The deployment accuracy of deployable structures is affected by temperature and flexibility. To obtain the higher accuracy, various measures such as the optimization design and the control process are employed, and they are all based on deployment dynamics characteristics of deployable structures. So a precise coupled thermo-structural deployment dynamics analysis is important and necessary. However, until now, only a one-dimensional thermal effect is considered in the literatures because of simplicity, which reduces the accuracy of the model. Therefore, in this paper, a new model coupling mechanical field with a temperature field is presented to analyze the deployment dynamics of a deployable structure with scissor-like elements (SLEs). The model is based on the absolute nodal coordinate formulation (ANCF) and is established via a new locking-free beam element whose formulation is extended to account for the two-dimensional thermally induced stresses due to the heat expansion for the first time. Namely, in the formulation, the thermal influences are along two-dimensional directions, the axial direction and the transverse direction, rather than along a one-dimensional direction. The validity and precision of the proposed model are verified using a flexible pendulum example. Finally, the dynamics of a linear deployable structure with three SLEs modeled by the element is simulated under a temperature effect.

Over the past three decades, deployable structures have been extensively used in space missions because they own the properties of high stiffness, low mass, and small folding volume [

In order to achieve higher accuracy, the precise deployment dynamics characteristic of deployable structures is essential because it is fundamental to the optimization design [

As the absolute nodal coordinate formulation grows and is verified in many fields [

To this end, a new model coupling mechanical field with a temperature field of a deployable structure with scissor-like elements (SLEs) based on ANCF is presented. The model is established by a new planar locking-free shear deformable beam element inspired from the gradient-deficient element which alleviates the thickness locking. In order to simulate the deployment dynamics characteristic accurately, the formulation of the shear deformable beam element is extended to contain the thermal stress due to heat expansions. Note that the temperature field is uniform and the heat expansions are along two-dimensional directions; that is to say, the thermal loads are applied in an axial direction and a transverse direction, which is a great improvement compared with that of Li and Wang [

This paper is organized as follows. The coupled thermo-structural formulation of the locking-free shear deformable beam element with two-dimensional thermal loads is discussed in Section

In this paper, using a two-dimensional shear deformable beam element [

The position vector of an arbitrary point on the beam element.

An arbitrary point

The matrix

The velocity at any material point can be obtained:

Thus, the kinetic energy of the beam element is expressed as follows:

Before deducing the strain energy, in order to bring the thermal effect into the formulation of the beam element, some assumptions are introduced. The material is isotropous, and the temperature field is homogeneous along the beam element. So the strain vector due to the change of temperature can be written as [

The strain tensor can be written as a vector form because of its symmetry:

So the strain vector considering the coupled thermo-structural effect can be obtained:

In terms of constitutive relation, the stress vector related to the strain vector is given by

The elastic forces can be deduced by using the strain energy

Finally, according to Lagrange’s equation [

A simple example that a rectangular pendulum moves from the horizontal rest position under the effect of gravity (

Planar rectangular pendulum with a change of the temperature field.

Material properties and geometry sizes of the pendulum.

Characteristic parameters | Values |
---|---|

Length |
1 |

Cross-sectional area ^{2}) |
0.02 × 0.02 |

Material density ^{3}) |
7750 |

Elasticity modulus ^{3}) |
2 × 10^{8} |

Poisson’s ratio |
0 |

Thermal expansion coefficient ^{−1}) |
2.85 × 10^{−5} |

Figure

The displacement of point a.

Difference value.

So the proposed model is used to assemble the deployable structure with SLEs under the thermal effect. The dynamics equations with the constraint conditions can be expressed as follows:

In this section, a deployable structure with three scissor-like elements considering the coupled thermo-structural effect is discussed. As shown in Figure _{1}. The material property and geometry size of the structure are given in Table

The linear array deployable structure with three SLEs.

Material properties and geometry sizes of the deployable structure.

Characteristic parameters | Values |
---|---|

Length |
2 |

Cross-sectional area ^{2}) |
0.02 × 0.02 |

Material density ^{3}) |
2750 |

Elasticity modulus ^{2}) |
7 × 10^{10} |

Poisson’s ratio |
0 |

Thermal expansion coefficient ^{−1}) |
2.3 × 10^{−5} |

Temperature profile with time.

Figure _{1}a_{2}, rod b_{2}a_{3}, and rod b_{3}a_{4} in the changing temperature field as shown in Figure

Temperature effect on the deflection of rods.

Rod b_{1}a_{2} deflection with time

Rod b_{2}a_{3} deflection with time

Rod b_{3}a_{4} deflection with time

The motion diagrams of a representative point a_{4} selected to stand for the dynamics characteristics of the deployable structure are shown in Figure _{4}. The thermal effect results in additional vibrations emerging in acceleration, which can be observed by comparing the thermo-flexible curve with the flexible curve. Besides, referring to Figures

Deployment dynamics characteristics of the mechanism.

Displacement with time

Velocity with time

Acceleration with time

In this paper, the dynamics analysis of the deployable structure with three SLEs considering the coupled thermo-structural effect based on ANCF is presented.

In order to establish the precise model coupling of the mechanical field with the temperature field of the mechanism, the formulation of a new locking-free beam element with the additional slope vector is extended to account for the thermally induced stresses in two-dimensional directions, that is, the axial direction and transverse direction

It shows that the new beam element is free of locking phenomena as a result of a good agreement between the calculating results of the simple pendulum and the simulation results of the ANSYS within a large overall motion. Then, the validity and the precision of the proposed thermo-structural model are demonstrated through the further margin calculation: the difference of the proposed model is approximately identical to the difference of ANSYS, whereas the difference of the 1D thermo-structural model has a relatively large gap with the difference of ANSYS. Therefore, the proposed model is more precise than the 1D thermal-structural model. On the other hand, it also demonstrates that the development of the new two-dimensional thermo-structural model is significant

In the final simulations of the deployable structure, by contrasting the thermo-flexible deflection with the flexible deflection, it is found that the main influence of temperature is that the change of temperature leads to many vibrations: for the deflection of rods, many small vibrations arise because thermally induced transverse strain results in varying moment of inertia related to the cross-sectional area besides the change of the length of rods resulting from thermally induced axial stress. On the other hand, for the deployment dynamics characteristics, temperature has a huge impact on deployment acceleration. The thermal effect will result in additional vibrations emerging in acceleration of an arbitrary point on the deployable structure, and these vibrations will become larger perspicuously at inflection points of temperature change

For the new beam element, the displacement field is defined as

The nodal displacement conditions shown in Figure

Thus, the polynomial coefficients can be derived from (

By substituting (

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

This work was supported by the National Natural Science Foundation of China (grant number 51175422).