Figure 3: Equivalent 1-DOF system
The diagram above shows the equivalent systems of the two single degree of freedom systems used in the first part of the experiment. The bottom floor mass includes the mass of the top floor which is clamped to it.
Experiment
Using the rig described previous, a series of tests were performed with the objective of demonstrating the phenomenon of resonance.
First each floor was clamped in turn, in order to create two 1DOF systems. An impulse force was then applied to the system to induce a free vibration. Values of frequency and amplitude (outputted as an amplified voltage) were then recorded using virtual bench computer software. The recorded data was then used to find the natural frequencies of the two 1DOF systems and hence stiffness and damping ratios could be found.
In the second part of the experiment, a forced vibration was induced using the motor attached to the bottom floor. The complete two degree of freedom system was analysed. The motor was set to rotate at a low speed, where the rotational speed is equivalent to the frequency of the excitation force on the system. Once the responses reached a steady state, the software could be used to measure the amplitude of the vibration in Vrms and the excitation frequency. The speed of the motor was slowly increased until readings for all resonant frequencies had been noted. To obtain useable results for this experiment, the amplitude reading was converted to meters, from root mean square Volts. A graph of frequency and amplitude could then be plotted.
Experimental Analysis of Results
Analysis of the top floor 1-DOF system
The graph of amplitude against frequency was used to find the natural frequency. To calculate it, the time period of a number of vibrations was measured and then divided by the number of cycles used. The inverse of this number is the frequency (Hz).
Natural frequency: ƒn1 = 10.53Hz
This is converted into radians per second by multiplying by 2π: Ωn1 = 66.14 rad/s
In order to find the stiffness of the struts, k1, the following equation is required:
k1 = m2. (Ωn1)2
k1 = 4.3.(66.14)2 = 18.81 Kn/m
The damping of the struts, c1 and c2, can be found using the damping ratio which can be found using the logarithmic decrement method as the system is an underdamped case and therefore amplitude decays logarithmically. The logarithmic decrement is the difference between the number of successive amplitudes. There is no need to convert the amplitude to displacement amplitude as the logarithmic decrement is a ratio and hence the magnitude is irrelevant. The logarithmic decrement is:
= 0.2393
= 0.1270
= 0.015084
This value can now be used to find since, 0.015084 =
-
= 0.019547
As: c1 = 1.2m2.ωn1
c1 = 11.118
Analysis of the top and bottom floors 1-DOF system
The same analysis was done with of the 1-DOF system with top and bottom floors clamped together. The results were:
Natural frequency: ƒn2 = 14.03Hz
Ωn2 = 88.14 rad/s
k2 = (m1+m2).(Ωn2)2
k2 = (20.8+4.3).(88.14)2 = 194.997 Kn/m
= 0.0537
= 0.0195
= 0.01842
2 = 0.0216
As: c2 = 2.2(m1+m2).ωn2
C2 = 95.574
Displacement amplitude vs. Frequency plot for 2-DOF system
As the results were obtained using the motor to change the frequency of the excitation force, the data that was given was an output of amplitude in root mean square Voltage which cannot be used analytically. The values therefore had to be changed into amplitude displacement (meters). To change the Vrms output into a displacement which can be plotted, the readings were turned into accelerations using the following equation, which is divided by ωn2 since the acceleration is integrated twice to change it into a displacement:
= ((Vrms*((1000*SQRT(2)*9.81)/(S*10)))/( ωn2))
Where S = the sensitivity of the accelerometer (97.7 for top and 99.8 for bottom)
Below is a table showing the new values for Amplitude:
Below is the plot of the results:
The values obtained for amplitude appear to be slightly inaccurate. One of the reasons for this could be that the Vrms values were provided were inaccurate due to human error. From the graph of displacement amplitude against frequency, the natural frequencies can be found. The 2-DOF system has two resonant frequencies and these are shown on the graph as peaks. The values are:
fn1 = 9.5
fn2 = 16
These frequencies are approximations, since the graphs of the two masses don’t show definite peaks at the same point, i.e. for the first frequency.
Theoretical Analysis of Results
Derivation of Equations of Motion
To derive the equations of motion, Lagrange’s energy method is used, where:
T (Kinetic energy) = T(q1,...qn, q1,...qn)
V (Potential Energy) = V(q1,...qn, q1,...qn)
D (Dissipation function) = D(q1,...qn, q1,...qn)
Qi = (where i=1,...,N)
For a free vibration induced system with 1DOF, the following workings can be applied to derive the equations of motion:
The generalised coordinate, q is x where:
T=2 , V = , D = 2
Substituting these values into Lagrange’s equation Qi, gives the simple equation of motion:
For an undamped case, the equation of motion above can be written as follows:
Sub in the following values:
Ωn= ζ=
When undamped, ζ = 0, therefore the damping term in the equation of motion is disregarded. The roots of CE are:
Ωn Ωnj
The general solution then becomes:
C1 and C2 are both constants which can be calculated by using the initial conditions of the system.
The diagram below is a representation of the motor with an eccentric mass. F is the horizontal force exerted by the mass mr on the machine. The displacement in the horizontal direction is y1 the rotational speed is ω. The radius of the disc is e.
Figure 4 – Diagram of eccentric mass rotating round a centre
Using Newton’s law, the force component can be derived:
Figure 5 – Diagram showing the forces exerted on the two floors.
The forces can then be equated to form the equations of motion as follows:
The result of this analysis can be written in matrix form as follows:
+
The equation of motion above can be used in the eigenvalue analysis of the free, un-damped, 2-DOF system. This analysis can be used to estimate the natural frequencies of the 2-DOF system, using the given mass values and the values of k1 and k2 found in the analysis of the experiment.
Eigenvalue Analysis of free undamped 2-DOF system
Assuming it is possible to have harmonic motion of m1 and m2 at the same frequency and phase angle then:
The amplitudes are substituted into the equations of motion with changing to, giving:
The right hand side of the equation is disregarded as this method is only used in free vibration. To find the characteristic/frequency equation, the determinant of this set of equations should be found:
=
=0
Using these frequency equations, ωn can be found by inputting the experimentally determined values for stiffness, using :
Now solve the quadratic using the obtained values for stiffness:
89.44Y²-4975307.7Y+3667893570 = 0
Y = 54880.06, 747.2576
Ωn = 234.26, 27.336 (rad/s)
When these results are compared to the experimental results of 59.69 and 100.53, these theoretically derived figures are very dissimilar. Either there has been a miscalculation in the theoretical analysis, or the experimental data is very unreliable
Theoretical displacement amplitude vs. Frequency plot
The equations of motion are used to create the displacement amplitude vs. Frequency plot. This can be done by equating the equation to a force f(t):
+
The GS for f(t):
As previously stated:
and
All this gives:
+
=
In order to find the amplitudes X1 and X2, the inverse function must be found:
=
=
=
Therefore X1 and X2 are:
and
If we move F to the left-hand side to get X/F, which is the amplitude for a given force, giving:
and
No damping is assumed in this theory.
Discussion
The 2-DOF model was a satisfactory representation of the building over the frequency range considered. It helped on the understanding of forces exerted on a building containing rotary machinery. The experiment only gave forces considered in the horizontal direction, whereas in a real-life scenario the forces would be in other directions. To give more reliable data it would be better to experiment in many degrees of freedom, but practicality suggests that the 2-DOF is the best choice.
The 1-DOF model was useful, and proved to be a successful method for finding the stiffness and damping ratio of the system. This is because the damping was legible in this system. When applied to the 2DOF equations of motion, the values for stiffness gave realistic values for the resonant frequencies. When compared to the values calculated theoretically there was quite a difference, and therefore there was an error somewhere in the experiment.
There are many possible sources of error. Firstly, the accelerometers that were used are very sensitive instruments and therefore the signals can be distorted easily by very small interferences such as people talking, moving around in the room, or, traffic from outside shaking the building. Secondly, there were two rigs being used in the laboratory and as they were vibrating at high amplitudes, one could have affected the other. In addition, human error could have affected the results, for example misread values could have had an effect on the relationships.
Overall the experimental results do not compare well with the theoretical analysis. The eigenvalue analysis produced results for resonant frequencies that were within very different with the experimental values. This shows that the experiment provided far from an accurate model for determining the natural frequencies.
