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Dynamics-based nonsingular interval model and luffing angular response field analysis of the DACS with narrowly bounded uncertainty

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Abstract

This paper develops a dynamics-based nonsingular interval model and proposes a first-order composite function interval perturbation method (FCFIPM) for luffing angular response field analysis of the dual automobile cranes system (DACS) with narrowly bounded uncertainty. By using the nonsingular interval model to describe a structure parameter with bounded uncertainty, the reasonable lower and upper bounds can be obtained, which is quite different from the traditional interval model with approximate bounds only from a large number of samples. Firstly, for the DACS with deterministic information, the inverse kinematics is analyzed, and the dynamic model of the DACS is established based on the virtual work principle and the inverse kinematics. Secondly, considering the nonsingularity of the dynamic response curves, a dynamics-based nonsingular interval model is introduced. Based on the nonsingular interval model, the interval luffing angular response vector equilibrium equation of the DACS is established. Thirdly, a first-order composite function interval perturbation method is proposed. In the FCFIPM, the composite function vectors are expanded by using the first-order Taylor series expansion, based on the differential property of composite function and monotonic analysis technique, the lower and upper bounds of the interval luffing angular response vector of the crane 1 and crane 2 of the DACS are determined. The first case is to investigate the deterministic kinematics and dynamics of the DACS with a given trajectory. The second case is provided to illustrate the detailed implementation process of constructing a dynamics-based nonsingular interval model. Finally, some numerical examples are given to verify the feasibility and efficiency of the FCFIPM for solving the luffing angular response field problem with narrowly interval parameters.

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Abbreviations

D :

The length of \(A_1 A_2 \)

d :

The length of payload \(C_1 C_2 \)

\(L_i \) :

The length of lifting arm \(A_i B_i \)

\(\gamma _i \) :

The luffing angular of lifting arm \(A_i B_i \)

\({{{\varvec{r}}}}_{A_i} \) :

The position vector of joint point \(A_i \) in the base frame \(\left\{ B \right\} \)

\(\dot{{\varvec{r}}}_{A_i}\) :

The velocity vector of joint point \(A_i \)

\({{\varvec{r}}}_{B_i} \) :

The position vector of joint point \(B_i \) in the base frame \(\left\{ B \right\} \)

\(\dot{{\varvec{r}}}_{B_i}\) :

The velocity vector of joint point \(B_i \)

\({\varvec{a}}_{B_i } \) :

The acceleration vector of joint point \(B_i \)

\({{\varvec{r}}}_{C_i } \) :

The position vector of joint point \(C_i \) in the base frame \(\left\{ B \right\} \)

\({\dot{{\varvec{r}}}}_{C_{i}}\) :

The velocity vector of joint point \(C_i \)

\({{\varvec{a}}}_{C_i } \) :

The acceleration vector of joint point \(C_i \)

\({{\varvec{r}}}_{O_i } \) :

The position vector of centroid \(O_i \) of lifting arm \(A_i B_i \) in the base frame \(\left\{ B \right\} \)

\({{\varvec{a}}}_{O_i } \) :

The acceleration of centroid \(O_i \) of lifting arm \(A_i B_i \)

\({{\varvec{r}}}_{O_p } \) :

The position vector of centroid \(O_p \) in the base frame \(\left\{ B \right\} \)

\({{\varvec{v}}}_{O_p } \) :

The velocity vector of the origin \(O_p \)

\({{\varvec{a}}}_{O_p } \) :

The acceleration vector of centroid \(O_p \)

\({{\varvec{r}}}_{C_i }^p \) :

The position vector of joint point \(C_i \) in the moving frame \(\left\{ P \right\} \)

\({{\varvec{J}}}_{{{\varvec{w}}}_{A_i B_i } } \) :

The partial angular velocity matrix of lifting arm \(A_i B_i \)

\({{\varvec{J}}}_{{{\varvec{v}}}_{B_i } } \) :

The partial velocity matrix of joint point \(B_i \)

\({{\varvec{J}}}_{{{\varvec{w}}}_p } \) :

The partial angular velocity matrix of the payload

\({{\varvec{J}}}_{{{\varvec{v}}}_{C_i } } \) :

The partial velocity matrix of joint point \(C_i \)

\(S_i \) :

The length of hoisting rope \(B_i C_i \)

\(\beta _i \) :

The rotation angle of hoisting rope \(B_i C_i \)

\(\dot{\beta }_{i}\) :

The angular velocity of hoisting rope \(B_i C_i \) with respect to the lifting arm \(A_i B_i \)

\(m_i \) :

The mass of lifting arm \(A_i B_i \)

\({\varvec{R}}\) :

The rotation matrix from moving frame \(\left\{ P \right\} \) to base frame \(\left\{ B \right\} \)

\({\varvec{R}}'\) :

The deviation of \({\varvec{R}}\) with respect to time

\({\varvec{R}}''\) :

The deviation of \({\varvec{R}}'\) with respect to time

\(\theta \) :

The rotation angle of \(\left\{ P \right\} \) relative to \(\left\{ B \right\} \)

\(\dot{\theta }\) :

The deviation of \(\theta \) with respect to time

\(\ddot{\theta }\) :

The deviation of \(\dot{\theta }\) with respect to time

y :

The Cartesian coordinates of the origin \(O_p \) along the y-axis.

z :

The Cartesian coordinates of the origin \(O_p \) along the z-axis.

\({{\varvec{F}}}_{A_i B_i } \) :

The inertia force of lifting arm \(A_i B_i \) respecting to joint point \(A_i \)

\({{\varvec{M}}}_{A_i B_i } \) :

The inertia moment of lifting arm \(A_i B_i \) respecting to joint point \(A_i \)

\({{\varvec{F}}}_p \) :

The inertia force of payload respecting to point \(C_1 \)

\({{\varvec{M}}}_p \) :

The inertia moment of payload respecting to point \(C_1 \)

\({\varvec{\tau }}\) :

The driving torque vector of the DACS

\({\varvec{\tau }}_1 \) :

The driving torque that impose on lifting arm \(A_1 B_1 \)

\({\varvec{\tau }}_2 \) :

The driving torque that impose on lifting arm \(A_2 B_2 \)

\({{\varvec{J}}}_{\mathrm{DACS}} \) :

The kinematic Jacobian matrix of the DACS

\({{\varvec{J}}}\) :

The dynamic Jacobian matrix of the DACS

\({{\varvec{S}}}_i \) :

The matrix of the ith crane

\({{\varvec{T}}}_i \) :

The vector of the ith crane

\({\varvec{\gamma }} \) :

The luffing angular response vector.

\({{\varvec{y}}}\) :

The interval parameter vector

\(y_r \) :

The interval variable

\({{\varvec{S}}}_i \left( {{{\varvec{K}}}_i \left( {{\varvec{X}}} \right) } \right) \) :

The composite function matrix of the ith crane

\({{\varvec{T}}}_i \left( {{{\varvec{K}}}_i \left( {{\varvec{X}}} \right) } \right) \) :

The composite function vector of the ith crane

\({{\varvec{y}}}^{{\varvec{c}}}\) :

The midpoint value of the interval parameter vector \({{\varvec{y}}}\)

\(y_r^c \) :

The midpoint value of interval parameter \(y_r \)

\(\Delta y_r \) :

The interval radius of interval parameter \(y_r \)

\({{\varvec{S}}}_i^c \) :

The midpoint value of composite function vector \({{\varvec{S}}}_i \left( {{{\varvec{K}}}_i \left( {{\varvec{y}}} \right) } \right) \)

\(\Delta _1 {{\varvec{S}}}_i^I \) :

The deviation interval of composite function vector \({{\varvec{S}}}_i \left( {{{\varvec{K}}}_i \left( {{\varvec{y}}} \right) } \right) \)

\({{\varvec{T}}}_i^c \) :

The midpoint value of composite function vector \({{\varvec{T}}}_i \left( {{{\varvec{K}}}_i \left( {{\varvec{y}}} \right) } \right) \)

\(\Delta _1 {{\varvec{T}}}_i^I \) :

The deviation interval of composite function vector \({{\varvec{T}}}_i \left( {{{\varvec{K}}}_i \left( {{\varvec{y}}} \right) } \right) \)

\({\varvec{\gamma }} _i^c \) :

The midpoint value of interval luffing angular response vector \({\varvec{\gamma }}_i^I \)

\({{\Delta }} _1 {\varvec{\gamma }} _i^I \) :

The deviation interval of interval luffing angular response vector \({\varvec{\gamma }}_i^I \)

\(\overline{{\varvec{\gamma }}_i } \) :

The upper bound of interval luffing angular response vector \({\varvec{\gamma }}_i^I \)

\(\underline{{\varvec{\gamma }}_i }\) :

The lower bound of interval luffing angular response vector \({\varvec{\gamma }}_i^I \)

\(D^{c}\) :

The midpoint of length of \(A_1 A_2 \)

\(d^{c}\) :

The midpoint of length of payload \(C_1 C_2 \)

\(L_1^c \) :

The midpoint of length of lifting arm \(A_1 B_1 \)

\(L_2^c \) :

The midpoint of length of lifting arm \(A_2 B_2 \)

\(\Delta D\) :

The interval radius of interval variable D

\(\Delta d\) :

The interval radius of interval variable d

\(\Delta L_1 \) :

The interval radius of interval variable \(L_1 \)

\(\Delta L_2 \) :

The interval radius of interval variable \(L_2 \)

\(D_F \) :

The interval change ratio of interval variable D

\(d_F \) :

The interval change ratio of interval variable d

\(L_{1F} \) :

The interval change ratio of interval variable \(L_1 \)

\(L_{2F} \) :

The interval change ratio of interval variable \(L_2 \)

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (51575150 and 51605126).

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  1. School of Mechanical Engineering, Hefei University of Technology, 193 Tunxi road, Hefei, 230009, China

    Bin Zhou, Bin Zi & Sen Qian

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  1. Bin Zhou
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Correspondence to Bin Zi.

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Zhou, B., Zi, B. & Qian, S. Dynamics-based nonsingular interval model and luffing angular response field analysis of the DACS with narrowly bounded uncertainty. Nonlinear Dyn 90, 2599–2626 (2017). https://doi.org/10.1007/s11071-017-3826-1

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