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With the assumed mode method, the closed form dynamic model of two-link flexible manipulators is derived by using the Lagrangian approach.
Then applying Galerkin's method, the closed form solutions are presented for the truncated conical shell with simply supported boundary conditions subjected to three types of thermal loading.
In the existing version of the method, the closed loop system involves a state observer that makes the solution to the robust MPC optimization problem a time consuming step.
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Compared with the existing method, the closed-form method requires no initial position or iterations.
While the previous proposed method is the iteration-based method, the closed-form location estimator is developed in this section.
Using the separation of variables method, the closed-form natural frequencies are successfully obtained in this work.
Based on the unit-load method, the closed-form compliance equations of the flexure hinge are derived, and the motion accuracy is investigated.
According to the Lyapunov method, the closed-loop adaptive control system is proven to be globally stable, with tracking errors converging to a neighborhood of zero.
By utilizing Lyapunov's direct method, the closed-loop tracking errors and the NN estimated weights are demonstrated to be uniformly ultimately bounded.
Based on circular motion assumption and harmonic balance method, the closed-form formulae to estimate the steady-state responses are derived, which are then compared with the results calculated by the group-preserving scheme.
The underlying principle of CEh is the same as for any genetic clustering method: the closer the match in the numerical patterns present in two genotype files is, the closer is the inferred genetic relationship between them.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com