The task of this experiment was to develop a model of a car going over a speed bump and to use the model to optimize the speed at witch the car went over that bump. The first simulation model was Model A where the car was given as a single mass and the shocks were composed of a spring and a damping element. The second simulation model was Model B where the car and shocks are modeled as in A but the tire is also modeled as a mass with a spring and damping element. Each of these models was tested on three different bumps modeled by cosine waves and on a discrete representation of a measured bump on the Swarthmore Campus. Three different car velocites were used to determine the responses to the various bumps by each suspension system model. Both automotive suspension models were simulated with Matlab's Simulink toolkit, but Model A was also simulated numerically with a Runge-Kutta differential equation solver method.

The speed bumps outside Whittier Place on the Swarthmore Campus were measured using a 6 foot long measuring device that had movable dowels spaced 6 inches apart. After checking the measuring beam was level, the amount of protrusion of each dowel down to the road surface was measured, resulting in a discrete representation of the cosine-shaped speed bump. Four different speed bumps were measured in this way. In the analysis, however, only the speed bump with the best data was used to analyze the various responses of the suspension systems.

The rest of the analysis procedure involved writing a makeroad.m Matlab function to generate road sections according to the equation provided in the lab instructions (link below).

The procedure followed in the laboratory did not deviate significantly from the procedure described in the experiment description, which can be found at: http://palantir.swarthmore.edu/maxwell/classes/e12/S04/labs/lab04/

Data and Matlab Methods

• bumps.txt

• do_modelb.m - Generate plots for the Model B Simulink simulation results for the Y and Z.
• makeroad.m - Create a section of road (speed bump) given a height, length, and velocity.
• model_a.mdl - Simulink file for Model A.
• model_b.mdl - Simulink file for Model B.
• init_modelb.mdl - Initalizes workspace variables in Matlab for the Model B Simulink simulation.
• whittierbump.m - Generate the road section from the actual speed bump data collected from Whittier Place.

Velocity: 0.25 m/s

Velocity: 0.5 m/s

Velocity: 1 m/s

Road Height=0.10m, Road Length=0.4m

Velocity: 0.5 m/s

Velocity: 1 m/s

Velocity: 5 m/s

Road Height=0.15m, Road Length=2m

Velocity: 1 m/s

Velocity: 5 m/s

Velocity: 50 m/s

Road Height=0.05m, Road Length=0.01m

Velocity: 0.25 m/s



Velocity: 0.5 m/s



Velocity: 1 m/s

Road Height=0.10m, Road Length=0.4m

Velocity: 0.5 m/s



Velocity: 1 m/s



Velocity: 5 m/s

Road Height=0.15m, Road Length=2m

Velocity: 1 m/s



Velocity: 5 m/s



Velocity: 50 m/s

part 7: whittier place speed bumps.

Using Model A at 4 different velocities

Velocity: 5 m/s

Velocity: 10 m/s

Velocity: 25 m/s

Velocity: 50 m/s

Using Model B at 5 different velocities

Velocity: 5 m/s



Velocity: 10 m/s



Velocity: 15 m/s



Velocity: 25 m/s



Velocity: 50 m/s

Runge-Kutta Analysis

The Matlab source codes for the Runge-Kutta differential equation solver are given below. The resulting plots for the various road shapes follow.

• rk2solver.m
• vfunk.m
• xdotfunc.m
• xfunct.m

Road (height=0.05m, length=0.1m, varying velocities)

Road (height=0.1m, length=0.4m, varying velocities)

Road (height=0.15m, length=2m, varying velocities)

Reducing the Vertical Accelerations Felt by Automobiles.

Discussion - Using Model B, we tried to reduce the accelerations felt by the car for a small bump (2cm x 10cm long) at 65mph (~29 m/s). Compared to the normal parameters, we attempted to reduce the maximum acceleration by a factor of two. Note that in the graphs below we are plotting the acceleration of the Y (tire) and Z (car) positions, not the positions. The goal is to reduce the acceleration of the Z by about a factor of two.

The standard response to the a 2 x 10 cm bump at 65 mph using the standard parameters are shown below.

Item	Value
Mass of the car	450kg
Spring constant for the car	35kN/m
Damping constant for the car	7.3kN/(m/s)
Mass of the tire	18kg
Spring constant for the tire	70kN/m
Damping constant for the tire	233N/(m/s)

Suspension Parameters Adjustment - To reduce the accelerations felt by the car for a small bump at 65 mph (~29 m/s), we could increase the mass of the car, reduce the spring constant, or decrease the damping coefficient. Increasing the mass of the car yielded the most reduction in acceleration, but in real life it seems unreasonable to double the mass of the vehicle simply to reduce vertical accelerations over speed bumps (consider the added fuel expense!). Reducing the spring constant would decrease the vertical acceleration, but reducing it too much would cause the tires to lift off the ground after a bump, which would result in loss of vehicle control. Decreasing the damping coefficient would cause the automobile to oscillate vertically after going over even a small bump, which is entirely undesirable. In the end, it seems best to increase the mass of the car as much as reasonable, and decrease the damping slightly. Considering this options, the simulations suggested that a carr mass of about 650 kg and a damping of 5000 N/(m/s) were fruitful in reducing the vertical accelerations felt. Graphs of the results are given below. Note that the Z acceleration is approximately half that of the original test.

Simulink: Reduced Vertical Acceleration Response to 2x10 cm bump at 65 mph (29 m/s)

Y Acceleration:
Z Acceleration:

Real Car and Truck Speed Bump Data and Commentary

The following graphs show the actual response of a car and a truck going over the Whittier place speed bumps described earlier in the lab. The data was acquired with a video camera, with a computer vision recognition system detecting some colors attacted to the hubcap and the car's frame. The brown lines show the tire displacement, while the pink lines show the car or truck frame displacement.

In general, the tires and the car frame are displaced about the same amount for all the different speeds. However, as the car speeds up over the bump the displacement of the car frame start to lag the displacement of the tires. The car also tends to oscillate much more after leaving the bump at lower speeds.

For the truck, the displacement of the car frame is much larger than the displacement of the tires. Also, neither the car frame nor the tires oscillate as much as the car did. This means that the suspension on the truck is much more damped than the suspension on the car. Again as with the care, the truck frame displacement starts to lag the tire displacement as the truck speeds up and the truck tends to oscillates more at slower speeds.

conclusions and future work.

Model A shows that as the speed at which the car goes over the bump goes up the vertical displacement of the car goes down. Also, as the speed increases the period of oscillation goes up. This means that even though the car is not on the speed bump for as long a time the effects of the bump are felt for about the same amount of time. Model B gives about the same results as A except the car oscillates much more. As the speed of the car increases the vertical displacement of the car goes down as in Model A.

The nastiness of the bump depends on several factors. The spring constant and the damping constant are two of the most important ones. If the spring is too stiff then the bump is "nasty". Similarly, the damping should be small to decrease the upward acceleration but, if it's too small then there will be a lot of oscillation after the bump.

Finally, the speed of the car is the biggest factor. The faster the car, the nastier the bump is.

If the spring constant and the damping constant are optimized then the optimum speed should minimize vertical acceleration as well as the oscillation after the bump. We believe that obeying the speed limits at all times makes the bumps significantly less nasty.

The laboratory experiments performed confirmed our theories and equations.

Future work will involve more labs in the future.

Nastiness, be it emotional or physical, is supplementary to the status of nature in which speed bumps tend to find existence by virtue of god. That being as it may (since it is - corbum portis umeros) there is more to come in the conclusions of future labs, regardless of bumps, both on the roads and in life, though mortal it is. Ooh engineering! Life can be so tragic sometimes: you're here today and you're here tomorrow. On the topic that might regard a bumpy path along a country road (Whittier place, clearly), the nastiness of a bump finds itself directly proportional to square of the mass, or rather, more succinctly, the height of the bump. Also involved in the equation is the length thereof.

Note: No bunnies were harmed in the making of the bumps.

Professor Maxwell's Lab Assignment Page:
http://palantir.swarthmore.edu/maxwell/classes/e12/S04/labs/lab03/