Automobile suspension analog; feedback system

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.

Setting 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.

Feedback system

A feedback system is a system in which a part of the output is returned to the input and used to control the output. The input to the process is the reference input combined with the fed-back signal.

 r(t) --> (+)--e--> [ Process ] --> y(t)
           ^ -                        |
           |                          |
           +------[ Feedback ]<-------+

• Negative feedback subtracts the output from the input; it reduces error and gives stability.
• Positive feedback adds the output; it increases the effect and may cause instability (e.g. microphone squeal).

Example: room thermostat. The desired temperature is the reference. A sensor measures the actual room temperature and the difference (error) switches the heater on or off. When the room is below the setting, the heater runs; when it is above, it stops. This keeps the temperature near the set value despite disturbances.

The suspension is also a feedback system: the spring and damper forces depend on the displacement x and velocity x', which are the outputs fed back to the input of the mass, as shown in the analog circuit where -(K/M)x and -(D/M)x' return to the summer.