1、Kinematic Analysis Of Complex Gear MechanismsAbstractThis paper presents a general kinematic analysis method for complex gear mechanisms. This approach involves the null-space of the adjacency matrix associated with the graph of the mechanism weighted by complex coefficients. It allows to compute th
2、e rotational speed ratios of all the links and the frequency of all the contacts in this mechanism (including roll bearings). This approach is applied to various examples including a two degrees of freedom car differential.Keywords: Kinematic analysis, gear train, graph theory, car differential1 Int
3、roductionThe research explained in this paper takes its source in the domain of the Health and Usage Monitoring Systems (HUMS) of helicopters. Nowadays, a lot of studies are done on such systems 1. A very important part of studies to improve the performances of HUMS concerns the vibration analysis o
4、f the transmission, and especially of the Main Gear Box (MGB). The aim of these studies is to identify defaults on the MGB using vibration analysis. In fact, each default of contact between the different links of a complex system, as a MGB, can generate an harmonic disturbance at a precise angular f
5、requency in the vibratory signal.In that domain, a few researches have led to the use of Kalman filters on angularly sampled signals 2. Such filters can provide a good estimation of the magnitude and phase of an harmonic component in a signal when its frequency is well-known 3. So, to create the dyn
6、amic Kalman model, a very good knowledge of (angular) frequencies of all the contacts in this mechanism (including roll bearings) is required. To determine these frequencies, rotational speed ratios of the various links of the mechanism is required first of all.There are a lot of kinematic analysis
7、approaches for different types of gear trains. The tabular method is commonly used but can involve a lot of calculation, and cannot give the velocities of elements whose rotation axis are not on the input/output axis 4. The vector analysis method gives very good results for bevel gears, but is very
8、complex and can lead to human mistakes 5. The graph theory method can be easily computerized, and can give the velocities of all elements of the gear trains. It can also be adapted for bevel gears 6. It has been studied by Nelson in 7 so as to find the angular velocities of all links in bevel epicyc
9、lic gear trains. It is also limited to gear trains whose input and output axes are co-linear.In this paper, a new kinematic analysis method, based on the work of Nelson, is introduced. Its objective is to list all the mechanical contacts between all elements in the transmission system (ball-bearing,
10、 gears.) and for each of these contact, to find its angular apparition rate, that is the number of times this contact appears for one revolution of the input shaft. Of course, to solve this problem, a general tool to compute all the speed ratios between the links of the transmission is required.Ther
11、e is a few advantages to this method. The most important of them is that it is possible to analyze very complex mechanisms, as long as its internal composition is known. For example, it is possible to deal with a system whose input and output axes are not co-linear. Systems with several degrees of f
12、reedom, as a car differential can also be studied with this method.The first section presents the kinematic analysis method. In the second section, examples are presented to demonstrate the interest and the generality of this method: asimple epicyclic bevel gear train and a car differential.2 Kinema
13、tic Analysis Method2.1 Speed ratio matrixIn this section, the kinematic analysis method is introduced. First, it is important to understand that the difference with the kinematic analysis methods already existing, consists in the introduction of complex numbers in the definition of each link of the
14、mechanism.The first step of that method is to build the table T of mechanism links and joints. For a mechanism with N links this table is a N N table representing the kinematics graph of the mechanism. Only mechanisms with turning pairs (revolute joints) or gear pairs are considered here. The elemen
15、t (i, j) of T denotes the interaction of the link i on the link j. The table T is built following a few rules: Turning-pairs are noted (p) for the elements (i, j) and (j, i). For gear pairs, the element (i, j) is equal to and (j, i) to , where and are respectively the number of tooth on elements i a
16、nd j. (respectively ) is the angle between the axis of rotation and the axis of the gear tooth of element i (respectively j) in contact with element j (respectively i), counted in the positif clockwise and so that | | /2 for an external gear. Otherwise, elements of T stay empty.Lastly the reference (carrier) link k is also required for each gear pair. The reference link is the link in which t
