Vibrating Beam Lab Report

3B5 Vibrating Beam Laboratory

Name: Barry O’Connor Date: 04/01/2016

Student Number: 13321346 Demonstrator: Kun Zhao

Introduction:

In this experiment we studied the response of a beam which is fixed at one point by a hinge, given an initial displacement. Our aims of this lab were to predict and to observe how damping, stiffness and inertia determine the overall displacement of the beam due to variances in initial conditions and harmonic loading. After the beam is given the initial displacement no external force acts on the system, due to this the amplitude gradually get smaller and smaller with time as shown in graphs in the report. A huge part of the experiment was to understand how the damper influences the response of the beam

The computer is running LabView which is measuring the displacement from the centre of the beam from the point l2 from the rigid point. The LVDT displacement transducer sends the data to the computer which makes a displacement versus time plot for us to analyse.

We were allocated by the lab demonstrator into groups of 3 and this is the procedure we followed:

1. Measure the free vibration displacement time-history response of a rotational spring–mass–dashpot system subjected to a manually applied initial displacement.

2. Determine the inertia, stiffness and damping constants for use in a single degree of freedom model of the vibrating beam system, and compare the predicted displacement time response to the measured behaviour of the beam when subjected to an initial displacement and allowed to oscillate freely. Optional: change the damping ratio in the model and assess the effect of this in the predicted time-history displacement

The equivalent standard form is:

(x_1 ) ̈+[2ξω_n ] (x_1 ) ̇+[ω_n^2 ] x_1=0

This is the standard equation of motion of a single degree-of-freedom vibrating system in which 휔푛 and 휉 are known as the natural frequency and damping ratio respectively. These are important parameters in a mechanical system as they characterise the dynamic response. By inspection of [7] and [8], the theoretical natural frequency, 휔푛 (푟푎푑 푠 −1 ), is given by ω_n=√((kl_4^2)/(I_beam+Ml_1^2+m_motor l_2^2 ))

For the calculation of the damping factor 휉, the ratio of 푛 successive oscillation amplitudes (푋0,1, …푋푛) is ln x_0/x_n =nδ=2πξn where 훿 is the logarithmic decrement. The logarithmic decrement (훿) can be measured experimentally and used to calculate 휉, which is easier than measuring 퐶.

The standard solution to equation [8] when 휉 < 1 (underdamped) is given by x_1 (t)=Ae^(-ξω_n t) cos⁡(tω_n √(1-ξ^2 )+∅_1 where 퐴 and 휙1 are determined from the initial conditions of displacement and velocity. This equation describes a decaying oscillatory motion observable in the