Analog method for automobile suspension

Analog method

An analog computer is a computing device that represents each variable by a continuously varying physical quantity (usually voltage) and solves equations by connecting electronic units that behave like the mathematical operations. The analog method solves a differential equation by building a circuit whose voltages obey the same equation (an analogue of the system) and recording the output.

Importance of differential equations

Continuous physical systems (mechanical, electrical, thermal, biological) change with time, and their laws are written as differential equations. These equations let us predict the response to any input, the stability of the system and the effect of parameter changes, and they are the standard model for control and design. Because exact solutions are often impossible, they are usually solved by simulation, either on an analog computer or numerically on a digital computer.

Automobile suspension problem

A car body of mass M is supported by a spring of stiffness K and a shock absorber (damper) with coefficient D. Let x be the displacement of the body from its equilibrium position and F(t) the external force from the road.

      F(t)
        |
     +-----+
     |  M  |
     +-----+
      |   |
      K   D
      |   |
  ================ wheel / road

Forces: inertia M x'', damping D x', spring K x. Newton's second law:

M x'' + D x' + K x = F(t)

Solving for the highest derivative:

x'' = F(t)/M - (D/M) x' - (K/M) x

Analog computer circuit

Two integrators in series produce x' and x from x''. Each integrator inverts the sign.

 F/M ------------\
 -(D/M)x' -------- Sum --> [Int 1] --> -x' --> [Int 2] --> x
 -(K/M)x --------/            |                         |
     ^                        |                         |
     |                        v                         v
     |                     [pot D/M]              [Inverter]
     |                        |                         |
     |                  back to Sum                   -x
     +------------------[pot K/M]<---------------------+

Steps:

1. Integrator 1 takes the sum x'' and gives -x'. Its initial condition is -x'(0).
2. Integrator 2 takes -x' and gives +x. Its initial condition is x(0).
3. Output -x' goes through potentiometer D/M and returns to the summing input, giving the term -(D/M) x'.
4. Output x passes through an inverter to give -x, then through potentiometer K/M, giving -(K/M) x.
5. The forcing term F/M is applied at the input of integrator 1.
6. The output x is recorded against time.

Adjusting the D and K pots changes the damping and stiffness, and the displayed curve shows whether the ride is oscillatory, critically damped or over-damped (D^2 compared with 4 K M). Amplitude and time scaling are applied so that the voltages stay within the machine's range.