Please show all steps ### System Description**Figure Overview**The image presents a mechanical system comprising two masses, \( M_1 \) and \( M_2 \), connected by a spring with constant \( K \). The system rests on a frictionless surface, allowing free horizontal movement. A damper \( D \) is attached to \( M_1 \), and an external force \( f(t) \) acts on \( M_2 \).**Variables and Directions**- \( x_1(t) \): Position of mass \( M_1 \)- \( x_2(t) \): Position of mass \( M_2 \)- The spring between \( M_1 \) and \( M_2 \) extends or compresses as the masses move.- \( f(t) \): The force applied to \( M_2 \), acting in the direction of \( x_2 \) movement.### Tasks**a. Derive the Differential Equations**To model the motion of this mechanical system accurately, you will need to establish differential equations based on Newton’s second law. Consider forces from the spring, damper, and any external applies (e.g., \( f(t) \)).**b. State Space Representation**Convert the described system into a state space model. Define the state variables, where:- \( x_1 = x_1 \)- \( x_2 = \dot{x}_1 \) (velocity of \( M_1 \))- \( x_3 = x_2 \)- \( x_4 = \dot{x}_2 \) (velocity of \( M_2 \))Utilize sensors to measure the velocities of \( M_1 \) and \( M_2 \), aiding the transition into a comprehensive state space form. This representation is crucial for analyzing system dynamics and control design.

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Given the system: x:(t) x,(t) f(t) M1 M2 frictionless surface a Derive the differential equations to describe the motion of the system. b. Convert the multiple degree of freedom system into state space with the following state variable definitions with a sensor measuring the velocity of M1 and another sensor measuring the velocity of M2: x1 = x1 x2 = i1 X3 = X2 X4 = i2

Given the system: x:(t) x,(t) f(t) M1 M2 frictionless surface a Derive the differential equations to describe the motion of the system. b. Convert the multiple degree of freedom system into state space with the following state variable definitions with a sensor measuring the velocity of M1 and another sensor measuring the velocity of M2: x1 = x1 x2 = i1 X3 = X2 X4 = i2

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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