دو روش تجزیه و تحلیل حساسیت برای سیستم های چند عضوی همراه با تصادم
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|25541||2000||25 صفحه PDF||سفارش دهید||محاسبه نشده|
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Mechanism and Machine Theory, Volume 35, Issue 10, 1 October 2000, Pages 1345–1365
In this paper, two semi-analytical methods of sensitivity analysis, the Direct Differentiation Method and the Adjoint Variable Method, are extended to multibody systems with collisions. Elementary impact theory is used for the description of collisions, i.e. the duration of the impact, configuration changes during the impact, wave propagation and so on are neglected. The variables of state are discontinuous at the time of impact. As a consequence, the design derivatives of the variables of state and hence the adjoint variables are also discontinuous. An algebraic relationship between the adjoint variables before and after the impact is derived. The number of adjoint equations are not increasing with the number of design variables whereas the number of equations in the Direct Differentiation Method depends linearly on the dimension of the design space. This is not only true for the two methods as they are known from the literature but also for the extended methods. Although the implementation of the extended Adjoint Variable Method in a numerical simulation package is rather elaborate and cumbersome, it has been done successfully as is shown with the aid of a circuit breaker mechanism with three contacts. The numerical efficiency of the new approach is discussed in case of this example system.
Sensitivity analysis theory was originally developed in optimal control theory but the Direct Differentiation Method and the Adjoint Variable Method have also extensively been applied in structural mechanics and in multibody system dynamics . Since sensitivity analysis methods provide gradient information that is required by local optimization strategies, they have very often been discussed in the context of design optimization . In the dimensional synthesis of linkages for example, a rather common task is the determination of the geometry of a linkage such that one link performs a prescribed motion. This task is called motion generation and can be formulated as a nonlinear optimization problem introducing a measure for the deviation of the motion of the current linkage from the prescribed motion. This error must be minimized choosing suitable values of the geometry parameters. In this context, sensitivity analysis denotes the computation of the derivatives of the error with respect to the geometry parameters. With the aid of these design derivatives a direction for a redesign can be found. In the present paper, however, sensitivity analysis is discussed for integral-type performance measures evaluating the transient dynamic behaviour of mechanisms. In  it was shown that such performance measures are discontinuous for most technologically relevant mechanisms, if collisions are described by one of the elementary impact theories. For this reason, local strategies are not suited for the optimization of the transient dynamic behaviour of mechanisms with collisions. Global strategies embodying stochastic elements should be applied instead. However, this does not mean that there would be no need for the extension of existing sensitivity analysis methods to systems with collisions. Clustering methods, for example, combine stochastic elements with local optimization procedures can be used for the design of mechanisms with collisions. In  the Adjoint Variable Method has been applied to mechanisms with intermittent motion. Collisions have been modelled by inserting a stiff spring between two impacting bodies. The major drawback of this model is a numerical one. The differential equations of motion can become stiff during the impact interval. In this respect, the description of collisions by an elementary impact theory is preferable. Only algebraic equations have to be evaluated at the times of impact which is numerically much more efficient. Underlying assumption of all elementary impact laws such as Newton’s or Poisson’s law  is that the duration of the impact can be neglected and that the system configuration does not change during the impact. Integration of the differential equations of motion over the impact interval yields an algebraic relationship between the momenta of the system before and after the impact and the impact impulses. The unknown impact impulses can then be computed with the aid of the impact law. This paper is concerned with the extension of the Direct Differentiation Method and the Adjoint Variable Method to systems with collisions that are described by an elementary impact law. An integral-type objective function of the form equation(1) where has been proposed by Haug  for the evaluation of the performance of a multibody system. Here, x are the variables of state and the upper index 2 denotes evaluation at time t2. The first term on the right-hand side of Eq. (1) involves only the state of the mechanism at the terminal time t2 and the values of the design variables b. The second term evaluates the state over some period of motion . The initial time t1 of the evaluation and the terminal time t2 may be given implicitly by the corresponding states of the system at these times: equation(2a) equation(2b) The initial state of the system is assumed to be given at a fixed time t0, t0≤t1, as a function of the design b: equation(3) Note that the initial time of the system simulation t0 and the initial time of the evaluation t1 need not necessarily be identical. In the following we will assume that the system dynamics will be governed by a first order differential equation of the general form equation(4) The paper is laid out as follows. In Section 2, the Direct Differentiation Method and Adjoint Variable Method are described as they are known from the literature. In Section 3, both methods are extended to systems with collisions. As an example, a circuit breaker mechanism is presented in Section 4. Both extended methods are applied to this system and compared with respect to their numerical efficiency.
نتیجه گیری انگلیسی
In the present paper, two semi-analytical methods of sensitivity analysis, the Direct Differentiation Method and the Adjoint Variable Method, have been extended to mechanisms with collisions. Collisions have been taken into account in the framework of elementary impact theory neglecting the duration of impact, configuration changes during the impact, phenomena such as wave propagation and so on. It has been shown that the extension is straightforward when treating the algebraic impact equations in the same way as initial conditions. A characteristic feature of the adjoint equations is that the same matrices appear as in the original equations, but in their transposed form. The number of the adjoint equations is equal to the number of differential equations of motion. The adjoint equations, however, must be integrated backward in time. Their number is not increasing with the dimension of the design space whereas the number of basic equations in the Direct Differentiation Method is linearly increasing with the number of design variables. This is true for the original methods as well as for the extended methods. The implementation of the extended Adjoint Variable Method in a numerical simulation package for multibody systems is elaborate but has been done successfully as is shown by the results for a circuit breaker mechanism with three contacts. Unfortunately, the authors are not able to draw general conclusions regarding the numerical efficiency of both extended methods. In case of the circuit breaker, however, it seems that the approximation of the design derivatives by finite differences is quite able to compete with the semi-analytical methods. Only when the differential equations of motion become stiff and their integration takes long due to evaluation of the Jacobian the CPU time for the extended Adjoint Variable Method is lowest. The performance of the extended Direct Differentiation Method is not satisfying, see Table 2. Certainly, further investigations are necessary in order to assess the efficiency of the presented methods. Since the number of adjoint equations is not increasing with the number of design variables it can be expected, however, that for high dimensional design spaces the Adjoint Variable Method turns out to be efficient. This was found at least in structural optimization regarding the conventional Adjoint Variable Method. It should be noted, however, that typically the number of design variables is much lower in mechanism design than in structural design. This is true in particular for mechanisms with collisions. They are often quite complicated although the degree of freedom is low as in case of our example system.