A novel study on Kepler’s law and inverse square law of gravitation

Zhang, Bingzhan; Zhen, Shengchao; Zhao, Han; Huang, Kang; Deng, Bin; Chen, Ye‐Hwa

doi:10.1088/0143-0807/36/3/035018

Cited by 10 publications

(9 citation statements)

References 11 publications

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“…where M ( q , t ) ∈ R n × n is a positive definite inertia matrix, q ·· ∈ R n is the acceleration vector of the discrete unconstrained mechanical system, and Q ( q , q · , t ) is the imposed forces on the discrete unconstrained mechanical system, which can include gravitational force, centrifugal force and the control force input. 25 In addition, q · ∈ R n denote the velocity of the system, thus equation (2) contains the initial displacement and velocity information.…”

Section: Udwadia–kalaba Equationmentioning

confidence: 99%

Application of Udwadia–Kalaba approach to semi-active suspension control of a heavy-duty truck

Huang

Xian

et al. 2019

Proceedings of the Institution of Mechanical Engineers, Part D:

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As an important component of the chassis system, the vehicle suspension plays a significant role in vehicle comfort, stability, and safety. In dynamic modeling of a semi-active suspension, traditional method is a complicated process, the actuator force is difficult to solve. In the process of model simplification, the influence of the vehicle-axle and balanced shift quality on the vehicle dynamics is not considered. For this reason, applying the Udwadia–Kalaba approach to dynamic modeling and control for a three-axle semi-active truck suspension can establish a simple and accurate equivalent discrete system. By fixing the displacement and acceleration of the body, the simulation is conducted with real excitations between the road and six wheels, and the actuator control forces of the semi-active suspension are obtained under the constraint by applying the Udwadia–Kalaba approach. The contrast in the analysis results from the Udwadia–Kalaba approach and the traditional Lagrange multiplier method indicate that the Udwadia–Kalaba equation can obtain the semi-active suspension actuator forces without introducing additional Lagrange multipliers and that the actuator forces acquired with the Udwadia–Kalaba approach are equal to those of the traditional Lagrange multiplier method.

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Section: Udwadia–kalaba Equationmentioning

confidence: 99%

Application of Udwadia–Kalaba approach to semi-active suspension control of a heavy-duty truck

Huang

Xian

et al. 2019

Proceedings of the Institution of Mechanical Engineers, Part D:

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“…18,[22][23][24] Udwadia and colleagues [24][25][26][27] also successfully addressed the dynamics and control of nonlinear uncertain systems using the fundamental equation to provide considerable reference value for the basic research and the further application of this method. Many researchers have studied and established dynamical model in some application fields by employing the Udwadia-Kalaba approach, such as satellite systems, 28 industry mechanical arm, 29 parallel manipulator, 30 flexible multibody systems, 31 machine fish, 14 heavenly bodies' movements (especially on Kepler's law and the inverse square law of gravitation), 32 and falling cat's movements. 33 In view of its strong attachment to the surfaces, high moving speed and simple control system, climbing robot based on dual-cavity structure and wheeled locomotion mechanism is studied in this article.…”

Section: Introductionmentioning

confidence: 99%

Concise method to the dynamic modeling of climbing robot

Liu

2017

Advances in Mechanical Engineering

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With the aim of dynamic modeling of the climbing robot with dual-cavity structure and wheeled locomotion mechanism, a succinct and explicit equation of motion based on the Udwadia-Kalaba equation is established. The trajectory constraint of the climbing robot, which is regarded as the external constraint of the system, is integrated into the dynamic equation dexterously. A modified numerical method is considered to reduce the errors because the numerical results obtained by integrating the constrained dynamic equation yield the errors. The trajectories are almost coincident by comparing the modified numerical value and the theoretical value. The driving torques required to guarantee the climbing robot to move along the given trajectory are obtained explicitly, which overcomes the disadvantage of obtaining dynamical equation from traditional Lagrange equation by Lagrange multiplier. The simulations are performed to demonstrate that the dynamical equation established by this method with brevity and accuracy is in accordance with reality status.

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“…Many researchers have since used Udwadia-Kalaba's method to solve problems in the dynamics analysis of multibody systems. [15][16][17][18][19][20][21][22][23] Schutte and Dooley 11 utilized the Udwadia-Kalaba approach to study the periodicity of tethered satellite motion. Huang et al used the Udwadia-Kalaba formulation to obtain the closed-form equation of the motion of a human body with joint friction.…”

Section: Introductionmentioning

confidence: 99%

“…Zhen et al 22 adopted the Udwadia-Kalaba approach to study the movements of a falling cat and explained in detail the surprise phenomenon, in which a cat always manages to right itself and lands safely on its feet when dropped at rest with its feet pointing up. Zhang et al 21 applied the Udwadia-Kalaba approach and deduced that the motion orbits of heavenly bodies are always conic based on the law of universal gravitation and angular momentum being conservative; they also verified that the motion of a heavenly body complies with the inverse square law.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis and tracking trajectory control of a crane

Huang

Shunqiang

Zhen

et al. 2016

Proceedings of the Institution of Mechanical Engineers, Part E:

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This paper introduces a method to solve the crane motion trajectory control problem. A dynamic model is proposed based on the Udwadia-Kalaba equation, which can be solved without extra parameters, such as the Lagrange multiplier. The motion trajectory of a crane is used as a constraint (referred to as trajectory tracking constraint). To satisfy the system trajectory, a method to calculate the driving conditions on the basis of the above conditions is proposed. A 2D plane dynamic model of a crane is established. Five stages of crane movement are obtained. Simulation is performed with Matlab. Simulation results simulation show that the Udwadia-Kalaba equation can be well applied to trajectory tracking control of cranes.

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A novel study on Kepler’s law and inverse square law of gravitation

Cited by 10 publications

References 11 publications

Application of Udwadia–Kalaba approach to semi-active suspension control of a heavy-duty truck

Application of Udwadia–Kalaba approach to semi-active suspension control of a heavy-duty truck

Concise method to the dynamic modeling of climbing robot

Dynamic analysis and tracking trajectory control of a crane

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