Among all the fields in solid mechanics the methodologies associated to multibody dynamics are probably those that provide a better framework to aggregate different disciplines. This idea is clearly reflected in the multidisciplinary applications in biomechanics that use multibody dynamics to describe the motion of the biological entities, or in finite elements where the multibody dynamics provides powerful tools to describe large motion and kinematic restrictions between system components, or in system control for which multibody dynamics are the prime form of describing the systems under analysis, or even in applications with fluid-structures interaction or aeroelasticity.
This book contains revised and enlarged versions of selected communications presented at the ECCOMAS Thematic Conference in Multibody Dynamics that took place in Lisbon, Portugal, which have been enhanced in their self-containment and tutorial aspects by the authors. The result is a comprehensive text that constitutes a valuable reference for researchers and design engineers and helps to appraise the potential of application of multibody dynamics to a wide range of scientific and engineering areas of relevance.
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The goal of the present example is to identify a set of parameters such that a measurable output matches a desired trajectory. The external force vector results in.
The algebraic constraint equation. Instead of using virtual test data, a measured trajectory from a test-bench can be used here. In Fig. The large deviation between the initial and the optimal solution results from the incorrect values of the parameters to identify. The reduction of the cost function during the optimization is shown in Fig. It can be seen that the error is reduced significantly with increasing number of iterations. The convergence of the four parameters over the iterations is shown in Fig.
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It can be seen that the linear stiffness parameter c E 1 converges very fast, whereas the cubic damping parameter d H 2 converges slower. In this paper, we show a new approach for the computation of the gradient of a cost function associated with a dynamical system for a parameter identification problem. We present the discrete adjoint method for an implicit discretization scheme and the required Jacobian matrices in detail for the HHT-solver as a representative of a widely used implicit solver in multibody dynamics.
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Note that the discrete adjoint system depends on the integration scheme of the system equations. The presented method has two main advantages in comparison with the traditional adjoint method in the continuous case see, e. The computation of the adjoint variables depends only on the recursive iteration scheme used to solve the system equations. Hence, only a system of algebraic equations has to be solved successively. The second advantage is that in combination with the HHT-solver, the cost function may also depend on the accelerations, if the discrete adjoint method is used.
The reason is that the accelerations are included in the state vector of the solver method. Hence, the Jacobian matrices that are necessary for the discrete adjoint computations remain similar to the Jacobian matrices that are required for the HHT-solver. Otherwise, in the continuous case, the accelerations are not included in the state vector, but have to be expressed by the motion equations in the cost function, which lead to complex Jacobian matrices [ 12 ].
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The straightforward and efficient considerations of the acceleration in the cost function have the advantage that the measured signals from acceleration sensors can be used directly for parameter identification in practice. Due to the simple use and low price of acceleration sensors, this strategy is a promising approach in the field of parameter identification.
The theory described in this paper is a powerful tool for parameter identification in time domain. In most cases the results lead to a best-fit solution, which means that high-frequency components with low amplitudes are not considered.
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However, the discrete adjoint method can also be used to identify parameters influencing the system at special frequencies. Hence, the basic idea is to compute the Fourier coefficients for the relevant oscillations and include the amplitude spectrum in the cost function. National Center for Biotechnology Information , U. Multibody System Dynamics. Multibody Syst Dyn. Published online Nov 3. Author information Article notes Copyright and License information Disclaimer.
Corresponding author. Received Mar 28; Accepted Oct Abstract The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. Keywords: Adjoint method, Discrete adjoint method, Parameter identification. Introduction In the last few years the complexity of the multibody systems has grown tremendously.
Discrete adjoint method for implicit time integration methods The discrete adjoint method is an elegant and efficient way to compute the gradient of a cost function. Application to the HHT solver In this section the implicit iteration scheme 3 is specified for the HHT-solver, which is a widely used time integration method in multibody system dynamics. The discrete adjoints for a simple harmonic oscillator In this section the discrete adjoint gradient computation is shown on a simple academic example. Example: engine mount As an illustrative example, we consider an engine mount, which is installed in every commercial car.
Open in a separate window. Model of the engine mount [ 10 ]. Time-dependent input force F t. Conclusion In this paper, we show a new approach for the computation of the gradient of a cost function associated with a dynamical system for a parameter identification problem. References 1. Alexe M. On the discrete adjoints of adaptive time stepping algorithms.
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