Design of a Car Suspension System - Math Problem Example

Running Head: Design of a car suspension system Design of a car suspension system [Student Name] [Course Title] [Instructor Name] [Date] Executive summary The design of car suspension considered a car that weighing 2 tons and only damping coefficient and spring stiffness was adjusted to determine their impact. The Matlab was used to generate results and the results show that when stiffness is increased the oscillation reduces while damping coefficient changes produces similar results. This is due occurrence of an error due to the spring constant is changes. This means that changing damping constant of suspension system and springs constant the response of car suspense system. Introduction Car suspension system is interconnected parts of a car that hard in the car motion as well as reduces passenger discomfort caused by various barriers on the road. These parts include shock absorbers, stabilizers, springs, tires and air tires. When designing a car suspension system, one considers a simple sock absorber which is an important device in a car. The sock absorber or a suspension system will help in preventing socks which result from barriers such as drag forces irregular road sections, vibrations from engines and wheels and other factors. In designing suspension systems this factors should be taken into consideration to improve the convertibility of drivers and passengers of vehicles. The car suspension system designed relay on signal to estimate the impact of road barriers on the motion of the car and comfort of vehicle use. This is because the car suspension system produces signals which are as a result of oscillation of the chassis system. Once there is contact between the road and the car suspension system, signals are produced which can be analyzed to produce output signals. In this case Matlab has been used to design vehicle suspension system. The initial stiffness constant is taken as 3000 N/m(K), the damping constant of suspension system as 30 N.m/s and weight of car as 2 ton. The relationship is that when there changes in damping coefficient, there is a corresponding changes in the force. 1. Kind of filter to be used When designing the car suspension system, a low pass filter is used because of its ability to indentify irregular signals within an image thus eliminating necessary features from the analysis. Low pass filters allows high frequency filters to pass through. The following is an example of an original image that is filtered using a low pass filter to produce a filtered image. The image brightness has been reduced and its quality improved without shadows. There are differences between a low pass filter and a high pass filter that is a high pass filter allows some sections of the image to be pronounced 2. Mathematical modelling In this case a simple suspension system is been designed in consideration of the effect of dumper the impact of the mass between the motioned barriers and the impact of the mass when it collided with the barriers. In modeling one consider a simple case of a damper system with stiffness k, damping coefficient, C and mass M as shown in the diagram below From the consideration above there are two motion of the wheel and road that is (/2) and collission motion that is (/2). This brings us to the forming differential equation governing the motion of the chassis that will solve the Mx=u-K(y1-y2-y10)-Mg-Lo(i1-i2) =Jx + Ziu + Z2r +c1 Where , c1 = constant vector J=, Z1=, Z2=. The suspension ratio, R = 2 (a). Differential equation- In determining the deferential equation determining the motion of the chassis Hooke’s law is used because of the existence of the spring which can be compressed or stretched and the following formula is applicable. F=maf=k Whereby F is the magnitude of the force used on the spring, while is the distance of the spring that is stretched or compressed, k is the spring constant. The spring constant is the stiffness of the spring therefore the following are differential equations governing the movement of chassis. Integrating the equation and fixing the dimensions, we get the following equations. M = M = where h(t) = x(t) And kx(t) = ky(t) + The above differential equation that governs the motion of the chassis is inconsistence with Hooke’s law. 2(b). the frequency response- in determining frequency response of the derived linear constant-coefficient differential equation using the equation derived above is as follows. kx(t) = ky(t) + x(t) = y(t) = H (j) = k (= k ( + m k (= k ( + k = k ( + k = (k - 2(c). the main purpose of designing is minimizing the amplitude of the up and the down motion of the car. In the design it was discovered that the amplitude of the car depended on the frequency response that is the spacing of the barrios in the road such as bumps. The amplitude of the cars up down motion does depend on frequency response. However resonant frequency gives a great concern because it be an result of an error. Resonance occurs when a system is driven at one of the natural frequencies; the result vibration is large in amplitude compared to when the driving frequency is not close to any of the natural frequencies. Resonance may cause structural damages due to large amplitude which is caused by vibration. This is calculated as shown below, is calculated as shown below; 3). When the customer comes back complaining various parameters will be considered for adjustment which will be mass, damping coefficient or the stiffness of the spring. Changing mass will not have much impact on the amplitude since the vibration that produces signals are determined by shock absorbers that have the damping coefficient. Once mass is changed force also changes meaning there is no impact. However signal production which causes passengers discomfort is as a result of damping coefficient. 4a) changing of linear constant-coefficient differential equation- In this case a new component will be introduced which will affect the differential equation that was formed previously, this equation will take the form of kx(t) = + ky(t) + c ( where the new component part of the equation is c ( in the equation x and y presents the movement of the vehicle and the new component within the system. 4b) the new frequency response after introducing a shock absorber that moves through viscous fluid is shown below k = and c = 2 x(t) = + y(t) + 2( 2 4c) weight of a car to turn Plot of graphs of frequency response with varying wn and zeta values 4d). Step command Graphs with varying wn and zeta values These are the graphs: 5. The design criteria for the car suspension system The cut‐off frequency related to the natural frequency When the cut off frequency is equal to natural frequency the movement will be very sluggish and the car response will be maximum because of the bumpy road. This is so because resonance occurs when the vibrations are very high producing large amplitude. This cannot be compared with when the vehicle is being driven in a cut off frequency which is lower or close to the natural frequency. When the cut off frequency is lower than the natural frequency, it means the passengers are comfortable since the ride is smooth. This is called a soft ride. When the cut off frequency is greater than the natural frequency the ride is said to be very smooth and very soft since the car response is low The effects of varying the damping ratio –This was done using the following matlab codes Increasing the damping ratio will lead to decreased comfort due to increased oscillation or vibration changes thus changing the frequencies for an compensated time. This uncomfortable oscillatory response will make the passenger. When decreasing the damping ratio there will be reduced uncomfortable oscillatory response leading to increased comfort. It shows that the relationship between damping coefficient and the stiffness of the spring is inverse that is an increase in damping coefficient will lead to a decrease in the number of oscillation that are experienced by the passengers. Once there are changes in the damping ratio there is need to consider the cause of an error if it is a shock absorbers then they should be replaced. The damping ratio affects the time‐domain of system are again important thus affecting smooth ride of a car. In the design various changes have been made to spring stiffness and damp coefficients. From analysis it can be noted that the ffirst Transfer function was G=. Matlab was used to assess frequency response and to simulate the system transient response. System performance specification was based upon closed-loop time domain criteria of rise time and damping ratio. The following output graph was produced; . From the above graph it can be noted that the acceleration is low when encountering the barriers as the frequency is high at first and reduces. It can be noted that the amplitude of the graph made is high initially then it changes to small oscillation before going flat. The oscillation shows how the vibration affects passengers during the use of the car. It means that there is discomfort at the beginning. In this case damping coefficient was 35 and the stiffness of the spring was considers to be 3000. Looking at the second case where damping coefficient is changed the transfer equation obtained is as following G=. It will be noted that the step response graph made has many oscillations as compared to the first graph meaning that the change in damping coefficient leads to changes in the vibration of the chassis when facing the obstacle. The change in damping coefficient ensures that oscillation or vibration changes thus changing the frequencies for a compensated time. It shows that the relationship between damping coefficient and the stiffness of the spring is inverse that is an increase in damping coefficient will lead to a decrease in the number of oscillation that are experienced by the passengers. The graph above shows the step response of a signal when there is a change in the damping coefficient. We can note that there is a blur in the standing wave when the spring is moving, however when the spring start moving there is large oscillation. There are quick waveforms after eruption of one second. The upward and downward of movement of the signal is a clear indication that the effect of barrier is to reduce oscillation. This confirms that the relationship between the damping coefficient and spring stiffness is inverse. In order to determine the impact of changing both damping coefficient and spring stiffness from the original figures, damping coefficient was maintained at 15 while spring stiffness was taken as 4500. The following transfer function was formed G=. From the function the following signal plot was made. Looking at the plot above, it will be noted that the amplitude of the vehicle has large waveforms which reduce as time goes by. After reaching 1.2 seconds the waveforms becomes very small. This means at the start the vibration is very high but as the acceleration of the car increases the vibration reduces. That means the settled time is lower when there is no peak overshoot. It can be concluded that the change in damping coefficient and the stiffness of the spring the signal graph takes the shape of the graph were the stiffness cannot change. 6. The optimum parameters for the car suspension system The optimum parameters is selected from the criteria is natural frequency and wn=10,zeta=0.1where the passengers will be comfortable. At this point it can be noted that the parameters can be changed in damping coefficient which is in form of shock absorbers. 7. Summarizing your findings The design aimed to make a simplified car suspension system which can work and make the customers comfortable. The model used a spring, a damper, mass and the road. The mathematical model has been used to formulate differential equation that governs the motion of the chassis. It as also been used to determine leaner constant-coefficient differential equation. The equations have been used to develop Matlab codes for solving effects of varies damping ratio, relationship of cut-off frequency and natural frequency and impact of mass. This has helped understand the topic of signals and system from modeling the relvant equations to using Matlab. References Eisner, H. (2008). Essentials of Project and Systems Engineering Management. Hoboken, N.J.: John Wiley & Sons. Hsu, K, Soong, R, Chen, K & Lan, T, (2013). A Computerized Approach to the Design of Automobile Suspension System Jadlovska, A., Katalinic, B.; Hrubina, K., & Wessely, E. (2013). On Stability of Nonlinear Systems and Application to Apm Modeling, DAAAM International Scientific Book., Likaj, R., Shala, A., Bruqi, M. & Bajrami, XH (2014). Optimal design and analysis of vehicle suspension system, DAAAM International scientific book. Proulx, T. (2011). Rotating Machinery, Structural Health Monitoring, Shock and Vibration: Proceedings of the 29th Imac, a Conference on Structural Dynamics, Springer, 2011. London: Print. Robichaud, J. M. (2010). Reference Standards for Vibration Monitoring and Analysis. Saint John, NB Canada. n.d. Sun Lu, 2002. Optimum design of “road-friendly” vehicle suspension systems subjected to rough pavement surfaces. Applied Mathematical Modelling, Volume 26, issue 2002 Read More

 

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