Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3

in #steemstem6 years ago (edited)

In the last few posts, we've been studying the mathematical modelling of car suspension systems (i.e. mass-damper-spring systems). So far, we've explored the under damped, and critically damped settings for the shock absorber. In this post, we'll have a look at the suspension behaviour (at least from a mathematically point of view) when the shock is over damped.

Suspension.jpg
Figure 1. Common suspension architecture of sports and racing cars (source: Wikimedia Commons)

Let's recap the equation of motion for the mass-damper-spring system...

s1.png

where:

  • s23.png = sprung mass [kg]
  • s24.png = damping coefficient [Ns/m]
  • s25.png = spring constant [N/m]

with these parameters, we can have 3 different cases (or behavioral characteristics)...

Case I < 0 Under damping
Case II = 0 Critical damping
Case III > 0 Over damping

Case III: Over damped

From the last post, the parameters of our study suspension are...

  • sprung mass m = 300kg
  • spring constant k = 1200N/m
  • initial displacement q13.png = 0.15m
  • initial velocity q27.png = 0m/s

For the system to be critically damped, we needed to tune the damper up to...

s11.png

Now, any damping coefficient over this value is considered to be over damped.

Let's see what happens to the behaviour of our suspension system when we tune the shock absorber (damper) up to, say...

t1.png

This will be quite an over damped system! Now, the schematic of the quarter-suspension system (we're just looking at one corner of the car for now) is shown in Figure 2 below...

t33.png
Figure 2. Suspension schematic with parameters required for over damping

Substituting these values into the equation of motion (1)...

t2.png

The characteristic equation to (2) is...

t3.png

The roots of equation (3) are...

t4.png

Numerically, the roots are...

t5.png

By equation (3) of Post #3, the general solution is...

t6.png

The first derivative of the general solution is...

t7.png

Ok. Now that we have the general solutions, let's apply the initial conditions so that we can see what this system behaves like. The given initial displacement may be a reasonable depiction of a car being driven off a curb.

Let's apply the velocity initial condition first...

t8.png

Now applying the displacement initial condition...

t9.png

t10.png

So the particular solution is...

t11.png

The behaviour of this particular solution is depicted in Figure 3 below...

t12.png
Figure 3. Behaviour of the suspension system with t34.png and t35.png

Exploring the initial values problem

Let's now explore the solution with a few different permutations for the initial conditions...

For this part of the post, let's leave everything in algebraic terms for now to make calculations more efficient. So starting with the general solution...

t36.png

Applying the displacement initial condition...

t14.png

Applying the initial velocity condition...

t15.png

And solving for A...

t16.png

Alright, now that we have templates for our coefficients, let's explore what happens when we vary the initial conditions.


Condition #2, with t18.png = 0.15m and t19.png = -0.6m/s...

t17.png

t20.png

The particular solution to this set of initial conditions is...

t21.png


Condition #3, with t18.png = 0.15m and t19.png = -0.3m/s...

t22.png

t23.png

The particular solution to this set of initial conditions is...

t24.png


Condition #4, with t18.png = 0.15m and t19.png = 0.3m/s...

t25.png

t26.png

The particular solution to this set of initial conditions is...

t27.png


All of the above solutions are superimposed in Figure 4...

t28.png
Figure 4. Various particular solutions to the over damped car suspension problem.


Let's see how the suspension behaves with zero initial displacement, but a non-zero initial velocity. For instance...

Condition #5, with t18.png = 0 and t19.png = 0.3m/s...

t29.png

t30.png

The particular solution to this set of initial conditions is...

t31.png

A few solutions showing the behaviour of the mechanism with t18.png = 0 and varying values of t19.png is shown in Figure 5 below...

t32.png
Figure 5. Various particular solutions to the over damped car suspension problem.

Discussion about the above case

As you can see, after a sufficiently large amount of time t, or as t13.png, the mass (car body) is restored to its resting position.

Now, for all intents and purposes, the car body is restored to the equilibrium position in about 10s, which for most real-world applications is a very sluggish response indeed. This can be very dangerous as I'll explain...

At best, the suspension (and all of the components attached to it such as the wheels and brakes) may not return to its neutral (equilibrium) position fast enough to absorb the next bump, which after several bumps, makes the car feel like it has no suspension, making for a very rough, bumpy ride.

At worst, the car may hit a bump such that it becomes airborne, and because of the lag in returning to the equilibrium position, the wheels may remain out of contact with the ground for longer than normal. In a car, if the wheels are not in contact with the ground, the driver has no control of the car.

Damping and Mountain Bikes

Mountain biking is a favorite hobby of mine. Now, for a practical application and explanation of how changing the damping characteristics affects the handling of a mountain bike, watch Seth's YouTube video from 3:44 below.

Note, in mountain bike speak, "damping" is synonymous with "rebound"...

In the video, when Clint had is rebound set to the slowest setting (maximum "over damping") he was 9s slower on a short circuit and said that the ride felt very harsh after a few bumps because the suspension packed up and would not return to the neutral position in time for the next bump.


Credits:

All equations in this tutorial were created with QuickLatex

All graphs were created with www.desmos.com/calculator


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Second Order Differential Equations

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  5. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 1
  6. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 2
  7. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3

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Feel free to ask me any math question by commenting below and I will try to help you in future posts.

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It's nice to see something from the Engineering. But, I have the advice for you: try to write it from the perspective of the car/ bike enthusiasts. Something like MacPherson vs double wishbone or pull-rod vs push-rod.

Pros and cons, settings, what those systems can/can't do...

Or some racer vs ricer tuning stupidities.

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This is pour dynamic modeling

You would make a a grate lecturer

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