![Dynamic
load
Stationary
load
[13] and [14]
Moving
load
Random
[18]
Harmonic
[15] and [16]
Constant
[15]
Distributed
[17]
According to the position of dynamic load, can classify to two type:
moving dynamic load and stationary dynamic load.](https://image.slidesharecdn.com/presentation-170405121310/85/DYNAMIC-RESPONSE-OF-SIMPLE-SUPPORTED-BEAM-VIBRATED-UNDER-MOVING-LOAD-7-320.jpg)
![Mathematical Methods
deal with differential equation produced
Exact
solution
Double Laplace
transformation
[23]
Fourier
series
[17], [19] and[20]
Green
function
[21] and [22]
Approximation
solution
Range-kutta
[24]
Finite element
[25]
Many methods are applied to determine dynamic responses under moving load ;
some yield to exact solution, and the others yield to approximate solution which
has an excellent agreement with the exact solution.](https://image.slidesharecdn.com/presentation-170405121310/85/DYNAMIC-RESPONSE-OF-SIMPLE-SUPPORTED-BEAM-VIBRATED-UNDER-MOVING-LOAD-8-320.jpg)
![Element Standard Used
Copper(Cu) 58.0-63.0 57.98
Tin(Sn) 0.5-1.5 1.02
Zinc(Zn) 33-40 35.8
Iron (Fe) 0.5 0.519
Aluminum(Al) 0.8 0.341
Silicon(Si) 0.05 0.031
Table shows compare on with stander brass alloy type c85700. According to
this table, it can be seen that the chemical composition of the used brass
alloy c85700 is in conformity with that for the standard alloy[29].](https://image.slidesharecdn.com/presentation-170405121310/85/DYNAMIC-RESPONSE-OF-SIMPLE-SUPPORTED-BEAM-VIBRATED-UNDER-MOVING-LOAD-16-320.jpg)
DYNAMIC RESPONSE OF SIMPLE SUPPORTED BEAM VIBRATED UNDER MOVING LOAD
- 1. DYNAMIC RESPONSE OF SIMPLE SUPPORTED BEAM VIBRATED UNDER MOVING LOAD By: Sadiq Emad Sadiq Supervision by: Dr. Ali Raad Hassan Republicof Iraq Ministry of Higher Education and Scientific Research University of Technology MechanicalEngineering Department
- 2. الرحيم الرحمن هللا بسم طْعُي َفْوَسَلَوَكُّبَر َيك ىَضْرَتَف العظيم العلي هللا صدق
- 3. 1- introduction Beam is typically described as a horizontal structural element supports vertical load. It having one dimension (length) which is many times greater than its other dimensions (width and depth). Beams are one of the most fundamental structural and machine components Buildings bridge, robotic and aircraft they are most important applications of beam in the field of mechanical engineering and civil. In many of these applications, beams are subjected to dynamic load, this load excite the vibration of the beam structure, which causes durability concerns.
- 4. three theories Timoshenko theory ratio of length span to the high of beam( L/th ≤ 10) , so called a thick beam, so the effects of rotary inertia and shear deformation should be considered Rayleigh theory takes the effect of rotary inertia Euler-Bernoulli theory neglects the effects of rotary inertia and shear deformation and is applicable to an analysis of thin beams where (L/th≥10) To investigate the behavior of these vibrations There are
- 5. According to the position of dynamic load, can classify to two type: moving dynamic load and stationary dynamic load. Moving load the origin of transverse vibration of structure under moving load can brief upon the breakdown accident of the Stephenson’s bridge across river Dee Chester in England in 1847, when the carriages of local passenger train fell in river during pass the bridge, it pushes the engineers for investigation of moving load problem.
- 6. The importance of this problem is manifested in numerous applications, in bridges, guide ways, rails, roadways, pipelines, overhead cranes, and cableways
- 7. Dynamic load Stationary load [13] and [14] Moving load Random [18] Harmonic [15] and [16] Constant [15] Distributed [17] According to the position of dynamic load, can classify to two type: moving dynamic load and stationary dynamic load.
- 8. Mathematical Methods deal with differential equation produced Exact solution Double Laplace transformation [23] Fourier series [17], [19] and[20] Green function [21] and [22] Approximation solution Range-kutta [24] Finite element [25] Many methods are applied to determine dynamic responses under moving load ; some yield to exact solution, and the others yield to approximate solution which has an excellent agreement with the exact solution.
- 9. 2 - Aims Of Study : 1- Calculation the natural frequencies for a simply supported beam. 2- Determination of the dynamic response of a simply supported beam vibrating under moving load. 3-Evaluating the dynamic bending stresses for simply supported beam traversed under moving load . 4-Displaying the effect of speed and magnitude of moving load on the dynamic response and dynamic bending stresses for mid span. 5-Achieving finite element modeling by ANSYS software to simulate the transverse vibration under moving load for a simply supported beam.
- 10. 3-Theoretical presentation Assume a differential beam element subject to external load and the free body diagram as shown Apply the translational equilibrium equation and rotation equilibrium equation EI 𝜕4 𝑦(𝑥,𝑡) 𝜕𝑥4 + 𝜌𝐴 𝜕2 𝑦(𝑥,𝑡) 𝜕𝑡2 = 𝑓(𝑥,𝑡)
- 11. Eigen value and Eigen function of modal analyses. Fourier series presentation of moving load Dynamic response of beam vibrated under moving load three stages required to solve the partial differential equation and obtain the dynamic response of beam vibrated under moving load as illustrated
- 12. 3.1 Eigen value and Eigen function of modal analyses. for simply supported beam the eigen-value and eigen-function of modal analyses are 𝑦 𝑥,𝑡 = 𝑛=1 ∞ sin 𝑛𝜋 𝑙 𝑥 (𝐴 𝑛 cos 𝜔𝑡 + 𝐵𝑛 sin 𝜔𝑡) , 𝜔 𝑛 = 𝑛𝜋 2 𝐸𝐼 𝜌𝐴𝐿4 3.2 Fourier series presentation of moving load 𝑓(𝑥,𝑡) = 2𝑝 𝑙 𝑛=1 ∞ sin 𝑛𝜋𝑢 𝑙 𝑡 sin 𝑛𝜋 𝑙 𝑥
- 13. 3.3Dynamic response of beam vibrated under moving load 𝑦 𝑥,𝑡 = 2𝑝𝑙3 𝐸𝐼𝜋4 𝑛=1 ∞ 1 𝑛4 ∗ 1 1 − 2𝜋𝑢 𝑙𝜔 𝑛 2 2 sin 𝑛𝜋 𝑙 𝑥 𝑠𝑖𝑛 2𝜋𝑢 𝑙 𝑡 − 2𝜋𝑢 𝑙𝜔 𝑛 sin 𝜔 𝑛 𝑡 The above equation represents the dynamic response of transverse vibration of a uniform Euler -Bernoulli beam traversed under moving load. Simply supported boundary conditions are considered
- 14. 4-Experimental work The practical part of this study includes all steps for experimental work , arranged in the form of sections as illustrating in Figure dynamic stresses due to moving load Force transverse vibration of beam under moving load Free transverse vibration of beam Design and manufacturing of carriage beam manufacturing tensile test material selection and chemical composition
- 15. 4.1 material selection and chemical composition : In this work, brass alloy type c85700 has been chose to study the dynamic stress by moving load .to check up the chemical composition of the selected alloy, a chemical analysis has done by Oxford FOUNDRY-MASTER Xpert, see Figure (4.2). This test was done in Central Organization for Standardization and Quality Control Baghdad Iraq.
- 16. Element Standard Used Copper(Cu) 58.0-63.0 57.98 Tin(Sn) 0.5-1.5 1.02 Zinc(Zn) 33-40 35.8 Iron (Fe) 0.5 0.519 Aluminum(Al) 0.8 0.341 Silicon(Si) 0.05 0.031 Table shows compare on with stander brass alloy type c85700. According to this table, it can be seen that the chemical composition of the used brass alloy c85700 is in conformity with that for the standard alloy[29].
- 17. 4.2 tensile test : The aim of the tensile test for brass alloy (c85700) is to evaluate the values of ultimate tensile strength, yield strength, % elongation and Young's Modulus of the selected metals. The dimensions of the tensile specimens that used are shown in Figure below :
- 18. Figure illustrated frizzing processing for beam 4.3beam manufacturing
- 19. 4.4 Design of Carriage: The design and implementation of carriage should satisfy the practical requirements of moving load problem, which are : 1. Minimum attached area between the carriage and beam to decrease the friction. 2. Simple and easily assembling and disassembling with beam. 3. Rolling motion should not cause sliding between the beam and carriage during motion 4. There are no adverse effects on the beam surface during motion. 5. The possibility of moving with any speed. 6. Can carry any load into the elastic reign. 7. Carriage designed in minimum size to consider it and the load as concentrated load.
- 20. Final shape of carriagedrawing in SOLDWORKS
- 21. 4.5 Free transverse vibration of beam The vibration test involves studying the fundamental natural frequency for the beam
- 22. 4.6 Force transverse vibration of beam under moving load Force transverse vibration of uniform beam by moving load test involves studying of the dynamic response due to the moving load. In this test, ,three different moving loads are applied, each one of them travels at uniform speeds over beam span, see Table Masses(kg) Load(N) Speed(m/s) 4 39.24 0.15 39.24 0.20 39.24 0.25 6 58.86 0.15 58.86 0.20 58.86 0.25 8 78.48 0.15 78.48 0.20 78.48 0.25
- 23. The rig of this test manufactures to complete the practical requirements illustrated in Figure
- 24. The subsystem consists of dc motor, pulley, metallic string, carriage, and power supply, as shown in Figure
- 25. Measuring Unit: It consists of minor power supply , data acquisition, accelerometer, wire and on/off switch see Figure
- 26. Where the labview program installed and by help of sound and vibration tools in this program the single and double integration operation will be done , which yield to in axel form tables and curves of velocity and amplitude vs time
- 27. 4.7 dynamic stresses due to moving load The purpose of the experimental work is to estimate the dynamic bending stress at mid span beam from force transverse vibration due to moving load . Through the observation of the Figure (4.29), the moving load position is obtained according to variables (a and b ) instantaneously .the moving load is located at the three prospects as illustrated 1-moving load travels before mid-span beam (0≤a≤ L/2), 𝑀 = 𝑝𝑏𝑥 𝑙 − 𝑝(𝑥 − 𝑎) Rearrangement yields 𝑝 = 2𝑀 𝑎
- 28. Substitutes the pervious equation in dynamic response equation yield to 𝑀 = 𝐸𝐼𝜋4 𝑦 𝑥,𝑡 𝑎 4𝑙3 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 Where summation = 𝑛=1 ∞ 1 𝑛4 ∗ 1 1− 2𝜋𝑢 𝑙𝜔 𝑛2 2 sin 𝑛𝜋 𝑙 𝑥 𝑠𝑖𝑛 2𝜋𝑢 𝑙 𝑡 − 2𝜋𝑢 𝑙𝜔 𝑛 sin 𝜔 𝑛 𝑡 2-moving load travels after mid-span beam (L/2≤a≤L), 𝑀 = 𝑝𝑏𝑥 𝑙 Rearrangement yields : 𝑝 = 2𝑀 (𝑙 − 𝑎) Substitutes the pervious equation in dynamic response equation yield to 𝑀 = 𝐸𝐼𝜋4 𝑦 𝑥,𝑡 𝑙 − 𝑎 4𝑙3 𝑠𝑢𝑚𝑚𝑎𝑡𝑖𝑜𝑛 )
- 29. The experimental result of previous test (y(x,t)) is applied to obtain the dynamic bending moment. The flexure formula is employed to convert the dynamic bending moment to dynamic bending stresses as shown, 𝜎𝑏𝑒𝑛𝑑𝑖𝑛𝑔 = − 𝑀𝑦 𝐼
- 30. The obtained results from the numerical and experimental work are discussion in this section. The results are divided into three parts 5.1 Free Transverse Vibration of Beam 5- RESULTS AND DISCUSSION Analysis of the experimental signal by sigview software. Experimental response in time domain
- 31. 5.2 Fore Transverse Vibration of Beam Under Moving Load. The experimental and numerical results were obtained when a constant moving load with a uniform speed passed through beam span from lift to right. In this presentation the result of one case will illustrated and the another case will shown in table. Experimental dynamic response of mid-span when load 39.24 N travel at 0.25 m/s Numerical dynamic response of mid-span when load 39.24 N travel at 0.25 m/s
- 32. Load Speed (m/s) Maximum static deflection (m) Maximum dynamic deflection (m) The rate of increase % (N) Experimentally Numerically Experimentally Numerically 39.24 0.15 -0.0022 -0.002244391 -0.0024 2.017757425 9.090909091 39.24 0.2 -0.0022 -0.002520416 -0.0027 14.5643527 22.72727273 39.24 0.25 -0.0022 -0.002878441 -0.0035 30.83824176 59.09090909 58.86 0.15 -0.0033 -0.003327476 -0.00374 0.83260953 13.33333333 58.86 0.2 -0.0033 -0.003482576 -0.0041 5.532598664 24.24242424 58.86 0.25 -0.0033 -0.003556374 -0.005 7.768923564 51.51515152 78.48 0.15 -0.0044 -0.004595223 -0.005 4.436876061 13.63636364 78.48 0.2 -0.0044 -0.00460624 -0.0056 4.687273403 27.27272727 78.48 0.25 -0.0044 -0.0048 -0.007 9.090909091 58.09090909
- 33. 5.3 Dynamic bending stresses of beam due to moving load : As previous slide the result of dynamic bending stresses of beam vibrated under moving load for one case will illustrated and the another case will shown in table Numerical dynamic bending stresses of mid- span when load 39.24 N travel at 0.25 m/s Experimental dynamic bending stresses of mid-span when load 39.24 N travel at 0.25 m/s
- 34. Load (N) Maximum static bending stresses(MPa) Maximum dynamic bending stresses(Mpa) The rate of increase % Experimentally Numerically Experimentall y Numerically 39.2 -18.57 -24.7897 -25.46 33.49326871 37.10285407 39.2 -18.57 -27.3856 -28 47.4722671 50.78082929 39.2 -18.57 -34.0552 -30.708 83.38826064 65.3634895 58.9 -27.9 -35.4536 -38 27.07383513 36.20071685 58.9 -27.9 -37.8756 -42 35.75483871 50.53763441 58.9 -27.9 -45.8542 -46 64.35197133 64.87455197 78.5 -37.181 -44.52 -50.8 19.73857615 36.62892337 78.5 -37.181 -49.713 -56 33.70538716 50.6145612 78.5 -37.181 -54.0084 -60.91 45.25806191 63.82023076
- 35. 6 - Recommendations for Future Work The following recommendations are suggested for future work: 1. To investigate the resonance regain, the length of beam should be extended to provide more speed for moving load. 2. Studying the influence of various boundary conditions and different cross section area on the dynamic deflection and dynamic stress . 3. A comparative study can be investigated among the concentrated moving load, distributed moving load and harmonic moving load . 4. Re-petition this work with a cracked beam or a composite beam. 5. In the proposed work, the vibration can be suppressed by connecting with suitable control system