Laboratory Experiment #3
Cantilever Beam Vibration

April 20, 2009

Nathan Empson
Lab Section 12
Group Partners:
Bruce Yang
Vaisakh Kumar
David Hanson
Steven Meyer
Course Instructor: Dr. Stephen Conlon
Lab TA: Chanrashekhar Tiwari
The key objective of this, the third and final structures laboratory session, is to observe and document the behavior beam placed under vibration via both analytical and experimental methods. The study of vibrations is vital to almost all products of aerospace engineering. Rockets undergo large vibration loads upon lift-off; planes see great vibration loads in flight. Prior to performing the experiment, theoretical computations were completed to find such parameters as mode shapes, node locations, and natural frequencies. In order to obtain accurate data for experimental analysis, several tools were used including: a single axis accelerometer, and LabView, a signal amplifier, a shaker and more. The most relevant data revealed the effects of mode shapes and tip mass on the natural frequency of the beam.

In this laboratory session, experimenters were most interested in observing the reaction of a beam as it is placed through a wide frequency of vibrations. Prior to beginning the experiment, theoretical natural frequencies should first be calculated in order to aid in constructing an accurate experiment setup. The derivation begins with the governing differential equation for transverse displacement of a cantilever beam:
p(x)=d^2/(dx^2 ) (EI (d^2 w)/(dx^2 )) (1)
The loading of such a dynamically stimulated beam is as follows:
p(x)=-?A (?^2 w)/(?t^2 ) (2)
Equations (1) and (2) may then be combined to form what is known as the “wave equation” for a beam.
d^2/(dx^2 ) (EI (d^2 w)/(dx^2 ))+?A (?^2 w)/(?t^2 )=0 (3)
By separating the variables and assuming that the mass per unit length is constant, we then arrive at a trigonometric equation allowing for the plugging in of boundary conditions to approach a characteristic equation for a cantilever beam with or without a tip mass. The boundary conditions, as soon shown, vary slightly for a beam without a tip mass compared to one with a tip mass.
w(x)=C_1 sinßx+C_2 cosßx+C_3 sinhßx+C_4 coshßx (4)
The four above constants of equation (4) are now solved knowing the boundary conditions for a cantilever beam without a tip mass:
w(0) = 0; No deflection at the root
w'(0) = 0; Slope of beam at the root is zero
EIw”(L) = 0; Bending moment at the tip is zero
EIw”'(L) = 0; Shear force at the tip is zero
The result of plugging in the above constants into equation (4) is the characteristic equation that will be evaluated in MatLab (see appendix) to find the roots and thus the natural frequencies of the beam.
cosßL=-1/coshßL (5)
From the roots of equation (5), the values of ßL that allow the equation to be valid, the natural frequencies may then be solved for. In this laboratory experiment, we were only interested in the first four natural frequencies. This is again reflected in the MatLab code of the appendix.
?_n=?(ß_n L)?^2 v(EI/(?AL^2 )) (6)
The characteristic equation for the tip mass case is found in the exact same way as equation (5) was derived, by plugging in the respective boundary conditions and evaluation the constants of equation (4). The boundary conditions for a cantilever beam with a tip mass are as follows:
w(0) = 0; Deflection at the root is zero
w'(0) = 0; Slope of beam at the root is zero
EIw”(L) = 0; Bending moment at the tip is zero
EIw”'(L) = -?2Mtipw(L); Non-zero shear
The characteristic equation for a tip mass case, the counterpart of equation (5), is then found to be:
1+cosßLcoshßL+ßL M_tip/M_beam (sinhßLcosßL-sinßLcoshßL)=0 (7)
The roots are found in a similar way to the tip mass less case and were utilized in solving for theoretical natural frequencies. The results for the natural frequencies may be found in comparison to the experimental values in tables (3) and (4).
The last equation used in analyzing the vibrated beams is that which describes the critical damping ratio. The critical damping ratio was determined experimentally by first exciting the beam and then recording the decay of the amplitude of displacement versus time. The below equation describes the critical damping ratio.
?=1/?_n (ln?(A_o/A_n ))/((t_n-t_o)) (8)
Experimental Procedure
The objective of this laboratory session is to analyze the response of a cantilever beam when it is placed under vibration. In order to amply monitor the beam, several electronic instruments were used. Pictured below, in figure 1 is a schematic representation of the overall experimental setup. The computer begins the system by outputting the desired frequency of oscillation. This signal then goes through and amplifier and is passed through to a shaker. The beam, fixed upon the shaker, will then vibrate according to the vibrations imposed by it. A single axis accelerometer is mounted at the tip of the beam to monitor its response to the vibrations input via the shaker. The data gathered from the accelerometer is amplified and then passed back to ultimately both the oscilloscope and computer so that the vibrations may be observed on both pieces of equipment.

Figure 1. Experiment Electronics Setup
The equipment was turned on and allowed sufficient time to warm up. During this time key specimen dimensions were taken. These dimensions, some given and some measureable, are given below.

Table 1. Important Specimen and Experimental Properties
Specimen Property Value (English) Value (SI) Total Length 18.75 in 0.47625 m Gage Length 17.5 in 0.4445 m Beam Width 0.998 in 0.0253492 m Beam Thickness 0.123 in 0.00312 m Young’s Modulus (E) 10.0 x 106 psi 6.895 x 1010 Pa Moment of Inertia (I) 1.5476 x 10-4 in4 6.4416 x 10-11 m4 Beam Mass 0.00654 slug 95.5 g Accelerometer Mass 0.000308 slug 4.5 g
With the properties of the beam known and the equipment warmed up, a sine wave of very low amplitude and frequency was applied. Knowing that at resonance the ratio of output to input will be a maximum, the lab group proceeded in increasing the frequency until the first, or fundamental, resonance was reached. At this time the laboratory group could easily note the effect that increasing the input amplitude had on the behavior of the beam. The signal amplifier was turned off and a free-decay trace was captured.

Figure 2. LabView GUI Screenshot
A sine wave was once again input with low amplitude, and of frequency higher than that of the first resonance. The input frequency was slowly increased until the second natural frequency was determined. The signal amplifier was again turned off, and a generous amount of salt was distributed through the length of the beam. When the signal amplifier was again turned on, the salt settled to the node points, points of zero displacement. These values were recorded, and can be seen in comparison with the theoretical results in figures (5) and (6). This process of finding the resonance, pouring salt, measuring node points, and recording a free-decay trace was performed for two more frequencies, making four in total.

Results and Discussion

Figure 3: Functions of ßL vs. ßL Plotted to Find Characteristic Equation Roots
With the characteristic equation derived for a cantilever beam both with and without a tip mass, found in equations (5) and (7) respectively, the roots were necessary to be found either graphically or numerically. Each of the equations were manipulated in such a way as to equate to cos(ßL). This allowed for them to be overlain on the same plot, as in figure 3, constructed via MatLab. The blue and green lines signify the solution for a cantilever beam. The blue and red lines correspond to the solution that includes a tip mass. It should be noted that each intersection of the red and blue lines occurs at a smaller ßL value that the intersection of the blue and green. This is proof that the theory states the natural frequencies of a cantilever beam with a tip mass are lower than an identical beam without a tip mass.
Table 2: Theoretical Roots of Characteristic Equations

As seen in the table above, the difference between the theoretical roots for the two cases is quite marginal. This is due to the fact that the mass of the accelerometer is small in comparison to the beam itself. With the roots solved for, and all other parameters known, these values may be plugged in for ßL in equation (6). Evaluating this equation for the natural frequencies yields the following, in comparison with the experimental values.

Table 3: Percent Error of Experimental Resonance Frequencies Without A Tip Mass

Table 4: Percent Error of Experimental Resonance Frequencies With A Tip Mass

Tables 3 and 4 show that the percent error between the theoretical and experimental frequencies varied from single digits to nearly thirty. Neither of the theoretical values had a significantly better percent error on average, which leads to the conclusion that considering such a small tip mass in this system is not very beneficial. The expected result was that the tip mass theoretical data would be extremely close to the measured values while the model without a tip mass would be of slightly higher frequencies. Unfortunately this leads to the conclusion that the data gathered in the lab was not of a high degree of accuracy. This could be due to a number of sources, which are discussed in the conclusion.

Figure 4: Theoretical and Experimental Resonance Frequencies
The above figure graphically depicts the variance of natural frequency with the mode number. Note from equation (6) that the natural frequencies are dominated by a (ßL) term. Figure 4 accurately depicts this quadratic function. Due to the value of the frequency growing exponentially it is intuitive that the error also grows exponentially, while the percent error remains on the same order of magnitude.

Figure 5: Theoretical Mode Shapes & Node Locations

Figure 6: Theoretical vs. Experimental Node Locations
Figures 5 and 6 of the previous page each compare the theoretical node locations (points of zero displacement of the beam) with the actual or measured node locations. To reiterate, the node locations were found via distributing salt along the length of the beam, vibrating the beam, and then measuring the location where the salt collected. The very method is intrusive, and no doubt is a source of error. By placing the salt on the beam we are introducing an extra mass. Furthermore, there was difficulty in measuring the node location because the salt was still dispersed throughout a fairly large length, sometimes close to an inch. A best estimation was used in producing the experimental results. Figure 7, below, illustrates this source of error and aids in understanding the experimental node determination process.

Figure 7: Mode 2 Node during Experiment

Figure 8: Average Critical Damping Ratio vs. Mode
Figure 8 has relatively few data points, rendering it hard to come to any solid conclusions. It may be said that each of the mode shapes has a similar critical damping ratio. The critical damping ratio is a measurement of how the disturbances of the beam fade away after the initial vibration input of the shaker.

Figure 9: Mode 1 Figure 10: Mode 2

Figure 11: Mode 3 Figure 12: Mode 4

In future experiments it would be advantageous to analyze the response of cantilever beams with a wider value of tip masses. The tip mass used in the experiment was quite small relative to the mass of the beam itself, and showed little effect on the natural frequency and mode shapes.
Theoretically, a cantilever beam with a tip mass should always experience resonance at lower frequencies that that of a cantilever beam without a tip mass.
A non-intrusive way of experimentally measuring the node locations would eliminate a source of error. The method utilized in this laboratory session of placing a large amount of salt onto the beam alters the beam itself. This undoubtedly had an impact on the results of the experiment, and was a source of error.
At one point during the experiment the single axis accelerometer was detached from the beam itself. The detaching and reattaching of the accelerometer was probably another source of error. It may not have been reattached in the same location, yielding a slight variation in the way the data was collected. Furthermore, the accelerometer (the tip mass) was not perfectly on the end of the cantilever beam as assumed so by the theoretical calculations made in deriving equation (7).
MatLab Code
% AERSP 305- Lab 3: Vibrations
% Nathan Empson
% The goal of this program is to derive natural frequencies and mode shapes for a cantilever beam.
% The governing characteristic equation, obtained via the B.C’s is
% 1+cos(BetaL)*cosh(BetaL)=0.This equation is later manipulated to yield
% natural frequencies (BetaL values for which the equation is satisfied).


g(i)= -1 / (cosh(x(i)));
h(i+1) = ((-x(i)*mtip/mbeam) * (sinh(x(i)) * cos(x(i)) – sin(x(i)) * cosh(x(i)))-1)/cosh(x(i));

%By limiting the loop to p<5, it will run until 4 natural frequencies are
%found. The loop finds locations where f(i) and g(i) overlap eachother.
%This iteration is for the first 4 natural frequencies of a cantilever beam
%void of any tip mass.
BetaL(p)= x(i);
Beta(p) = x(i)/L;

while (p<5)
x(i+1) = x(i) + 0.0025;
f(i+1) = cos(x(i+1));
g(i+1)= -1 / (cosh(x(i+1)));
h(i+1) = ((-x(i+1)*mtip/mbeam) * (sinh(x(i+1)) * cos(x(i+1)) – sin(x(i+1)) * cosh(x(i+1)))-1)/cosh(x(i+1));

if (abs((f(i+1) – g(i+1)))<0.001)
BetaL(p)= x(i+1);
Beta(p) = x(i+1)/L;



figure(‘Name’,’Natural Frequencies Plot’,’NumberTitle’,’off’)
plot (x, f,x, g, x, h)
xlabel (‘Beta * L’)
ylabel (‘Amplitude’)
legend (‘cos(BL) ‘,’-1/cosh(BL)’,'(-BL*Mtip/Mbeam) *[sinh(BL)cos(BL) – sin(BL)cosh(BL)]-1 / cosh(BL)’)
title (‘Graphical Determination of Cantilever Beam Natural Frequencies’)
W1(i) = (cos(Beta(1)*x(i)) – cosh (Beta(1)*x(i))) + (-cos(BetaL(1)) – cosh(BetaL(1))) / (sin(BetaL(1)) – sinh (BetaL(1))) * (sin(Beta(1)*x(i)) – sinh(Beta(1)*x(i))) * (-cos(BetaL(2)) – cosh(BetaL(1))) / (sin(BetaL(1)) + sinh(BetaL(1)));
W2(i) = (cos(Beta(2)*x(i)) – cosh (Beta(2)*x(i))) + (-cos(BetaL(2)) – cosh(BetaL(2))) / (sin(BetaL(2)) – sinh (BetaL(2))) * (sin(Beta(2)*x(i)) – sinh(Beta(2)*x(i))) * (-cos(BetaL(2)) – cosh(BetaL(2))) / (sin(BetaL(2)) + sinh(BetaL(2)));
W3(i) = (cos(Beta(3)*x(i)) – cosh (Beta(3)*x(i))) + (-cos(BetaL(3)) – cosh(BetaL(3))) / (sin(BetaL(3)) – sinh (BetaL(3))) * (sin(Beta(3)*x(i)) – sinh(Beta(3)*x(i))) * (-cos(BetaL(2)) – cosh(BetaL(3))) / (sin(BetaL(3)) + sinh(BetaL(3)));
W4(i) = (cos(Beta(4)*x(i)) – cosh (Beta(4)*x(i))) + (-cos(BetaL(4)) – cosh(BetaL(4))) / (sin(BetaL(4)) – sinh (BetaL(4))) * (sin(Beta(4)*x(i)) – sinh(Beta(4)*x(i))) * (-cos(BetaL(2)) – cosh(BetaL(4))) / (sin(BetaL(4)) + sinh(BetaL(4)));
%This loop takes each of the first four natural frequencies and determines
%the displacement caused via each one of these modes. The points of
%interest will be when these plots cross zero on the y-axis, signifying a
W1node(q) =1;
W2node(r) = 1;
W3node(s) = 1;
W4node(t) = 1;
while (x(i)<L)
x(i+1) = x(i) + 0.001;
W1(i+1) = (cos(Beta(1)*x(i)) – cosh (Beta(1)*x(i))) + (-cos(BetaL(1)) – cosh(BetaL(1))) / (sin(BetaL(1)) – sinh (BetaL(1))) * (sin(Beta(1)*x(i)) – sinh(Beta(1)*x(i))) * (-cos(BetaL(1)) – cosh(BetaL(1))) / (sin(BetaL(1)) + sinh(BetaL(1)));
W2(i+1) = (cos(Beta(2)*x(i)) – cosh (Beta(2)*x(i))) + (-cos(BetaL(2)) – cosh(BetaL(2))) / (sin(BetaL(2)) – sinh (BetaL(2))) * (sin(Beta(2)*x(i)) – sinh(Beta(2)*x(i))) * (-cos(BetaL(2)) – cosh(BetaL(2))) / (sin(BetaL(2)) + sinh(BetaL(2)));
W3(i+1) = (cos(Beta(3)*x(i)) – cosh (Beta(3)*x(i))) + (-cos(BetaL(3)) – cosh(BetaL(3))) / (sin(BetaL(3)) – sinh (BetaL(3))) * (sin(Beta(3)*x(i)) – sinh(Beta(3)*x(i))) * (-cos(BetaL(3)) – cosh(BetaL(3))) / (sin(BetaL(3)) + sinh(BetaL(3)));
W4(i+1) = (cos(Beta(4)*x(i)) – cosh (Beta(4)*x(i))) + (-cos(BetaL(4)) – cosh(BetaL(4))) / (sin(BetaL(4)) – sinh (BetaL(4))) * (sin(Beta(4)*x(i)) – sinh(Beta(4)*x(i))) * (-cos(BetaL(4)) – cosh(BetaL(4))) / (sin(BetaL(4)) + sinh(BetaL(4)));

%This section of the program takes values of X, displacement along the
%major axis of the beam, at which the displacements are very near zero.
%Post-program analysis may be necessary if the program falsely
%identifies two very close points both as nodes.
if abs (W1(i+1)) <0.001 && x(i) > 1
W1node(q) = x(i);
if abs (W2(i+1)) <0.001 && x(i) > 1
W2node(r) = x(i);
if abs (W3(i+1)) <0.001 && x(i) > 1
W3node(s) = x(i);
if abs (W4(i+1)) <0.001 && x(i) > 1
W4node(t) = x(i);


figure(‘Name’,’Mode Shapes’,’NumberTitle’,’off’)
plot(x, W1, x, W2, x, W3, x, W4, x, z, 14, z , ‘o’ , 8.25, z , ‘o’ ,6.5, z , ‘o’ ,14.25, z , ‘o’ ,11.5, z , ‘o’ ,16.5, z , ‘o’ )
%Plots the mode shapes, and the nodes, denoted by circles, both theory and experimental.
xlabel (‘X (inches)’)
ylabel (‘Displacement ‘)
legend (‘Mode 1′,’Mode 2′,’Mode 3′,’Mode 4’)
title (‘First Four Modes of Cantilever Beam Resonance’)