The Spring and Damper Mechanism as the Major Principle Applied in the Design of the Safety Apparatus - Lab Report Example

Name Course Tutor Date Introduction It is a fact that accidents occur on a day to day basis be it on roads or railways. Over years various developments to avoid damages caused by accidents have been on an increase thereby sparking a lot of debates in the public domain. The most notable discoveries of the twentieth century were actually the use of springs and damping devices as a means of reducing impact felt due to resultant momentum especially on wheels and other mechanical components. Further improvements were made to the interior whereby injuries due to undesirable movements such as loss of control by operators could be inhibited. In this case the use of airbags became handy as their capability to act as dampers when triggered could avoid damage or loss of lives. It is due to these findings that this study seeks to unveil the means by which moving masses can be protected from damaging after hitting barriers in such circumstances such as head on collision. In order to come up with a system that has the ability of both protecting the mass in movement and at the same time damp the resultant forces such as vibrations, Newton’s second law of motion comes in handy. From Newton’s second law of motion it is observed that momentum change is directly proportional acceleration as long as a harmonious state of motion is maintained. This law takes into consideration that momentum is a product of velocity and mass of the object in motion and that the direction in which the mass is moving is parallel to the applied force. By the end of this report, a safety device shall be designed to indicate how all the forces mentioned shall be overcome by mechanical damping. Stating Newton’s second law of motion: . Figure 1: Graphic presentation of the Problem. Mechanical Expressions i. Natural Frequency The natural frequency is vibration caused due dissipation of energy accumulated by momentum in a moving mass. It is an important parameter in this study as it gives a projection of what happens at the destination or after the mass hits the spring damper system. It is also important to note that when the natural frequency is at an elevated degree, it actually results to resonance which is amplified vibration in other terms. In order to calculate natural frequency the following derivations should be made: The momentum of a moving object is equal to the weight, where displacement is positive to the direction of acceleration. Therefore, the forces acting on the mass are In the equation above, is the spring constant. This only applies in single degree of movement which is discussion. Applying mass in the Newton’s second law of motion: But Considering the circular frequency then it can be concluded that: (In a state of harmonious mass motion) ……. Also …………………………. Substituting in above, the following equation is obtained: Thus the natural frequency is established as: ……………………………………..... ii. Energy Expressions It is paramount to note that the energy equations shall be of great importance when calculating the potential energy that is contained in the spring coupled by the momentum to be applied on the moving mass. In accordance to Hooke’s law, the elasticity of a material is directly proportional to the load applied on it. Therefore in this case: Where is the elongation or compression of the spring and is the spring constant. Since there is change in the elongation value, on integration: or, iii. Damping Damping in a vibrational mechanism is usually achieved by introduction of frictional forces which in turn transfer the motion away through intermolecular interactions. The damper is meant to absorb the force in order to slow down the moving mass on impact. For harmonic vibrations the energy equation is stated as: The frictional force acting on the damper shall be stated as: Where is the coefficient of damping. For all forces acting on a harmonic vibrating mass to balance, then: On differential simplification of the above equation: Breaking down the above equation, it is evident that angular frequency for un-damped vibration of a moving mass shall be: The expression for the damping ratio shall be expressed as: Apparatus Design The project design shall be based on the third law of Newton that relates to action reaction. In his definition, Newton stated that there is an equal to every action force exerted, but rather in an opposite direction. This force is known as the reaction force. It follows that: Where is the action force and is reaction force that is equal but acting in an opposite direction. Therefore, in the design of the system that is going to protect the mass from damaging, it is a requirement that these two forces be considered. For the sake of experimental setup, considering we have to protect as moving mass of 5,000 kilograms, then the safety apparatus’ threshold reaction force shall be, , where is the top acceleration of the mass. For example if the maximum acceleration of the mass such as a vehicle or a train is then the force shall be 50,000 Newton. The apparatus shall be set in such a way that it absorbs the shock on hard impact just the same as the shock absorbers are built. This is based on the fact that, the mass to be protected has to be equipped with proper safety apparatus to counter the reaction the damper and the spring should nullify the forces to be encountered. To illustrate this, the damper should possess the capability to damp up to the mass’ maximum attainable force of 50,000 Newton. The figure 2 below shows the suggested design setup of the safety apparatus to be used in absorbing the reaction energy as a result of a collision. Figure 2: Safety apparatus design. Experimental Setup The experimental setup shall be entirely dependent on the pioneer spring mass experiment as shown in figure 3 below. The main aim of this experiment though shall be; to establish the perfect spring and damper types suitable for the kind of safety apparatus. The suitability of these gadgets should be in line with their ability to counter the reaction force. Figure 3: A typical experiment to establish spring and damper constants suitable for Safety Apparatus design. This experiment uses the following apparatus: Three springs of different spring constants, three dampers with different damping constants and lastly three different masses. It is assumed that the acceleration is and this is due to the gravitational pull. After setting up the apparatus as shown above, at the beginning of each experiment the initial point of reference is marked and noted down. A mass such as is then hanged in the spring and damper connection and the impacting change in length noted down. The results are then tabulated and tabulated for vibration analysis. The experiment matrix is shown below: The Constant Parameters The Constant Parameters The Constant Parameters 1.8 14 0.1 1.8 14 0.2 1.8 14 0.3 1.8 0.1 14 1.8 0.1 16 1.8 0.1 20 14 0.1 1.8 14 0.1 3.0 14 0.1 1.2 Table 1: The experiment matrix Where the respective masses are, assuming that the masses of the springs are negligible. The damping constant shall be 0.8 for the under-damped spring, 1.0 for the correctly damped spring and 1.2 for the over-damped spring. Lastly the spring constants of the springs to be used for demonstration purposes are as follows: All these assumptions are made depending on suitability of the material that is available. Discussion The force applied by the mass is the product of its mass and the gravitational acceleration which occurs naturally. Since the action force is equal to the reaction force it is then true to state that: the force exerted by the masses shall require a combination of a spring and a damper with the same reaction capability in order to counter these forces. In design it is therefore important to indicate that the masses used in the demonstration shall have a comparison index or ratio to the 50,000 Newton moving mass. From the following calculation it is evident that the force at which the mass shall have a head on collision shall be reduced due to the spring damper combination in the design of the safety apparatus. Without a spring and damper: Force for Calculating the conversion ratio for the prototype of 50,000 Newton: In a case where the action to be countered is equal to then the reaction has to be. Considering the positive force the reaction force shall be In order to establish the compression distance that will be caused by the force, then for each combination above shall be calculated as shown below as long as the harmonious motion is maintained: For combination 1: therefore which is a considerable elongation or compression ratio in cases whereby practical application is required. The vibrations to be damped also have to be calculated in order to establish the best damping coefficient to be applied. According to the results above, the figure is considerably small thus damping for up to 0.2 of natural frequency shall mean that if a spring has to be attached together with the damper, the vibrations to be eliminated are close to nil. The series of spring in terms of their constants to be used in prototype design according to the conversion notation shall be as follows: i. ii. iii. In order for the safety mechanism to be safe, it is paramount that the damping ratio to be equal to 1. This means that the system has to be sufficiently damped to avoid damage on the moving mass or a backward force for that matter. Therefore equating the rest of the values against the damping ratio: , According to the above matrix, if are the best combinations, then the damping coefficient shall be: The correct damping coefficient is 1.789, for, while and shall be under-damped and over-damped respectively for the same combination. Conclusion The major principle applied in the design of the safety apparatus is the spring and damper mechanism. In this case, the momentum is converted into vibrations by the spring and absorbed by the damper. It is however important to note that the device should be damped correctly in order to avoid any damages on the moving mass. This shall be of great benefit to such states whereby barriers are erected on an impromptu basis and train stoppage at the end of the rail. Read More