MechanicalSystems

 Q: What kinds of systems are addressed by MechanicalSystems? MechanicalSystems deals with the kinematics and dynamics of rigid-body systems. It is actually two separate but highly parallel packages—one for two-dimensional and one for three-dimensional kinematics. Any number of bodies may be combined by constraints to form a mechanism of arbitrary complexity. The model may then be solved numerically to find the position, velocity and acceleration of each body. If external forces are applied to the mechanism, MechanicalSystems can find the resulting reaction forces applied to each body in the model. Q: What coordinate systems are used by MechanicalSystems? By default, MechanicalSystems uses reference-point coordinates. The position of each two-dimensional body is represented by the coordinates of its origin and the direction angle of the local axis (x, y, c). The position of each three-dimensional body is represented by the coordinates of its origin and four Euler parameters or quaternion elements (x, y, z, qo, qi, qj, qk). However, user-defined generalized coordinates may be assigned to any body and freely mixed with other bodies represented by a reference point. Using generalized coordinates can significantly reduce the complexity and increase the performance of a model. Q: What are the symbolic capabilities of MechanicalSystems? MechanicalSystems generates all constraint equations and equations of motion symbolically. In general, the resulting mechanism models do not have closed-form solutions, so they are solved numerically. For those special cases that have symbolic solutions, Mathematica's built-in symbolic capabilities may be used to manipulate and solve the constraint equations. Q: Can MechanicalSystems solve inverse kinematics problems? Yes, it can. MechanicalSystems has a very flexible inverse kinematics (or design synthesis) module. Any number of mathematical conditions may be specified and applied at any number of different mechanism configurations. MechanicalSystems will adjust a specified set of design variables, attempting to find a design that satisfies all of the conditions. The mathematical conditions may involve positions, velocities, accelerations or joint reaction forces. Q: Does MechanicalSystems do dynamic simulations? MechanicalSystems does both inverse (where the motion is specified and the reaction forces are sought) and forward (where the applied forces are specified and the motion profile is sought) dynamic simulations. MechanicalSystems uses a variable-order, adaptive, Adams-Bashforth solution method for forward dynamic simulations. For 3D models using angular coordinates, MechanicalSystems uses a special integration block that integrates in angular coordinate space and then does an exact transformation to Euler parameter space. This method provides highly accurate simulations with large time-steps for models that would otherwise be limited by the integration of sinusoids representing their angular orientation. Q: Can MechanicalSystems handle simulation problems involving three-dimensional gyroscopic forces? Yes, it can. Because of the difficulty in tracking the rapidly changing coordinates of the gyroscopic bodies, gyroscopic problems (where some bodies in the model are spinning at a much higher frequency than the motion of interest) suffer in accuracy and efficiency when three-dimensional reference-point coordinates are used. MechanicalSystems provides two alternative methods to deal with such problems. First, special gyroscopic constraints can be used to efficiently represent bodies that spin about one of their principal axes. Second, and more generally, generalized angular coordinates may be used to represent the spinning body regardless of its symmetry or balance. Q: Can MechanicalSystems handle problems involving friction? Yes. The external forces applied to each body in a MechanicalSystems model are specified as symbolic functions of position, velocity or the reaction forces at constraints. In this environment, friction at a mechanism joint is simply an applied force that is a function of the position of the body and the reaction force at the joint (and possibly the velocity of the body). MechanicalSystems does not attempt to provide any particular friction model—the nature of the friction model is up to the user. Note that the symbolic expressions given to specify friction forces may have discontinuities, so it is possible to represent such things as the sign change in static friction upon a direction reversal. Q: What documentation is provided with MechanicalSystems? MechanicalSystems comes with complete documentation that automatically fully integrates with the searchable Mathematica help browser. The documentation contains several examples of complete mechanism models, which may be used as templates for your own models. A printed manual is also included, except with download versions of the product. Documentation is also available online. Q: What do I need to run MechanicalSystems, and how do I order it? MechanicalSystems 2.2.0 requires Mathematica 10 or greater and is available for all Mathematica platforms. MechanicalSystems can be purchased from Wolfram Research or your local reseller. Pricing and ordering information is available in our online store. Q: Where can I get help if I have technical questions about MechanicalSystems? For assistance in installing or operating MechanicalSystems, contact our Technical Support department or your local reseller.