vector is

(15.121)

The transfer equation is

(15.122)

The transfer matrix is

(15.123)

Where

15.20.3 Planar Rigid Body with Multiple Input Ends and One Output End

The state vector is

(15.124)

The transfer equation is

(15.125)

The transfer matrix is

(15.126)

where the submatrices are the same as in Equation (15.102).

15.20.4 Planar Rigid Body with One Input End and Multiple Output Ends

The state vector is

(15.127)

The transfer equation is

(15.128)

The transfer matrix is

(15.129)

where the submatrices are the same as those in Equation (15.123).

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

The transfer equation is

(15.130)

The transfer matrix is

(15.131)

where

15.21.2 Spatial Rigid Body with Multiple Input and Multiple Output Ends

The state vector is

(15.132)

The transfer equation is

(15.133)

The transfer matrix is

(15.134)

where

is determined by Equation (7.84), are determined by Equation (7.77), and m is the mass of the rigid body. f_C is an external force acting on the mass center of the rigid body.

15.21.3 Spatial Rigid Body with Multiple Input Ends and One Output End

The state vector is

(15.135)

The transfer equation is

(15.136)

The transfer matrix is

(15.137)

where the submatrices are the same as in Equation (15.134).

15.21.4 Spatial Rigid Body with One Input End and Multiple Output Ends

The state vector is

(15.138)

The transfer equation is

(15.139)

The transfer matrix is

(15.140)

where the submatrices are the same as in Equation (15.134).

15.24.1 Fixed Hinge whose Inboard Body is a Rigid Body and whose Outboard Body is a Euler–Bernoulli Beam Moving in a Plane

The state vectors are

(15.147)

The transfer equation is

(15.148)

The transfer matrix is

(15.149)

where all elements are the same as in Equation (8.135).

15.24.2 Fixed Hinge whose Inboard and Outboard Bodies are Euler–Bernoulli Beams Moving in a Plane

The state vectors are

(15.150)

The transfer equation is

(15.151)

The transfer matrix is

(15.152)

where

θ_O is the orientation angle of the output end of the fixed hinge, that is, the orientation angle of the body‐fixed coordinate system of its outboard beam. is the eigenvector of the inboard beam. The n is the highest order of the modes of the beam connected with the fixed hinge. The other elements are the parameters of the outboard beam, and their meanings are the same as in Equation (8.116).

15.24.3 Fixed Hinge whose Inboard Body is a Euler–Bernoulli Beam and whose Outboard Body is a Rigid Body Moving in a Plane

The state vectors are

(15.153)

The transfer equation is

(15.154)

The transfer matrix is

(15.155)

where the meanings of all the elements are the same as in Equation (8.135).

15.25.1 Fixed Hinge whose Inboard Body is a Rigid Body and whose Outboard Body is a Beam Moving in Space

The state vectors are

(15.156)

The transfer equation is

(15.157)

The transfer matrix is

(15.158)

where

is the kth natural frequency, is determined by Equation (7.84), are determined by Equation (7.77), and is the coordinate transform matrix. EI, l and are the bending stiffness, length and mass per unit length of the beam, respectively. and (f_2z and f_2y) are distributing external torques (forces) with respect to the body‐fixed coordinate system. Y^j(x₂) and Z^j(x₂) are the eigenvectors of the outboard beam.

15.25.2 Fixed Hinge whose Inboard and Outboard Bodies are Euler–Bernoulli Beams Moving in Space

The state vectors are

(15.159)

The transfer equation is

(15.160)

The transfer matrix is

(15.161)

where

Ω_k is the kth natural frequency, is determined by Equation (7.84), are determined by Equation (7.77), and is the coordinate transform matrix. EI, l and are the bending stiffness, length and mass per unit length of the beam, respectively. Y^j(x₂) and Z^j(x₂) are the eigenvectors of the outboard beam. f_2z and f_2y ( and ) are distributing external forces (torques) with respect to the body‐fixed coordinate system.

15.25.3 Fixed Hinge whose Inboard Body is a Euler–Bernoulli Beam and whose Outboard Body is a Rigid Body Moving in Space

Deleting the row elements corresponding to generalized coordinates in Equation (15.161), the transfer equation and transfer matrix of a fixed hinge whose inboard body is a beam and whose outboard body is a rigid body moving in space can be obtained.

The state vectors are

(15.162)

The transfer equation is

(15.163)

The transfer matrix is

(15.164)

where the meanings of all the elements are the same as in Equation (15.161).

15.26.1 Smooth Hinge whose Inboard and Outboard Bodies are Euler–Bernoulli Beams Moving in a Plane

The state vectors are

(15.165)

The transfer equation is

(15.166)

The transfer matrix is

(15.167)

where

u_4,1, u_4,2, , u_4,10 are the elements of the transfer matrix of the outboard beam. EI is the bending stiffness of the beam, is the mass per unit length of the beam and l is the length of the beam. f_2,y(x₂, t) are the distributed external forces acted on the beam in the y₂ direction and m′ is the distributed external torque acted on the beam. and are determined by Equation (7.62).

15.26.2 Smooth Hinge whose Inboard Body is a Rigid Body and whose Outboard Body is a Euler–Bernoulli Beam Moving in a Plane

The state vectors are

(15.168)

The transfer equation is

(15.169)

The transfer matrix is

(15.170)

where the meanings of all the elements are the same as in Equation (15.167).

15.26.3 Smooth Pin Hinge whose Inboard Body is a Euler–Bernoulli Beam and whose Outboard Body is a Rigid Body Moving in a Plane

The state vectors are

(15.171)

The transfer equation is

(15.172)

The transfer matrix is

(15.173)

where the meanings of all the elements are the same as in Equation (7.219).

15.27.1 Smooth Ball‐and‐socket Hinge whose Inboard Body is a Rigid Body and whose Outboard Body is a Euler–Bernoulli Beam Moving in Space

The state vectors are

(15.174)

The transfer equation is

(15.175)

The transfer matrix is

(15.176)

where

Ω_k is the kth natural frequency, is determined by Equation (7.84), are determined by Equation (7.77) and is the coordinate transform matrix. l and are the length and mass per unit length of the beam, respectively. Y^j(x₂) and Z^j(x₂) are the eigenvectors of the outboard beam. f_2z and f_2y ( and ) are distributing external forces (torques) with respect to the body‐fixed coordinate system.

15.27.2 Smooth Ball‐and‐socket Hinge whose Inboard and Outboard Bodies are Euler–Bernoulli Beams Moving in Space

The state vectors are

(15.177)

The transfer equation is

(15.178)

The transfer matrix is

(15.179)

where the meanings of all the elements are the same as in Equation (15.176).

15.27.3 Smooth Ball‐and‐socket Hinge whose Inboard Body is a Euler–Bernoulli Beam and whose Outboard Body is a Rigid Body Moving in Space

The state vectors are

(15.180)

The transfer equation is

(15.181)

The transfer matrix is

(15.182)

where the meanings of all the elements are the same as in Equation (15.176).

15.28.1 Elastic Hinge whose Inboard and Outboard Bodies are Beams Moving in a Plane

The state vectors are

(15.183)

The transfer equation is

(15.184)

The transfer matrix is

(15.185)

where

For example

K₁ and K₂ are the stiffness coefficients of the nonlinear spring, and are the torsional stiffness coefficients of the nonlinear rotary spring. l is the length of the beam. is the mass per unit length of the beam, f_2,y is the distributed external force acted on the beam in the y₂ direction and m′ is the distributed external torque acted on the beam. Y^k(x₂) is the eigenvector of the outboard beam, is the eigenvector of the inboard beam, and are determined by Equation (8.109), and and are determined by Equation (7.62).

15.28.2 Elastic Hinge whose Inboard Body is a Rigid Body and whose Outboard Body is a Euler–Bernoulli Beam Moving in a Plane

The state vectors are

(15.186)

The transfer equation is

(15.187)

The transfer matrix is

(15.188)

where the meanings of all the elements are the same as in Equation (15.185). In the computation should be used.

15.28.3 Elastic Hinge whose Inboard Body is a Euler–Bernoulli Beam and whose Outboard Body is a Rigid Body Moving in a Plane

The state vectors are

(15.189)

The transfer equation is

(15.190)

The transfer matrix is

(15.191)

where and the meanings of all the other elements are the same as in Equation (15.185).

15.29.1 Elastic Hinge whose Inboard and Outboard Bodies are Euler–Bernoulli Beams Moving in Space

The state vectors are

(15.192)

The transfer equation is

(15.193)

The transfer matrix is

(15.194)

where is a matrix and is a matrix.

have the same meaning as in Equation (8.191). is the torsional stiffness of the rotary springs, is the stiffness of the springs, and K_x, K_y and K_z and are the stiffness coefficients of the linear springs and the torsional stiffness coefficients of the rotary springs, respectively.

15.29.2 Elastic Hinge whose Inboard Body is a Rigid Body and whose Outboard Body is a Euler–Bernoulli Beam Moving in Space

The state vectors are

(15.195)

The transfer equation is

(15.196)

The transfer matrix is

(15.197)

where the meanings of all the elements are the same as in Equation (15.194).

15.29.3 Elastic Hinge whose Inboard Body is a Euler–Bernoulli Beam and whose Outboard Body is a Rigid Body Moving in Space

The state vectors are

(15.198)

The transfer equation is

(15.199)

The transfer matrix is

(15.200)

where the meaning of is the same as in Equation (9.227) and the meanings of and are the same as in Equation (9.223).

15.30.1 Vibration System under Real‐time Control

The controlled vibration system is shown in Figure 15.15. The system is mounted on the lumped mass k, where the controlled force acting on the displacement, velocity and acceleration of lumped mass k is

(15.201)

The transfer equation of the lumped mass k under real‐time control can be obtained as

(15.202)

where

(15.203)

m_p is the mass of the lumped mass p. F_p is a simple harmonic external force acting on the lumped mass p with frequency Ω.

15.30.2 Controlled Branched System

A controlled branched system is shown in Figure 9.32. The state vectors of each connection point are

(15.204)

(15.205)

(15.206)

, , , , , and have the same form as , and have the same form as , and , , , and have the same form as .

(15.207)

(15.208)

The overall transfer equation is

(15.209)

The state vector is

The transfer matrix is

(15.210)

where

15.31.1 Vibration System under Real‐time Control

For the controlled vibration system shown in Figure 15.15, linearizing Equation (15.201) yields

(15.211)

The transfer equation of the controlled lumped mass p is

(15.212)

where

(15.213)

(15.214)

m_p is the mass of the lumped mass p. F_p is the external force acting on the lumped mass p.

For the delay controlled system, the control force can be seen as an external force related to the previous time motion state. By adding the control force into the external force submatrix of the corresponding transfer matrix, the controlled system can be considered as a system without control. The control force F_p,c in Equation (15.201) is a function of the motion quantities , and :

(15.215)

where τ is the delay time. Adding the control force F_p,c into the element u₂₃ of the transfer matrix of the element p, we obtain

(15.216)

15.31.2 Controlled Flexible Manipulator System

The controlled planar flexible manipulator system assembled by hub 1 and flexible arm 3, featuring surface‐bonded piezoceramics and piezofilms, is shown in Figure 9.42a. m₁, r₁, , and are the mass, gyration radius, moment of inertia, desired orientation angle and desired orientation angular velocity of the hub 1, respectively. θ₁ and are the actual orientation angle and actual angular velocity, respectively. EI₃, A₃, l₃, b and t_b are the bending stiffness, cross‐section area, length, width and thickness of flexible arm 3, respectively. E_a, l_a, t_a and d₃₁ are the piezoelectric elastic modulus, length, thickness and strain constant of the segmented piezoelectric ceramic (PZT) actuator. K_p and K_v are the proportional gain coefficient and velocity gain coefficient of the servomotors, respectively. τ₀(t) is the control torque of the motor, K_ai is the gain coefficient of the segmented PZT actuator, V_i is the driven voltage applied to segmented PZT actuator i and u is the deformation of the flexible arm. Y^k(x₂) is the kth generalized eigenvector describing the deformation of the flexible arm. , is the position of each piezofilm sensor/PZT on the corresponding flexible arm in the body‐fixed reference frames. Adopting the proportional‐differential (PD) controller and modal velocity feedback control on the PZT actuators, the transfer matrices of the rigid body and beam under control are derived as follows.

Considering only the control moment related to its feedback state, the transfer matrix of the rigid body under control can be obtained as

(15.217)

where

The meanings of u₄₁, u₄₂, u₄₅, u₄₆, u₅₇ and u₆₇ are the same as in Equation (7.101).

Considering the control moment of the PZT actuators, the transfer matrix of the beam under control is

(15.218)

Where

(15.219)

Considering the distributed moment of the PZT actuators, combining the control equation of PZT actuators and the numerical integration procedure, if the highest order of the modes considered is , then the state vectors of the fixed hinge connected to the beam under control can be defined as

(15.220)

The transfer equation is

(15.221)

The transfer matrix is

(15.222)

where

Adopting the PD controller and modal velocity feedback control on the PZT actuators, the transfer matrix of the equivalent control element is

(15.223)

where is the feedback parameter matrix related to the feedback from beam 3 to hub 1. For the modal velocity feedback control, that is

(15.224)