Executive summary
Demonstrate the patterns and relationships that stiffness and resonant frequencies follow under different circumstances on an air track. Stiffness is a measure of the resistance of a material to deformation under applied force. Resonant frequencies are the frequencies that a system appears to oscillate at greater amplitudes.
Content
Introduction pg.2
Theoretical calculations and background information pg.2
Experimental design and procedure pg.4
Analysis result and conclusion pg.7
Reflection to other experiment pg.9
Introduction
The purpose of this dynamic Lab is to measure the stiffness and resonant frequencies of a coupled oscillator on an air track. Four experiments will be taken in order to see the behaviour and performance of the rubbers connecting the trolleys. Calculate theoretical results by using background information. Comparisons of theoretical and experimental results will be done to see errors and find conclusions.
Resonant frequencies are the frequencies that a system oscillates at greater amplitudes. This type of oscillations is what makes systems to vibrate many times. On the other hand, stiffness is a measurement of the ability a material have to extend without deformation. Low stiffness can result in failure of a system and high stiffness is required in the design of systems that deformation should be at its minimum. So both two factors are important in the performance of a system, in order to perform efficiently and without any failure.
Theoretical calculations and background information
A system working at resonant frequencies means that the amplitude of oscillations is high. This mainly is a disadvantage for a system. However these “disadvantages” can be used in engineering to achieve appropriate outputs that co-operate with high amplitudes.
Advantages and disadvantages of resonant frequencies
Disadvantages
* Fatigue failure of components. * Annoyance, sound and disturbance. * Damaging systems close to it. * Decreasing the efficiency of a system.
Advantages
* Effective for appropriate designs and systems. * Clock operation system * Music * Vibrating systems
The second factor tested in the four experiments taken, was stiffness. When a force is applied to a body, the body deforms. The measurement of the ability the body has to resist deformation is called stiffness. Stiffness is a constant force factor when Hooke’s law is applied in a system, as result stiffness will affect the output result constantly. Stiffness is a factor that will affect the operation of a system again. High stiffness can either benefit or harm a system. As a result specific stiffness should be chosen for each operation.
Also mode shape is investigated in the fourth experiment. Mode shape demonstrated the expected curvature of a mass vibrating at a particular mode. Since mode shape is multiplied by a function that varies with time, mode shape describes the curvature of vibration at all points in time. However magnitude of curvature will change. Factors affecting the mode shape are the boundary conditions and the shape.
Theoretical calculations are been done in order to find stiffness and resonant frequencies.
Experiment one (spring constant)
In the first experiment we are using F=k*x in order to find the constant K. experimental procedure is needed using a vertical stand to measure the extension and input it in the formula.
Experiment 2 (1 DOF system)
The second experiment has to do with a 1-DOF system. One trolley and rubber bands are used. Natural frequency of the oscillator is calculated theoretically in order to compare it with the experimental.
Natural frequency:
Experiment 3 (2 DOF system)
In this case we have an extra trolley and an extra rubber band fitted in the system in order to produce a 2-DOF system that is approximately symmetrical. To achieve this, set rubber band 2 connected with mass 2.then connecting spring is connected to mass 2 and mass 1 and mass 1 is connected to the shaker. Natural frequencies are calculated theoretically using the formula bellow:
Experiment 4 (asymmetric coupled oscillator)
Finally an asymmetric coupled oscillator is tested. An extra rubber band is connected parallel to rubber band 1. By using the mechanical impedance matrix to be equal to 0 at resonance, calculate the natural frequencies f1 and f2 of each shape. The procedure is shown below:
Experimental design and procedure
Four experiments are going to take place in order to find frequencies under different circumstances (e.g. 2 DOF systems). Excel operates as a very simple simulator and performs calculations and demonstrates relationships in graphs.
Apparatus
* An air track allowing 2 similar masser (trolleys) to travel horizontally with negligible friction; mass trolley= 0.5kg * 4 similar rubber bands (band in position 1 will connect the trolley with an electromagnetic shaker). * An accelerometer that is attached to each mass. The two signals from these are fed to a data acquisition system (DAP board) that a display providing you with plots of acceleration against time in DAP view. * Vertical stand together with loads (for experiment 1)
Image 1.1 Air track connected with the shaker and the DAP view
DAP view
DAP view is a software that is connected to the masses. A display will provide you with graphs of acceleration against time. Using the graphs and the tools that the software provides you, you can record data and find frequencies over time.
Experiment one (spring constant)
In the first experiment you are required to find the force deformation behaviour of the rubber bands under different loads. Using a vertical stand (image) you can calculate the spring constant K for each and. For each rubber band, different loads (max of 0.6 kg) are to be hanged from the vertical stand. Extension of the rubber under tested weight is recorded. Using the following equation you can calculate K.
F= k*x => k=F/x
Where F (N) is the force, k (N/m) is the spring constant and x (m) is the extension.
Image 1.2 vertical stand.
Spring constant results:
Table 1.1 shows the factors and calculations in order to find the constant spring factor (K).
Experiment 2 (1 DOF system)
The second experiment is using the same rubber bands, but now they are tested on the air track system. Two rubber bands are used together with one trolley. Set the system to operate at free oscillation and record the natural frequency using the DAP view that is connected with the air track.
Then set the system into forced oscillation using the small shaker. Calculate resonance by direct observation of the system.
Table 1.2 rubber bands properties
In order to get the best results out of DAP view, recordings are been taken within a 5T period in order to be more accurate. As a result the frequency recorder in DAP view is multiplied by 5. Then to find the frequency, the following formula is used:
Frequency= 1/ (f/5)
Table 1.3 DAP view period and calculated frequency. 5T (DAP View) Hz | frequency Hz | 2.075 | 2.409638554 | 2.1375 | 2.339181287 |
Experiment 3 (2 DOF system)
The third experiment is done using a second trolley and an additional rubber band. Create an approximately symmetrical 2-DOF system and using the shaker, calculate the two resonant frequencies and mode shapes empirically.
Table 1.4 mode shapes and resonant frequencies mode shapes | resonant frequencies (Hz) | 0.579 | 3.454231434 | 0.582 | 3.436426117 |
Resonant frequencies are found from the dap view. By changing the frequency and observing the behaviour of the system from dap view, you can recognise the resonant frequency when DAP view is at its higher amplitude. After recording the resonant frequency you can use it to calculate the mode shape. Equation bellow is used to calculate the mode shape:
Resonant frequency= 1/(fr/2)
For a 2T DAP view.
Experiment 4 (asymmetric coupled oscillator)
By adding an extra rubber band parallel with the first rubber band you can create an asymmetric coupled oscillator. Resonant frequencies and mode shapes are found experimentally like experiment 3. I lost the experimental results and I couldn’t find them for experiment 4. Sorry!
Analysis result and conclusion
Both theoretical and experimental results should be taken in order to come to conclusions and compare the theoretical results that appear to be perfect with the experimental results that factors not calculated in the theory affect the final results.
The main error that usually appears in an experiment is the calibration of apparatus in experiments usually affects the results. Firstly rubber bands have been used many times and some (very little) permanent deformation might occur previously to them. This of course will affect the spring constant values. Also air track should be cleaned before used and make sure that nothing will go against the trolleys and produce friction between them.
Also in the DAP view, you have to make sure that you don’t change the scale factor over the experiment since it changes the time- base. Results in the DAP view should be taken carefully in order to record the best wanted value. Finally experiments should be repeated and find the average values in order to be the more accurate you can.
Conclusions
From the first experiment where spring constant is our investigation point, we can see that as force increases, extension increases too. A graph is plotted to show the behaviour of the force against extension. A linear and proportional pattern is observed as you can see below:
Graph 1.1 force vs extension
In the second experiment, where a 1-DOF system is tested, we can see that the theoretical value is less than the experimental one. This is mainly due to the calibration of the apparatus and friction that is said to be negligible in the theoretical results. Also in an unforced oscillation there is no damping function, unlikely the forced oscillator.
In the third experiment that a 2-DOF system is tested, we can see that resonant frequencies are higher than the natural frequencies that are calculated theoretically. This is expected since resonant frequency is the higher frequency a mass can experience. Also we can see that f1 is less than f2 and that the resonant frequencies calculated are equal for each case.
Also by comparing the natural frequency of 1-DOF with a 2-DOF system we can see that a 2-DOF system has a higher natural frequency at position 1 and a lower at position 2. This can be used as a mechanical advantage for machines that require such a system.
Finally in experiment four the mode shape and resonant frequencies should be differ again. Since I have lost my experimental result for experiment 2, I will assume that result again should be differing from the theoretical one due to inefficiency of results.
Coming to a conclusion with background knowledge, research and examples previously observed from engineers, having a low resonance frequency is not efficient. When the frequency of the driver reach the resonance frequency of the pinion, failure will take place to the system. An example of such failure is the Tacoma Bridge, where the air frequency reached the resonant frequency of the bridge as a result of the bridge oscillating at high amplitudes and collapsing. So resonance is a very important factor for engineering design since our expectations is to have the perfect output that will last long.
Reflection to other experiments
Another 4 experiments where carried for the dynamics labs
Rubber bush
The objective of this experiment is to investigate the performance of the rubber bush for soft and stiff position. Damping ratio and transmissibility is investigated while the rubber bush tries to isolate vibrations between one structure and another.
Cussons Oscillator
The objective of this experiment is the investigation of a simple harmonic motion for free and damped oscillation. Relationship between damping magnitude and damping ratio can be demonstrated.
Engine Dynamics
The objective of this experiment is the investigation of the effects of engine mass and piston balancing in an engine. Frequencies higher than natural frequencies are investigated in order to to effectively de couple the internal mechanics from the spring force. Altering the pistons allows you to calculate parameters that will help you to demonstrate patterns and relationships of the experiment.
Torsional vibration
The objective of this experiment is to calculate the torsional rigidity of a stainless steel beam. Investigate natural frequencies and resonance in free, force and coupled torsional vibration. Resonance patterns can be demonstrated when a rotational vibration is transmitted along the rod. Other parameters such as modes, phase shift and damping may be investigated and result in patterns and relationships.
Conclusion of reflection of other experiments
Doing the experiments you can see that altering parameters in engineering can benefit you to achieve the best and desire output. Machines with the same concept but with a different output can be achieved by using the above relationship that will show you the best parameters to use in your engineering design.
