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Vibration Testing and Analysis

(a) Investigate the free response of a vibrating system in the time and frequency domains using experimental method (Refer to Lab 2: Vibration analysis and modeling of a cantilever beam)

i. Investigate the time and frequency domain response of the cantilever beam when perturbed (without the mass attached to the end of the beam).
ii. Attach the mass at the end of the beam and determine the damped natural frequency of the cantilever beam .

(b) Analyses the response data of the vibrating systems utilizing MATLAB software

i. Identify the dominant damped natural frequencies and plot time and frequency domain responses of cantilever beam with/without mass attached.
ii. Using the time domain response, determine the natural frequency and damping ratio using log decrement method.
(c) Investigate the forced response of a vibrating systems identifying natural frequencies and associated mode shapes

i. Perform a frequency sweep using the signal generator starting from 5Hz to 300Hz. As you increase the frequency, determine the first three damped natural frequencies and their associated vibrating shapes where the amplitudes of vibration increase significantly.
ii. Develop your understanding of the concepts of magnification factor – Perform a frequency sweep from 1Hz to three times the first damped natural frequency. Take regular readings of frequency and accelerometer amplitude and try to recreate graph for magnification factor.

(d) Mathematical modeling of the cantilever beam

Theoretically model the cantilever beam, which you have used in your experiment, as one degree of freedom system. You should consult your lecture notes and tutorial examples regarding one degree of freedom modelling and utilise your critical reasoning skills for analysing how to model these systems. Calculate the natural frequency of the beam with/without mass attached.

Assume the beam is in a fixed-free constrained condition. You should also utilise the following information

E – Young’s modulus for mild steel – 200GPa
ρ – the density for the beam, for mild steel – 7850Kgm-3

(e) Verification and validation of the results by comparing the experimental and the analytical results.

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Principal Stress Determination

To conduct a laboratory experiment to measure strains on the surface of the cantilever subject to a lateral load using a strain rosette.

To process the experimental data, carry out analysis to determine the principal stresses using different methods and write a summative report.

(a) Experimental data processing to capture the proportional behaviour between the weight applied and the strain for each strain gauge. You can only work on either loading or unloading case.

(b) Use the last digit of your MMU ID number as weight in Newton (if your ID number ends ‘0’, use 10N for the weight) and estimate the corresponding strain values and using the results on the ‘best-fit’ line.

(c) Construct a strain Mohr’s circle.Graphically determine the principal strains and the 1st principal direction relative to the shaft axis (longitudinal) direction.Also determine the maximum shear strain.
Calculate the principal strains, the 1st principal direction and the maximum shear strain using the formula method to verify the solution from the Mohr’s circle.

(d) Calculate the experimental principal stresses using the generalized Hooke’s law. Furthermore, calculate the maximum shear stress.

(e) Calculate the theoretical principal stresses and the maximum shear stress by applying the beam bending theory.

(f) Result Comparisons

1. Create a finite element model of the cantilever beam using Solid works, simulate the scenario when the weight W’ (i.e. the value in deliverable (c)) is applied. Find out the principal stresses from your finite element model in Solid works Simulation.

2. Compare the experimental result, the theoretical solution and the FEA prediction showing the percentage difference for each of the principal stresses and the maximum shear stress. Comment on the likely error sources causing the differences between the experimental, the theoretical and the FEA results.

(g) Preliminary investigation by simulation

Since the nominal value of the Young’s modulus provided is a common book value for mild steel rather than being obtained from testing the actual batch, it is of interest to examine the sensitivity of the strain results from Solid works simulation to its variation. It is suggested that a tolerance of the Young’s modulus be considered.

Make comments on the likelihood that the discrepancy between the simulation results and the experimental results of the principal strains be caused by the error in the nominal value of the Young’s modulus.

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