The stiffness constant associated withthe bending kf can be calculated by the ratio ∥F∥/∥u∥. Then,we studied the variation of this ratio with respect to each ofthe model parameters. Each one of the parameters was taken assingle variable while keeping constant the others. A quadratic dependence on h, a linear dependence on b, and a reverse lineardependence on ˜ a were observed. This is expressed in the followingequation:kf = βh2 b˜ a, (8)were β is a proportionality constant. Since parameters ˜ a and bdepend on the geometry of the triangular faces of the part, thevalue of the kf constant is calculated for each of the springsassociated to the bending between faces.Rewriting Eq. (4) with the expression for themembrane energygiven by Eq. (7), we have:ET = 12kmiu2i+jkf ,ju2j. (9)By defining kmf ,j = kf ,j/km and absorbing the constants thatresult as a common factor one gets:ET =iu2i+jkmf ,ju2j. (10)3.2.3. Minimum energy valueWhen displacements are applied to some of the surface pointsof a real elastic object, this object becomes deformed in aspecific way. From a physics perspective, this means that thepoints of the system adopt a configuration of minimum energy,which represents the elastic potential energy stored in the objectdue to its deformation. Taking into account this principle, theproposed method applies an optimization process to determinethe configuration that presents theminimumvalue of the system’senergy function. Then, such configuration represents the nodeposition on the deformed polygonal model.As an example, let us consider a one-dimensional systemmade of two linear springs with stiffness constants {k1, k2}and assembled as shown in Fig. 3. The nodes are identified as{n0, n1, n2}; the {x0, x1, x2} positions represent the configurationwhen the system is idle, e.g. non-deformed. If n0 remains fixedand a displacement ∆2 is applied to the n2 node, then, the n1node will move to the new x′1 position. Taking into account therestrictions x′0 = x0 and x′2 = x2 + ∆2, the deformed systembecomes represented by a newconfiguration {x′0, x′1, x′2}, forwhichthe minimum value of the system’s energy function is calculated.In terms of deformations {u1, u2}, the system’s energy functionis expressed by:E = 12k1u21 + 12k2u22. (11)Taking into account the relationship between deformations anddisplacements {∆1,∆2} around the initial position of equilibriumwe have:u1 = ∆1, u2 = ∆2 − ∆1. (12)The energy function is rewritten as:E = 12(k1 + k2)∆21 − k2∆2∆1 + k2∆22. (13) Since the applied displacement ∆2 is a constant, the energy is aquadratic function of the ∆1 variable. The minimum value of thisfunction Emin is the new equilibriumpoint x′1 of the n1 node, whichcan be determined through aminimization process of the system’senergy function.As shown in this example, starting from an equilibrium initialposition and then applying a displacement to a point (or a set ofpoints) of the system, such system adopts a new configuration ofequilibrium to which the system’s energy value is minimum.4. Modeling deformationsThe deformation process is necessary to carry out the non-rigid alignment, and then a comparison, of the acquired datamodel against its CAD model. Such comparison determines if thereal part meets the expected specifications defined in advance byengineering design considerations.In order to simulate the deformation of a real part, thetransformation applied to the virtual model needs to be adeformation based on physics. This section describes the proposedmethod to calculate the deformation of a CADmodel of deformableparts; such method is based on the spring model described in theprevious section. In summary, the process inspection method iscomposed of three stages: in the first stage, a polygon reductionis applied to the initial part’s CAD mesh in order to reduce thecomputational burden; the second phase consists at minimizingthe system’s energy function by which a simplified deformedmodel is obtained; finally, in the third stage, an interpolationprocess is used to obtain a deformed model with all the points ofthe initial CAD model mesh. Fig. 4 illustrates the proposed processby using a general scheme which is explained in detail in thefollowing sections.In order to apply the proposed method, it is assumed thatthe part’s surface is represented by a polygonal model oftriangular faces. 弹性质量系统变形检测英文文献和中文翻译(3):http://www.751com.cn/fanyi/lunwen_34627.html