Chapter 3

This chapter looks at mathematical modeling of mechanical systems and electrical systems.

Impeadence : is the measure of the opposition that a circuit presents to a current when a voltage is applied.

3-2 Mathematical Modeling of Mechanical Systems

To obtain the equivalent spring constant from the above image

In parallel

$$k_1x + k_2x = F = k_{eq}x \\\\ k_{eq} = a_1 + k_2$$

in series

$$k_{eq} = \frac{1}{\frac{1}{k_{1}} + \frac{1}{k_2}}$$

A dashpot is a device that provides viscous friction. The equation is modeled identically to the spring coefficients except that the derivative is the

The Viscous coefficient $ b $ follows a similar pattern

In parallel

$$f = b_{eq} (\dot{y} - \dot{x}) \\\\ b_{eq} = b_1 + b_2$$

In Series

$$b_{eq} = \frac{1}{\frac{1}{b_{1}} + \frac{1}{b_2}}$$

Mathematical Modeling of Electrical Systems

• Kirchoffs current law (Node Law)
  • the algebraic sum of all currents entering and leaving a node is zero.
  • The sum of currents entering a node is equal to the sum of currents leaving the same node
• Kirchoffs Node law (Loop Law)
  • At any given instant the algebraic sum of the voltages around a loop in an electrical circuit is zero
  • The sum of the voltage drops is equal to sum sum of the voltage rises around a loop

LRC Circuit

- The inductance L (henry)
- The resistance R (ohm)
- The capacitance C (frand)

Operational Amplifier (Op amp)

• The input into the minus terminal is inverted and input to the positive is not. The opamp is defined by

$$e_0 = K(e_2-e_1) = -K(e_1-e_2)$$

Where the e's are input voltages