STAFFORDSHIRE UNIVERSITY
Assignment 2: Vibration Analysis of Tower Rig
Submitted by: Mohamed Humaid Al-Badri (09032170) Email:
[email protected] Award Title: Mechanical Engineering Module Title: Applied Structural Integrity Module Code: CE00449-7 Submitted to: Prof. Peter Ogrodnik
1 Submission Date: 21/12/2011
Abstract
The vibration is a natural phenomenon that occurs as results of affected force. This report is based on the lab experiment for tower rig which consist of four floors and apply some vibration load in each floor to determine the natural frequencies and modes of the tower. Also, the tower is modeled in Ansys as 3D and 2D model, and calculated the frequencies by classic theoretical. Then, we got the results for each method and made the compression between these results for each mode.
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Table of Contents:
Table of Contents: ......................................................................................... 3 List of Figures: ............................................................................................... 4 List of Tables:................................................................................................. 4 1.0 Introduction: ........................................................................................... 5 2.0 Experiment of Tower Rig: ........................................................................ 5 2.1 Aim of the Experiment: ....................................................................... 5 2.2 Apparatus and Procedure: .................................................................. 6 2.3 Readings from Experiment: ................................................................. 8 3.0 Theoretical of the experiment: ............................................................... 9 3.0 Ansys Analysis: ...................................................................................... 11 3.1 3D Ansys Analysis: ............................................................................. 11 3.2 (Line Body) 2D Ansys Analysis: .......................................................... 13 4.0 Results and Conclusions: ....................................................................... 16 5.0 References: ............................................................................................ 19 6.0 Appendix A: ........................................................................................... 20
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List of Figures: Figure 1 Tower rig model .............................................................................. 7 Figure 2 Bodies masses for tower rig [2] ..................................................... 10 Figure 3 3D Ansys model boundary conditions ........................................... 12 Figure 4 Different modes of 3D model........................................................ 13 Figure 5 Line Body model boundary conditions.......................................... 14 Figure 6 Mesh refinements for line body .................................................... 15 Figure 7 Different modes of line body ........................................................ 16 Figure 8 Compression results for tower rig ................................................. 17
List of Tables: Table 1 Experiment readings ......................................................................... 8 Table 2 Frequencies for Theoretical calculations ........................................ 11
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1.0 Introduction: The vibrations of the floors have become a significant design consideration for engineers in order to avoid any failure in their structural design. The study of vibrations for a cretin construction needs a lot of information to be considered in the analysis. At the same time, the analysis should be more accuracy or otherwise the construction will be failed. The use of accurate predictive model is sensitive laboratory and manufacturing equipment in these structures [1]. For prediction of the vibrations the FEA is an impractical technique as it is too computationally intensive for most full scale structures [1]. In this experiment we will analysis and simulate the harmonic response of a structure known a tower rig under the impression of load. The harmonic response can be defined as the steady state response of the system to the application of load. We will obtain the graphical representation of the response of the system, which is basically amplitude for deformations and accelerations for various frequencies. The peak of frequencies gives us an indication of the sustainability of our design against fatigue [2].
2.0 Experiment of Tower Rig: 2.1 Aim of the Experiment: The aim of the experiment is to study, understand, analysis and measure the vibration models for a four story tower rig by using the spectral analyzer. The vibration analysis needs to understand mode 5
shapes for lightweight construction and reduce oscillation in flexible structures [1].
2.2 Apparatus and Procedure: The main components of the experiment are tower rig, accelerometer, oscilloscope, transducer and electrical motor. The tower rig consists of four floors, each side welded to two structural steels as shows in Figure 1. Each floor represented here by steel plate, we will call them plate 1, 2, 3, and 4. Also, there is two accelerometer attached on the base of the rig (plate 1) and the other is moved from one plate to another to record the frequency response. The oscilloscope is an electric device that allows signal to be viewed and used to measure the amplitude of the signal wave shape and frequency of the system. The electric motor is used to apply a load on the floor one (plate 1) and allow the tower to vibrate in order to take the readings for some intervals of frequency. There are some assumptions for the experiment:
The system consider is equilibrium,
The friction and the damping of the system are negligible,
The stiffness of the spring to be same as structural steel, and
The system moves in one direction only (1 axis).
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Figure 1 Tower rig model
The experiment starts by switch on the electric motor which produces a reciprocating motion force to the plate 1, and fixing the accelerometer in the same plate in order to decide the suitable frequency for the first mode which is no vibration on the plate 1. Then, repeat the same method but this time moving the accelerometer from one plate to another to decide the suitable frequency for each floor (plate), and take the readings for frequencies and amplitudes from oscilloscope.
The tower rig dimensions:
Height = 700 mm
Width = 150 mm
Length = 200 mm
Dimension of the plate = (200 × 150 × 25) mm
Dimension of the 4 steel rods = (700 × 12 × 3.2) mm
Center to center between plates = 225 mm
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2.3 Readings from Experiment: The readings are decided to be taken at frequencies/ modes 5, 8, 12 and 25 Hz because at these values the vibration of one plate is almost zero while the other plate vibrate. The readings taken from the experiment at for all plates are as shown in Table 1.
Table 1 Experiment readings
Frequency
2.030 3.010 4.000 5.010 5.520 6.000 7.000 7.590 8.030 9.000 10.020 11.040 11.470 12.010 13.020 14.000 15.010 16.020 17.060 18.080 19.040 20.010 21.000 22.300 23.000 24.200 25.000 25.500
Plate 1
Plate 2
Plate 3
Plate 4
CH1
CH 2
CH1
CH 2
CH1
CH 2
CH1
CH 2
(V)
(V)
(V)
(V)
(V)
(V)
(V)
(V)
5.200 5.200 1.320 1.280 1.000 1.000 0.760 0.780 1.000 3.520 2.080 1.360 1.360 1.180 0.900 1.080 0.940 1.020 2.560 2.320 2.120 2.080 1.640 1.520 1.480 1.400 1.480 1.360
0.240 0.432 0.148 0.084 0.148 0.184 0.256 0.384 0.640 1.600 1.600 1.000 0.860 0.580 0.460 0.240 0.200 1.840 2.040 1.240 1.000 0.760 0.340 1.320 1.120 0.980 0.960 0.940
5.200 5.040 1.360 1.360 1.080 0.960 0.760 0.640 1.000 3.440 2.440 1.600 1.460 1.220 0.900 1.020 0.900 0.940 2.240 2.360 2.160 2.200 1.720 1.640 1.520 1.360 1.320 1.280
0.232 0.416 0.188 0.160 0.180 0.136 0.104 0.960 0.300 0.760 0.920 0.800 0.700 0.700 0.780 0.940 0.960 2.440 1.400 0.480 0.080 0.480 1.640 1.440 0.840 0.540 0.448 0.400
5.200 4.800 1.240 1.440 0.960 1.040 0.700 0.640 1.040 3.520 2.360 1.680 1.440 1.360 1.160 1.060 0.940 0.960 2.400 2.280 2.120 2.040 1.520 1.640 1.480 1.400 1.360 1.320
0.204 0.560 0.256 0.196 0.248 0.296 0.344 0.312 0.440 0.640 0.440 0.200 0.100 0.060 0.180 0.320 0.500 1.760 1.640 1.000 1.000 1.200 2.400 1.160 0.600 0.288 0.216 0.176
5.200 4.320 1.320 1.400 1.000 1.000 0.860 0.700 1.080 3.600 2.200 1.580 1.420 1.440 1.000 1.080 1.040 0.860 2.400 2.200 2.080 1.880 1.520 1.560 1.560 1.400 1.400 1.400
0.220 0.400 0.264 0.276 0.316 0.312 0.392 0.464 0.680 1.280 1.320 0.920 0.740 0.640 0.700 0.740 0.780 2.120 1.520 0.760 0.640 0.640 1.080 0.520 0.280 0.120 0.080 0.064
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3.0 Theoretical of the experiment: The vibration models equations of this experiment are derived based on Newton’s second law where:
(1) And (2) Where a is acceleration and kx is based on Hooke’s law, based on Hartog’s work [2]. The frequency f can be expressed as:
(3) Where T is the time and defined as:
(4) So, the frequency f can be defined as:
(5) The velocity and acceleration of motion according to circular frequency are defined as:
(6) And acceleration
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(7) The tower rig model can be divided into 4 bodies mass as shown in Figure 2, affected by force F, and the total value of this force can be defined as:
Figure 2 Bodies masses for tower rig [2]
(8)
(9) By repeating for the other 3 mass bodies, so we will have 4 equations which they can be solved in matrix form to get the unknown values. The mass of the floor (plate) can be determined by:
(10)
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Since the density of the steel is 7850 kg/m3 and using the dimensions of the plate, so the mass of each plate can be calculated. Also, the moment of inertia and stiffness coefficient can be defined as:
(11)
(12) Where E is the Young’s Modulus and equal 2 × 1011 Pa. By using the matrix equations, the natural frequencies can be calculated for the tower rig, and it is found as shown in Table 2.
Table 2 Frequencies for Theoretical calculations
Sr. No
W2
F (HZ)
1
503.827
3.57
2
4247.133
10.37
3
9982.175
15.90
4
15025.482
19.50
3.0 Ansys Analysis: 3.1 3D Ansys Analysis: We designed the tower rig as 3D model in Ansys using the same dimensions (700 × 200 × 150) mm and then we applied the boundary conditions. The force is applied in the bottom plate in the longitude direction as shown in Figure 3. Also, the displacement is applied in 11
the bottom surface of the same plate. The harmonic responses for both deformations and accelerations are taken in order to compare with the experimental and theoretical results. Some of these harmonic responses are shown in Appendix A.
(b) Displacement
(a) Applied Force
Figure 3 3D Ansys model boundary conditions
The calculations done by Ansys are at 4 different modes of frequency. The modes for 3D model are 5, 8, 12 and 25 HZ and they are shown in Figure 4.
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(a) 1st Mode
(b) 2nd Mode
(c) 3rd Mode
(d) 4th Mode
Figure 4 Different modes of 3D model
3.2 (Line Body) 2D Ansys Analysis: We designed the tower rig as 2D model in Ansys and then we applied the boundary conditions. The force is applied in the bottom plate in the longitude direction as shown in Figure 5. Also, the displacement is applied in the bottom surface of the same plate.
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Figure 5 Line Body model boundary conditions
Since the mesh refinement plays a critical role in the calculations and accuracy of the model, we refine the default mesh for 2D line body model at sizes 15mm, 10mm and 5mm as shown in Figure 6.
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(a) Default mesh
(b) 15mm mesh
(c) 10mm mesh
(d) 5mm mesh
Figure 6 Mesh refinements for line body
Then, after some trails of calculations, we found that the suitable size for the mesh for 2D model is 10mm, which can gives optimum results. The calculations done by Ansys are at 4 different modes of frequency. The modes are 5, 8, 12 and 25 HZ and they are shown in Figure 7.
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(a) 1st Mode
(b) 2nd Mode
(c) 3rd Mode
(d) 4th Mode
Figure 7 Different modes of line body
4.0 Results and Conclusions: From the above analysis, we plotted the experiment, 3D Ansys and 2D line body for the four plates as shown in Figure 8.
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(a) First Plate
(b) Second Plate
(d) Fourth Plate
(c) Third Plate
Figure 8 Compression results for tower rig
According to the compression results, the experimental values for the first and second plates matches the results obtained from Ansys for 3D and line body. Also, the 3D and line body results are very close to each other. For the third plate, the experimental values still close to the line body results, but the values of 3D are little bit far from
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experimental results and 2D Ansys analysis. For the fourth plate, also the results of experiment and 2D are close but 3D is not matching. It is observed that, there are some difference in the results between the experimental and Ansys 3D and 2D, and this is due to the human error and accuracy of the modeling in Ansys. Also, the mesh refinement affects the accuracy of the results. In addition, the experiment values obtained are based on circumstances of set up, level of noise and understanding of steps.
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5.0 References: [1] S. G. Kelly, Fundamental of Mechanical Vibrations, McGraw – Hill, New York (2000). [2] W. T. Thomson, Theory of Vibration with Application (4th ed.), Nelson Thornes Ltd. Cheltenham (2003). [3] C. T. F. Ross, Advanced applied stress analysis, Ellis Horwood Limited, New York (1987).
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6.0 Appendix A: Harmonic Responses for 3D Ansys Model:
Figure A1.1: 3D Model Ground Floor
Figure A1.2: 3D Model First Floor
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Figure A1.3: 3D Model Second Floor
Figure A1.4: 3D Model Third Floor
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