Kinematics Analysis Using Matlab Simmechanics - Report Example

KINEMATICS ANALYSIS USING MATLAB SIMMECHANICS Student University KINEMATICS ANALYSIS USING MATLAB SIMMECHANIC’S Introduction Kinematics refers to the branch of classical mechanics that involves the descriptions of the motion of objects, points and systems of bodies without any consideration to its causes. Its study is referred to as geometry of motion. In order for a motion to be described, a study of trajectory of lines, points and their differential properties such as acceleration, and velocity should be undertaken. In addition, kinematics is used in astrophysics in describing the motion of celestial bodies and systems and in robotics, biomechanics and mechanical engineering, it is used in describing the motion of systems which are composed of joined parts such as the robotic arm, the skeleton of a human being and the engine [Rus09]. As a result of this, the study of kinematics can be abstracted into mathematical functions which can be used for the dynamic analysis of the systems under study. For example, the use of unit circle in representing the rotation of elements in the complex plane and use of planar algebra in representing shear mapping of classical motion in the space and absolute time in order to represent the Lorentz transformations of the relativistic time and space [PPT07]. With the use of geometric transformation in describing the movement of the components of any mechanical system simplifies the derivation of the equations of motion of the system which is central to the analysis of its dynamicity [Mer96]. Also, kinematic analysis refers to the process of measuring the quantities of kinematics which are used in describing the motion of system. By use of this process, the range of movement of a certain mechanism can be found and working in reverse leads to the designing of a mechanism that gives the desired degree of range of motion. In this paper, a simulation of a double mass spring damper using Matlab SimMechanic as a computer tool in analyzing the kinematics of systems and objects is being undertaken. Matlab SimMecahnic’s Matlab SimMechanics is a modelling environment for engineering design and simulation in block diagram format for inflexible bodily machines and bodies and their motions by use of the ordinary Newtonian dynamic forces and torques [The07]. Double mass Spring Damper A double mass spring damper is a system comprising of two springs, two masses and two dampers in which each of the elements of the components is of equal magnitude. In Figure 1, the arrangement shows the double mass spring damper of the systems to be modeled in Matlab SimMechanics. From the system above, the SimMechanics environment blocks were defined. Each block defined was a library function of the SimMechanics which has particular meaning in the SimMechanics software. After the definition of the blocks to be used, they were then dragged to the model window and arranged in such a manner to ensure proper connection and visualization of the system. The flowing is an explanation of the various blocks diagram used and their role in the model deigned. A Support is used in supporting the bodies into place in order to shows the fixed points of the objects. This supports are of mechanical in nature. This support prevents the body from falling to the ground. Figure 2 shows the block diagram of translational mechanical reference point which is used in SimMechanics as a support in modelling mechanical systems. Figure 2: Translational Mechanical reference In Matlab SimMechanics mass is represented by a block. By use of the block in the system with the right connection in relation to what is being measured, the output results can be achieved. Figure 3 shows the symbol for the block diagram for the mass which is used in SimMechanics. Figure 3: Translational mass The damper controls the speed of the oscillating objects. It does so by either reducing the speed or increasing the speed of the object. It is measured in terms of newton per meter per second. The change in the oscillation of the object due to the effect of a damper is inversely proportional to the value of the damping factor. Figure 4 shows the translational damper used in the Matlab SimMechanics as a modelling tool of mechanical systems. Figure 4: Translational damper A spring is a helical devices that has the capability of being compressed and stretched without losing its shape and length when operated within its operational range. It is commonly used to absorb movements of objects and exerts and maintain tension within limits. Figure 5 shows the symbol of a translational spring used in the Matlab SimMechanics used in modelling of mechanical systems. Figure 5: Translational spring The sensor was incorporated in the systems in order to read the physical variations of the system under study. In this case, the sensor was used to read the speed and physical position of the masses in relation to their movement and their relative position to one another. Figure 6 shows the Matlab SimMechanics block diagram for the sensor. Figure 6: Motion sensor The incorporation of the converters in the modeling of SimMechanics systems helps in converting Simulink signals to physical signals in order for the scope to read and relay them for visualizations. Figure 7 shows the symbol for the block diagram of a converter used in SimMechanics. Figure 7: PS-Simulink Convertor The scope is used in displaying the physical quantity of a simulation in order to be visualized. This gives the result of simulation of the model formed from the system given. Figure 8 shows the symbol for the scope of a SimMechanics. Figure 8: Scope Procedure for Modeling of the System After the dragging of the various components of the required model of the double mass spring damper, the components were connected with respect to their role in the system. This ensured the formation of the model required in the analysis of the problem of a double mass spring damper. Figure 9 shows the full connection of the system of the double mass spring damper. Figure 9: Connected components of the Double Mass Spring Damper After all the above was done, the system was simulated. After the simulation, the result of the simulation was graphed as an output for analysis. Results The following are the diagram from the output of the system after simulation. Figure 10 shows the velocity of the combined masses and figure 11 shows their relative position due to their displacement from the fixed point. Figure 10: Output velocity of the combined masses Figure 11: Position of masses from the reference point Discussion and Analysis From the system modelled, the forward and inverse dynamic analyses was done. From the analyses, the system demonstrated that the velocity and position of the masses in relation to the force which was generating the motion was directly proportional to each other. From the results obtained, the combined masses are shows to be stationary at the beginning until the 4th seconds when they are displaced from their original position. In addition, the change in velocity also occurs at the forth second for the first time. Shows that the masses are displaced whenever there is a disturbance and they keep on oscillating until there is an external force to counteract their motion. With the change in hardware of the system, the output of the system in relation to the position, velocity and accelerating also changes. From the experiment, the change in either the spring constant or the damping coefficient affected he system as follows; When the spring stiffness was changed, the amplitude displacement of the masses changed inversely proportional to the change in the stiffness. By the change in the spring stiffness to 2000 N/m, the output of the simulation due change in relation to the position of the combined masses is shown below. Figure 12 shows the decrease with amplitude with increase in stiffness. Figure 12: change in amplitude as a result of change in spring stiffness. By changing the damping coefficient of the damper at the Translational Damper block to 700 N/m/s, the speed of the masses changes in an inverse proportion to the maximum point and then back to its original position. Figure 13 shows the effect of increasing the damping coefficient to 700 N/m/s on the object. Figure 13: Effect of change in damping coefficient in relation to the position of the masses The error and instability caused in the system is due to the abnormal configuration of the position of the masses. With the proper set of the position of the mass, the output of the system will be correctly obtained. Figure 14 shows the right position of the masses with respect to the reference point of the masses. Figure 14: the position of the masses with change in their sizes Conclusion SimMechanic is a helpful tool of modelling and simulating systems without derivation of relevant equations like in Simulink. With the use of SimMechanics, the study and analysis of the behavior of objects and systems can be done easily by inclusion of mechanical subsystems into Simulink models. This is helpful for the non-experts by visualizing the tools of modelling and interpretation s of the results. In addition, the modeling and designing of systems using SimMechanics is faster in designing since the there is no derivation of equations which consumes more time. Despite its advantages of not deriving equations prior to its modelling, it has disadvantages in that it has limited physical modelling approach and a restricted Simulink environment. But the SimMechanic approach in simulation of systems is well suited for most practical problems of technical computing. Therefore, this approach should be embraced in the industries. References Rus09: , (Russel, 2009), PPT07: , (P, 2007), Mer96: , (Merz, 1896), The07: , (The Mathworks, Inc, 2007), Read More