Engineering Feasibility Study: Vehicle Shock Absorption System

Transcription

1 Engineering Feasibility Study: Vehicle Shock Absorption System Neil R. Kennedy AME40463 Senior Design February 28,

2 Abstract The purpose of this study is to explore the possibilities for the springs and dampers involved in a suspension system of an autonomous vehicle to be utilized in all-terrain situations. The main design variables will be the spring rate and damping coefficients, with state variables for the situation being the weight of the vehicle, the terrain that could be seen, the force transferred from the shock movement to the vehicle, and the vehicle ride height (a.k.a. ground clearance). The methods used in analysis of the system were a single mass-spring-damper model, and also the use of ordinary second order differential equations. Another purpose of the study is to generate the ideal spring rate and damping coefficient to be used in a suspension system for the final product - and may change the current suspension system on the proof of concept prototype. 1 Engineering Analysis As an all-terrain vehicle, the autonomous hazardous-material-seeking vehicle will require a relatively high performance suspension system. The suspension that is on the proof-ofconcept prototype is made up of both front and rear independent suspension systems with a coiled spring and damper to act as a shock absorber for each wheel. The design variables used in this engineering trade study are the spring rate of these coil springs, and also the damping ratio of the damper within the shock. The state variables that will directly result from these design variables are transferred force to the vehicle, along with the ability to quickly dampen out uneven terrain. Furthermore, ride height will also be considered, but more as a secondary variable. 1.1 Research The research conducted prior to performing any analysis involved limiting the design space based on vehicle dimensions and cost. Utilizing known retailers such as McMaster-Carr, it was possible to limit the spring rate below 15 lbs/in. based on cost and pre-existing shock dimensions - namely shock diameter. Furthermore, the full vehicle weight had to be estimated. The chassis alone weighs 4.08 lb f, and it is possible to estimate the enclosed vehicle body and other equipment to weigh roughly 10 lb f. This enables an overall vehicle weight estimate of 14 lb f, which equates to lb m. It is important to distinguish between lb f as units of weight and lb m as units of mass for all calculations 1.2 Methods Used The main methods used to model this suspension system is a single spring-mass-damper system (Fig 1) and the accompanying second order differential equation (Eqn 1). mẍ = bẋ + kx (1) In Equation 1, m is the mass of the vehicle, b is the damping coefficient of the shock absorber, k is the spring rate of the coil spring, and x is the measure of the spring displacement, and 2

3 Vehicle Mass x k b Unsprung Mass Figure 1: Mass-Spring-Damper System Model its associated velocity and acceleration derivatives. This equation allows for overdamped, critically damped, and underdamped situations. The main goal of this study will be to achieve a critically damped system to allow the suspension to recover as quickly as possible. The overdamped, critically damped, and underdamped solution forms to Equation 1 are the following, respectively where r is the respective roots of the equation (Eqns 2-4): x(t) = c 1 e r 1t + c 2 e r 2t (2) x(t) = c 1 e rt + tc 2 e rt (3) x(t) = c 1 e λt cos(µt) + c 2 e λt sin(µt) (4) In order to achieve critical damping, it is necessary to relate spring rate and the damping coefficient. Namely, in solving for the roots of the differential equation, the radical in the quadratic equation must be zero (Eqns 5, 6) r = b ± b 2 4mk 2m (5) b = 2 km (6) This means that the spring rate must first be determined in order to calculate the ideal damping coefficient. The simplest model to compare these results was to fix the shock deflection, which gives an initial position. This initial value was fixed at one inch for this study. Also, the initial velocity is assumed to be zero because the study is focused on recovery from the vehicle going over an obstacle, and when the wheel is directly above the obstacle, its velocity in the 3

4 direction of wheel travel is zero. Furthermore, since the spring rate is necessary to determine the damping coefficient, an acceptable spring rate will be determined by the force transferred upon striking an obstacle. In order to model the transferred force when the vehicle hits uneven terrain, the main equation is relatively simple - it is the force exerted by the spring under a predetermined displacement (Eqn 7). F orce = kx (7) This simulation only involves the spring force, because it assumes that the wheel is at the top of an obstacle and therefore has no velocity. This nulls the damping effects to show only the instantaneous force exerted on the vehicle. Upon minor inspection of Equation 7, it becomes obvious that the lowest spring rate is ideal for lowering the transferred force. However, ground clearance is another consideration for the suspension, and will also affect the optimal choice of springs. This means that there must be a transferred force of approximately 14 lb f into the vehicle with no deflection from impacts. However, since there is a linear relationship between displacement and spring force, it is possible to install springs onto the shocks already deformed to maximize ride height and still not effect the differential equations or transmitted force equations. This is also due to the fact that the weight of the vehicle and transferred force are in opposite directions. 2 Results These results were gained utilizing MATLAB to graph the governing equations shown and derived in the previous section. To begin, the spring rate had to be calculated, or at least narrowed. As expected, an increase in spring rate corresponds directly to an increase in force transferred to the vehicle (Fig 2). Although it becomes a simple situation to use the smallest possible spring rate, this spring also will not support the vehicle. A much better solution was to utilize a spring that could support the weight of the vehicle, and still limit transferred force to the vehicle from additional displacement from the wheel deflection. As a readily-available estimate, a spring with a spring rate of 10 lb f /in allows a simple calculation of pre-loading a spring by 1.4 in. in order to achieve 14 lb f and undergo no deflection at rest. Furthermore, this would allow for a much more manageable transferred force of roughly 10.7 lb f for the front shock travel of 1.07 inches. For the rear shocks, with a travel of 1.68 inches, the forces can be up to 16.8 lb f. Both of these are more extreme cases, and are still manageable through typical bracketing and other methods of securing electronics and equipment. As can be seen with a larger spring rate of 12 lb f /in, these forces begin to increase dramatically with a higher spring rate - reaching above 20 lbs f with maximum rear shock travel (Fig 2). This spring rate of 10 lb f /in will be used in further calculations. The next step of the trade study was to determine the damping coefficient and study the motion of the vehicle upon hitting an obstacle. Utilizing Equation 6, it was possible to determine that an ideal damping coefficient with a spring rate of 10 lb f /in is Utiliz- 4

5 Force (lb f ) k = 0.5 k = 1.0 k = 1.5 k = 5.0 k = 7.5 k = 10.0 k = 12.0 Transferred Force vs. Displacement Displacement (in.) Figure 2: Transferred Force vs. Spring Displacement for varied Spring Rates ing this information, the overdamped, critically damped, and underdamped situations were plotted using MATLAB (Fig 3). It can be seen that the overdamped situation requires nearly 4 seconds to reach equilibrium again, the underdamped situation requires approximately 1.5 seconds, and the critically damped system requires less than 1 second to reach equilibrium again. This clearly shows the advantages of calculating a critically damped system with the associated spring rate. 3 Conclusions The reasons for choosing a spring rate of 10 lbs f /in were addressed in the Results section. However, there is the issue of the model being a single mass-damper-spring system. The actual vehicle will have four wheels and shocks supporting the entire weight of the vehicle. However, this weight could not be divided into four equal parts due to the different mounting angles where each shock is positioned. However, it is a very simple conclusion to draw that a lower spring constant can be used to lessen the transferred force, and even allow for more wheel articulation while still maintaining a high ground clearance. The spring rate could be as low as 2.5 lb f /in if depressed prior to installation to maximize ride height. Furthermore, these springs would be much easier to work with on a small scale due to the lower forces necessary to install and manipulate them. However, spring rates closer to 4.0 lb f /in should be used in order to ensure sufficient vertical force. With this newly-applied model, the damping coefficient also must be changed. Utilizing Equation 6, it is possible to determine that b = 2.64 with k = 4.0 lb f /in in order to be critically damped. This is still utilizing the possibility of one wheel supporting the entire mass of the vehicle (0.435 lb m ) due to the uneven weight distribution and relatively extreme terrain where the vehicle will operate. 5

6 Motion of Mass Spring System with 1 in. Intial Deflection (k = [lb/in.]) Overdamped (k = 5 b = 4.17) Critically Damped (k = 10 b = 4.17) Underdamped (k = 15 b = 4.17) Displacement (in.) Time (sec) Figure 3: Damped Motion vs. Time (k = 10 lb f /in) The largest conclusion drawn from this trade study shows the strong correlation between the damping ratio and the spring rate in order to achieve critical damping. In the final product, and possibly in the proof-of-concept prototype, these will require close study and material selection. For the proof-of-concept prototype, it would be a much simpler process to test for the damping coefficient and match a spring coefficient. In final product design it would be necessary to accurately and specifically design the two as a system to work together to dampen impact forces. 6