First-order Approximation Coupling Model Research Articles

Nonlinear dynamic modeling and decoupling for rigid–flexible coupled system of spacecraft with rapid maneuver

In this paper, the rigid–flexible coupled model of a spacecraft composed of a rigid platform and two flexible solar arrays is investigated. Considering the rapid maneuver of a spacecraft, the first-order approximation coupling model should be adopted. The nonlinear dynamic equations, which remain the second-order coupling terms of axial displacement caused by the transverse motion of solar arrays, are obtained by using the Hamilton principle. Then the global mode method is adopted to obtain the mode shapes of the linearized model. The global mode shapes are proved to be orthogonal and used to discrete the continuum first-order approximation coupling model. Then the rigid–flexible coupled model of the spacecraft is decoupled by global mode shapes. The model is validated by comparing with that obtained from the finite element method. The simulation results demonstrate that the global mode method has the advantage of lower dimensions but higher accuracy. The analysis of parameters variation and dynamic responses shows that the first-order approximation coupling model is more accurate than the linearized model and has a broader scope of application than that of the zero-order approximation coupling model.

Dynamic Analysis of Planar 3-RRR Flexible Parallel Robots with Dynamic Stiffening

In consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of flexible beam, the first-order approximation coupling model of planar 3-RRR flexible parallel robots is presented, in which the rigid body motion constraints, elastic deformation motion constraints, and dynamic constraints of the moving platform are considered. Based on the different speed of the moving platform, numerical simulation results using the conventional zero-order approximation coupling model and the proposed firstorder approximation coupling model show that the effect of “dynamic stiffening” term on dynamic characteristics of the system is insignificant and can be neglected, and the zero-order approximation coupling model is enough precisely for catching essentially dynamic characteristics of the system. Then, the commercial software ANSYS 13.0 is used to confirm the validity of the zero-order approximation coupling model.

A HIGH-ORDER RIGID-FLEXIBLE COUPLING MODEL OF A ROTATING FLEXIBLE BEAM UNDER LARGE DEFORMATION

The dynamic modeling theory of a flexible beam which is rotating in a plane is further studied, and the highorder coupling dynamic model is investigated here. Both the transversal deformation and the longitudinal deformation of the flexible beam are considered. And the non-linear coupling deformation term, also known as the longitudinal shortening term caused by transversal deformation, is considered here. The high-order terms related to the non-linear coupling term are retained, which are ignored in the first-order approximation coupling modeling. Thus, we can get the rigid-flexible coupling dynamic equations of the system. The high-order coupling model can not only be used in small deformation case, but also in large deformation case. Then simulations, compared to the absolute nodal coordinate formulation and the first-order coupling model, are given to prove the validity of the high-order coupling model. And the results show that the high-order coupling model can make up for the deficiency of the first-order approximation coupling model in the large deformation situation.

Dynamic Modeling and Analysis of Hub-Tapered Beam System

The dynamic modeling of a hub-tapered flexible beam system which is rotating in a plane is investigated. In the modeling, the high-order terms related with the nonlinear coupling term are retained, which are ignored in the first-order approximation coupling modeling. So, this model can also be called a high-order approximation coupling model, which cannot only be used in the case of small deformation but also in the case of large deformation. Solving for this mathematical model, we can obtain the dynamics of the system. Examples are given to indicate the differences between the high-order approximation coupling model and the first-order approximation coupling model. Also, the natural frequencies are studied through the transversal bending vibration analysis. The width ratio and the height ratio of the tapered beam are shown to have great influences on the transversal bending natural frequencies.

Analyses of rigid-flexible coupling dynamic properties of a rotating internal cantilever beam

The rigid-flexible coupling dynamic properties of an internal cantilever beam attached to a rotating hub are studied in this paper. Based on the accurate description of non-linear deformation of the flexible beam, the first-order approximation coupling model is derived from Hamilton theory and assumed mode method, taking into account the second-order coupling quantity of axial displacement caused by transverse displacement of the beam. The simplified first-order approximation coupling model which neglects the effect of axial deformation of a beam is presented. The simplified model is transformed into dimensionless form in which dimensionless parameters are identified. Firstly, the dynamic response of an internal cantilever beam is compared with that of an external cantilever beam, which are both in non-inertia system. Then, the stability of an internal cantilever beam is analyzed. Finally, the convergence of critical rotating speed of an internal cantilever beam is analyzed. Generally, it is pointed that an internal cantilever beam has a dynamic softening phenomenon, which is different from the dynamic stiffening phenomenon of an external cantilever beam. The critical ratio of the internal radius to the length of the beam for unconditional stability and the critical rotating speed of conditional stability of an internal cantilever beam are derived. When the first natural frequency decreases as the rotating speed increases, the dynamic system of the internal cantilever beam is conditionally stable. As the number of modes increases, the critical rotating speed of an internal cantilever beam decreases, and it has a convergent value.

旋转内接微梁的动力学建模及稳定性分析

The dynamic properties of the micro beam are obviously different from those of the traditional macro beam due to the size effect of material on a micro scale. The rigid-flexible coupling dynamic properties analysis of an internal micro beam attached to a rotating hub is studied in this paper. Based on the accurate description of non-linear deformation of the flexible beam, the first-order approximation coupling model are derived from couple stress theory and Hamilton theory, taking the second-order coupling quantity of axial displacement caused by transverse displacement of the beam into account. The simplified first-order approximation coupling model which neglects the effect of axial deformation of a beam is presented. The simplified model is transformed into dimensionless form in which dimensionless parameters are identified. Firstly, the dynamic response of an internal micro beam is compared with that of an external micro beam, which are both in non-inertia system. Then, the influence of size effect on the dynamic properties of an internal micro beam is studied. Generally, it is pointed out that an internal micro beam has dynamic softening phenomena, which is different from the dynamic stiffening phenomena of an external micro beam. The critical ratio of the internal radius to the length of the beam for unconditional stability and the critical angular speed of conditional stability of an internal micro beam are derived. The size effect results in the descending of vibration amplitude and the increasing of vibration frequency and critical angular speed of conditional stability of the micro beam. As the number of modes increases, the critical rotating speed of an internal cantilever beam decreases, and it has a convergent value.

Active control and experiment study of a flexible hub-beam system

The first-order approximation coupling (FOAC) model was proposed recently for dynamics and control of flexible hub-beam systems. This model may deal with system dynamics for both low and high rotation speed, while the classical zeroth-order approximation coupling (ZOAC) model is only available for low rotation speed. This paper assumes the FOAC model to present experimental study of active positioning control of a flexible hub-beam system. Linearization and nonlinear control strategies are both considered. An experiment system based on a DSP TMS320F2812 board is introduced. The difference between linearization and nonlinear control strategies are studied both numerically and experimentally. Simulation and experimental results indicate that, linearized controller can make the system reach an expected position with suppressed vibration of flexible beam, but the time taken to position is longer than expected, whereas nonlinear controller works well with precise positioning, suppression of vibration and time control.

Complete geometric nonlinear formulation for rigid-flexible coupling dynamics

A complete geometric nonlinear formulation for rigid-flexible coupling dynamics of a flexible beam undergoing large overall motion was proposed based on virtual work principle, in which all the high-order terms related to coupling deformation were included in dynamic equations. Simulation examples of the flexible beam with prescribed rotation and free rotation were investigated. Numerical results show that the use of the first-order approximation coupling (FOAC) model may lead to a significant error when the flexible beam experiences large deformation or large deformation velocity. However, the correct solutions can always be obtained by using the present complete model. The difference in essence between this model and the FOAC model is revealed. These coupling high-order terms, which are ignored in FOAC model, have a remarkable effect on the dynamic behavior of the flexible body. Therefore, these terms should be included for the rigid-flexible dynamic modeling and analysis of flexible body undergoing motions with high speed.

Frequency Characteristics of a Flexible Hub-beam System with Arbitrary Settling Position of Attached Mass

In this article, the frequency characteristics of a flexible hub-beam system with an attached mass in an arbitrary position are investigated using a first-order approximation coupling (FOAC) model. Three kinds of damping are considered in the model: Damping located at the hub bearing, structural damping caused by the beam material, and damping caused by the air. Cases with both known and unknown large motion of the system are considered. First, the FOAC model is presented with an unknown large motion, using the Hamilton principle and finite element discretization method. Then the dynamic equation for a non-inertial system is presented by neglecting the large motion of the system, and the frequency characteristics of the system are studied using numerical simulations. Simulation results indicate that, for the case with a known large motion, the response frequency of the beam is degressive as the settling position of the mass moves from the fixed end to the free end of the beam. The response frequency of the beam decreases with increasing inertia of the hub. Damping affects only the dynamical equilibrium position of the flexible beam and has little effect on the response frequency of the beam. For the case with an unknown large motion, the response frequency of the beam does not show an obvious degressive trend as the settling position of the mass moves from the fixed end to the mid-point of the beam, but it becomes degressive when the mass moves from the mid-point to the free end of the beam. Damping affects not only the final position of the system, but also the vibration amplitude of the flexible beam. However, the response frequency of the beam is barely affected by the damping.

Optimal tracking control of a flexible hub–beam system with time delay

In this paper, a new technique of time-delay compensation is proposed for active control of a flexible hub–beam system. The first-order approximation coupling (FOAC) model proposed recently for dynamics of hub–beam systems is used to verify the applicability of this technique. The FOAC model is first linearized to obtain a linearized equation. The linearized equation with time delay is then transformed into a standard form with no time delay by a particular integral transformation. The time-delay controller is designed based on this standard equation using the classical optimal tracking control theory. Since the controller is a function of modal coordinates, a modal filter is presented to estimate the modal coordinates from physical sensor measurements. The effectiveness of the proposed technique for time delay is demonstrated by numerical simulations. Simulation results indicate that a very small time delay may result in instability of the control system if it is not compensated in control design. The proposed time-delay controller is effective in controlling the system even when the maximum time delay for stability without time-delay compensation is greatly exceeded. Moreover, for the system without time delay, the proposed time-delay controller may possibly obtain much better control effectiveness than the controller without time delay.

Active control of a flexible hub-beam system using optimal tracking control method

In this paper, active control of a flexible hub-beam system is investigated using optimal tracking control method and a first-order approximation coupling (FOAC) model. Using the Hamilton theory and assumed mode method (AMM) for discretization, the FOAC model considering second-order coupling quantity of axial displacement caused by transverse displacement of beam is first investigated. Subsequently a simplified FOAC model (or the SFOAC model) which neglects the effect of axial deformation of beam is presented in this paper. Active position control for the system is then studied by optimal tracking control theory. Since the controller obtained is a function of mode coordinate, an extraction method of modal coordinate from actual physical measurements is developed. Numerical simulations are carried out to demonstrate effectiveness of the proposed method. Simulation results indicate that the SFOAC model is valid for dynamic description of flexible hub-beam system and thus is available for control design. Modal filter proposed is effective in extracting modal coordinate from physical measurements. Desired position of system may be achieved by the proposed controller. Residual vibration of beam may be suppressed by this controller as well.

Modeling study of a flexible hub–beam system with large motion and with considering the effect of shear deformation

For a flexible hub–beam system with large motion, the first-order approximation coupling (FOAC) model is developed currently, which is verified numerically and experimentally to be valid for dynamic description of the system. In the FOAC model, the assumption of Euler–Bernoulli beam is adopted with neglecting the effect of shear deformation of flexible beam. So this model is only available for the case of slender beam. When the beam is short in length direction, shear deformation is a factor that may have biggish effect on system dynamics. In this paper, dynamic modeling of flexible hub–beam system with considering the effect of shear deformation is investigated. Firstly, based on the assumption of Timoshenko beam and using Hamilton's principle, a coupling dynamic model with considering shear deformation is established, in which geometric stiffness term and axial foreshortening effect caused by the transverse deformation of beam are considered in the modeling. Subsequently, the dynamic model is discretized using finite element method (FEM). Finally numerical simulations are carried out to demonstrate the effectiveness of proposed dynamic model. Simulation results indicate that, dynamics characteristics of the hub–beam system using the Timshenko beam hypothesis and the Euler–Bernoulli beam hypothesis are almost identical when the beam is a slender one. For this case, shear deformation has little effect on system dynamics. But when the beam is short in length direction, shear deformation may have biggish effect on system dynamics. This effect becomes larger as the width to length ratio increases.

Dynamic Analysis of a Hub‐Beam System by a First‐Order Approximation Coupling Model

In this paper, a first‐order approximation coupling (FOAC) model is investigated to analyze the dynamics of the hub‐beam system, which is based on the Hamilton theory and the finite element discretization method. The FOAC model for the hub‐beam system considers the second‐order coupling quantity of the axial displacement caused by the transverse displacement of the beam. The dynamic characteristics of the system are studied through numerical simulations under twos cases: the rotary inertia of the hub is much larger than, and is close to, that of the flexible beam. Simulation and comparison studies using both the traditional zeroth‐order approximation coupling (ZOAC) model and the FOAC model shows that when large motion of the system is unknown, possible failure exists by using the ZOAC model, whereas the FOAC model is valid. When the rotary inertia of the hub is much larger than that of the beam, the result using the ZOAC model is similar to that using the FOAC model. But when the rotary inertia of the hub is close to that of the beam, the ZOAC model may lead to a large error, while the FOAC model can still accurately describe the dynamic hub‐beam system.

Model study and active control of a rotating flexible cantilever beam

For a dynamic system of a rotating flexible cantilever beam, the traditional model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the beam are uncoupled. This traditional hybrid coordinate model is referred as the zero-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of a flexible cantilever beam with a high rotational speed. In this paper, a first-order approximation coupling model is presented to analyze the dynamics of rotating flexible beam system, which is based on the Hamilton theory and the finite element discretization method. The proposed model for the system considers the second-order coupling quantity of the axial displacement caused by the transverse displacement of the beam. The dynamic characteristics of the rotating beam system when using the zero-order approximation coupling model are compared with those when using the first-order approximation coupling models through numerical simulations. In addition, the applicability of the two dynamic models for control design are studied by using the classical optimal control method. Simulation and comparison studies show that, for the case without control for the system, there exists big difference between the result using the zero-order approximation coupling model and that using the first-order approximation coupling model even for the case of small angular velocity of the system. The larger is the angular velocity, the bigger is the difference. Vibration frequency of the beam by using the first-order approximation coupling model is higher than that by using the zero-order approximation coupling model. When the angular velocity of the system is close to or is larger than the fundamental frequency of the beam without rotation motion, the zero-order approximation coupling results in a wrong result, while the first-order approximation coupling model is valid. For the case with control for the system, the applicability of the zero-order approximation coupling model can be much broadened. The critical angular velocity of the system for validity of the zero-order approximation coupling model is much larger than that without control for the system. The first-order approximation coupling model is available not only for the case of small angular velocity but also for the case of large angular velocity of the system, and is applicable to the cases with or without control for the system.

Dynamic analysis of a flexible hub-beam system with tip mass

For a dynamic system of a rigid hub and a flexible cantilever beam, the traditional hybrid coordinate model assumes the small deformation in structural dynamics where axial and transverse displacements at any point in the beam are uncoupled. This traditional hybrid coordinate model is referred as the zeroth-order approximation coupling model in this paper, which may result in divergence to the dynamic problem of some rigid–flexible coupling systems with high rotational speed. In this paper, characteristics of a flexible hub-beam system with a tip mass is studied. Based on the Hamilton theory and the finite element discretization method, and in consideration of the second-order coupling quantity of the axial displacement caused by the transverse displacement of the beam, the rigid–flexible coupling dynamic model (referred as the first-order approximation coupling (FOAC) model in this paper) and the corresponding model in non-inertial system for the flexible hub-beam system with a tip mass are presented firstly, then the dynamic characteristics of the system are studied through numerical simulations under twos cases: the large motion of the system is known and is unknown. Simulation and comparison studies using both the traditional zeroth-order model and the proposed first-order model show that even small tip mass may affect dynamic characteristics of the system significantly, which may result in the largening of vibrating amplitude and the descending of vibrating frequency of the beam, and may affect end position of the hub-beam system as well. The effect of the tip mass becomes large along with the increasing of rotating speed of large motion of the system. When the large motion of the system is at low speed, the traditional ZOAC model may lead to a large error, whereas the proposed FOAC model is valid. When the large motion is at high speed, the ZOAC model may result in divergence to the dynamic problem of the flexible hub-beam system, while the proposed second model can still accurately describe the dynamic hub-beam system.