The car body and its occupants are represented by the mass block, which is suspended (hence “suspension”) on a spring that connects it to the ground (the tire/wheel is assumed for simplicity to be a solid object in this simplistic model, though with THAT, we certainly could add the “unsprung” mass of the wheel, hub, tire, etc., as well as a spring and a damper to model the tire’s compliance) with a shock absorber (aka “damper” or “dashpot”).
We can construct a FBD (Free Body Diagram — how I remember this term after almost 50 years is beyond me because I haven’t done an FBD since my university days) where all forces must be in equilibrium if the car is just sitting there; i.e., no net force results from the system. Therefore, the force of the Mass (Fm) must equal the forces of the damper (Fd) and of the spring (Fs):
Fm + Fd + Fs = 0
May the Forces be With You
We know that the force on a mass (m) is given by Newton’s 2nd Law of Motion:
Fm = ma, where a is acceleration
We also know that the force by a spring with spring constant k is given by Hook’s Law as:
Fs = ky, where y is vertical distance or displacement
And the force in a damper with a damping coefficient d is given by Nobody’s Law (why is that?) as:
Fd = dv, where v is the vertical speed
About Your Car’s Differential…
Recall that speed v is the first derivative of position y′ and that acceleration a is the second derivative of position y′′ (with apologies for the lame Lagrange’s notation with apostrophes because Newton’s notation of derivatives is impossible on Electronic Design’s typesetter system. It’s barely capable of technical, scientific, and engineering scrawl beyond what the bunch of monks in the back room of Microsoft can do with their type sets. Now I know why Bob Pease was missing the equations in his blog on automobile suspension Analog Computing, here).
So, the differential equations are written as follows:
Fm = my′′
Fd = dy′
Fs = ky
Fm + Fd + Fs = 0 then becomes, by substitution:
my′′ + dy′ + ky = 0
And rearranging to solve for the highest derivative becomes:
my′′ = – ( dy′ + ky )
or
y′′ = – ( dy′ + ky )/m
Which is the form I have to leave it in because of the lame editing software created by a bunch of y′′′s at Microsoft.
Then THAT Rabbit Comes Out of the Hat
All of this math and physics stuff pretty much tracks with what’s in THAT’s Section 9.2 writeup. That’s good, because the equations are based on physical laws governing the operation of each component of our crude suspension system. And in engineering and science, there’s no room for lay public opinion on things like spring changes in length. Make Engineering Great Again.
Looking at Section 9.2 of the THAT user guide, we then see a giant, unexplained, leap to the Analog Computer topology that reflects our y′′ = – (dy′ + ky)/m equation in Figure 2.
