# First and Second Order Systems - Classical Analysis

• mm391
In summary, the conversation discusses a first order system with a time constant τ = c/k and how to find the response for the unforced case with initial condition y(0)=y0. Clarification is given on the differential equation used to represent the system.
mm391

If we have the following fist order system:

τ.dx/dy+y(t)=x(t) where τ=c/k where "k" is the spring stiffness and "c" the linear damper coefficient and τ is a time constant.

For the unforced case x(t)=0, we need to write down an expression for the response when the initial condition is y(0)=y0. ANd how do we sketch the repsonse?

I have to clarify something before giving any advisement.

Are you sure the differential equation is what you have provided or is it:

τ.dy/dt + y(t) = x(t)

The reason I ask is the above equation represents a first order spring-damper system where τ = (dampening coefficient/spring stiffness), as you stated.

## 1. What is a first order system?

A first order system is a type of dynamic system that can be described by a first-order differential equation. It has one energy storage element and one independent energy source, making it a simple system to analyze. Examples of first order systems include RC and RL circuits.

## 2. What is a second order system?

A second order system is a type of dynamic system that can be described by a second-order differential equation. It has two energy storage elements and one independent energy source, making it more complex to analyze than a first order system. Examples of second order systems include mass-spring-damper systems and RLC circuits.

## 3. What is classical analysis?

Classical analysis is a mathematical method used to analyze the behavior of first and second order systems. It involves using Laplace transforms to convert differential equations into algebraic equations, which can then be solved using traditional algebraic methods. This allows for the determination of system responses such as steady-state and transient responses.

## 4. What is the difference between first and second order systems?

The main difference between first and second order systems is the number of energy storage elements and independent energy sources they have. As mentioned before, first order systems have one of each, while second order systems have two energy storage elements and one independent energy source. This difference also affects the complexity of their mathematical analysis and their responses to input signals.

## 5. What are the applications of first and second order systems?

First and second order systems have a wide range of applications in various fields of science and engineering. They are commonly used to model and analyze electrical and mechanical systems, such as circuits, motors, and suspension systems. They are also useful in control systems, signal processing, and other areas that involve dynamic behavior.

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