Re: Spline Standards And Spline Calculation.pdf
Roan Distilo
shaft splines and serrations are ridges, or teeth-type keys, that are an integral part of the shaft that mesh with grooves in a mating hub to transfer torque and rotational motion. For example, a bevel gear mounted on a shaft might use a male shaft spline that matches the female spline on the gear, as shown below.
Although a splined shaft looks like having a series of shaft keyways with keys pushed in, splines are considerably stronger than the keyed joint as the keyways weaken the shaft and reduce its torque-carrying capacity.
Depending on their relative axial movement, splines and serrations can also be grouped as fixed splines or flexible splines. As the name suggests, a fixed spline is a joint that does not move axially, such as gears, pullers, turbine wheels, etc.
These have straight and parallel tooth flanks, as shown in the figure below, and as per various standards, number of teeth can vary from 4 to 12. They can transmit higher torque than involute splines and serrations because of their large tooth thickness from the minor to the major diameter of the profile. But might fail due to fatigue due to stress concentration in the root of the flanks.
Naturally, it lacks centring ability because of the straight flanks, forcing it to rely on the major and minor diameter fits to manage the centring. Because of the straight-sided face, there will be a line of contact and surface contact will only exist after some wear.
Involute splines are very common and similar to internal and external involute gear teeth. They are comparably stronger than the parallel spline because of the lower stress concentration factor and have better surface quality. Involute splines can be produced by gear manufacturing techniques and have the ability to self-centre under load.
Involute splines are made with pressure angles 30o, 37.5 and 45o and can include between 60 and 100 splines per the American National Standard. Involute splines can be either Side fit or Diameter fit.
Serrations also have straight flanks but are angled, as shown in the figure below. The biggest advantage of the serrations is that the angles flanks centre the shafts and the hub resulting in self-centring splines. Flank angles are generally between 50o and 90o.
The main disadvantages of serrations are due to comparably small teeth, it can only be used for low torque applications. These are only used for non-axial moving applications. Like straight-sided splines, there will be line contact and wear.
The load is equally distributed if the transferring load is purely radial torsion and the torsional radial load is in the middle of the spline length. But if, for example, a bevel gear is used, this will put some unwanted axial loads into the spline.
If there are any axial or radial shock loading on the element connected, then care should be taken to support the external axial and radial shock loads to increase the joint life. This should also be considered during the calculations using the spline application factor.
Life factors for splines under wear conditions are based on the number of revolutions of the spline joint, not reversible cycles. Wear life factor only applies to flexible or sliding spline compressive stress calculations, as each time the spline slide back and forth, it wears the teeth.
I am looking for ways to evaluate exactly (i.e. analytically or semi-analytically) integrals of the type:$$\int_-\infty^+\inftyB_i^k(u)e^-\frac(u-\mu)^22\sigma^2du,$$where $B_i^k$ is a spline of order $k$, an element of the B-Spline basis for the linear space of splines of order $k$ on knots $\t_i\$, defined as usual recursively by:$$B_i^k(x)=\fracx-t_it_i+k-t_iB_i^k-1(x)+\fract_i+k+1-xt_i+k+1-t_i+1B_i+1^k-1(x),$$with$$B_i^0(x)=\begincases 1 & x\in [t_i;t_i+1) \\ 0 & \textotherwise \endcases$$Of particular interest would be the case of $\mu=0, \sigma=1$.
I am aware of the Gauss-Hermite quadrature :$$\int_-\infty^+\inftyf(x)e^-\fracx^22\approx \sum_i=1^n w_i f(x_i),$$where $x_i$ are the roots of a Hermite polynomial of order $n$ and $w_i$ are the associated weights. Importantly, the approximation sign can be replaced by an exact equality when $f$ is a polynomial of degree $\leq 2n-1$. (There are versions where the integral is with respect to $e^-x^2$ instead of $e^-\fracx^22$, by changing the type of Hermite polynomial employed).
My question is : is there such an exact equality formula for B-spline basis functions?I am looking to express the integral at the beginning of this question as a sum analogously to the Gauss-Hermite quadrature.
The problem seems to be that even though $B_i^k$ is known to have finite support, it is not itself a polynomial: each of the restrictions $B_i^k_(t_j;t_j+1)$ is a polynomial, without the full function being a polynomial. Otherwise, the answer would have been a trivial application of the Gauss-Hermite quadrature. Is is possible that there is a Gauss-Hermite-type quadrature for integration domains that are compact intervals (as opposed to integration domains that are $\mathbbR$) ?
Since a B-spline is a piecewise polynomial function, the question is whether there exists an exact equality formula for the integral $\int_-a^bu^pe^-u^2/2du$. This integral equals an elementary function of $a$ and $b$ for $p$ an odd integer, while for $p$ an even integer it contains error functions. In general the B-spline will contain both even and odd powers, so no "exact equality formula" in terms of elementary functions will be forthcoming.
To overcome the topological constraints of non-uniform rational B-splines, T-splines have been proposed to define the freeform surfaces. The introduction of T-junctions and extraordinary points makes it possible to represent arbitrarily shaped models by a single T-spline surface. Whereas, the complexity and flexibility of topology structure bring difficulty in programming, which have caused a great obstacle for the development and application of T-spline technologies. So far, research literatures concerning T-spline data structures compatible with extraordinary points are very scarce. In this paper, an efficient data structure for calculation of unstructured T-spline surfaces is developed, by which any complex T-spline surface models can be easily and efficiently computed. Several unstructured T-spline surface models are calculated and visualized in our prototype system to verify the validity of the proposed method.
With a series of excellent mathematical and algorithmic properties, non-uniform rational B-splines (NURBS) has been widely used in the field of computer aided geometric design for representing curves and surfaces. Nevertheless, in modern industry, complex engineering models comprised of multiple NURBS patches are always not watertight because of the existence of gaps and overlaps along the interfaces of trimmed NURBS surfaces. Thus, T-splines were firstly proposed by Sederberg et al. [1, 2] in 2003 to conquer the limitations of NURBS in practical engineering applications.
As a generalization of NURBS, T-splines introduce T-junctions and extraordinary points into its control mesh. Theoretically, a T-spline surface can represent any arbitrarily shaped model no matter how complicated its topology structure is. Compared with NURBS, the advantages of T-splines can be reflected in the following aspects. Firstly, a NURBS surface is defined in a rectangular topological grid. It requires a large number of superfluous control points to maintain the topological shape while implementing refinement. This shortcoming can be overcome by T-splines which can achieve local refinement without introducing an entire row of control points. In addition, it is difficult to represent a complex model with a single NURBS surface and the gaps along the common boundary of two NURBS surfaces are unavoidable. T-splines provide a promising way to breakdown these barriers. In ref. [3], multiple trimmed NURBS patches are merged into a single watertight T-spline surface. Li et al.[4] studied the linear independence of T-spline blending functions and proposed the notion of analysis-suitable T-splines. Analysis-suitable T-splines satisfy a simple topological requirement and their blending functions are linear independent [4,5,6]. So far, T-splines have been used in many fields such as geometric modeling [7,8,9], isogeometric analysis [10,11,12,13,14,15] and shape optimization [16,17,18].
In complex T-spline models, the extraordinary points are always indispensable. T-splines containing extraordinary points are called the unstructured T-splines [14]. When encountering an unstructured T-spline surface, the knot interval vectors about the vertexes around the extraordinary points are ambiguous. More details about the concept of extraordinary points are presented in section 2. Some methods have been developed to deal with the problems caused by extraordinary points [14, 19, 20]. In the template method proposed by Wang et al. [19], gap-free T-spline surfaces are generated by inserting zero-interval edges around the extraordinary points. Liu et al. [20] proposed a knot interval duplication and optimization method to obtain local knot vectors. In ref. [14], Scott et al. introduced a linear interpolation scheme to calculate Bzier control points from T-spline control points, which is easy to understand and implement.
Since T-spline surfaces have flexible topology, constructing a robust and efficient data structures of T-splines for storing and further data processing is a challenging topic. Asche et al. [21] presented a T-spline data structure implementation based on a half-edge (HE) data structure and implemented the algorithms with CGAL geometry programming library. Lin et al. [22] developed the so-called extended T-mesh which can be represented in an obj-like format file and converted into the face-edge-vertex data structure conveniently. With this method, each vertex in the extended T-mesh has a knot coordinates, which cannot solve the situtation with extraordinary points. Xiao et al. [23] also proposed a set of new T-spline data models to obtain better data storing and operating efficiencies. However, all the T-spline data structures mentioned above cannot deal with the T-splines with extraordinary points, i.e., unstructured T-splines. To the best knowledge of the authors, there are no public research papers or open sources which directly present the suitable approaches to handle the unstructured T-splines from a view of programming implementation.
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