Rigid-Flexible Coupling Dynamics Analysis of Boom-Hoisting System of Wind Power Crane

Abstract

Based on the slenderness of wind power crane boom and the complexity of working environment, a mathematical model of rigid-flexible coupling dynamics including wind load and vibration characteristic analysis is carried out for the boom-hoisting system. The equivalent spring-damping system, spatial pendulum system and elastic double-force rod are used to simulate the elastic vibration of boom, the swing of lifting weight and the elastic vibration of wire rope, respectively. The kinematic characteristics of each component are described using a hybrid coordinate system. Simulation of wind loads through an instantaneous wind model consisting of the average and pulsating wind. Derivation of dynamic model for wind power crane under wind load based on the Lagrange equation. Taking a certain type of wind power crane as the research object, based on the derived mathematical model and ADAMS model, the response curves of boom and lifting weight are solved in MATLAB and ADAMS respectively for the rotary working condition, to verify the rationality and accuracy of the mathematical model and to obtain the influence of wind load on the swing angle of lifting weight. The analysis results provide a certain theoretical basis for crane structure design, control system design and wind turbine hoisting.

1 Introduction

Wind power crane is a type of lattice boom crawler crane, which has irreplaceable importance in the field of engineering machinery as the cornerstone of wind turbine installation. The working environment of wind power crane has the characteristics of large lifting scale and long-term operation in strong wind environments [1], which will inevitably cause changes in the deflection and oscillation angle of the lifting weight during hoisting operations. In the early stages, cranes use rigid body dynamics and kinematic elasticity analysis methods to estimate the swing trend of the lifting weight, and it tends to be a large slenderness ratio, which makes the boom vibration gradually obvious during hoisting, so that the coupling effect between the swing of the lifting weight and the boom vibration cannot be ignored.

Multi-body dynamics has made great progress in the field of cranes; however, the development of rigid-flexible coupling dynamics in boom cranes still has numerous problems. Fan et al. [2] established a four-degree-of-freedom dynamics model for the lifting weight of a truck crane, and analyzed the effects of rope length and rotational angular acceleration on the swing of the lifting weight. Ouyang et al. [3] established a nonlinear dynamic model of rotary crane considering the double pendulum effect, and deduced that the swing angle has stronger coupling under the double pendulum effect. All of the above models characterize the dynamics of the lifting weight in terms of a rigid-body dynamics model, ignoring the coupling effect of the boom vibration on the swing of the lifting weight. In order to more realistically reflect the dynamic characteristics of the boom-lifting system, Zheng et al. [4] used the multi-body system transfer matrix method to establish a rigid flexible coupling model of a truck crane, which better reflects the coupling relationship between boom vibration and lifting weight deflection and oscillation, but the equation is too complicated and difficult to solve. Yan et al. [5], Kwang-Phil et al. [6] and Jerman et al. [7] used modal analysis and the finite segment method to describe the elastic vibration of the boom, but it is difficult to apply them to the model with medium and large deformation. Anja et al. [8] utilized a Euler–Bernoulli beam equivalent armature and derive a Euler–Lagrange flexible multibody system model, and the simulation results have a high degree of matching with the experimental measurements.

In summary, scholars at home and abroad have conducted numerous studies on the dynamic characteristics of boom cranes, but the existing dynamic models still have problems in the description of system structure and component motion. Based on this, this paper establishes a mathematical model of rigid-flexible coupling dynamics (hereinafter referred to as the mathematical model) including wind load for the boom-hoisting system. The equivalent spring-damping system, spatial pendulum system and elastic double-force rod are used to simulate the elastic vibration of the boom, the swing of the lifting weight and the elastic vibration of the wire rope, respectively. The kinematic characteristics of each component are described using a hybrid coordinate system. The wind load borne by the model can be simulated using an instantaneous wind model. Establishing a rigid-flexible coupling dynamic model for the rotation, variable amplitude and lifting motion of wind power crane based on the Lagrange equation. Numerical analysis software is applied to solve the response curves of the model under corresponding inputs, and compare them with the ADAMS model results to verify the rationality and accuracy of the mathematical model.

2 Mathematical Modeling of Dynamical Equations

Figure 1 shows the boom-lifting system of wind power crane, and Fig. 2 shows the abstract rigid-flexible coupling dynamic model of wind turbine crane. The global coordinate system o–xyz is established at the center of the crane turntable.

Fig. 1

Boom-hoisting system

Fig. 2

Rigid-flexible coupling dynamic model

In order to simplify the establishment process of Lagrange equations, the following assumptions are made for the crane: (1) Neglect the mass of the wire rope; (2) The lifting weight is simplified as a mass; (3) Neglect the friction between the various institutions in the system; (4) Consider only the stable static wind load on the horizontal plane. Based on the research content of this paper, the Lagrange equation is as follows:

$$\frac{d}{dt}\left( {\frac{{\partial {\varvec{L}}}}{{\partial \dot{\user2{q}}_{{\mathbf{i}}} }}} \right) - \frac{{\partial {\varvec{L}}}}{{\partial {\varvec{q}}_{{\mathbf{i}}} }} + \frac{{\partial {\varvec{F}}_{{\mathbf{c}}} }}{{\partial \dot{\user2{q}}_{{\mathbf{i}}} }} = {\varvec{Q}}_{{{\mathbf{out}}}}$$

(1)

2.1 Dynamic Analysis of Boom

As in Fig. 2, the equivalent spring-damper system concentrates the boom mass at the equivalent mass point (m_r) at the head of the boom; taking the variable amplitude rotation center as the origin, the inertial coordinate system o₁–x₁y₁z₁ at the tail of the boom is established; based on the projection of the boom on the rotary plane at o₁, the following-rotation coordinate system o₁–x₂y₂z₂ is established; and with the spatial position of the boom as the reference, the following-rotation coordinate system o₂-x₃y₃z₃ is established at the equivalent mass point (m_r) to describe the spatial position of the head of the boom. Using the follower coordinate system o_j–x_iy_iz_i, it is known that the vector diameter of the equivalent mass point (m_r) in the global coordinate system o–xyz is given by

$${\varvec{r}}_{{\mathbf{r}}} = {\varvec{R}}_{{\upalpha }} \left[ {{\varvec{R}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) + {\varvec{r}}_{{\mathbf{a}}} } \right]$$

(2)

where, ${\varvec{r}}_{{\mathbf{b}}} = [\begin{array}{*{20}c} {l_{{\text{b}}} } & 0 & 0 \\ \end{array} ]^{{\text{T}}}$ is the position vector of the mass point when the boom is not deformed in the coordinate system o₂–x₃y₃z₃, $l_{{\text{b}}}$ is the length of the boom; ${\varvec{r}}_{{\mathbf{a}}} = [\begin{array}{*{20}c} a & 0 & 0 \\ \end{array} ]^{{\text{T}}}$ is the position vector of the amplitude center o₁ in the coordinate system o–xyz, a is the linear distance between the amplitude center o₁ and the center of the turntable o; ${\varvec{u}} = {\varvec{u}}({\varvec{t}}) = [\begin{array}{*{20}c} {u_{{\text{x}}} {(}t{)}} & {u_{{\text{y}}} {(}t{)}} & {u_{{\text{z}}} } \\ \end{array} (t)]^{{\text{T}}}$ is the vibration displacement of the mass point in the coordinate system o₂–x₃y₃z₃ at time t; ${\varvec{R}}_{{{\varvec{\upalpha}}}}$ is the coordinate transformation matrix between the coordinate system o₁–x₂y₂z₂ and o₁–x₁y₁z₁; ${\varvec{R}}_{{{\varvec{\upbeta}}}}$ is the coordinate transformation matrix between the coordinate system o₁–x₂y₂z₂ and o₂–x₃y₃z₃.

The vectorial diameter r_r of the mass point (m_r) is a first-order derivative with respect to time t, and its velocity in the inertial system can be obtained as

$$\dot{\user2{r}}_{{\mathbf{r}}} = {\varvec{R}}_{{{\varvec{\upalpha}}}}{\varvec{\varLambda}}_{{{\varvec{\upalpha}}}} \left[ {{\varvec{R}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) + {\varvec{r}}_{{\mathbf{a}}} } \right] + {\varvec{R}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}{\varvec{\varLambda}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) + {\varvec{R}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{{\varvec{\upbeta}}}} \dot{\user2{u}}$$

(3)

where, ${\varvec{\varLambda}}_{{{\varvec{\upalpha}}}} = \dot{\alpha }(t){\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}$ is the angular velocity correlation matrix of the boom rotation, $\alpha (t)$ is the rotation angle of the boom at time t; ${\varvec{\varLambda}}_{{{\varvec{\upbeta}}}} = \dot{\beta }(t){\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}$ is the angular velocity correlation matrix of the boom luffing, $\beta (t)$ is the luffing angle of the boom at time t.

Let ${\varvec{R}}_{{{\mathbf{\beta r}}}} = {\varvec{R}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}),\,\,{\varvec{r}}_{{{\mathbf{bu}}}} = {\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}$. From Eq. (3), the kinetic energy of the equivalent spring-damped system is given by

$${\varvec{T}}_{{\mathbf{r}}} = \frac{1}{2}\dot{\user2{r}}_{{\mathbf{r}}}^{{\text{T}}} m_{{\text{r}}} \dot{\user2{r}}_{{\mathbf{r}}} { = }\frac{1}{2}\left[ {\begin{array}{*{20}c} {\dot{\alpha }} & {\dot{\beta }} & {\dot{\user2{u}}^{{\text{T}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{{\mathbf{\alpha \alpha }}}}^{{\mathbf{r}}} } & {{\varvec{M}}_{{{\mathbf{\alpha \beta }}}}^{{\mathbf{r}}} } & {{\varvec{M}}_{{{\mathbf{\alpha u}}}}^{{\mathbf{r}}} } \\ {{\varvec{M}}_{{{\mathbf{\beta \alpha }}}}^{{\mathbf{r}}} } & {{\varvec{M}}_{{{\mathbf{\beta \beta }}}}^{{\mathbf{r}}} } & {{\varvec{M}}_{{{\mathbf{\beta u}}}}^{{\mathbf{r}}} } \\ {{\varvec{M}}_{{{\mathbf{u\alpha }}}}^{{\mathbf{r}}} } & {{\varvec{M}}_{{{\mathbf{u\beta }}}}^{{\mathbf{r}}} } & {{\varvec{M}}_{{{\mathbf{uu}}}}^{{\mathbf{r}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{\alpha }} \\ {\dot{\beta }} \\ {\dot{\user2{u}}} \\ \end{array} } \right]$$

(4)

where, ${\varvec{M}}_{{{\mathbf{\alpha \alpha }}}}^{{\mathbf{r}}} { = }m_{{\text{r}}} {\varvec{R}}_{{{\mathbf{\beta r}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{{\mathbf{\beta r}}}}$; ${\varvec{M}}_{{{\mathbf{\alpha \beta }}}}^{{\mathbf{r}}} = \left( {{\varvec{M}}_{{{\mathbf{\beta \alpha }}}}^{{\mathbf{r}}} } \right)^{{\text{T}}} = m_{{\text{r}}} {\varvec{R}}_{{{\mathbf{\beta r}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}} {\varvec{r}}_{{{\mathbf{bu}}}}$; ${\varvec{M}}_{{{\mathbf{\alpha u}}}}^{{\mathbf{r}}} = \left( {{\varvec{M}}_{{{\mathbf{u\alpha }}}}^{{\mathbf{r}}} } \right)^{{\text{T}}} = m_{{\text{r}}} {\varvec{R}}_{{{\mathbf{\beta r}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}$; ${\varvec{M}}_{{{\mathbf{\beta \beta }}}}^{{\mathbf{r}}} = m_{{\text{r}}} {\varvec{r}}_{{{\mathbf{bu}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}} {\varvec{r}}_{{{\mathbf{bu}}}}$; ${\varvec{M}}_{{{\mathbf{\beta u}}}}^{{\mathbf{r}}} = \left( {{\varvec{M}}_{{{\mathbf{u\beta }}}}^{{\mathbf{r}}} } \right)^{{\text{T}}} = m_{{\text{r}}} {\varvec{r}}_{{{\mathbf{bu}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}^{{\text{T}}}$; ${\varvec{M}}_{{{\mathbf{uu}}}}^{{\mathbf{r}}} = m_{{\text{r}}} {\varvec{I}}_{{{\mathbf{3}} \times {\mathbf{3}}}}$, ${\varvec{I}}_{{{\mathbf{3}} \times {\mathbf{3}}}}$ is the unit matrix. Let the boom-head mass correlation array be M_r.

The damping of the equivalent spring-damped system is given by

$${\varvec{F}}_{{{\mathbf{cr}}}} { = }\frac{1}{2}\dot{\user2{u}}^{{\text{T}}} {\varvec{C}}_{{\mathbf{r}}} \dot{\user2{u}}$$

(5)

where, ${\varvec{C}}_{{\mathbf{r}}} = diag(c_{{\text{x}}} ,c_{{\text{y}}} ,c_{{\text{z}}} )$ is the damping coefficient matrix of the equivalent system, $c_{{\text{i}}} = 2m_{{\text{r}}} \xi_{{\text{r}}} \omega_{{{\text{nr}}}}$.

The equivalent spring-damped system considering its elastic potential energy, it is given by

$${\varvec{V}}_{{\mathbf{r}}} = \frac{1}{2}{\varvec{u}}^{{\text{T}}} {\varvec{K}}_{{\mathbf{r}}} {\varvec{u}}$$

(6)

where, ${\varvec{K}}_{{\mathbf{r}}} { = }diag(k_{{\text{x}}} ,k_{{\text{y}}} ,k_{{\text{z}}} )$ is the matrix of stiffness coefficients of the equivalent system, $k_{{\text{i}}} = F_{{\text{i}}} /u_{{{\text{ri}}}}$.

2.2 Dynamic Analysis of Lifting Weight

As in Fig. 2, the spatial pendulum system concentrates the mass of the lifting weight at the equivalent mass point (m_q); at the equivalent mass point (m_q), the coordinate system o₁–x₂y₂z₂ is used as the reference, and the follower coordinate system o₁–x₄y₄z₄ is established so that it is always parallel to the coordinate system o₁–x₂y₂z₂, which is used to describe the spatial position of the lifting weight. Using the rotation coordinate system o_j–x_iy_iz_i and the body coordinate system o₁–x₄y₄z₄, the vector diameter of the equivalent mass point (m_q) in the global coordinate system o–xyz is given by

$${\varvec{r}}_{{\mathbf{q}}} = {\varvec{R}}_{{{\varvec{\upalpha}}}} \left[ {{\varvec{R}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) + {\varvec{l}}_{{\mathbf{q}}} + {\varvec{r}}_{{\mathbf{a}}} } \right]$$

(7)

where, ${\varvec{l}}_{{\mathbf{q}}} = \left[ {\begin{array}{*{20}c} {\left( {l + \delta } \right)\sin \theta_{x} \cos \theta_{y} } & {\left( {l + \delta } \right)\sin \theta_{y} } & { - \left( {l + \delta } \right)\cos \theta_{x} \cos \theta_{y} } \\ \end{array} } \right]^{{\text{T}}}$ is the spatial position vector of the lifting weight in the follower coordinate system o₁–x₄y₄z₄, $l = l(t)$ is the length of the wire rope at time t, $\delta = \delta (t)$ is the elastic deformation of the wire rope at time t, let ${\varvec{\phi}}({\varvec{t}}) = \left[ {\begin{array}{*{20}c} {l(t)} & {\delta (t)} \\ \end{array} } \right]^{{\text{T}}}$; $\theta_{{\text{x}}} = \theta_{{\text{x}}} (t),\theta_{{\text{y}}} = \theta_{{\text{y}}} (t)$ is the oscillation angle in the amplitude plane and the deflection angle in the rotary plane, respectively, let ${\varvec{\vartheta}}({\varvec{t}}) = \left[ {\begin{array}{*{20}c} {\theta_{{\text{x}}} (t)} & {\theta_{{\text{y}}} (t)} \\ \end{array} } \right]^{{\text{T}}}$.

The vectorial diameter r_q of the mass point (m_q) is a first-order derivative with respect to time t, and its velocity in the inertial system can be obtained as

$$\begin{aligned} \dot{\user2{r}}_{{\mathbf{q}}} & = {\varvec{R}}_{{{\varvec{\upalpha}}}}{\varvec{\varLambda}}_{{{\varvec{\upalpha}}}} \left[ {{\varvec{R}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) + {\varvec{l}}_{{\mathbf{q}}} + {\varvec{r}}_{{\mathbf{a}}} } \right] + {\varvec{R}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}{\varvec{\varLambda}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) \\ & \,\,\,\,\,\, + \,{\varvec{R}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{{\varvec{\upbeta}}}} \dot{\user2{u}} + {\varvec{R}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{\mathbf{f}}} \dot{\user2{\phi }} + {\varvec{R}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{\mathbf{J}}} \dot{\user2{\vartheta }} \\ \end{aligned}$$

(8)

where, ${\varvec{R}}_{{\mathbf{f}}} = \left[ {\begin{array}{*{20}c} {\sin \theta_{x} \cos \theta_{y} } & {\sin \theta_{y} } & {{ - }\cos \theta_{x} \cos \theta_{y} } \\ {\sin \theta_{x} \cos \theta_{y} } & {\sin \theta_{y} } & {{ - }\cos \theta_{x} \cos \theta_{y} } \\ \end{array} } \right]^{{\text{T}}}$ is the wire rope velocity matrix; ${\varvec{R}}_{{\mathbf{J}}} = (l + \delta )\left[ {\begin{array}{*{20}c} {\cos \theta_{x} \cos \theta_{y} } & 0 & {\sin \theta_{x} \cos \theta_{y} } \\ { - \sin \theta_{x} \sin \theta_{y} } & {\cos \theta_{y} } & {\cos \theta_{x} \sin \theta_{y} } \\ \end{array} } \right]^{{\text{T}}}$ is the swing velocity matrix of lifting weight.

Let ${\varvec{R}}_{{{\mathbf{\beta q}}}} = {\varvec{R}}_{{{\varvec{\upbeta}}}} ({\varvec{r}}_{{\mathbf{b}}} + {\varvec{u}}) + {\varvec{l}}_{{\mathbf{q}}} + {\varvec{r}}_{{\mathbf{a}}}$. From Eq. (8), the kinetic energy of the spatial pendulum system is given by

$$\begin{gathered} {\varvec{T}}_{{\mathbf{q}}} = \frac{1}{2}\dot{\user2{r}}_{{\mathbf{q}}}^{{\text{T}}} m_{{\text{q}}} \dot{\user2{r}}_{{\mathbf{q}}} \, \hfill \\ \;\;\;\;\;{ = }\frac{1}{2}\left[ {\begin{array}{*{20}c} {\dot{\alpha }} & {\dot{\beta }} & {\dot{\user2{u}}^{{\text{T}}} } & {\dot{\user2{\phi }}^{{\text{T}}} } & {\dot{\user2{\vartheta }}^{{\text{T}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{{\mathbf{\alpha \alpha }}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{\alpha \beta }}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{\alpha u}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{{\varvec{\upalpha}}}{\varvec{\phi}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{{\varvec{\upalpha}}}{\varvec{\vartheta}}}}^{{\mathbf{q}}} } \\ {{\varvec{M}}_{{{\mathbf{\beta \alpha }}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{\beta \beta }}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{\beta u}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{{\varvec{\upbeta}}}{\varvec{\phi}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{{\varvec{\upbeta}}}{\varvec{\vartheta}}}}^{{\mathbf{q}}} } \\ {{\varvec{M}}_{{{\mathbf{u\alpha }}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{u\beta }}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{uu}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{u}}{\varvec{\phi}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\mathbf{u}}{\varvec{\vartheta}}}}^{{\mathbf{q}}} } \\ {{\varvec{M}}_{{{\varvec{\phi}}{{\varvec{\upalpha}}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\varvec{\phi}}{{\varvec{\upbeta}}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\varvec{\phi}}{\mathbf{u}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{\user2{\phi \phi }}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{\user2{\phi \vartheta }}}^{{\mathbf{q}}} } \\ {{\varvec{M}}_{{{\varvec{\vartheta}}{{\varvec{\upalpha}}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\varvec{\vartheta}}{{\varvec{\upbeta}}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{{\varvec{\vartheta}}{\mathbf{u}}}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{\user2{\vartheta \phi }}}^{{\mathbf{q}}} } & {{\varvec{M}}_{{\user2{\vartheta \vartheta }}}^{{\mathbf{q}}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\dot{\alpha }} \\ {\dot{\beta }} \\ {\dot{\user2{u}}} \\ {\dot{\user2{\phi }}} \\ {\dot{\user2{\vartheta }}} \\ \end{array} } \right] \hfill \\ \end{gathered}$$

(9)

where, ${\varvec{M}}_{{{\mathbf{\alpha \alpha }}}}^{{\mathbf{q}}} { = }m_{{\text{q}}} {\varvec{R}}_{{{\mathbf{\beta q}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}} {\varvec{R}}_{{{\mathbf{\beta q}}}}$;${\varvec{M}}_{{{\mathbf{\alpha \beta }}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\mathbf{\beta \alpha }}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} { = }m_{{\text{q}}} {\varvec{R}}_{{{\mathbf{\beta q}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}} {\varvec{r}}_{{{\mathbf{bu}}}}$;${\varvec{M}}_{{{\mathbf{\alpha u}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\mathbf{u\alpha }}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{R}}_{{{\mathbf{\beta q}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}$;${\varvec{M}}_{{{{\varvec{\upalpha}}}{\varvec{\phi}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\varvec{\phi}}{{\varvec{\upalpha}}}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{R}}_{{{\mathbf{\beta q}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}} {\varvec{R}}_{{\mathbf{f}}}$; ${\varvec{M}}_{{{{\varvec{\upalpha}}}{\varvec{\vartheta}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\varvec{\vartheta}}{{\varvec{\upalpha}}}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{R}}_{{{\mathbf{\beta q}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upalpha}}}}^{{\text{T}}} {\varvec{R}}_{{\mathbf{J}}}$; ${\varvec{M}}_{{{\mathbf{\beta \beta }}}}^{{\mathbf{q}}} = m_{{\text{q}}} {\varvec{r}}_{{{\mathbf{bu}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}} {\varvec{r}}_{{{\mathbf{bu}}}}$;${\varvec{M}}_{{{\mathbf{\beta u}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\mathbf{u\beta }}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{r}}_{{{\mathbf{bu}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}^{{\text{T}}}$; ${\varvec{M}}_{{{{\varvec{\upbeta}}}{\varvec{\phi}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\varvec{\phi}}{{\varvec{\upbeta}}}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{r}}_{{{\mathbf{bu}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}^{{\text{T}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{f}}}$; ${\varvec{M}}_{{{{\varvec{\upbeta}}}{\varvec{\vartheta}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\varvec{\vartheta}}{{\varvec{\upbeta}}}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{r}}_{{{\mathbf{bu}}}}^{{\text{T}}}{\varvec{\varGamma}}_{{{\varvec{\upbeta}}}}^{{\text{T}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{J}}}$;${\varvec{M}}_{{{\mathbf{uu}}}}^{{\mathbf{q}}} = m_{{\text{q}}} {\varvec{I}}_{{{\mathbf{3}} \times {\mathbf{3}}}}$; ${\varvec{M}}_{{{\mathbf{u}}{\varvec{\phi}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\varvec{\phi}}{\mathbf{u}}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{f}}}$; ${\varvec{M}}_{{{\mathbf{u}}{\varvec{\vartheta}}}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{{\varvec{\vartheta}}{\mathbf{u}}}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{R}}_{{{\varvec{\upbeta}}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{J}}}$; ${\varvec{M}}_{{\user2{\phi \phi }}}^{{\mathbf{q}}} = m_{{\text{q}}} {\varvec{R}}_{{\mathbf{f}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{f}}}$;${\varvec{M}}_{{\user2{\phi \vartheta }}}^{{\mathbf{q}}} = \left( {{\varvec{M}}_{{\user2{\vartheta \phi }}}^{{\mathbf{q}}} } \right)^{{\text{T}}} = m_{{\text{q}}} {\varvec{R}}_{{\mathbf{f}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{J}}}$; ${\varvec{M}}_{{\user2{\vartheta \vartheta }}}^{{\mathbf{q}}} = m_{{\text{q}}} {\varvec{R}}_{{\mathbf{J}}}^{{\mathbf{T}}} {\varvec{R}}_{{\mathbf{J}}}$. Let the boom-head mass correlation array be M_q.

The damping of the spatial pendulum system mainly considers its internal and external damping of the system; the internal damping is replaced by the equivalent damping, and the external damping is mainly the air friction damping, then

$$\left\{ {\begin{array}{*{20}c} {{\text{Internal damping: }}{\varvec{F}}_{{{\mathbf{cq}}}}^{1} { = }\frac{1}{2}\dot{\user2{\vartheta }}^{T} {\varvec{C}}_{{\varvec{\vartheta}}} \dot{\user2{\vartheta }}} \\ {{\text{External damping}}:{\varvec{F}}_{{{\mathbf{cq}}}}^{2} { = }\mu_{{\text{g}}} \dot{\user2{\vartheta }}^{T} {\varvec{M}}_{{\user2{\vartheta \vartheta }}}^{{\mathbf{q}}} \dot{\user2{\vartheta }}} \\ \end{array} } \right.$$

(10)

where, ${\varvec{C}}_{{\varvec{\vartheta}}} { = }diag(c_{\vartheta } ,c_{\vartheta } )$ is the damping coefficient matrix of the equivalent system, $c_{\vartheta } = 2m_{q} \xi_{q} \omega_{nq}$, $\mu_{{\text{g}}}$ is the air damping coefficient.

The potential energy of the system mainly considers the gravitational potential energy of the lifting weight and the elastic potential energy of the wire rope, then

$$\left\{ \begin{gathered} {\text{Lifting weight}}:V_{{\text{q}}}^{{1}} = m_{{\text{q}}} g[\left( {l_{{\text{b}}} + u_{{\text{x}}} } \right)\sin \beta + u_{{\text{z}}} \cos \beta - \left( {l + \delta } \right)\cos \theta_{{\text{x}}} \cos \theta_{{\text{y}}} ] \hfill \\ {\text{Wire rope: }}V_{{\text{q}}}^{{2}} = \frac{1}{2}k_{{\text{g}}} \delta^{2} \hfill \\ \end{gathered} \right.$$

(11)

where, $k_{{\text{g}}}$ is the wire rope stiffness coefficient, $k_{{\text{g}}} = n\varepsilon E_{{\text{g}}} A_{{\text{g}}} /l$.

2.3 Model of Wind Load

“Crane Design Code” GB/T3811-2008 in the provisions of the crane working state of the wind load calculation formula [9] as follows

$$P_{{\text{w}}} = CpA = \frac{1}{2}\rho_{{\text{a}}} v^{2} CA$$

(12)

where, C is the wind coefficient; p is the working state of the calculated wind pressure; A is the actual windward area of the crane;${\rho }_{{\text{a}}}$ is air density; v is the instantaneous wind speed.

The instantaneous wind speed consists of the mean wind speed ($\overline{v}$) and the turbulent wind speed ($v_{{{\text{turb}}}}$) obeying the Gaussian distribution, i.e. $v = \overline{v} + v_{{{\text{turb}}}}$. The turbulent velocity is modeled as follows

$$\left\{ \begin{gathered} \frac{{dv_{{{\text{turb}}}} }}{dt} = \frac{1}{{T_{{\text{v}}} }}v_{{{\text{turb}}}} + m_{{{\text{wind}}}} \hfill \\ T_{{\text{v}}} = \frac{10.5z}{{\overline{v}}} \hfill \\ \end{gathered} \right.$$

(13)

where, m_wind is the standard distribution white noise generator; T_v is the time constant; z is the height of the mass center of the entity.

2.4 Dynamic Model of Boom-Hoisting System

Taking the generalized coordinates as $q_{{\text{i}}} = \left[ {\begin{array}{*{20}c} {\dot{\alpha }} & {\dot{\beta }} & {\dot{\user2{u}}^{{\text{T}}} } & {\dot{\user2{\phi }}^{{\text{T}}} } & {\dot{\user2{\vartheta }}^{{\text{T}}} } \\ \end{array} } \right]^{{\text{T}}}$. Substituting Eqs. (4)–(6) and Eqs. (9)–(12) into Eq. (1) and simplifying it, the system dynamics equation is given by

$$\user2{M\ddot{q}}_{{\mathbf{i}}} + \user2{C\dot{q}}_{{\mathbf{i}}} + {\varvec{Kq}}_{{\mathbf{i}}} = {\varvec{Q}}_{{{\mathbf{out}}}} + {\varvec{Q}}_{{\mathbf{v}}}$$

(14)

where, ${\varvec{M}} = \left[ {\begin{array}{*{20}c} {{\varvec{M}}_{{\mathbf{r}}} } & O \\ O & O \\ \end{array} } \right] + {\varvec{M}}_{{\mathbf{q}}}$ is the matrix of system stiffness coefficients; ${\varvec{C}} = diag(0,0,c_{{\text{x}}} ,c_{{\text{y}}} ,c_{{\text{z}}} ,0,0,c_{\vartheta } ,c_{\vartheta } )$ is the matrix of system damping coefficients; ${\varvec{K}} = diag(0,0,k_{{\text{x}}} ,k_{{\text{y}}} ,k_{{\text{z}}} ,k_{{\text{g}}} ,0,k_{\vartheta } ,k_{\vartheta } )$ is the matrix of system stiffness coefficients, $k_{\vartheta } = m_{q} g(l + \delta )$; ${\varvec{Q}}_{{{\mathbf{out}}}} { = [}\begin{array}{*{20}c} {T_{M\alpha } (t)} & {T_{M\beta } (t)} & 0 & {T_{uy} (t)} & {T_{uz} (t)} & {F_{l} (t)} & 0 & {T_{\theta x} (t)} & {T_{\theta y} (t)} \\ \end{array} ]^{{\text{T}}}$ is the system input signal, ${\varvec{Q}}_{{\mathbf{v}}}$ is the system inertial force.

In summary, a seven-input nine-degree-of-freedom underdriven rigid-flexible coupling dynamic mathematical model is established. Among them, 3 degrees of freedom describe the rotation, variable amplitude and lifting motions of the system; 3 degrees of freedom describe the boom head vibration of the boom; 1 degree of freedom describes the elastic vibration of the wire rope; and 2 degrees of freedom describe the spatial swing of the lifting weight. The numerical solution of the multi-body dynamics Eq. (14) is realized using the fourth-order Lunga-Kuta method and MATLAB.

3 Establish ADAMS Simulation Model

Based on a certain type of wind power crane, the flexible boom model is established by ANSYS, as shown in Fig. 3. In ADAMS, the CABLE module is used to establish the lifting wire rope; the boom tension plate is equivalently replaced by two wire ropes with larger stiffness; the rigid body turntable, chassis and support plate of boom are modeled by SOLIDWORKS; the contact force is set between the self-built ground and the lifting weight, and the flexible boom in ANSYS is used to replace the rigid boom in the original ADAMS model. The simulation model is shown in Fig. 4.

Fig. 3

Boom in ANSYS

Fig. 4

ADAMS rigid-flexible coupling model

4 Example Analysis

In the actual hoisting operation, there is only one of the three working conditions of rotation, variable amplitude and lifting motion of the wind power crane. This paper takes the rotary condition as an example, and analyzes the dynamic response curves of the above mathematical model and simulation model in MATLAB and ADAMS.

In this paper, when verifying the rationality and accuracy of the mathematical model, both models do not apply wind load. The rotary signal of the boom is shown in Fig. 5. When verifying the effect of wind load on the swing of the lifting weight and vibration of the boom head, the wind load is applied in the mathematical model, and the rotary signal of the boom is shown in Fig. 6. The acceleration and deceleration processes of the rotary signal are relatively smooth during the actual rotary operation, and the sinusoidal signal is used to approximate the simulated acceleration model during the solution process [9].

Fig. 5

Rotary signal 1

Fig. 6

Rotary signal 2

4.1 Model Parameter

For the above type of crane, the boom consists of 14 section booms, with a total length of 130 m when fully assembled. In the case analysis, the wire rope multiplication is 8 and the boom amplitude angle is 75°. The structural parameters of the equivalent spring-damping system of the boom are obtained by using ANSYS and ADAMS analysis results, as shown in Table 1. The equivalent stiffness of the elastic two-force bar is k_g = 9.07 × 10⁶ N m⁻¹.

Table 1 Structural parameters of the equivalent spring damping system

For wind load, the standard of crawler crane stipulates that the working wind speed of lattice boom crane should satisfy: when the boom length is more than 50 m, the wind speed should not be more than 9.8 m/s [10]. The wind speed in grade 5 is 8.0–10.7 m/s. In this paper, the influence of wind load on swing angle is studied using five-stage average wind speed of 9.35 m/s.

4.2 Dynamic Response Analysis of the System

Figure 7 shows the Z-direction (the direction perpendicular to the boom in the variable amplitude plane) response characteristic curve of the boom head at different moments when the lifting weight is 50 t. In 0–50 s, the lifting weight is placed on the ground in the ADMAS model, the boom head vibrates freely, and the Z-direction displacement of the boom head is 1000 mm when it is stable. In 50–60 s, the lifting weight of the ADAMS model is lifted off the ground, and at the same time, the boom head of the mathematical model starts to vibrate freely under the action of the crane weight of 50 t. In 60–250 s, the two models do the free vibration without excitation and with damping. When the rotary signal of Fig. 7 is applied at 250 s, the mathematical model shows a more obvious stimulated vibration with increased amplitude in the acceleration and deceleration stages, and the ADAMS model also has a corresponding trend but is not obvious. At the end of the rotation at 268 s, the vibration of the two models gradually decays to 0, and the Z-direction deformation of the models at the time of stabilization is 1767.3 mm and 1764.6 mm, respectively. In ANSYS, applying the boundary conditions and the 50 t lifting weight to the boom, the Z-direction deformation of the boom head is u_z = 1798.94 mm, as shown in Fig. 8; the comparison of the deformation results verifies the accuracy of the model construction.

Fig. 7

Z-direction vibration of boom head

Fig. 8

Ansys results

Figures 9 and 10 describe the oscillation angle in the variable amplitude plane and the deflection angle in the rotary plane of the two models under the action of the signals in Fig. 5, respectively. The oscillation angle in the variable amplitude plane is mainly caused by the centrifugal force during rotation of the lifting weight and the Coriolis force of the lifting weight deflecting on the rotary plane. From the graph, it can be seen that the response gradually increases to the peak value in the uniform motion stage, at this time, the oscillation angle of the ADAMS model is 2.10° and that of the mathematical model is 2.31°; the oscillation angle of the mathematical model increases by 15.2% compared with that of the ADAMS model at the maximum value of the difference. The mathematical model has a superior following to the ADAMS model, but the mathematical model decays faster than the ADAMS model due to the consideration of the effect of air resistance. The deflection angle of the rotary plane is mainly caused by the inertia force during rotation of the lifting weight, and the amplitude is obtained in the acceleration and deceleration phases. The maximum deflection angles of the two models are 2.54° and 2.60°, respectively, and the deflection angle of the mathematical model is 14.1% higher than that of the ADAMS model at the maximum difference.

Fig. 9

Oscillation angle of the lifting weight in variable amplitude plane

Fig. 10

Deflection angle of the lifting weight in rotary plane

4.3 Dynamic Response of the System Under Wind Load

Based on the description in Sect. 4.1, the instantaneous wind direction is specified to be 30° and blowing from y₋ to x₊ in the global coordinate system o–xyz. Due to the specificity of the model in this paper, the change of the windward area of the model during rotation is not considered. Using the wind load model in Sect. 2.3 to solve the wind load signals of the boom head and the lifting weight, the lifting weight is 10 t during the wind load impact analysis. The rotary signal is shown in Fig. 6.

Figure 11 shows the Z-direction vibration of the boom in the mathematical model without and with wind load. The model applies wind load at 100 s, and then the amplitude of the boom vibration increases and the equilibrium position is shifted downward by 21 mm, which is also consistent with the ANSYS model results.

Fig. 11

Comparison of Z-direction deformation of boom head in mathematical model without and with wind load applied

Figures 12 and 13 describe the swing angles of the lifting weight in the absence and presence of wind loads. In the absence of wind load, the maximum deflection angles of the crane in the variable amplitude plane and the rotary plane are 5.19° and 4.29°, respectively. In comparison with the results of the analysis in Sect. 3.2, it shows that the swing angle of the lifting weight increases greatly under the conditions of increased rotary speed and reduced lifting weight. In the case of wind load, the balance position of the lifting weight is shifted to the zero position, and the oscillation angle in the variable amplitude plane and the deflection angle in the slewing plane increase by 0.19° and 0.29°, respectively. The change in oscillation and deflection angles under wind load is small, but it has caused a large positional deviation for the ultra-long flexible boom crane that needs precise positioning. Based on the simulation data in this paper, the maximum offset of the lifting weight is 385 mm, which makes it difficult to ensure the accuracy of the hoisting if no intervention is made in the actual operation.

Fig. 12

Oscillation angle of the lifting weight in variable amplitude plane under wind load

Fig. 13

Deflection angle of the lifting weight in rotary plane under wind load

5 Conclusion

In this paper, based on the wind power crane with a large slenderness ratio, a rigid-flexible coupling dynamic mathematical model is established by using the equivalence method, and constructing the ADAMS rigid-flexible coupling model. The accuracy and rationality of the mathematical model are verified by the numerical simulation analysis of the two models. Based on this, the transient wind model is used to simulate the effect of wind load on the swing of the lifting weight. The results obtained are as follows:

• (1) Based on the Lagrange equation, the rigid-flexible coupling dynamics equation is derived by using the equivalent spring-damper system, the elastic two-pole system and the space pendulum system. Compared with the response of the ADAMS model, the results show that the mathematical model better simulates the swing of the lifting weight, the elastic vibration of the boom, the elastic vibration of the wire rope and the rigid-flexible coupling effect of the system.

• (2) Applying the instantaneous wind model to the mathematical model, the boom vibration and the swing of the lifting weight are offset to zero. In the case of light load and high wind level, it is necessary to stop the operation in time according to the field situation so as to avoid the inestimable danger.

• (3) This paper fully considers the influence of nonlinear effects on the swing of the lifting weight, improves the calculation accuracy of the model, and provides a certain theoretical basis for the subsequent design of the crane structure and the design of the anti-sway control system.