Question:
Task
a. Experimental investigation of a cantilever beam on free and forced vibration response.
b. Analyse the experimental reponse data of a cantilever beam utilising matlab software
i. Within the matlab environment import the vibration data you saved in the laboratory for the cantilever beam. Identify dominant damped natural frequencies and plot time and frequency domain response for the beam with/without mass attached.
ii. Using the time domain reponse, determine the damped natural frequency and damping ration and using log decrement method.
c. Modelling the cantilever beam by mathematical formula and by finite element simulation using an industry standard software such as solidworks.
i. Theoretically model the cantilever beam you have used in your experiment as one degree of freedom system. Calculate the natural frequency of the beam with/without mass.
ii. Research Euler-Bernoulli beam theory and utilising the physical measurements of the cantilever beam calculate theoretically the first 3 natural frequencies of the beam. You should also calculate the theoretical mode shapes associated with each of these natural frequencies.
ii. Model the cantilever beam using solidworkds and extract the first three natural frequencies. and their associated transverse mode shapes for the cantilever beam.
Assume the beam is in a fixed free constrained condition. You should also utilise the following information:
E-young’s modules for mild steel 210GPa
p- the density for the beam, for mild steel 7852 kgm3
d. Validation of the FE simulation resultrs by comparing the eperimental and the analytical results.
Answer
Attendance Record
You may be required to sign an attendance sheet in the laboratory. Please note that this is for the unit teaching team’s local records only and does not replace the requirement for on-line self-registration for participating students
For engagement monitoring purposes, completion of the lab will be only be registered when you upload your individual lab report.
Aim
The aim of this experiment is to: Investigate the free/Forced response of a vibrating system in the time and frequency domain
Associated Learning Activities
This experiment is supported by the following unit activities and resources:
Lecture/lab/tutorial classes
Formative assessment activity
Background Reading
Free response of a mass/damper & mass/spring/damper systems (Study book Unit1 Lect2) Mass/spring/damper systems & magnification factor (Study book Unit2 Lect1)
Further Reading http://lists.lib.mmu.ac.uk/items/875A60FD-DF00-6C16-776A–
81AA43F45094.html?equellassotoken=55112727:mmusso:1474971772781:wlSSgaOlipk9%2bK87z8 WyHg%3d%3d
Summative Assessment
This lab experiment forms part of the learning for the following learning outcome(s) and assessment(s) in this unit:
| Learning Outcome | Assessed by: |
Learning Outcome 1: analyse linear SDOF
Assessment 1 – Coursework
mechanical systems in both the frequency domain and the time domain
Learning Outcome 2: process data collected Assessment 1 – Coursework using standard equipment and transducers
References
Mechanical Vibration 5th edition. S. S. Rao. Prentice Hall, 2011.
About this resource
| Document last updated: | September 2016 | Note that printed or emailed copies of this | |
| document may be out of date. Always | |||
| Next scheduled review date: | June 2017 | ||
| check Moodle for the latest version. | |||
| Comments and feedback on this document should | Naveed Ashraf | ||
| be sent to the author(s): | |||
School of Engineering Undergraduate Programmes 2016-17
Vibration analysis and modelling of a cantilever beam
6E5Z2102 Solid Mechanics and Dynamics
Introduction
The main purpose of this lab is:
To investigate the free response of a vibrating system in the time and frequency domain.
To develop your understanding of forced vibrating systems identifying natural frequencies and associated vibrating shapes.
To develop your understanding of the concepts of magnification factor.
Theory
In this laboratory, we will look at the free response of a cantilever beam system with and without the additional mass of the accelerometer. We will investigate the relationship between the driving frequency acting on the system, its natural frequency and the amplitude of acceleration. Finally, we will consider the implications of the results in relation to the Magnification Factor (M) for the system.
You will apply your skills in computational analysis to analyse and present the time domain and frequency domain responses of the vibrating systems using MATLAB software. Furthermore, you will undertake self-directed learning to theoretically model the experimental system using one degree of freedom. Recall the definitions for the main concepts:
Free vibrations – Total energy of the vibrations remains same over the time such oscillations are called free vibrations. In free vibrations the amplitude of oscillations remains same. But in reality it is not possible in real systems the energy of vibrations is dissipated to surroundings over the time and the amplitude decays to zero such dissipation of energy of vibrations called damping.
Forced vibrations – Forced vibrations occur when the object vibrates forcefully at particular frequency by applying a periodic input of force then such vibrations are called Forced Vibration. This makes the right hand side of our equation of motion non-zero, see equation 1 with a sinusoidal forcing function. Where Fo is the amplitude of excitation, the forcing frequency and t time.
When a system’s natural frequency is close to the excitation frequency of the forcing function it will respond with maximum amplitude i.e. resonate. When systems resonate, they are subject to cyclical loading which will lead to fatigue failure. Hence, as a design engineer it is important to understand these fundamental concepts so that undesirable vibration leading to fatigue failure can be avoided.
Magnification Factor – The ratio of the dynamic to static amplitude of motion. equation 2, where X represents displacement and the static deflection which would occur if a static load of amplitude
F0 was applied to the system. r ( ) is the frequency ratio and the damping ratio. Figure 2 illustrates this relationship graphically.
Figure 1 above illustrates the experimental setup. The shaker transmits a constant sinusoidally varying force to the beam. The amplitude of this force is set by the signal generator and should remain fixed at all times. The frequency of the force is also set by the signal generator and can be varied during the experiment. The vibrations are measured by the accelerometer. This reading can be integrated electronically by the charge amplifier to give velocity or position measurements.
- Measure the dimensions of the cantilever beam using the equipment provided and complete Table 1.
| Beam width | Beam length | Beam depth | |
| Measurement | (mm) | (mm) | (mm) |
1 20 500 50
2
3
Mean Value
| Table 1: Cantilever beam dimensions | ||
| Identify the sampling frequency of the data acquisition system | fs = | Hz |
| Measure the mass of the given magnet | M = | Kg |
(Note: Neglect the mass of the attached accelerometer and blue tach)
The beam has been fixed at one end with the other end free to move, Figure 1. The beam can be considered to be in a fixed-free clamped condition if deflected its free response will exhibit transverse vibrations.
- Without the mass (magnet) attached to the end of the beam, run the Lab VIEW program vi, deflect the beam and record the free response of the system. Investigate the time and frequency domain response of the system when perturbed (i.e. deflect and observe the free response). When you deflect the beam the vibrations you induce can perturb the clamped rod inducing vibrations in it. You can use the cursors on the frequency domain graphs to check the locations of peaks between perturbations.
What is the damped natural frequency of the beam associated the first transverse mode (this will be the frequency which has the largest magnitude in the frequency domain)?
When you are comfortable with using the software record 3 seconds of the beam free response, save the time domain data to a file named beam_without_mass.txt. To save the data select the Save Data button, ensure you save data that exhibits a relatively smooth trace in the frequency domain.
Describe the time domain response of the system
Ans: Under damped
- Now attach the mass (magnet) at the end of the beam and determine the damped natural frequency of the system .Save the time domain data to a file named beam_with_mass.txt.
- Vibrating shapes and damped natural frequencies – Now remove the attached mass (magnet) and perform a frequency sweep using the signal generator. Start at 5Hz and work up to 300Hz. As you increase the frequency, try to identify three damped natural frequencies where the amplitudes of vibration increase significantly. Sketch the shape of the beam (on Figure 3), showing the “nodes” where there is no movement and also “antinodes” were the displacement is largest. To identify the position of the nodes, try holding a pen tip loosely and running it along the length of the beam. Nodes will cause little vertical displacement of the pen.
-
- Forced vibration plots – Perform a frequency sweep from 1Hz to three times the first damped natural frequency 1(approximately 40Hz). Take regular readings of frequency and accelerometer amplitude in Table 2.
School of Engineering Undergraduate Programmes 2016-17
Vibration analysis and modelling of a cantilever beam6E5Z2102 Solid Mechanics and Dynamics
Try to recreate graphs for magnification factor as shown in Figure 2.
Excitation Accelerometer Excitation Accelerometer frequency amplitude frequency amplitude 2 0.18 22 0.378 4 0.38 24 0.321 6 0.292 26 0.356 8 0.410 28 0.308 10 0.950 30 0.265 12 7.95 32 0.269 14 1.68 34 0.322 16 0.795 36 0.41 18 0.596 40 0.409 20 0.65
Table 2: Experimental data for the system acceleration response as a function of forcing frequency
