Optimization of natural frequency of vertical storage tower using FEA

Optimization of natural frequency of vertical storage tower using FEA

Mr. S S. Hatwalne1, Prof. A. S. Dhekane 2. Mr. Vinaay Patil 3

1. Asst.Prof., SITS Pune -41. 2.Prof., DYPC E Pune -41.

3. Head, Vaftsy CAE Pune , India.

.

Abstract – A vertical storage tower is typically used to store liquids or fine powders. To maximize the storage capacity, these columns are usually very tall typically over 27 meters. They are typically susceptible to wind loads, as the bending caused is much greater at these heights. The design problem is that, a vertical storage vessel is to be constructed near an agitator used in the process. The agitator is to generate vibrations at a certain frequency. If the natural frequency of the storage column matches that of the agitator, then it would result in a certain failure of the vessel at resonance. In addition to that, there is an inlet nozzle placed near the top of the column, the nozzle and the piping connected to it might also be susceptible to the vibration. The objective of the project will be to optimize the design and increase the natural frequency of tower in a manner that all failures can be avoided.

1. INTRODUCTION

   Tall cylindrica l stacks and towers may be

   susceptible to wind-induced oscillat ions as a result of vortex shedding. This phenomenon, often referred to as dynamic instability, has resulted in severe

   oscillations, excessive deflections, structural damage, and even failure. Once it has been determined that a vessel is dynamica lly unstable, either the vessel must be redesigned to withstand the effects of wind –

   induced oscillations or e xternal spoilers must be

   added to ensure that vortex shedding does not occur.

   The deflections resulting fro m vorte x shedding are perpendicular to the direction of wind flo w and occur at relatively lo w wind velocities. When the natural period of vibration of a stack or column coincides with the frequency of vortex shedding, the amp litude of vibration is greatly magnified. The frequency of vortex shedding is related to w ind velocity and vessel dia meter. The wind velocity at which the frequency of vortex shedding matches the natural period of vibration is called the critical wind velocity. [1]

   Wind-induced oscillations occur at steady, moderate wind veloc ities of 20-25 miles per hour. These oscillat ions commence as the frequency of vortex shedding approaches the natural period of the stack or column and are perpendicular to the

   prevailing wind. La rger wind velocities contain high velocity random gusts that reduce the tendency for vortex shedding in a regular periodic manner. A convenient method of relating to the phenomenon of wind e xcitation is to equate it to flu id flow around a cylinder. In fact this is the exact case of early discoveries related to submarine periscopes vibrating wildly at certain speeds. At low flow rates, the flow around the cylinder is laminar. As the stream velocity increases, two symmetrical eddies are formed on either side of the cylinder. At higher velocities

   vortices begin to break off fro m the main strea m,

   resulting in an imba lance in forces exe rted fro m the split stream. The discharging vortex impa rts a fluctuating force that can cause movement in the vessel perpendicular to the direction of the stream.

   Historica lly, vessels have tended to have many fewer incidents of wind-induced vibration than stacks. [1] There is a variety of reasons for this:

   1. Re latively thic ker wa lls.

   2. Higher first frequency.

   3. External attachments, such as ladders, platforms and piping, that disrupt the wind flow around the vessel.

   4. Significantly h igher da mping due to:

      1. Internal attachments, trays, baffles, etc.

      2. External attachments, ladders, platforms, and piping.

      3. Liquid holdup and sloshing.

      4. Soil.

      5. Foundation.

      6. Shell material.

      7. External insulation.

2. DESIGN PROBLEM

   The challenge posed is that the agitator and the vertical co lu mn are required to be on the same platform. Due to this the Column is susceptible to vibrations from the agitator. The frequency generated by the agitator is directly proportional to the rotations per minute (rp m). If the freq generated by the agitator matches with the natural frequency of the Vert ical Colu mn then such a resonance will cause the column to vibrate at ma x a mp litude and may even result in Failure. The issue can be resolved by increasing the natural frequency of the column. Natura l freq is a function of the mass (m) and the stiffness (k). If we

   optimize these parameters using structural modifications we can increase natural freq and then we can operate the agitator at higher rpm reducing the production operate the agitator at higher rpm reducing the production time. [9]

   Challenge while increasing the frequency of vertical co lu mn is:

   Frequency, f =

   So if we try to increase the stiffness by increasing stiffeners then weight also increases so no real diffe rence in frequency. [6]

   conditions of a structure change, its modes will change. For instance, if mass is added to a structure, it will vibrate differently. To understand this, we will ma ke use of the concept of single and multip le- degree-of-freedo m system. [2]

   We nd the equation of motion using modal analysis, v1 and v2 be the eigenvectors of the matrix K. These vectors are orthogonal (unless they correspond to the same eigenvalue, in wh ich case they should be made orthogonal). If they have also been normalized, then they form an orthonormal set. Now lets dene the matrix o f eigenvectors P to consist of these orthonorma l e igenvectors. In an equation this is

   P = [ v1 v2 ]

   This matrix is an orthogonal matrix (as its columns are orthonorma l). Such matrices have the convenient property that PT P = I. Matrix mode shapes S is defined as

   Figure 1: Block diagram of column, agitator and Stiffeners modifications

3. MODAL ANALYSIS

   A modal analysis determines the vibration characteristics (natural frequencies and corresponding mode shapes) of a structure or a mach ine component. It can serve as a starting point for other types of analyses by detecting unconstrained bodies in a contact analysis or by indicating the necessary time-step size for a transient analysis, for e xa mple . In addit ion, the modal-analysis results may be used in a downstream dynamic simu lat ion emp loying mode-superposition methods, such as a harmonic response analysis, a random vibration analysis, or a spectrum analysis. The natural frequencies and mode shapes are important parameters in the design of a structure for dynamic loading conditions. [6]

   Modes are inherent properties of a structure, and

   are determined by the material properties (mass, damping, and stiffness), and boundary conditions of the structure. Each mode is defined by a natural (modal or resonant) frequency, modal da mping, and a mode shape (i.e. the so-called modal para meters). If e ither the materia l properties or the boundary

   S = P

   x (t ) = q(t) = Pr(t) = Sr(t)

   Using all the equations we can rewrite the system of diffe rential equations to (t) + r (t) = 0 Where the matrix is given by = PT P =

   So we re ma in with differentia l equations

   + r = 0

   + r = 0

   The deferential equations have been decoupled! They dont depend on each other, and therefore can be solved using simple methods. The two decoupled equations above are called the modalequations. Also the coordinate system r (t) is called the modal coordinate system.

4. NATURAL FREQUENCY AND MODE SHAPES (EIGENVALUES AND EIGENVECTORS)

   If the shell is of constant diameter and thickness for its full length, the period of vibration maybe easily found from the work of C E FREESE. The graph is given by Author fro m the general formula for the period of the first mode of vibrat ion of a cantilever beam [1]:

   Lets take a mult i-degree of freedo m system. For generalizing the method for determining its natural frequencies and mode shape the differentia l equations of motion for the system is, [3] [m1 1 + (k1 + k2) x1 ] – k2 x2 = 0

   -k2 x1 + [m2 2 + (k2 + k3) x2] k3 x3 = 0

   -k3 x2 + [m3 3 + (k3 + k4) x3 ] k4 x4 = 0 —— [A]

   ———————————————- = 0

   – kn xn-1 + (mn n + kn xn) = 0

   For the principal mode of vibration, let us assume the solution as,

   x1 = X1 sin t x2 = X2 sin t

   x3 = X3 sin t ———————————– [B]

   —————-

   xn = Xn sin t

   Substituting equations (B) in equations (A ) and canceling out the common term sin t

   [(k1 + k2) m1 2] X1 k2X2 = 0

   -k2X1 + [(k2 + k3) m22] X2 k3X3 = 0

   -k3X3 + [(k3 + k4) m33] X3 k4X4 = 0

   ——————————————– = 0

   -knXn-1 + (kn mn 2) Xn = 0

   For the above equations, the solution, other than X1

   =X2 =X3 = .Xn = 0 is possible only when the determinant composed of the coefficients of Xs vanishes, or

   = 0

5. COMBINED RECTILINEAR AND ANGULAR MODES

   diffe rential equations of the system are written for motions in the x and directions by taking the forces and the mo ments in the respective directions acting on the system.[5]

   Figure 2: Analysis of a combined rectilinear and angular mode system

   M = – k1 (x l1) k2 (x + l2)

   J = k1 (x l1) l1 k2 (x + l2) l2 ————– [C]

   Letting,

   The body having mass M and moment of inertia J, supported as shown in the fig 2(a) and capable of oscillating in the directions x and . Let, at any instant, the body be displaced through rectilinear

   distance x and angular distance as shown in fig.

   2(b) at this instant taking be sma ll, springs k1, and

   ——————————— [D]

   k2 are compressed through (x-l1) and (x+l2) respectively, beyond their equilibriu m position. The

   and re me mbering that J = , the equations [C] reduces as below

   + a x = b

   + c = ) x

   Here b = coupling coeffic ient,

   If b = 0, then two equations are independent of each other and therefore the two motions, rectilinear and angular, can exist independently of each other with their respective natural frequencies a and . Thus for the case of uncoupled system when b = 0, i.e. k1l1

   = k2l2, the natural frequencies in the rectilinear and angular modes respectively, are [3], [4]

   =

   =

6. FINITE ELEMENT ANALYSIS

   The finite ele ment mode shapes of various frequencies helps in understanding the cylinders vibration behavior and also assists in choosing the frequency range and mode shapes of interest, selecting suitable accelerometer and impact ha mme r, selecting locations for accelerometer place ment, etc. The placement of the accelerometer is very critica l in obtaining valid data. The data obtained wont be useful if the accelerometer is placed at a node point of the structures mode shape for any frequency within the frequency range of interest. The thin uniform c ircula r cylindrica l shell used for the Expe rimental Modal Analysis is modeled and analyzes for its frequencies using normal mode analysis in MSC.NASTRA N byAlzahabi, B.; Natarajan, L. K [2,3].

   The fin ite ele ment method (FEM), somet imes referred to as finite e le ment analysis (FEA), is a computational technique used to obtain approximate solutions of boundary value problems in engineering. Simp ly stated, a boundary value problem is a mathe matica l proble m in which one or more dependent variables must satisfy a differential equation everywhere within a known doma in of independent variables and satisfy specific conditions on the boundary of the domain. [9] Boundary value problems are also sometimes called fie ld proble ms. The fie ld is the domain of interest and most often represents a physical structure. The field variables are the dependent variables of interest governed by the diffe rential equation. The boundary conditions are the specified values of the field variab les on the

   boundaries of the field. Depending on the type of physical proble m being analyzed, the field variables may inc lude physical displacement; temperature, heat flu x, and fluid veloc ity to name only a few Vert ical columns have a few characteristics which make it difficult to design. Firstly the model has no natural roll-off at high frequencies and it is modally very rich. It is characterized by resonant peaks which dominate the dynamics besides the vertical column themselves; complete systems contain many other components such as stiffeners, supports, vibration isolators etc. [6]

   Stiffeners are attached to column to strengthen

   the panel against pressure loads, to reduce vibrations of the stiffen plate against buckling. Stiffeners must be spaced at some convenient spacing so that plate stress is less than allowable stress. Stiffener may also function as column supports as well as plate reinforce ment. Stiffeners are generally welded to the column a lthough there are other methods to join.

   1. MODELING OF VERTICAL TOWER

      Vertica l tower and supports are modeled in ANSYS workbench. There are 20 stiffeners attached to column. All these stiffeners are welded to stiffeners.

      Figure 3: FE model of vertical column

   2. MESHING OF THE MODEL

      Meshing is the method of dividing the model into the number of ele ment to obtain the good accuracy in the analysis. As the number of ele ments increase the accuracy of analysis increases .

      The meshing is done as second order meshing method using shell 93.

   3. SIZE OF STIFFENERS

   The size of stiffener is R/2 of largest diameter of the tower; width wise and R/4 length wise. The stiffeners equally separated at 15 inch distance along the three sides.

   Figure 4: FE mesh of the vertical tower model

   Figure 5: Boundary conditions

   Figure 6: Natural Frequency of vertical tower when no stiffeners are added0

   Figure 7: 5 stiffeners added

   Figure 8: 12 stiffeners added

   Figure 9: 15 stiffeners added

   Figure 10: 20 stiffeners added

7. RESULTS:

Figure 11: Graph showing increase in natural frequency

CONCLUSION:

The additional support structure is having an effect on the natural frequency of the Tower.

As the number of stiffeners increases the frequency is observing a corresponding increase

At around 14 stiffeners which is at a height of 210 inch, there seems to be a Threshold point, beyond which there is rapid rise in the frequency

After around 17 stiffeners, the trend seems to be again linear, although further study is needed to determine if there is another Threshold point.

Based on the study, having around 15 Stiffeners is ideal from both cost and frequency considerations.

REFERENCES:

1. C. E. FREESE Vibrat ions of Ve rtica l Pressure Vessels,Journal of Engineering for Industry

2. A lzahabi, B.; Natara jan, L. K., Non-Uniqueness in Cy lindrica l Shells Optimizat ion, InternationalConference on Computer Aided Optimu m Design of Structures, OPTI 2003, Detroit, 2003.

3. Alzahabi, B.; Natara jan, L. K., Frequency

   Response Optimization of Cylindrica l Shells usingMSC.NASTRAN, The 1st International Conference on Finite Ele ment Process, Lu xe mbourg City,LUXFEM, Lu xe mbourg, 2003.

4. Y. Dong, D. Redekop Structural and Vibrational Analsis of Liquid Storage Tanks, Transaction of SMiRT19, Toranto, August 2007

5. G. K Grover, Mechanical Vibrations, 8th addition,

Ne m Chand & Bros, Rurkee, India

1. S. S. Rao, Mechanica l Vibrat ions, Pearson Education, 3rd impression, 2009 , Published by Dorling Kindersley Pvt Ltd

2. Dennis Moss, Pressure Vessel design manual,

   Third edit ion, G.P Publication 244-245

3. Harris, C. M., Shock and vibrat ion handbook,Mc Graw Hill, New York, 3rd ed., 1988.

4. Robert D. Cook, FE modeling for stress analysis, University of Wisconsyn-madison, John wiley & sons 1995

5. Tushar V. Taka wale , Methods to Improve Natural Frequency of Rectangular Acoustic Ducts using modal analysis in ANSYS, Dept of Mechanical Engg,

   VJTI, Mu mbai

6. ANSYS 12.0.1 Tutorial He lp.