## Laplace Analysis of Linear Systems

The differentiation theorem can be used to convert differential
equations into *algebraic* equations, which are easier to solve.
We will now show this by means of two examples.

### Moving Mass

Figure D.1 depicts a free mass driven by an external force along
an ideal frictionless surface in one dimension. Figure D.2
shows the *electrical equivalent circuit* for this scenario in
which the external force is represented by a voltage source emitting
*volts*, and the mass is modeled by an *inductor*
having the value *Henrys*.

From Newton's second law of motion ``'', we have

Thus, given

- Laplace transform of the driving force ,
- initial mass position, and
- initial mass velocity,

*all*linear, time-invariant (LTI) systems. For nonlinear and/or time-varying systems, Laplace-transform analysis cannot, strictly speaking, be used at all.

If the applied external force is zero, then, by linearity of the Laplace transform, so is , and we readily obtain

^{D.3}Similarly, any initial velocity is integrated with respect to time, meaning that the mass moves forever at the initial velocity.

To summarize, this simple example illustrated use the Laplace transform to solve for the motion of a simple physical system (an ideal mass) in response to initial conditions (no external driving forces). The system was described by a differential equation which was converted to an algebraic equation by the Laplace transform.

### Mass-Spring Oscillator Analysis

Consider now the mass-spring oscillator depicted physically in Fig.D.3, and in equivalent-circuit form in Fig.D.4.

By Newton's second law of motion, the force applied to a mass equals its mass times its acceleration:

We have thus derived a second-order differential equation governing the motion of the mass and spring. (Note that in Fig.D.3 is both the position of the mass and compression of the spring at time .)

Taking the Laplace transform of both sides of this differential equation gives

To simplify notation, denote the initial position and velocity by and , respectively. Solving for gives

denoting the modulus and angle of the pole residue , respectively. From §D.1, the inverse Laplace transform of is , where is the Heaviside unit step function at time 0. Then by linearity, the solution for the motion of the mass is

If the initial velocity is zero (), the above formula reduces to and the mass simply oscillates sinusoidally at frequency , starting from its initial position . If instead the initial position is , we obtain

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Example Analog Filter

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Laplace Transform Theorems