Double Mass Spring Damper in Matlab simMechanics - Math Problem Example

Table of Contents Rational 2 Validation 3 Review of model 7 References 8 Appendix 9 Introduction The main aim of this paper is to develop and simulate double mass spring damper in Matlab simMechanics. Graphs will be made for the results showing changes in various parameters and the results highlighted. In designing double mass damper Hooke’s law will be applicable. Rational The rational of this paper is to design double mass spring damper as shown in the diagram below with the intention of creating a model that will show the impact of changing the stiffness of the spring and damping coefficient. The model will have two masses, two spring constants, and two damping coefficients. The masses will be held by springs whose weight will be ignored. It will be assumed that force will be exerted by external force and there will be displacement. The following are damping coefficients, mass and spring constants. Figure 1: double mass spring damper find the natural frequencies and mode shapes of spring mass system, which is constrained to move in the vertical direction. Solution: the equations of motion are given by: Validation The double mass spring damper will be done in both excel and in Matlab Simulink using the same parameters. Figure 2 shows Simulink model that has been built. The Simulink model is a closed-Loop Response Figure 2: : Block diagram The solution to above kinematical parameters for individual members of model Hooke’s law will be applicable. According to Hooke’s law the displacement of the ideal spring is proportional to the force exerted that is the deformation-change in size or shape-of the object is proportional to the magnitude of the force that causes the deformation. The extension or compression-the increase or decrease in length from the relaxed length-is proportional to the force applied to the ends of the spring. The assumption is that springs mass is ignored. The springs have constant k is called the spring constant for a particular spring as shown above diagram of 40500 N/m. the spring constant is a measure of how hard it is to stretch or compress a spring. A stiffer spring has a larger spring constant because larger forces must be exerted on the ends of the spring to stretch or compress it. It can be noted that second mass has a wave that is stable while the first mass has its wave not being stable. Stable waves occur when a wave is reflected at a boundary and the reflected wave interferes with the incident wave so that the wave appears to stand still. Suppose that a harmonic wave on a string, coming from the right, hits a boundary where the string is fixed. The equation of the incident wave is . The +sign is chosen in the phase because the wave travels to the left. The reflected wave travels to the right, sois replaced with and the reflected wave is inverted, so is replaced with. Then the reflected wave is described by , the motion of the string is described by ], Thus . Where and Every point moves in second mass with the same frequency as compared first mass due impact of the force. However, in contrast to first mass where every point does reaches its maximum distance from equilibrium simultaneously. In addition, different points moves with different amplitudes; the amplitude at any point is Figure 3: joint forces spring for mass M1 and M2 Figure above shows the individual spring oscillation as well as joint springs at time intervals of what we can note is that a standing wave is blur of moving spring, with points that never move halfway between points of maximum amplitude. The nodes are at the points where sin since, the nodes are located at thus the distance between two adjacent nodes is . The antinodes occur where which is exactly halfway between a pair of nodes; so nodes and antinodes alternate, with one –quarter of a wavelength between a node neighbouring antinodes. If the other end is fixed as well, then it too is node. The spring thus has two or more modes, with one at each end. The distance between each pair of nodes is . We notice that the higher frequency standing waves are all integral multiples of the fundamental; the set of standing wave frequencies makes an evenly spaced set. These frequencies are called the natural frequencies or resonant frequencies of the spring. Resonance occurs when a system is driven at one of its natural frequencies; the resulting vibrations are large in amplitude compared to when the driving frequency is not close to any of the natural frequencies. When this results is compared to excel Figure 4: Excel type of spring Figure 5: Excel output for waves of first and second spring The figure above shows the waves patterns of two springs on masses. The two ends are always modes since they are fixed in place. Notice that each successive pattern has one more node and one more antinodes than the previous one. The fundamental has the fewest possible number of nodes and antinodes. The following Matlab codes were used in designing the model; Review of model The aim of the paper was to design and simulate double mass spring damper using simulink of Matlab. Various changes were made to spring stiffness and damp coefficients. From the graphs produced, it can be stated that when stiffness is increased the oscillation reduces as shown from the results. It also can be noted that an error occurs when the spring constant is changed. In changing damping constant of suspension system and spring constant the response of Double Mass Spring Damper is improved. The graphs of Double Mass Spring Damper designed are also made up of the time and frequency domains which contain a single signal in each of these domains that is constituted by complex numbers. This is showed by both excel and Matlab output. References Eisner, H., 2008. Essentials of Project and Systems Engineering Management. Hoboken, N.J.: John Wiley & Sons. Franklin G.F., Powell J.D. & Emami-Naeini A., 2009. “Feedback Control of Dynamic Systems. New York: Prentice Hall. Nise N.S.2011. "Control Systems Engineering", New York: John Wiley. Fay, T. & Graham, S. D. 2003. Coupled spring equations, international . journal mathematics. Education science & technology. Appendix Excel and Matlab Read More