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4.3 Mathematical Analysis
For a cantilever beam subjected to free vibration, and the system is considered as continuous system in which the beam mass is considered as distributed along with the stiffness of the shaft, the equation of motion can be written as (Meirovitch, 1967),
(4.1)
Where, E is the modulus of rigidity of beam material, I is the moment of inertia of the beam cross-section, Y(x) is displacement in y direction at distance x from fixed end, É is the circular natural frequency, m is the mass per unit length, m = ÃÂÂÂA(x) , àis the material density, x is the distance measured from the fixed end.
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)
Fig. 4.1 (a): A cantilever beam
![](data:image/png;base64,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)
Fig. 4.1 (b): The beam under free vibration
Fig. 4.1(a) shows of a cantilever beam with rectangular cross section, which can be subjected to bending vibration by giving a small initial displacement at the free end; and Fig. 4.1(b) depicts of cantilever beam under the free vibration.
We have following boundary conditions for a cantilever beam (Fig. 4.1)
(4.2)
(4.3)
For a uniform beam under free vibration from equation (4.1), we get
(4.4)
with ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEgAAAAtCAIAAACCtrr5AAABK0lEQVRoge2X3RKEIAhGef+Xdq9yyAVSQAXHc7FTM9R+J/wpKIcCuwPM4oplI6gYAACYskUUq0oWt4hilStGXeuYw5cD51h5rBJ3DB4KWgkBob+zW0bFf6Pony1qisnjV71z2G5wms+UuIz7bYtd0w7Aicn1XNOIeksgNaSVfHNhKNL1o4E2ig1dsk2MXAmFeuFYMxSb5Zi7i4JOpVrMHWs6Rm4sXk2bTVf38WkKq9Ip1jm/Q8EM0PcIXBvJB0mMO+UuIfGJOY6bWDSoHeBPw1GMa6ydWWIJhiIOtDecBbo525+3nYnTaS/pVz+OWGLy8jM0QcK1CMQttP/RX7ElgPi2PTRTrtgSPt6Skoo5tqtcsRXIMlnFfK1KEDH5k0f3ERRCbAZXLBvHiv0AWT48sNTEnp4AAAAASUVORK5CYII=)
The mode shapes for a continuous cantilever beam is given as
(4.5)
Where
![](data:image/png;base64,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)
A closed form of the circular natural frequency Énf, from above equation of motion and boundary conditions can be written as,
(4.6)
Where
![](data:image/png;base64,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)
So,
First natural frequency
(4.7)
Second natural frequency
(4.8)
Third natural frequency
(4.9)
The natural frequency is related with the circular natural frequency as
(4.10)
where I, the moment of inertia of the beam cross-section, for a circular cross-section it is given as
(4.11)
Where, d is the diameter of cross section and for a rectangular cross section
(4.12)
Where b and d are the breadth and width of the beam cross-section as shown in the Fig. 4.2.
![](data:image/png;base64,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)
Fig. 4.2: Cross-section of the cantilever beam
![](data:image/png;base64,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)
Fig. 4.3: The first three undamped natural frequencies and mode shape of cantilever beam
Table 4.1 Material properties of various beams
Material
|
Density (kg/m3)
|
Young’s Modulus (N/m2)
|
Steel
|
7850
|
2.1Ãâ€â€1011
|
Copper
|
8933
|
1.2Ãâ€â€1011
|
Aluminum
|
2700
|
0.69Ãâ€â€1011
|
Table 4.2 Different geometries of the beam
Length,L,(m)
|
Breadth,b,(m
|
Depth,h,(m)
|
0.45
|
0.02
|
0.003
|
0.65
|
0.04
|
0.003
|
Example 4.1: Obtain the undamped natural frequency of a steel beam with l = 0.45 m, d = 0.003 m, and b = 0.02 m.
First natural frequency
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAZCAIAAABB17xmAAABCElEQVRYhe2XSw7DMAhEuf+l20XVqIqHGcC2GiXMKo0xn1dArb1aVPbvBK6uBiTUgIQakJAAZGZm6yGOPu0rmczJzAZ5Vz5HxBgHlWWsBeSVAZ89G5IbvB40w3HZ2YbegZ75x4gH/n4LoH10pHMZOkUHvp8ClBrRmmYAkdMaIL6VlnXQuCwJ4kKRpyipuzyx4xmPZyHLeRXAScsU9Pj1CwFKBV0LiN0Nevw98sqbGbHCRKdy5m9ygGSu8/3F8yskkN3rcAdhS+nuaAS+zOKCaxL2nWcjc/Y8w1i85aM/q7y8b69zzeT7eTog3r18UG+s6L+ecYa3p3YNPaXOshqQUAMSakBCDUjoDRmzu6R6sAV7AAAAAElFTkSuQmCC)
Second natural frequency
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGIAAAAZCAIAAABFImxbAAAA8UlEQVRYhe2Xyw7DMAgE+f+fbm8+hGVZB+fNHKIKORiPwGrs1wjY1QU8g9Yk0ZokWpNEosnMzNarNAeP8ww86NPCSFIwP8x4LsQnjDaKtoZxr0bcVyHXdAJTOkicrzlE050dKSOpBEuapoa2ztqWmUoY3Wtg5VQFKf4OTnVXNPllZEde2PiN362UWKfoaCpD5bZqTdKme6okczQ7dA/WJF4l9Y470xEMwrsJr0zTjXbQ3Yvsa4SoSUnz+taOuj7MIFYJy4KHfCXbCSLtytvy3Wy/OcGKwNGnTKn/dOFUH17dbfjQUSu0JonWJNGaJFqTxB+wOkUpGNGcPQAAAABJRU5ErkJggg==)
Third natural frequency
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGoAAAAZCAIAAABW9SyvAAABLElEQVRYhe1XQQ7DMAjj/5/OLl0VJdg4JV01FZ8qcIjnAuqsFRKwpwX8N8q+FMq+FMq+FAL7zMzsXouH+tZB4btnV+Nndo4E4lnuWzGschmDytlKzkdk5ZnfEug+mSz3k77j3aTzkfXKKwkjUCFM3OxdW7Qj5IfHeVnlrFPNvUAf/stA3YGCnK9UCCu7Eb4Nd3afAfDiZKOLfF08EkM0B6956fotCJfRkNL5nHDHOD9jn9KhrbNP5Ld179zUwgu+cBkZyaWfqtyV5/MjbjZln9h6uzo0P2tuHO2s1VFFz0dE0THvoI3TPZRdGk90tq9ApmGOowFCqtSPhlkQ9eQtGDvc9WXuPsR8G8YOdxjgW7/sa/w/78HolkKjRr8Q5UIKZV8KZV8KZV8KZV8KH7aYOFJ/6B6LAAAAAElFTkSuQmCC)
The above frequencies have to be modified since there is a mass in the form of an accelerometer at the free end of the continuous beam. By continuous approach the solution is difficult since with tip mass the boundary condition at free end is now time dependent. Now we would explain a simpler procedure by which corrections to the natural frequency could be made so as to get closer to the measured natural frequency. In case of non-contacting sensors, there would not be any correction required since there will not be any additional tip mass on the beam.
4.3.1 Consideration of the mass of accelerometer onto a continuous cantilever beam
1. First natural frequency - Let us consider the beam specimen as mass-less with stiffness k and has a discrete effective mass, meff , at the free end, which produces the same frequency as a continuous beam specimen without any tip mass. Hence, the natural frequency of discrete model of the beam without an accelerometer can be written as,
(4.13)
with
(4.14)
From which the effective mass at tip can be written as
(4.15)
with
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOoAAAA1CAIAAADAnuDjAAADYUlEQVR4nO2ZSbKcMBAF+/6Xxosf0aZFzVUaeblwtEES4lWChP25ANiWz+wJABAH+oKNgb4sH7AkPzWaJcf6NEmBBUGFWKDv+qBCLNA3w4CdwwV9BaBvBug7GeibYUx6qBAN+awDO9B3JtA3CfSdiVdfeaPGbd0ORrjZwqxeFKiLgG1Ne/mvZ6OmV5XVizJ1AX0zQN/JBDYP5G/uyNnI6RVm9a5Y7WRevdDX/uqFvl0I6Et+iMRG2x1V36qs3hWrHewcMozZOVxv09fxLzLQ9xfXxwD0recbjSWjsL7PIwfo64pOblab1fbJBqjV93h37yT1Lc9qoXDt+33yoHDKdaFvX+OcSdQ51FIbHddFvZDcrEdWS+irzpsMUfDSPpQwH7XZCpRHR56Vj5NTsrTMs1CRvDUQmskN1HA30vePwui4NsbojG2qkF7y1+AnKVcDdRmyL09v07dx95mSa2Wfqe99ivYHrmYqtv+qcS1tscmfpO8VjU4dNjaZWthHs3mZBUdn4BpbRhPOxoYt6dIDe3qdorOMXNIljEnfQVNJvEL2LUAJnaJTz1Z1CUNv2LnffaeS2MB10pd7/3XCO8/A7buiU8/KXQbkE9f3PiI7uqdOnWqQKcBcHFVcUt8BtF+X5A+i2+OrLj/pcA1q3Q33mnihHtGpZ2t7PQcR9Pvf7NnnosRtTpENaqdLToC8JXtL+2RiHb1X6VTpcHRyY+Hq8pgZlAfMNdBdYtdlVuZZ+730nYj65OTHVxp4xzpMX7IAA+6Fi3EvnndReEeWQlgvJrvLHVyffvrK67J3jV6ZKn3JTGo2D19xycR3r0S5vtwgzcF9E7tzv9nkHXlXdd/FzljynjQFKKkB9I0NdR+zbPNwHzQ4tYVx6auu+5y+R0b3R4m+gb7HBurC/v5QRRfGeae+XGLP9KBvHIu+FiNfq6+6ZeL+bJr5rhua7YGU6Pv5Re57EoK+zXFuoYvlc3KmLuT3B3lcFvRV+l7ijl91l+xoumigz5EM1vc8m7lvWe4VW5LPaSFmUPV1fXbIK+NL9JXdzX8nnBZiBnLP+j3FNXhudpsjH4b+NzQUr75kXN58Tgsxg6ovkBmfEqryH05fuGsE+k4G+u4FCvPDkbvSg0GpfoC7e4FqgY2BvmBjoC/YmH9MvIXIhfqy5QAAAABJRU5ErkJggg==)
or
![](data:image/png;base64,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)
Now, if we consider the mass of accelerometer, macc, at the free end of the beam, than the total mass at free end will be
(4.16)
So for the discrete beam with accelerometer, the theoretical fundamental natural frequency after considering the mass of accelerometer will be
(4.17)
Following the above procedure, other natural frequencies can be improved to account for the mass of the accelerometer. However, we need to obtain the effective stiffness of equivalent discrete system with additional supports at nodes (for the second natural frequency at one location and for the third natural frequency at two locations). The finite element method analysis is presented subsequently for the same.
Example 4.2 Obtain the fundamental natural frequency of beam by considering the mass of the sensor also.
Taking data from Tables 4.1 and 4.2 for steel,
![](data:image/png;base64,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) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE0AAAAWCAIAAAAKIobTAAAAwUlEQVRYhe2T2xLAEAxE/f9P60MvQxPZRYYO3adSs9kjEeIeCrMDDNLGnCFRrzttIo85xoilfvpY3ybQTcK4BMgMS4V7fenc568JnL7NhJ6S03diL08YsdGX44Szrb7bmI86cyPVnKEg20T1fPXQqCirp0sqtp2vWZCzdrDVJXlTcS4nHApo5cZJ5utpFzzgzwnjVol8OfYZO1ILJ9moKkkHhscek2e/9K3X7cH4srbgVAZhSo7x+jnX0s+5lnbhPAA/4iHczP37PgAAAABJRU5ErkJggg==)
, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHAAAAAZCAIAAABmeT2OAAABDElEQVRoge2V2w7EIAhE+f+f7j40TQwCjjK6N+apJcuAR+jKVaJK3t3Ar6mAkvVfQEVERLzXNUMdydh9oxSCDFDzPgrofqDyaLlSazVsaK2WmYJYsYDeieiEsmgGPpmT2KPxRNqH/s6IQM1OdgH1LpBVIuaC1z3xp0Rcds8qXvZg+rwIiKmf1uTH7RBQb7/AnlQQgb5j7nr0UK5nZ8ZNBeloEz6yoBn8dUrmFOO24+Wa7WbBDaSGFyKeYnY++ECHI2xmecHzE3odAwquPOgW/+wTvqFgJzqX2IfhvjSeyUj+FBygU0ua7yxTy8ulNN+ae89RerJ8SamAklVAySqgZBVQsgooWQWUrAJK1gs8hmc/xOEtgwAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
The mass of accelerometer is 4.8 gm = 0.0048 kg, so the total mass will be
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUIAAAAaCAIAAABNZcLlAAACuklEQVR4nO2a3Y7rIAyEUd7/lSv2otoowng8/ISEar6Loy4YY4gH0/SkLITYnPR0AEKIUSRjIbZHMhZieyRjIbanIuOUUkp1ead/bo7qp2B2zLNJF2wX8Mb7IRlfBRPqeJwnx3GklD6fT4cNbv9S9J5dZ7ttCf0cx4ED9kByZRoF5twxRnU2pz378Khl/PBMWUWYURPDPvUAhOHZhO3AVbWF9MOcOx6/ION3RpW5vAQ2eF3AW4ce+NretIpwdXfImJFNh3SB/Hj/wA/uwjSkwi5qeQ+DAgirrtdYaIO5pkrGYWWuXoM9/950wA/pqgqbCnmGWq7J5H3uc7t4xr7AWqtxa6MnHuyntWuKjHmDDm6qxtXG7+fwPszEkGtqt9+0q0dALGMmOap4xuDfPpg6M31GZsmzZFxtBwM75LpSxjhUnDxVcHmcLuNs9Ia/h/O36O67fXyijyd94fzqOTwjgCtGSxNn7GCZjEkndhS/h32rsO2k85Gns1jG2B7fk6vDm16YfQlkPDfpCxXNco6z9ikN21lulTGWYof/pgibZOyFyh9GmPfIOPyua+2/GzJTxuC8L4aQaeQ5BLmFk5KJrWlGEnLJy2Q83rVMxmAgY1zl2Ut1xxvp1jAy/BU6FzIu9tR7GH1gRXkP2Ov1PI/MeAfMHoY2jHKYrj7lMxECGzy2ScY8d/xubA2AfcdRwoTkUb+YZVOWw5IYAkRl/8RVDnhundFOYXvDpYXhhQlatQHLxzvj7ScO8o5V4HbrZFay5dr/pvBKXFVgduwZWKv9dRT+4apqeb4JBwGsqEg8yVTg8cfJz5h95QvxFMxrsxdlanEML5AxmEsaFi9hPxkXnx8pxSvvAkIw7HSpvlbjZTK2XLtunV2IWbwuU1UMhWjldTqxhVEIgZFUhNgeyViI7ZGMhdgeyViI7fkDdzR7XKeR+W0AAAAASUVORK5CYII=)
Hence the corrected natural frequency after consideration of mass of the accelerometer would be
![](data:image/png;base64,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)
which can be written as
![](data:image/png;base64,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)
2. Second natural frequency - Using FEM, we will find the second natural frequency of the cantilever beam (continuous system) having accelerometer mass at free end. The basic procedure is outlined here.
1. In the first step, the geometry is divided into a number of small elements. The elements may be of different shapes and sizes.
2. Then elemental equations are obtained for each element.
3. In the third step the elemental equations are assembled to yield a system of global equations.
4. Then assembled equations are modified by applying the prescribed boundary conditions.
5. The last step is to find the solution from these modified equations.
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)
Fig. 4.4 (a): A cantilever beam with a tip mass Fig. 4.4 (b): Disceretisation of the beam
into 3 elements
For a uniform beam, the elemental stiffness matrix (Tiwari, 2010; Dixit, 2009)
(4.18)
and the consistence mass matrix is given as
(4.19)
1. First step: Divide the given beam geometry into three small elements as shown in Fig. 4.4 (b). In Fig. 4.4(b), numbers 1, 2, 3 and 4 represent the nodes.
2. Second step: Construct the elemental equations. For the first element, from equations (4.18) and (4.19), we have
(4.20)
Here {X} represents the vector of nodal variables (i.e. at the first and second nodes). In this case node variables are displacement and slope, the right hand side vector {F}, represents the force.
Then for the second element
(4.21)
In this only nodal variables will change and for third element, the elemental equations
(4.22)
Here, 0.0048 is the mass of the accelerometer in kg at free end of the cantilever beam.
3. Third step: Assemble all the elemental matrices to form a global matrix. The length of the each element l = 0.45÷3 m and area is A = 0.002Ãâ€â€0.03 m2, mass density of the beam material à= 7850 Kg/m3, and Young’s modulus of the beam E = 2.1 Ã 1011 N/m. After substituting values of the l, ÃÂÂÂ, d, E, A in elemental equations (4.20), (4.21) and (4.22); assembled equations become,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
and for free vibration
![](data:image/png;base64,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)
Hence, above equations can be written as
(4.23)
By solving the eigen value problem of equation (4.23), we get Énf1 = 11.85 Hz and Énf2 = 74.711 Hz. If we take more elements then we will get very accurate results especially of the higher modes. For example, on taking 10 elements then the first and second natural frequencies are Hz and Hz. From experimental response plot (Fig 4.5) of the same beam, the first natural frequency is 11Hz and second natural frequency is 79Hz. Some deviation is expected especially in higher mode due to modeling error of actual test conditions.
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