Dampers used in the suspension system are designed to have different dampening rates for the rebound and compression. This gives two different gradients on a Force vs. Velocity graph where Cr is the slope of the rebound (Extension) in the negative region of the velocity and CB is the slope of the Bump (Compression) in the positive region of the velocity as shown in Figure 1. Also, on a Force-Displacement Diagram the area under the graph for the positive force represents the energy dissipated over compression and the area enclosed by the negative force represents the extension as shown in Figure 1. The negative values represent that the force is opposing the direction of motion. In this, damping coefficients are calculated both linearly and through energy dissipation. This damper is referred to as an idealized piecewise-linear damper system which is shown in Figure 2.

Figure 1:Force vs Displacement and Force vs. Velocity for a Spring-Damper System

Figure 2: Schematic Diagram of Mono-Tube Damper

Using a Suspension Dynamometer, the spring/damper is subjected to a sinusoidal displacement cycle where the frequency, compression, rebound and speed can be adjusted whereas the displacement is fixed. The compression damping absorbs all vibrations as the wheels move upward whereas the extension damping returns the wheel to its original position. This helps in making the journey of the cyclist as smooth as possible. [2]

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3. Results

Table 1 shows the 7 configurations which were used to evaluate the Spring-Damper System under the Suspension Dynamometer. Configuration 1includes a spring along with the damper whereas Configuration 2-5 do not have a spring. Configuration 6 and 7 are additional measurements used to be compared with Configuration 1 and Configuration 5 respectively.

Table 1: Configurations for the different sets

				Damper Settings
	Spring Constant	Air Pressure (psi)	Speed Setting	Compression	Extension
1	350 lbf-in / 61294 N/m	50	2	0	0
2	None		2	0	0
3				0	4
4				3.66	0
5			4	0	0
6	350 lbf-in / 61294 N/m		4	0	0
7	None		4	4	0

3.1. Velocity-Voltage Calibration Factor

The Suspension Dynamometer gives the velocity of the spring-damper system in the form of Voltage. Therefore, a calibration factor must be deduced that transfers the velocity from voltage to its S.I Unit meters per second. This was done by two methods; one method is by plotting the data given from the Suspension Dynamometer on a graph, shown in Figure 3, and the actual frequency was computed by finding the period of the sinusoidal curve. Using Equation 1, the frequency was calculated. Using Equation 2, the angular speed was calculated which was used to find the velocity in meters per second by multiplying it with the amplitude (A) as in Equation 3. Then, a calibration factor was calculated by taking the ratio of the velocity in voltage and the velocity in meters per second as shown in Equation 4.

$f$

$A$

Figure 3: Amplitudes vs. Time for Set 2

$f = \frac{1}{T} = \frac{1}{0.6773} = 1.476$

Hz Equation 1

$ω = 2 πf = 2 π * 1.476 = 9.276$

rad/sec Equation 2

$V = ωA = 9.276 * 12.62 \times 10^{- 3}$

= 0.11707 m/s Equation 3

$Callibration Factor = \frac{Velocity in m / s}{Velocity in Voltage} = \frac{0.11707 m / s}{1.892 V} = 0.0618$

Equation 4

Since this method is valid for a pure sine curve only, a different method was done by differentiating the displacement numerically and plotting it against the velocity in voltage. Figure 4 shows that relationship and the calibration factor was found to be 0.0605. However, this method is prone to noise ,therefore, an average of Method 1 and Method 2 was taken to represent the calibration factor to reduce uncertainties.

Figure 4: dx/dt – Velocity Graph for Configuration 2

The average calibration factor was found by averaging the values of both methods.

Method 1: 0.0618

Method 2: 0.0605

Thus, the average value was found to be 0.0612.

3.2. Spring Stiffness / Pre-Load

Using Configuration 1, a Force – Displacement graph was plotted, as shown in Figure 5, to calculate the spring stiffness. A line of best fit was plotted, and the slope was found to be 0.06988 kN/mm. Therefore, the spring constant is 69880 N/m. It was given that the spring constant in Table 1 is 61294 N/m. This shows that there is a discrepancy of 14%.

Figure 5: Force-Displacement for Configuration 1

According to the Equation of the line, the line is a straight line that does not pass through the origin and has a y-intercept. This means that the spring has a Pre-Load of 1.33 kN. Therefore, in further calculations of a Spring-Damper, this offset needs to be compensated when processing further results.

3.3. Spring – Damper Response at Different Speeds

On a similar graph, the responses of both Configurations 1 and 6 were plotted as a form of comparison. Figure 6 shows both configurations. Using configuration 6, the slope of the line was found to be 0.07229 kN/mm. Therefore, the spring constant is 72290 N/m. The change of speed has increased the discrepancy to 17.94%.

Figure 6: Force-Displacement graphs for Configuration 1 and 6

3.4. Compression and Rebound Rates

The Compression and Rebound Rates can be calculated graphically in two different ways. Either by integrating the two enclosed areas of the ellipse that are above and below the velocity axis in a Force-Velocity graph or linearly where the two slopes of the best fit lines are calculated of a Force-Displacement graph. Both methods will be used to calculate the rates in order to identify possible sources of error and inaccuracies.

The energy dissipation method is calculated by integrating the areas enclosed by the ellipses and halving their sum. This is given by the following equation:

$E_{Piece - wise, 1 - Cycle} = \frac{1}{2} (π C_{b} A^{2} ω + π C_{r} A^{2} ω)$

Equation 4

After finding the areas of both halves, it was noticed that there is a sudden change in force while there was no change in displacement. If this is to be regarded, then it will lead to significant inaccuracies in the compression rate. Therefore, the area of this rectangular gap is computed and subtracted from the area of the upper ellipse. This rectangular gap represents non-ideal behaviours such as the Coulombs friction at seals or due to thermal effects. Figure 7 (a)-(d) shows the Force-Displacement graphs for Configurations 2-5. Using a MATLAB script –Appendix A, the areas were computed and are tabulated in Table 3.

Figure 7 (a)-(d): Force-Displacement Graph for Configurations 2-5

Table 2:Energy Dissipated in Bound and Rebound and the respective Bound/Rebound Rates

	Total Energy (J)	Lower Ellipse Energy (J)	Upper Ellipse Energy (J)	Coulombs Friction Energy (J)	Upper Energy- Coulombs Friction (J)	C_B(Ns/m)	C_R (Ns/m)
2	10.9092	5.0042	5.9050	4.0186	1.8864	396.9	1052.8
3	27.9345	22.1268	5.8077	3.9050	1.9027	403.5	4692.2
4	11.753	4.7904	6.9626	4.0790	2.8836	611.5	1015.8
5	34.7721	27.2220	7.5501	4.7777	2.7724	587.9	5772.7

The other method is then used to calculate the compression and extension rates which is linearly through the Force-Velocity Graph. As stated above, the compression and extension rates are different as compression damping helps the suspension absorbs road irregularity as the wheel moves upward in the stroke and Rebound damping helps the suspension to return to the proper position, after a bump or other irregularity causes the fork to compress, in a smooth and controlled motion. By taking two lines of best fit, one for the negative region of the velocity and one for the positive region of the velocity, the Extension rate and Compression rate will be deduced respectively.

Figures 8 (a)-(d) shows the relationship between Force and Velocity for Configurations 2-5 respectively. The rates are then tabulated in Table 3.

Figure 8(a)-(d): Force-Velocity Graph for Configurations 2-5

Table 3:Compression and Extension Rates from F-V Diagram

	Slope of +ve Region	Slope of -ve Region	C_B(Ns/m)	C_R (Ns/m)
2	0.5845	3.3343	584.5	3334.30
3	0.5945	12.1386	594.5	12138.6
4	1.146	3.072	1146	3072.1
5	0.5432	9.8575	543.2	9857.5

Table 4 summarises the Bound and Rebound Rates using both methods, Linearly and using the energy dissipation method.

Table 4: Summary of Bound and Rebound Rates using the two methods

	Energy Dissipation Method from Force-Displacement graph		Linear Method from Force-Velocity graph
	C_B(Ns/m)	C_R (Ns/m)	C_B(Ns/m)	C_R (Ns/m)
2	396.9	1052.8	584.5	3334.30
3	403.5	4692.2	594.5	12138.6
4	611.5	1015.8	1146	3072.1
5	587.9	5772.7	543.2	9857.5

3.5. Damper Response at Different Compression Rates

Configuration 7 is an altered version of Configuration 5 where the compression rate has been increased from 0 to 4. This was done to compare the dampers response at different compression rates. Figure 9 shows the Force-Displacement of both configurations and Figure 10 shows the Force-Velocity of both configurations.

Figure 9: Force-Displacement for Configuration 5 and 7

4.Discussion

In Configuration 1, the spring was found to have a pre-load of 1.33kN which has caused an offset. Therefore, it was compensated for when the results where being processed. Moreover, the preload is what makes the suspension of the bicycle work. As this allows the bicycle to push the tires down on big jumps and makes travel over terrain easier. Also, under compression, the spring load increases the contact pressure which improves traction. [3]

The spring constant was calculated in configuration 1 by finding the slope of the linear region in the Force-Displacement graph, it was found to be 69880 N/m which has a discrepancy of 14%. The reason behind this may be defined as since the spring can’t be tested alone and the damper used has a spring-like characteristic, so the damper has contributed in changing this value. The force exerted has compressed the spring as well as the air in the damper as at high pressures, the compressibility effects of air in the damper becomes significant. This explains the discrepancy in the values as the compressed air increases the energy and gives a deviated spring constant. Moreover, when configurations 1 and 6 were compared in Figure 6, the lower ellipse of Configuration 6 was deformed which illustrated that at higher frequencies the air in the damper gets compressed more and has led to a greater discrepancy in the spring constant. This has illustrated the characteristics of the damper where it has spring-like characteristics at high frequencies as the air inside is compressible. [4]

The Coefficients of Bump and Rebound were found in two different methods, Linearly and using the Energy dissipation method using the Force-Velocity and the Force-Displacement graphs respectively. Table 4 shows that the coefficient of bump i.e. the damping coefficient upon compression using the two methods was approximately the same, whereas the coefficient of the extension was found to vary considerably between the two methods. This is because the negative region of the Force-Velocity Graphs followed a linear behaviour whereas the positive region had deviated away from the linear region as a result from the non-ideal behaviour, as shown in Figure 8 (a)-(d). Since the linear approximation methods finds the line of best fit for half-cycle where the slopes represent the damping coefficient of extension and compression, C_b was found to differ significantly. On the other hand, the energy dissipation method finds the area of the half-cycle and the damping coefficients are deduced. This is more accurate as it takes each set of data into account rather than taking the average best fit line.

The Force-Displacement and the Force-Velocity graphs shown in Figures 8 and 9 do not match the ideal graphs that are shown in Figure 1. As the damper was changing from an extension to a compression, the Force-Displacement graph shows there is a sudden rise in the force at a constant displacement which resulted in a rectangular area between the ellipses. Also, in the Force-Velocity graphs there was a gap between the two linear regions. This was due to the non-ideal behaviour of the Spring-Damper system such as coulombs friction which is the friction that occurs between the piston and seal as shown in Figure 2. Another non-ideal behaviour was the increase in the compression damping rate which was due to the increase in pressure as there was an axial load acting on the damper called the piston side force. [5]

Table 4 shows that the Rebound Coefficient in 5 is much larger than that of Configuration 2, this explains the response of the damper at different frequencies. At a high speed, the damper underwent agitation much faster than Configuration 2, which generated a high force and air in the damper was compressed. Since the force is related to the acceleration and this has led to the deficiency of oil flow. Also, Configuration 3 has a higher extension rate which has also increased agitation and has led to a high Rebound coefficient compared to configuration 2. [6]

When Configuration 5 and 7 were compared, Configuration 7 showed a greater area enclosed for both the upper and lower ellipse. Also, the rebound and bump slopes from the force-velocity graph were higher as the slope was steeper. This shows that at a higher compression turns, air is more readily compressed in the damper and this increases the energy giving inaccurate coefficients.

Although the procedure in calculating the coefficients included two methods, the two methods have resulted in varying values and the energy dissipation method was proven to be a better approach. A further step to improve accuracy is to add lubrication between the piston and seal to overcome the Columb’s Friction. Moreover, replace the damper with a one that has a higher damping effect and air inside doesn’t compress easily. Another step that must be done is to repeat each configuration more than once and take an average of the values to improve the accuracy

5.References

[1]Ngowmpo, R., 2018. University of Bath. [online] Moodle.bath.ac.uk. Available from: https://moodle.bath.ac.uk/pluginfile.php/785983/mod_resource/content/6/Section1_Elements_of_vibration_systems.pdf [Accessed 13 Nov. 2018].

[2]Pan, M., 2018. University of Bath. [online] Moodle.bath.ac.uk. Available from: https://moodle.bath.ac.uk/pluginfile.php/1253929/mod_resource/content/1/SolidMech3_Lab_guidance_notes-2018.pdf [Accessed 13 Nov. 2018].

[3]Anon, 2018. [online] Available from: http://accutuneoffroad.com/articles/spring-preload-matters/. [Accessed 17 Nov. 2018].

[4]Dixon, J., 2008. The Shock Absorber Handbook. Hoboken: John Wiley & Sons, Ltd.

[5]Reimpell, J., Stoll, H. and Betzler, J., 2001. The automotive chassis. Oxford: Butterworth Heinemann.

[6]Dixon, J., 2008. The Shock Absorber Handbook. Hoboken: John Wiley & Sons, Ltd.