Function modeling & Suspension System Comparison Analysis

AIM: To plot a given function and analyze the example 'Suspension System Comparison' using Simulink.

EQUATION: The given equation is as follows;

`y_x = (x-1).(x-3)^2.(x-9)^2`

where;

`y` = dependent variable

`x` = independent variable

OBJECTIVE:

1. To plot the given function using basic Simulink blocks.

2. To run the Simulink example 'Suspension System Comparison' and determine:

- Objective of the simulation along with similarities and differences between the three systems under study.

- Observations from the results.

- How the oscillating velocity source is created.

SOLUTION:

Part #1: A Simulink model for the given function is shown below.

Output #1: A plot for the given function is displayed below for t = 5 sec.

A plot for the given function is displayed below for t = 20 sec.

Part #2: A Simulink model for the Suspension System Comparison is shown below.

- Similarities:

1. A common velocity source is considered to be the input of all three sub-systems.

2. Tire stiffness is the same for all three sub-systems i.e. 2e5 N/m.

3. Sprung mass and Unsprung mass is also equal for all three sub-systems.

- Differences:

1. Three different springs are present viz. shock absorber, non-linear spring and variable spring.

2. This affects the force, displacement and stress generated in the body.

THEORY: Suspension is the system of tires, tire air, springs, shock absorbers and linkages that connects a vehicle to its wheels and allows relative motion between the two. Suspension systems must support both road holding/handling and ride quality, which are at odds with each other. The suspension also protects the vehicle itself and any cargo or luggage from damage and wear.

The different types of suspensions used in the example are:

Shock absorber: A shock absorber or damper is a mechanical or hydraulic device designed to absorb and damp shock impulses. It does this by converting the kinetic energy of the shock into another form of energy (typically heat) which is then dissipated. Most shock absorbers are a form of dashpot (a damper which resists motion via viscous friction).

Non-linear spring: Non-linear springs are helical coil springs that exert an inconsistent amount of force as it is under a working load or torque. This means that the force needed to travel one inch, millimeter, or degree might not double when it travels two inches, millimeters, or degrees like a linear spring would.

Variable spring: Variable pitch springs have coils that are closer together in some areas and more widely spaced in others.

Sprung mass: It is the portion of the vehicle's total mass that is supported above the suspension, including in most applications approximately half of the weight of the suspension itself. The sprung mass typically includes the body, frame, the internal components, passengers, and cargo, but does not include the mass of the components at the other end of the suspension components.

Unsprung mass: It is the mass of the suspension, wheels or tracks (as applicable), and other components directly connected to them, rather than supported by the suspension. It includes the mass of components such as the wheel axles, wheel bearings, wheel hubs, tires, and a portion of the weight of driveshafts, springs, shock absorbers, and suspension links. If the vehicle's brakes are mounted outboard (i.e., within the wheel), their mass is also considered part of the unsprung mass.

Output #2: Variation in Sprung Mass Position and Input Velocity distribution for or a range of time t=0-120 sec is shown below. For the initial 20 sec, the sprung mass has almost similar variation in all three cases. As input velocity is absent in 20-40 sec and 80-100 sec frame, the displacement is almost negligible in all cases. But during 40-60 sec, variable spring incurs the highest variation whereas other two are nearly constant. For 80-120 sec, variable spring experiences the least distortion. Overall, the shock absorber displays a fairly stable nature in the plot.

A plot of power spectral density (dBm/Hz) vs frequency (Hz) using a spectrum analyzer depicts the analysis of noise levels per frequency as shown below. As frequency increases, the power spectral density decreases.

- Oscillating Velocity Source: The Simulink model for an oscillating velocity source is shown below. It consists of three sine wave and pulse generator blocks each. These are merged in a Sum block and passed through a Simulink-PS converter to convert the input signal into a physical signal.

Details of Sine Wave block Equation Amplitude Frequency Sine Wave 0.1 sin(3t) 0.1 5 Sine Wave1 0.1 sin(15t) 0.1 15 Sine Wave2 0.1 sin(75t) 0.1 75

Details of Pulse Generator block Amplitude Period (sec) Pulse Width (% of period) Phase Delay (sec) Pulse Generator 1 120 100/6 0 Pulse Generator1 1 120 100/6 120*2/6 Pulse Generator2 1 120 100/6 120*4/6

CONCLUSION: From the above results, we can infer that:

- The shock absorber is stable and better than other types considering a diverse range of time analysis.

- Displacement is inversely proportional to the spring stiffness.

Hence, the shock absorber should be implemented as it provides adequate damping effect and efficiency of the vehicle body.

REFERENCES:

1. Suspension System Comparison

2. Suspension system

3. Shock absorber , Image

4. Non-linear spring , Image

5. Image

6. Sprung mass , Image

7. Unsprung mass , Image