Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3
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.
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...
where:
- = sprung mass [kg]
- = damping coefficient [Ns/m]
- = 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 = 0.15m
- initial velocity = 0m/s
For the system to be critically damped, we needed to tune the damper up to...
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...
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...
Figure 2. Suspension schematic with parameters required for over damping
Substituting these values into the equation of motion (1)...
The characteristic equation to (2) is...
The roots of equation (3) are...
Numerically, the roots are...
By equation (3) of Post #3, the general solution is...
The first derivative of the general solution is...
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...
Now applying the displacement initial condition...
So the particular solution is...
The behaviour of this particular solution is depicted in Figure 3 below...
Figure 3. Behaviour of the suspension system with and
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...
Applying the displacement initial condition...
Applying the initial velocity condition...
And solving for A...
Alright, now that we have templates for our coefficients, let's explore what happens when we vary the initial conditions.
Condition #2, with = 0.15m and = -0.6m/s...
The particular solution to this set of initial conditions is...
Condition #3, with = 0.15m and = -0.3m/s...
The particular solution to this set of initial conditions is...
Condition #4, with = 0.15m and = 0.3m/s...
The particular solution to this set of initial conditions is...
All of the above solutions are superimposed in Figure 4...
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 = 0 and = 0.3m/s...
The particular solution to this set of initial conditions is...
A few solutions showing the behaviour of the mechanism with = 0 and varying values of is shown in Figure 5 below...
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 , 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
First Order Differential Equations
- Introduction to Differential Equations - Part 1
- Differential Equations: Order and Linearity
- First-Order Differential Equations with Separable Variables - Example 1
- Separable Differential Equations - Example 2
- Modelling Exponential Growth of Bacteria with dy/dx = ky
- Modelling the Decay of Nuclear Medicine with dy/dx = -ky
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- Mixing Salt & Water with Separable Differential Equations
- How Newton's Law of Cooling cools your Champagne
- The Logistic Model for Population Growth
- Predicting World Population Growth with the Logistic Model - Part 1
- Predicting World Population Growth with the Logistic Model - Part 2
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First order Non-linear Differential Equations
- There's a hole in my bucket! Let's turn it into a cool Math problem!
- The Calculus of Hot Chocolate Pouring!
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Second Order Differential Equations
- Introduction to Second Order Differential Equations
- Finding a Basis for solutions of Second Order ODE's
- Roots of Homogeneous Second Order ODE's and the Nature their Solutions
- Modelling with Second Order ODE's: Undamped Free Oscillations
- Modelling Car Suspension with ODE's: Damped Free Oscillations Part 1
- Modelling Car Suspension with ODE's: Damped Free Oscillations Part 2
- Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3
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Following...
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.
This is pour dynamic modeling
You would make a a grate lecturer