Mass Spring Damper System - Assignment Example

Name Institution Lecturer Course Date Mass Spring Damper (MSD) System Introduction An MSD system is a crucial engineering component requiring critical understanding. An MSD is an experimental procedure of analysing vibrations in mechanical system through the application of a required damping effect on a vibrating system. Such an analysis is critical in mechanical and structural systems because it helps in the identification of the most appropriate damping effect that should be provided for the respective system to contain vibrations within the system within allowable and safe values. For example, an MSD system is used for analysing the dynamics of mechanical systems including the analysis of the dynamic stability of an aircraft and analysis of the dynamics of the suspension system of automobiles (Hinrichsen and Pritchard 16 & Yechout and Morris 303). Additionally, a mass-spring-dumping system is widely used in control systems in various applications. For example, an MSD system is used to control steel production wherein it controls the thickness of steel during rolling (Gopal 79). This project seeks to employ the decaying curve method to determine the damping coefficient, C, of various mechanical viscous damping systems (dashpots). Theory Vibration is a crucial mechanical element whose analysis is critical to practically all mechanical systems. For there to be vibration in a system, three crucial elements must be present: Stiffness: this is the element that stores the potential energy required for the vibration (Gopal 99). The stiffness component arises from the structural properties of a mechanical system under consideration. Stiffness is defined as how rigid or resistive a material or system is. In the case of a spring, the stiffness component prevents the spring from extending or bending, which is depend on the material properties of the spring. Accordingly, a spring made of high carbon steel, for example, will behave differently from a material made of low carbon steel with respect to stiffness and, therefore, vibration. In other components, such as a beam or a column, the stiffness component defines the component’s resistance to bending. Mass or inertia: the component that stores the kinetic energy for the vibrating system (Gopal 99). Mass also includes the force that initiates the vibration and which the stiffness component strives to resist. It is the interaction between mass/inertia and stiffness that keeps a system vibrating until stopped by the damping component. In other words, the mass/inertia component converts the potential energy from the mass component to kinetic energy for vibration. In a spring, such as spring used in automotives, the mass component is the mass of the automotive resting on the spring in which vibration is initiated when a moving automotive hits a ditch. Damping: the component that dissipates energy. Without the effect of the damping system, which is not possible in the real life situation, a vibrating system will continue to vibrate indefinitely. However, damping leads to gradual reduction in vibration through energy dissipation in the form of heat and sound, unless there is continuous application of vibration force. The damping effect in a vibrating system may arise from three factors, dry friction, structural damping and through the use of mechanical dampers (Gopal 99). Dry friction: dry friction arises from the friction between the moving parts in the vibrating system. The effect of friction is to dissipate energy, which is lost through heat and sound arising from the friction based on the law of conservation of energy “energy is neither created nor destroyed but transformed to other forms. Structural damping: this occurs when there is internal friction between the various layers of the vibrating system, which leads to heat production, a common scenario during vibrations. Structural damping also occurs when there is plastic deformation of the vibrating system, such as plastic deformation of a spring. The effect of plastic deformation is failure by the vibrating system, such as a beam, to return to its initial state after every cycle, which leads to gradual reduction in vibration. Such a scenario usually occurs when the force initiating vibration in a system exceeds the bearing capacity of the vibrating component, such as a load exceeding the elastic limit of a beam. Damping through use of mechanical viscous dampers: a mechanical viscous damping system is used to apply a required damping effect on a vibrating system through the determination of the damping value. The first two sources of damping, dry friction and structural damping cannot be eliminated completely because they are uncontrollable in any vibrating system. During experimental procedures, these two sources of damping are usually eliminated as much as possible and thereafter assumed to no longer affect the vibrating system. Therefore, during the analysis of vibrations under specified conditions, the effects of dry friction and structural damping on the system being analysed are usually ignored and the system considered un-damped if a mechanical viscous damping element is not used. Otherwise, the system is considered to be analysed under damped condition. Mathematical Expressions A damped vibratory system is comprised of four main components: 1. A spring 2. A dashpot (damping system) 3. The mass component 4. A tracing component to trace the behaviour of the vibratory system after vibration has been initiated Therefore, there are three principle factors of consideration, which include the spring constant factor (k), the damping constant factor (C) and the mass (M), which are related as shown in the following mathematical expression. A simplified and ideal MSD system is as shown in figure 1 Figure 1: An ideal MSD system showing a spring (k), damper (R) and a mass (m) being given an initial displacement (y) If the system shown in figure 1 is given a vertical displacement y and left to vibrate on its own, it will exhibit a free vibratory motion in which time will vary (time-varying function-y(t)). The vibratory motion can be represented using equation 1 as shown. (1) In order to solve for y (t), the function will be defined as , which lead to an auxiliary equation and respective solutions as: Therefore, (2) Substitute this in y(t), results to: Therefore, (3) However, (4) Therefore, equation (3) can be represented as: (5) Decaying curve method of determining damping coefficient As aforementioned, the effect of damping on a vibrating system is to reduce vibrations gradually. The effect of such a gradual reduction in vibrations is known as decaying of the vibratory system, which can be represented schematically using the graph in figure 2. As shown in figure 2, the amplitudes of the vibrations reduce gradually as time proceeds (gradual reduction of amplitude with time). Figure 2: Graphical representation of the effect of damping on a damped vibratory system Considering a situation as shown in figure 1, which shows a typical decaying curve, the ratio of amplitude Yo at t = to, to the amplitude Yn at time t = to + n, can be represented as: (6) As shown in figure, there is an element of logarithmic decrement in amplitude, , and which can be defined as: (7) Where n = number of complete cycles (peak to peak) Y0 = the amplitude (height) of the first cycle (from the start of counting n) Yn = the amplitude (height) of the nth cycle, which is the last cycle in n Eliminating from equations 6 and 7 above leads to: Therefore, (8) In equation 4, there is an element of natural frequency, which is defined as: (9) Equation 4 also introduces the element of spring constant, which is dependent on material properties (spring material) and the dimensions of the spring. Spring constant, K, is given thus: (10) Therefore, in collaboration with equations 9 and 10, equation 4 can be used to determine the damping coefficient. Experimental Setup and Procedure The experiment will be setup as shown in figure 3. Figure 3: Experimental setup Parts 1) Mass1 2) Mass2 3) Shaft/Rod 4) Base 5) Lever 6) Damper 7) Spring 8) Graph paper wound on drum 9) Motor 10) Shaft/rod connecting motor and tracing paper 11) Marker Mass-Spring-Damper System Diagram Different spring materials, such as rubber and steel, will be used. Different masses, such as 290gm, 500gm, 750gm, will be used, which will be changed to obtain the change in the model to prove that different material of springs and different masses affect the bouncing behavior of the spring. Requirements 1. Three rubber mountings (Damper-mass number 2 in figure 3) a) OD: 50 mm, Length: 80 mm, Wight: 290 gm. b) OD: 60 mm, Length: 120 mm, Wight: 500 gm. c) OD: 70 mm, Length: 140 mm, Wight: 750 gm. 2. Two compression Springs (1 Carbon steel spring and 1 Stainless Steel 304) Wire Diameter: 2 mm OD of spring: 36 mm Length: 145 mm Gap: 6 mm Weight: 125 gm. 3. Three M S Blocks (mass number 1) a) 14mmX14mmX14mm Square block; Mass: 20 gm b) 16mmX16mmX24.5 mm Rectangular Block; Mass: 40 gm c) 16mmX16mmX33 mm Rectangular Block; Mass: 60 gm Where OD = outside diameter Assumptions The experiment is based on the assumption that the damping effect only comes from the dampers used for the experiment. Therefore, it is assumed that damping effect from dry friction and structural damping are negligible. However, it is not possible to eliminate these two components completely but their consideration in the experimental calculations will lead to complexity during analysis. Additionally, it is assumed that during the execution of the experiment, the springs shall operate within their elastic limit such that there shall not be cases of plastic deformations. The springs shall be assumed to be made of pure stainless steel and carbon steel respectively. Therefore, shear modulus values for the springs will be assumed to be: For carbon steel spring, G = 0.4 E (where E = 2.0 x 105 N/mm2) (Duggal 4) G = 8.0 x 104 N/mm2 For stainless steel 304 spring, G = 7.9 x 104 N/mm2 (George and Shaikh 28) Experimental procedure The experiment will be setup as shown in figure three. The variables under considerations include the mass (mass 1 and mass 2), the damping constant (three MS blocks) and spring constant (carbon steel spring and stainless steel spring). A graph paper will be wound on the drum as shown in figure 3 (part 8). The drum shall then be placed in such a way that the pointer/marker just touches the graph paper and indicates some marks. Having setup loaded the experiment and attached the marker/pointer to the end of the lever (component 5 in figure 3), the damper will be pushed down by hand. The motor will be started and the drum allowed to rotate until it makes a straight line along the graph paper would around the drum. The damper will then be released so that the mass (mass 1) is free to vibrate (move up and down) owing to the action if the spring. The wave motion of the system is traced on the rotating paper wound around the drum. The system will be allowed to vibrate until the drum makes one revolution after which the system will be stopped. Using the same spring, the procedure will be repeated twice using the two remaining masses, each mass one run. The procedure will also be repeated using three different dampers after which the spring will be changed and the entire procedure repeated. In each run, the graph paper will be removed from the drum and a new paper put in place such that each run will have its own graph paper. These graphs will then be used for the various calculations for determining damping constant. Data collection and Calculations Data from the experiment will be collected and recorded. Spring parameters Wire Diameter (d): 2 mm Outer diameter of spring (d): 36 mm Length (l): 145 mm Gap: 6 mm Weight: 125 gm Number of turns: (N x d) + ((N-2) x gap) = length 2N + 6 (N-2) = 145 8N = 147 N = 18 turns From equation 9, Where: G = shear modulus of the spring material For carbon steel spring, G = 8.0 x 104 N/mm2 For stainless steel 304 spring, G = 7.9 x 104 N/mm2 D = spring’s mean diameter, which is determined as, D D = 36 – 2 D = 34 mm For stainless steel spring For carbon steel, Data processing The experiment will result into graphs showing decaying amplitude with time as shown. Figure 4: Sample graph showing decaying amplitude with time In each graph, a number of full cycles, n, will be selected for analysis. The amplitudes of the first and last cycles (Y0 and Yn respectively) will be measured and used for subsequent calculation aimed at determining the damping constant. Using equation 7, is determined thus, The value of found is then applied for determining using equation 8 wherein is determined thus, Then the natural frequency of the system,, is determined using equation 9 wherein it is determined as, Having determined and K, equation 4 is used to determine damping coefficient as, Where M is the mass (M1 + M2) This is repeated for all the trials and the values of C, K and M recorded in the table below. Table 1: Results and calculations Carbon steel spring K = M 1 = 290g Trial MS Block   C (N.s/m) 1 20 2 40 3 60 M 2 = 500g Trial MS Block   C (N.s/m) 1 20 2 40 3 60 M 3 = 750g Trial MS Block   C (N.s/m) 1 20 2 40 3 60 Stainless steel spring K = M 1 = 290g Trial MS Block   C (N.s/m) 1 20 2 40 3 60 M 2 = 500g Trial MS Block   C (N.s/m) 1 20 2 40 3 60 M 3 = 750g Trial MS Block   C (N.s/m) 1 20 2 40 3 60 Finally, the results will be used to fill the experiment matrix shown in table 2 Table 2: Matrix for the experiment The variable Constant parameters Constant parameters M1 = 290 g K1 = 0.0558 C1 = M2 = 500 g K1 = 0.0558 C1 = M3 = 750 g K1 = 0.0558 C1 = K1 = 0.0558 M1 = 290 g C1 = K2 = 0.0565 M1 = 290 g C1 = C1 = M1 = 290 g K1 = 0.0558 C2 = M1 = 290 g K1 = 0.0558 C3 = M1 = 290 g K1 = 0.0558 Discussion and Conclusion Is there any relationship between attached mass (M) and damping coefficient? Is there any relationship between damping coefficient and spring constant? Does the speed of the drum affect the accuracy of the results? Possible sources of errors Internal friction: There may be been internal friction within the spring material leading to energy dissipation in the form of heat, which adds to the damping factor. Friction between moving parts: There may be friction between the various parts moving relative to each other. For example, the interaction between the pointer/marker and the graph may result to friction, which will affect the accuracy of the results by adding onto the damping effect of the system. Also, there may be friction between the spring and the shaft, whose effect is to add to the damping effect of the mechanical damping system Inaccuracy in determining various measurements: inaccuracy in determining spring geometry, such as the outer diameter, spacing and wire diameter may result to inaccurate values of spring constant. Inaccuracy in the determination of masses may result to errors in the determination of damping constants. Lack of material homogeneity: the two springs are supposed to be made of carbon steel and stainless steel 304 respectively. Lack of material homogeneity may result to inaccurate results owing to the use of incorrect shear modulus values. Works Cited Duggal, S. K. Des of Steel Str, 2E. Tata McGraw-Hill Education. 2000. George, Geohy & Shaikh, Hasan. Introduction to Austenitic Stainless Steels. In. Khatak, H. & Baldev, Raj. (Eds.). Corrosion of Austenitic Stainless Steel: Mechanisms, Mitigation and Monitoring. Pangbourne, UK: Woodhead Publishing Limited. 2002. Gopal, M. Modern Control System Theory. New Age International. 1993. Gopal, M. Control Systems: Principles and Design. Tata McGraw-Hill Education. 2002. Hinrichsen, Diederich & Pritchard, Anthony J. Mathematical Systems Theory 1: Modelling, State Space Analysis, Stability and Robustness. Springer. 2011. Yechout, Thomas R & Morris, Steven L. Introduction to Aircraft Flight Mechanics: Performance, Static Stability, Dynamic Stability, and Classical Feedback Control. AIAA. 2003. Read More