D && 9.0 DYNAMIC ANALYSIS

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1 9.0 DYNAMIC ANALYSIS Introduction When a structure has a loading which varies with time, it is reasonable to assume its response will also vary with time. In such cases, a dynamic analysis may have to be performed which reflects both the varying load and response. If however, the frequency of loading is low compared with the natural frequency of the structure, then the response given by a static analysis under the instantaneous load may suffice. This assumption is normally applied when the frequency is less than one third of the lowest natural frequency. If the applied load varies rapidly, then a variety of solution technique may be employed which take into account inertial effects due to mass and damping effects. 9.1 TYPES OF ANALYSIS The most general dynamic analysis will solve the following equation, which gives the time dependant response of every node point in the structure by including inertial forces and damping forces in the equation. Inertial forces are the product of mass times acceleration and damping forces are the product of damping coefficient times velocity. The general equation of motion is therefore, [ M ]{ D&& } + [ C]{ D& } + [ K]{ D} = { F} where, in matrix form, [M] represents the structural mass, { } the nodal acceleration vector, [C] the structural damping matrix, { D & } the node velocity vector, [K] the structure stiffness matrix, {D} the node displacement vector and {F} is the applied time varying nodal load vector. This equation is a set of differential equations in matrix form for the dynamic response of a structure modelled with a finite number of degrees of freedom. D && However, the solution to this set of equations taken incrementally in time may involve thousands of static solutions to generate a complete time history response for the structure. This may be impracticable for any significant length of time cycle. Therefore it is of importance to examine the vibrational response to certain specific inputs. The three most common types of analyses are: 63

2 MODAL Analysis HARMONIC FREQUENCY RESPONSE Analysis TRANSIENT DYNAMIC Analysis Modal Analysis In many engineering applications, the natural frequencies of vibration are of interest. This is probably the most common type of dynamic analysis and is referred to as an eigenvalue analysis. In addition to the frequencies, the mode shapes of vibration which arise at the natural frequencies are also of interest. These are the undamped free vibration response of the structure caused by an initial disturbance from the static equilibrium position. This solution derives from the general equation by zeroing the damping and applied force terms. Thereafter, it is assumed that each node is subjected to a sinusoidal functions of the peak amplitude for that node. If the displacement vector {D} has the form { D} = { A}sin( ωt) where A is the amplitude of displacement for every node and ω is the frequency of vibration. Therefore, the velocity vector is and the acceleration is { D & } = { A} ω cos( ωt) { D & } = { A} ω sin( ωt) Substituting these into the general equation of motion yields the eigenvalue equation, ([ K] - λ[ M]){ A} = { 0} where the eigenvalue, λ, is equal to ω 2, and {A} is the eigenvector associated with each value of λ. The total number of eigenvalues or natural frequencies is equal to the total number of degrees of freedom in the model. Each eigenvalue or frequency has a corresponding eigenvector or mode shape. Since each of the eigenvectors cannot be null vectors, the equation which must be solved is ([ K] - λ[ M]) = { 0} The mode shapes are also of interest to the engineer. These are normalised to the maximum displacement of the structure. The input conditions which initiate the vibration control the amplitudes of vibration in any problem. 2 It is worth noting that we are normally only interested in the first few eigenvalues of the model. Indeed, since the finite element model is an approximation of the structure, then the higher eigenvalues and vectors are inaccurate. The theoretical solution implies that the 64

3 structure will vibrates in any mode shape indefinitely. However, since there is always some damping present in any structure, the vibrations eventually decay Frequency Response Analysis This type of analysis is of interest when the steady state response of a structure to a harmonic force input at a given frequency is required. For example, a support arrangement for a pump or computer fan mounting, and so on. The response may be needed for a range of frequencies. In a frequency response analysis, the frequency of the response to a harmonic input is also harmonic and occurs at the same frequency. The forcing function can be defined as: { F} = { F } e o iωt where F o is the peak force amplitude and ω is the harmonic frequency. The nodal displacement therefore has the form 65

4 with velocity { D} = { D } e o iωt { D & } = { D } iωe ω o i t and acceleration { D & } = { D } ω e o 2 iωt Substituting in the general equation of motion results in (- ω 2 [ M] + iω[ C] + [ K]){ D } = { F } o o which shows that the displacement, {D o } is clearly a function of frequency, damping and force amplitudes. Solving this over a discrete range of frequency inputs determines the vibration frequency response. The displacements calculated here define the deformed structural shape. This is not the same shape as the mode shape unless the frequency coincides with the natural frequency. In frequency response analysis, damping can often be ignored since most structures are lightly damped and this simplifies the solution. This allows all frequencies except natural frequencies to be calculated. If a natural frequency is used as an input, and no damping is present, the 66

5 solution fails due to numerical problems. However, this is not a problem since near natural frequencies are normally adequate Transient Response Analysis If the input function is not harmonic but an arbitrary time dependant function, then a transient response analysis must be performed. In this, the general equation of motion is solved but, in this case, the time scale of loading is such that damping or inertia effects are considered important. This type of analysis is used to determine the time-varying displacements, strains and stresses in a structure as it responds to a transient load. There are two basic approaches to transient analysis. The first involves solving the systems of equations by direct integration which involves the whole systems of equations and requires many time steps with a complete solution in each step. This can become a large computing task for moderate sized problems. The second approach is known as modal superposition which assumes that the response of the structure can be adequately represented by the lower natural frequencies of the structure. The complete response therefore, is the summation of the correct fractions of the low frequency mode shapes. Mathematically, this involves a transformation of the equation from nodal displacement co-ordinates into a set of modal co-ordinates. This results in much fewer equations, but results in an approximate solution being obtained. However, this has proven to be sufficiently adequate for most structural vibration problems. 9.2 MASTER DEGREES OF FREEDOM When a large set of equations are being considered, as employed in a dynamic analysis, then an approach known as static condensation or Guyan Reduction is used in most commercial programs. In this, the whole geometric model is developed, however, a set of Master Degrees of Freedom, is selected from the total number of degrees of freedom for the model on the basis that the master set is more dominant in representing the structural response. This implies that these master degrees of freedom (MDOF) will control the vibration and the remaining degrees of freedom become slaves because they follow the pattern defined by the response of the master nodes. The mathematical treatment is not covered here, however some features of the approach must be understood. Some programs expect the user to define the masters, whilst others 67

6 automatically select the set. Some codes allow a mixture of the two approaches. Once the master set is identified, the programs condenses the slave DOF from the full set of equations. This is done by discarding the mass associated with the slave set and setting up displacement relations between the masters and slave sets which become a function only of the stiffness components relating the DOF. This implies a reduced set of equations involving only the master DOF which are solved using one of a number of numerical procedures. Thereafter, once the displacements are solved for the master DOF, back substitution takes place for the slave DOF and a good representation of the deformed geometry is obtained. Selecting Masters DOF which describe the lower modes are usually selected as masters, since low frequency response is normally where most vibration problems occur. In addition, since mass effects are ignored for the slave set of nodes, the master DOF must be those areas with a larger fraction of the structural mass. When automatic selection is used, the user defines the total number of MDOF to be selected. This parameter is continually increased and the run repeated until sufficient accuracy is reached in the solution between variations of the total number. Masters should be well distributed throughout the structure and not clustered since they are associated with mass. An example of the selection of MDOF for a cantilever thin plate. In the above case, it is useful to think of the masters as those nodes which attract the mass of the surrounding material. Since there are six DOF available (3 translational and three rotational DOF) then point mass relates to translational DOF and inertia relates to rotational DOF. If it is considered that rotary inertia effects have less effect than mass effects, the rotational DOF can be ignored. If the stiffness for in-plane translation is much higher than 68

7 lateral translational DOF, then only out of plane DOF need be considered. This implies that the number of equations reduces by five. Further selection of masters based on the uniformity of the mesh would result in the master selection shown above. A fixed edge as shown must not have those DOF removed but must remain fully fixed. MODAL ANALYSIS CASE STUDY - Simple Tapered Cantilevered Beam A simple tapered cantilever beam is analysed to evaluate the first five eigenvalues and mode shapes. The steel beam is 250mm long, 2.5mm thick and its width varies from 50mm to 44mm at the free end. Several modelling methods are used together with varying mesh types and densities. Model Type 1 - Beam Elements The first model type contains both five and twenty element configurations. The coarseness of the mesh is apparent for Mesh 1a. 69

8 Model Type 2 - Plate Elements Mesh 2a 2x5 Element Plate Model Mesh 2b 4x10 Element Plate Model Results for the 2x5 plate model are within 2% of the beam model except for torsional mode 3 which increases by 8%. The 4x10 model produces a further 1% increase and smoother plots. This study shows that it is possible to obtain reasonably accurate results for the natural frequency and mode shape for this structure using relatively coarse models using wither beam or plate models. This is also true for finding eigenvalues and mode shapes for most structures. Coarse models usually provide good values for the lower eigenvalues which are the ones of most importance. 9.3 MODAL ANALYSIS IN ANSYS Modal analysis in Ansys is a simple extension of linear analysis. Firstly, the finite element model is created as normal within PREP7 and saved prior to entering the solution phase (/SOLUTION). Thereafter, the analysis type is selected (REDUCED or SUBSPACE) by using the MODOPT command. If a reduced analysis is chosen, then the master DOF must be selected using M and TOTAL commands. Thereafter the solution is executed using SOLVE. In both cases, once the solution has completed (/), the solution must be re-entered (/SOLUTION) and the expansion pass started (EXPASS,ON) afterwhich the number of modes to be expanded must be entered (MXPAND, number). Then expansion is executed using the SOLVE command. 70

9 Post-processing is performed in the usual manner however each frequency is stored using the SET command. Enter PREP7 Build Model Exit PREP7 Enter Solution /SOLUTION MODOPT,REDUCED Select Masters M,node,dof,0 eg. M,234,ALL,0 Use Autoselect TOTAL,number MODOPT,SUBSPACE,5 Specifying 5 modes No master selection required using this technique Choose whether full subspace or reduced analysis In both cases, the mode shapes must be expanded SOLVE SH /SOLUTION EXPASS,ON MEXPAND,number SH SOLVE CASE STUDY - Simple Turbine Blade The final example is that of a turbine blade which is analysed using both the reduced analysis and the full subspace analysis. The turbine blade model shown is somewhat simplified with the root shown as a solid block and the blade represented as a thin shell. Compatible elements are used. The solid section is formed using SOLID73 elements which have both translational and rotational degrees of freedom at each node. These elements are compatible with SHELL 63 elements used in the curved portion of the blade. The geometry is imported from an external source, in this case PATRAN, and comprises of a list of nodal points and the element connectivity. In modal analysis, the model creation is performed in a similar manner to that used for linear analysis. Full listings for both the reduced and full analyses are found on the class webpages using filenames BLADE1.INP and BLADE2.INP. 71

10 1 1 Z Y X Z Y X MODE 1 :: FREQ= MODE 2 :: FREQ= Z Y X Z Y X MODE 3 :: FREQ= MODE 4 :: FREQ=

11 1 MODE 5 :: FREQ= Z Y X Mode /Method Reduced The above table shows the differences in eigenvalue obtained from two of the available modal analysis options, reduced and full subspace methods. The full method gives more accurate answers, however, takes somewhat longer to solve. The reduced method provides acceptable results for the first two modes yielding results within 10% of the full solution. Increasing the total (TOTAL command) number of master degrees of freedom will produce a more accurate result. Full Subspace Animation of Displaced Shapes Since mode shapes represent the dynamic behaviour of a structure at a given frequency, it is useful to view these in animation form. ANSYS provides a pseudo-dynamic display by storing a number of frames as segments in memory and quickly redisplays these to provide dynamic motion of the deformed shape. Indeed, variations contours can also be displayed using this feature. A typical command sequence to produce animations is shown as follows /SEG,MULTI C*** Set up animation frames to repeat five times on displaced shape ANMAC,5,1,1,1 /SEG,OFF C*** Execute animation ANIM Using this sequence will generate a dynamic display. NOTE: With the advent of ANSYS 5.5 and later by using the MENU picks, animation can be viewed and saved as AVI files. (Plot Controls fi Animate, then choose item from list.) 73

12 File BLADE1.INP (c) D H Nash /PREP7 C*** PREP7 INPUT PRODUCED BY "PATANS" VERSION 2.2B-1 (c) DHNash 15 Feb 1996 /TITLE,Modal Analysis of a Simple Turbine Blade - Full Analysis R,1,2.5E-3 C*** NODAL COORDINATE DETION N, 1, E-02, E+00, E D, 228,UX, E+00,, 228, 1,UY,UZ,ROTX C*** End of model created in PATRAN c*** Enter solution step to evaluate eigenvalues /SOLUTION ANTYPE,MODAL C*** Choose reduced analysis type MODOPT,REDUC C*** Define 2 masters by hand at outer extremities of blade M,365,ALL M,257,ALL C*** Let ANSYS pick a further 200 MDOF TOTAL,200 SOLVE C*** Now re-enter and expand the mode shapes /SOLUTION C*** Switch on the expansion pass EXPASS,ON C*** Expass first five mode shapes MXPAND,5 SOLVE C*** Post process and look at displaced shapes /POST1 C*** Pick first analysis/first mode shape SET,1,1 /VIEW,1,1,1,1 PLDISP,2 74

13 File BLADE2.INP (c) D H Nash C*** use model creation as previous C*** End of model created in PATRAN SAVE c*** Enter solution step to evaluate eigenvalues /SOLUTION ANTYPE,MODAL C*** Choose reduced analysis type and work on 1st five modes MODOPT,SUBSPACE,5 SAVE SOLVE C*** Now re-enter and expand the mode shapes /SOLUTION C*** Switch on the expansion pass EXPASS,ON C*** Expass first five mode shapes MXPAND,5 SOLVE C*** Post process and look at displaced shapes /POST1 C*** Pick first analysis/first mode shape SET,1,1 /VIEW,1,1,1,1 PLDISP,2 75