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Mathematical models for analytical synthesis and software 3D realization
of spatial motion transformers:
Review of past and current research results
Dr. Emilia Abadjieva,
Assoc. Prof. in Department „ Scientific Computations”

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1. Scientific importance of the problem
The intensive development of industrial manufacture and the application of the different types gear transmissions define the necessity of forming an appropriate strategy for choosing the right type motion transformers from a functional view point with their effective realization and also from technological point of view. The choice of the optimal type gear mechanisms is connected with determining the common section of the upper defined two groups of requirements. This is realized by comparison between the exploitation properties of mechanical transmissions with their technological characteristics and with the manufacturing cost.
The science of the gear mechanisms together with the above problem is necessary to create the preconditions for finding out and studying new and little known characteristics of multibody systems transforming motions in accordance with preliminary given laws. This is a premise for development of scientific studies serving the synthesis and design of new types gear transmissions that would satisfy concrete industrial requirements. The solution of the second task is based on development of adequate mathematical models. These models describe the physics of the processes of motion transformation and based on them approaches to the synthesis and design of the corresponding transmissions.

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2. Usefulness and applicability of the obtained and expected results
Mathematical modeling of physical objects (processes and discrete systems bodies that realize them) through the application of scientific achievements in mechanics is the basis of modern scientific knowledge. The type of the developed theoretical analogue of the physical object is in direct dependence of the defined research goal.
The development of the mathematical models describing adequate spatial transformation of motions determine the scientific significance of studies. These models are applicable for the development of algorithms for analytical and software synthesis.
The accomplished and coming for realization studies can be grouped as follows:
-mathematical models and programs, accomplishing theoretical studies in order to establish new kinematic and strength characteristics of the developed transmissions;
-models and programs oriented to geometric and technological synthesis, including by the inovative 3D software technologies.
The developed and permanently improved theoretical apparatus is oriented towards the study of two types three links spatial mechanisms: hyperboloid gears, transforming rotation motions and spatial rack mechanisms, transforming rotation into rectilinear translation

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3. Kinematic interpretation of the approach to mathematical modeling
Kinematic theory of spatial motion transformations is in the base of the developed and applied by me three types mathematical models for synthesis of spatial gear mechanisms: upon a pitch contact point, upon a mesh region and a combination between the above models.
The main constructive elements –pitch configurations are defined through the special section of this theory. They are applicable in the models for synthesis upon a pitch contact point. The mentioned primitives are essential to determine the basic characteristics of transmissions, such as: size of the structure and type of their general geometry; sizes, longitudinal and profile orientation of active tooth surfaces; strength load of transmission, its efficiency, etc.

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3. Synthesis upon a pitch contact point
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Two basic task are solved, when :
Synthesis of pitch configurations;
Synthesis of active tooth surfaces.

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4. 3 D software realization of spatial gear pairs
Fig. Computer design of spatial gear drives of type Helicon and Spiroid: a ) geometric pitch configurations, b) CAD model

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4. 3 D software realization of spatial gear pairs
4. 3 D software realization of spatial gear pairs
Fig. 3D software model and 3D printed model of micro-module face gear pairs type Helicon, synthesized for the active joint of the leg of insect-type robot with offset 17 mm; gear ratio 41/2 (axial module mm)

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4. 3 D software realization of spatial gear pairs
Fig. 3D software model and 3D printed model of micro-module face gear pairs type Helicon, synthesized for the active joint of the leg of insect-type robot with offset 17 mm, gear ratio 78/1 (axial module 0.45 mm)

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5. Application of the International Standards solving the judicial dispute
Fig. Planet cylindrical gears from the tractor URSUS 1634 transmission
Fig. Indentation (BDS ISO 10825, pp. 27, 1997): The depressions in the active tooth flanks are caused by the foreign bodies (metal particles) during the meshing

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Application of the International Standards solving
the judicial dispute
Fig. Breakage (BDS ISO 10825, pp. 56, 1997): The both illustrations show, that from all teeth of the gears are broken edges or parts, that are close to the tips of the teeth. Those two gears are damaged due to inaccurate assembling, that caused non-uniform loading of the meshed teeth
Fig. Conical pinion and crown from the tractor URSUS 1634 transmission

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Thank you for your attention!