Free or natural vibrations are a result of random disturbances and remain until they are damped by either the systems own internal or other friction forces. These vibrations depend on the systems own balance between elastic and inertial forces which act during the oscillation. These factors give rise to the property known as the natural frequency. An example of free vibrations would be how a tuning fork will continue to vibrate following a knock.
Forced vibrations occur from externally varying forces and are independent of the balance between elastic and inertial forces which act during the oscillation. But when a forced frequency equals the natural one, resonance occurs. The most common example of a system which undergoes forced vibrations is one which includes an internal combustion engine. Usually the frequency of the vibratory oscillation caused by the engine shall pass through natural frequency. The less time the frequency of oscillation spends close to the natural frequency the better. This is why diesel engine vibrate more on start-up than petrol ones since the are slower to speed up.
Self-excited vibrations are sustained by the vibratory motion itself, hence; if the vibration ceased, so does the driving force. It is common for self excited vibrations to occur at one of the natural frequencies. One of the most famous examples of the destructive powers of self-excited vibrations is in the case of the Tacoma Narrows bridge Fig1.
Fig. 1
In this case the self excited vibrations were caused by a phenomenon known as vortex shedding which are predominantly caused by sharp, top and bottom edges of a structure situated in fluid flow. In the case of the Tacoma Narrows bridge it was its ‘I’ shape which gave rise to the vortices shown in fig. 2. These vortices are also known as Karman vortices and which, in the case of the Tacoma Narrows bridge, gave rise to Torsional flutter with a frequency which matched that of the bridges natural frquency, and subsequently led to the complete catastrophic failure of the bridge after amplitudes of +/- 45 degrees of the decking had been reached.
Fig. 2
Natural Frequency: there are a number of methods which can be used to work out the natural frequency. One method is by calculation, but this method is very difficult to apply over large and complex structures. But in our test model, it is relatively straight forward and will be explored in the calculations section of the report. One method is to subject it manually to varying forces. But this can take a long time and involve costly test facilities. And one of the most common methods is the transient test, because it is very quick and simple to perform.
Experimental work
Procedure:
Shown in fig. 3 is the test rig used to examine the amplitude of vibration and transmission of vibration to the foundation plate. A DC motor was used to allow the best possible variable speed control. This was mounted on a base plate that was then supported through a series of columns and compression springs to the foundation plate. Since the motor was very well balanced, the vibration was induced by the addition of various out of balance weights on the motors shaft. Fig.3
As the shaft of the motor rotated the out-of-balance weights gave rise to a vibratory force that caused the test rig to oscillate. The vertical oscillations were measure by a transducer which was connected to the test rig. The transducer detected the vertical displacement by producing a voltage which was displayed on an oscilloscope.
Measurements:
Firstly, the voltage displacement transducer was calibrated using a slip gauge between the stylus of the displacement transducer and the base plate. The voltage change was observed on the oscilloscope on removal of the slip ring. This allowed a range of calculations to be undertaken using the output electrical signal properties. For a range of rotational speeds above and below resonance, the amplitudes of vibration were calculated. The rotational speeds were worked out using the sinusoidal oscilloscope trace as the time base settings were known. There measurements and calculated results were entered into the table available in appendix A.
Also performed was a ‘bump’ or transient test. In this method the free or natural vibrations are measured by recording the ‘transient’ after striking the test rig. From the initial regular frequency of the transient, the natural frequency can be easily calculated. A wave form that was similar to the wave picture to the right, Fig 4 was observed on the oscilloscope. Fig.4
Calculations:
First of all we used the formula: ω2n=(1/2π)K/M to get the calculated natural frequency.
ωn=(1/2π)√(22900/9.815)
ωn=7.69 Hz
Here is an example of the calculations we applied to measurements:
In yellow are the measured results and in green are the constants. And in white are the calculated values.
From the bump test we calculated the natural frequency to be about 7.67Hz using the formula:
No of waves/(length of wave (cm)*scale (s/cm))= 4/(5.215*0.1)=7.67Hz
Results:
From the graph we can see that the natural frequency occurs at around an angular velocity of around 49 Rad/sec which is equivalent to around 7.8 Hz we can assume this the be the natural frequency since the centripetal force is much lower than the dynamic reaction, even at higher angular velocities. This value is very close to the calculated value of 7.69 Hz. The transient test gave us a value of 7.67Hz for the natural frequency.
Discussions & Conclusions
From the results of experiments it can be stated that the natural frequency at which resonance occurs is around about 7.7 Hz. It impossible to say whether or not the third result (marked in yellow in the appendices) gives the highest value for the dynamic reaction. I believe that we should have obtained more results around about 7.7 Hz so as to gauge whether or not the calculated values were more accurately. But unfortunately with the time available only a limited number of results could be obtained.
It is often important to know the damping ratio of a system in order to carry out further calculations. The test rig had a damping ratio of about 0.04 , which was calculated from the formula:
Ln(X0/Xn)=(2Pi*n*ξ)/√(1-ξ2)
The centripetal force at 1000rpm is 56.5N, as shown in the calculation below.
F=mrω2
=(5.15*10-3)*104.72
=56.476N
It should be noted that as the centripetal force of the disc increases exponentially the dynamic reaction asymptotically tends to zero. This shows that the further the frequency of vibration of the system is away from the natural frequency, the less the dynamic reaction Hence the more dynamically isolated the system has become, thus the term vibration isolation.. In industry it is usually not cost effective to construct a forced vibratory system that operates at incredibly high frequencies without reasons other than resonance. The general rule of thumb is to design the system to run at twice the natural frequency.
There is more than one way to implement vibration isolation on a system. One way already covered is to change the operating frequency, so it is out of range of the natural frequency. But in self exciting systems very little control over the applied forces are had. So the other option is to change the inertial and elastic properties of the system. In general, stiff light systems have high natural frequencies whereas soft heavy systems have low natural frequencies. To change the natural frequency of the system you can add or subtract weight or you can stiffen or soften the structure.
Appendices
