Tuned Mass Damper – A Coupled System - Engineering Assignment Help

Assignment Task

1. Introduction

A prototype tuned mass damper [1] is shown in Fig. 1.

The masses are supported on a frictionless horizontal plane.

The overall objective is to develop a tuneable damper that produces anti-phase signals to counteract mechanical vibrations, for example in a vehicle, or in a machine tool.

At this prototyping stage a state-space model is to be developed and assessed. This will use a simplified set of parameter values so that the mathematical procedures may be easily verified.

2. Assignment Tasks

• Derive the differential equations defining the motion of the masses, via Newton’s 2nd law, and hence construct the system matrix, A, and the state equation, where the state vector is:

• Taking all the parameter values as unity, determine the eigenvalues and eigenvectors, and hence obtain the spectral matrix, ????, the modal matrix, ????, and its inverse, ????−1
• Given the initial conditions: ????(0)=1; ????̇(0)=0;????(0)=0; ????̇(0)=0, determine expressions for the time response of each state variable.
• Verify your answers to the above via the Laplace transform.
• It has been suggested that connecting a damper, ????=1 N m−1s, between mass, ????1, and the left-hand datum (i.e., in parallel with the spring, ????1), will reduce the oscillations of both masses to zero as time, ????→∞, given the initial conditions specified above.

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