Finite Element Analysis of a Wind Turbine Tower

Author W. Gong Date 25/07/2070

Content

1. Prototype model introduction .......................................................................................... 3

2. Finite element model ....................................................................................................... 5

2.1 DIANA FEA program ............................................................................................... 5

2.2 Model setup ............................................................................................................... 5

3. Model verification ........................................................................................................... 8

3.1 Model geometry verification ..................................................................................... 8

3.2 Eigen modes analysis in DIANA .............................................................................. 8

3.3 Verification of eigen modes .................................................................................... 12

4. Static Analysis ............................................................................................................... 12

4.1 Static wind only ....................................................................................................... 13

4.2 Static wave only ...................................................................................................... 18

5. Dynamic Analysis ......................................................................................................... 19

5.1 Dynamic wind only ................................................................................................. 19

5.2 Dynamic wave only ................................................................................................. 31

6. Simulated earthquake spectral response analysis .......................................................... 31

7. Fatigue analysis ............................................................................................................. 35

8. Coupled wind-wave load analysis ................................................................................. 36

8.1 Static coupled wind-wave load analysis.................................................................. 37

8.2 Dynamic coupled wind-wave load analysis ............................................................ 40

1. Prototype model introduction In the summer of 2009, the first large-scale floating wind turbine “Hywind” was installed (Fig 1). Hywind is a spar-buoy concept with three catenary mooring lines. A technical drawing of the Hywind demo project is given in Fig 2. It consists of a wind turbine, a transition piece, a substructure, a mooring system and an electric cable. The wind turbine consists of blades, nacelle, hub, upper tower and middle tower. The transition piece is the middle tower and the lower tower. The total mass of Hywind is 5086 metric tonnes, excluding the mass of the mooring system. The load carrying structure of the turbine consists of three parts; the substructure, the transition piece and the tower. All are made from steel. The substructure is the lowest part with a length of 89 m and has constant diameter of 8.3 m. The transition piece (TP) has a length of 28 m and has a varying diameter. Between depth 4.5 < d < 11 m, the diameter of TP changes linearly from 8.3 m to 6.0 m. The diameter stays constant up to a height from MWL H = 11 m. In between 11 < H < 17 m the diameter varies linearly to a diameter of 5.0 m. The tower deck is positioned at H = 17 m. The tower length is 46.5 m. The diameter of the tower decreases linearly to the top from 5.0 m to 2.4 m. The wind turbine has a rated capacity of 2.3 MW. The rotor has a diameter of 82.4 m and the rotor hub towers 65 m above mean water level (MWL). The shortest distance from MWL to the rotor tip is 24.5 m. The total weight of the turbine is 138 tonnes. Adjustments have been made to suite the floating conditions, but the blades are standard.

Fig 1 Hywind in construction

Fig 2 Hywind Dimension

2. Finite element model 2.1 DIANA FEA program DIANA is a well proven and tested software package with a reputation for handling difficult technical problems relating to design and assessment activities in concrete, steel, soil, rock and soil-structure interaction. The program’s robust functionality includes extensive material, element and procedure libraries based on advanced database techniques, linear and non-linear capabilities, full 2D and 3D modeling features and tools for CAD interoperability. DIANA deals well with nonlinear analysis for concrete cracking formation, steel structure’s local stability analysis and foundation’s construction analysis. With the continuous demand for more efficient utilization of resources and materials, together with the increasingly complex nature of engineering structures, DIANA is becoming the software of choice for gaining a competitive advantage when tackling design and assessment work. DIANA uses Midas FX+ as its pre & post processor. Midas FX+ Modeler is a leading finite element analysis modeling tool, equipped with advanced geometric modeling functions and powerful mesh generation algorithms. It is capable of modeling any complex configuration encountered in civil/architectural structures and industrial facilities. 2.2 Model setup According to the prototype model, Hywind is in fact a floating offshore structure suspended by mooring lines. This brings a lot of complexities for the boundary conditions and the motion behaviour of the structure. Due to the limited period of this project, simplifications of the Hywind model are made. First, the structure is modeled as a bottom founded structure with pinned foundation. Second, loading conditions applied on the model are also simplified, basically composed of static and dynamic wind loads and static and dynamic wave loads. Also, minor details of the prototype are not considered much here either, etc. These simplifications would of course render the analysis result losing its accuracy compared with the real situation. However, as the purpose of this project is to familiarize the analysis process with FEM program DIANA as well as to get more general knowledge on analyzing the wind turbine tower structure, results of the analysis are preferred but of minor importance. Therefore, the above simplifications should be allowed here. Yet, on the other hand, the analysis results should be in accordance with all the simplified boundary and loading conditions of the modeled structure and the results should be interpretable to allow for better understanding. According to the above situation, details of the finite element model are introduced as following:

Fig 3 FEM model of Hywind tower in FX+

Geometry Geometry of the finite element model is modeled in accordance with the geometrical shape of the prototype. The tower is composed of different sections of conical and cylindrical parts. Thickness of the tower is 200 mm for the lower part height of 89m and 150mm for the above sections. The turbine nacelle and rotors are excluded in the model; however, the weight of these components is modeled as vertical force applied at the top of the tower. Material Tower is made of steel with material parameters as following: Young’s Modulus 2.0E+5 kN/mm2 Poison Ratio 0.3 Density 7.9E-9 t/mm3 Yield Model Von Mises Hardening Parameters 0.275 kN/mm2, 0; 0.330 kN/mm2, 0.025

Finite Element Type Two types of finite element are used for the model. The tower body is modeled using curved shell element and the bottom foundation and the top of the tower is modeled using flat shell element. For the curved shell element, Q20SH (quadrilateral curved shell, 4 nodes, linear) is applied and for the flat shell element, Q20SF (quadrilateral flat shell, 4 nodes, linear, mindlin) is applied. Boundary Condition As stated above, simplification converts the original prototype from a floating offshore structure into a bottom founded structure. Therefore, in the finite element model, boundary condition is simply modeled as pinned bottom nodes. The nodes of the tower bottom are restricted to have translational motions in the X, Y and Z directions.

Fig 4 Boundary condition of the FEM model

Loading Loading condition of the model is generally composed of the following:

1. Self weight of the tower 2. Turbine tower weight modeled as vertical pressure at the top of the tower 3. Wind loads (static & dynamic) 4. Wave loads (static & dynamic) 5. Simulated earthquake spectrum

3. Model verification 3.1 Model geometry verification Based on the descriptions stated above, the geometry parameters of the total tower model are: Total Area = 3.44153e+009 [mm²] Total Volume = 6.34156e+011 [mm³] Total Mass = 5009.83 [T] The total mass of the model is in consistence with the mass of the prototype structure and thus the model’s geometry is verified. 3.2 Eigen modes analysis in DIANA From DIANA’s analysis, the first 12 eigen modes of the structure and the respective eigen frequencies are computed. This eigen mode analysis is helpful for verification of the model on the first order and also useful for providing reference for the further dynamic analysis.

Fig 5(a) 1st eigen mode perspective view and top view

Fig 5(b) 2nd eigen mode perspective view and top view

Fig 5(c) 3rd eigen mode perspective view Fig 5(d) 4th eigen mode perspective view

Fig 5(e) 5th eigen mode perspective view Fig 5(f) 6th eigen mode perspective view

Fig 5(g) 7th eigen mode perspective view Fig 5(h) 8th eigen mode perspective view

Fig 5(i) 9th eigen mode perspective view Fig 5(j) 10th eigen mode perspective view

Fig 5(k) 11th eigen mode perspective view Fig 5(l) 12th eigen mode perspective view

In summary, the eigen frequencies of the first 12 modes are shown in Fig 6. Eigen analysis shows the structure’s twisting behaviour only starts from the 9th mode with eigen frequency of 7.58Hz. For the first 8th modes, structural behaviour is only flexural. Due to the axially symmetrical shape, each eigen mode pair has same characteristics for the X and Y directions.

Fig 6 First 12 modes eigen frequencies

3.3 Verification of eigen modes According to Laurens B. Savenije (December 2009), frequency parameters of fixed wind turbine tower structure are given in the following table 1. For fixed type, the first 3 eigen frequencies are about 0.3Hz, 0.3Hz and >0.9Hz according to the table. This is consistent with the eigen analysis computed as above and thus it can be approximately believed that the finite element model of the tower is sound, provided many simplifications are applied here.

Table 1 Natural periods of a floating and fixed turbine

4. Static Analysis After verification of the model, analysis of the structural behaviour subject to external loadings is performed. External loadings are mainly from two sources: wind and wave or atmospheric and hydraulic forces. Each force is composed of static (average) component and dynamic (fluctuating) component. Characteristics of the wind load and the wave load are conceptually similar but yet the sections of the tower that are subject to the loadings are different. Wind force is loaded on the turbine rotors and the tower body also and wave loading is majorly on the region of the tower body that is near the water surface. Here, firstly, the influence of the wind load and the wave load are studied separately.

Coupled impacts from the two will be analyzed in a later stage, after a better understanding of the separate contributions. For the wind impact, static wind force and dynamic wind force are analyzed: 4.1 Static wind only Load parameters: Wind speed: V=13m/s (Beaufort scale 7 wind speed) Impact area: surface area formed by the rotors (round surface with diameter D=82.4m) Drag coefficient: Cd=2.0 Resulted force magnitude: F=1163.79kN Force direction: X direction Loaded area: tower top Analysis results Structure’s deformation and the von Mises stress are computed according to this static wind force condition:

Fig 7(a) X direction displacement under static wind at 13m/s

Fig 7(b) Von Mises stress under static wind at 13m/s

From the analysis results, under the static wind force of wind speed 13m/s, the maximum displacement of the tower at the top is about 288 mm and the maximum von Mises stress at the tower bottom is around 0.0288 kN/mm2, which is far less than the yield stress of the material. In order to study the nonlinear behaviour of the tower, firstly the structural response of a larger force load is analyzed here linearly, which is going to provide a reference for the nonlinear result in the later stage: Load parameters: Wind speed: V=50m/s (Beaufort scale 12 wind speed) Impact area: surface area formed by the rotors (round surface with diameter D=82.4m) Drag coefficient: Cd=2.0 Resulted force magnitude: F=17215.78kN Force direction: X direction Loaded area: tower top Analysis results

Fig 8(a) X direction displacement under static wind at 50m/s

Fig 8(b) Von Mises stress under static wind at 50m/s

The above results show that under the static wind force of wind speed 50m/s, the maximum displacement is about 4.05 m and the maximum von Mises stress is 0.297 kN/mm2, which exceeds the material’s initial yielding stress. For the same conditions of wind speed 50m/s, nonlinear analysis is performed and the results are shown below:

Fig 9(a) nonlinear analysis X direction displacement under static wind at 50m/s

Fig 9(b) Nonlinear analysis Von Mises stress under static wind at 50m/s

The above shows that results of the nonlinear analysis differ from results of the linear analysis, even if the loading conditions are exactly the same, Nonlinear analysis gives a slightly larger deformation response of the structure while the stress condition is minor, with maximum von Mises stress of 0.279 kN/mm2, which is about 6% less than the result of the linear analysis. 4.2 Static wave only As stated above, for static analysis, the characteristics of the wind load and the wave load are similar and the different is the loading area on the structure. The static wind load is applied at the top of tower, shown in the previous section. However, for the static wave load, it should impact on the region around the water line, depending on the scale of the wave height. For reference, wave force on cylindrical structure can be computed as:

F = Fdrag + Finertia Fdrag = 0.5ρCdAu2 Finertia = ρCmVa

With sufficient wave load parameters (u, Cd, etc) resulted static wave force can be calculated according to the above formulas and static analysis of the wave load on the tower can be performed the same way as that of the wind load in DIANA. Here the static wave load analysis will thus not be redundantly performed again. 5. Dynamic Analysis 5.1 Dynamic wind only In addition to the static analysis in the former section, dynamic analysis of the tower is performed in this section. The dynamic wind load condition is modified via a small scale wind gust data (Fig 10). In order to produce a more continuous result, this small scale gust is copied several times to simulate a more variable condition. The simulated result is thus shown in Fig 11.

Fig 10 Small scale wind gust data by Laurens B. Savenije (December 2009)

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Fig 11 Simulated longer term wind gust data

Based on the above loading condition, response of the tower is computed and the results are shown below. For transient analysis in DIANA, 5 different time integration schemes are available. In order to give a better understanding of the different characteristics of these 5 different schemes, each of the schemes is applied based on the same loading conditions above. The size and number of time step is an important issue for performing transient analysis in DIANA. Initially, the input time step size is set as 0.1s. This means for the total duration of the 30s of the external loading, there would be 300 time steps. Also, as the first eigen mode period of the tower is about 2.16s. This means one period could contain about 20 results points which are regarded as sufficient for the analysis result. Results of the different time integration schemes are thus shown below respectively:

1) Linear Euler Backward Scheme:

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Fig 12(a) Top point displacement in X direction

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Fig 12(b) Bottom reaction force in X direction

2) Linear New Mark Scheme (Beta 0.25, Gamma 0.5):

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Fig 13(a) Top point displacement in X direction

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Fig 13(b) Bottom reaction force in X direction

3) Linear Wilson Theta Scheme (Theta 1.4):

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Fig 14(a) Top point displacement in X direction

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Fig 14(b) Bottom reaction force in X direction

4) Linear Hilbert Hughes Taylor Scheme (Alpha -0.1):

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Fig 15(a) Top point displacement in X direction

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Fig 15(b) Bottom reaction force in X direction

5) Linear Implicit Runge Kutta Scheme:

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Fig 16(a) Top point displacement in X direction

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Fig 16(b) Bottom reaction force in X direction

Conclusion: Based on the results from the above 5 different time integration schemes, the New Mark, Wilson Theta, Hilbert Hughes Taylor and the Implicit Runge Kutta schemes show very similar results with only the Euler backward scheme being significantly different from the other four. For the results from the first four schemes, it can be seen that the response of the tower has a period that is in accordance with the first eigen mode of the structure (frequency 0.462Hz, period 2.164s). The amplitude range of the first four schemes also share common characteristics. However, for the Euler implicit scheme, the results only reflect the characteristics of the external loading (compare Fig 11 and Fig 12) and the eigen mode characteristics of the structure itself are completely damped out in the computed results. Therefore, it can be concluded that using the first four schemes seem to be much safer when performing dynamic analysis and thus the following analysis will be using the New Mark scheme only for simplicity reason. Before performing nonlinear dynamic analysis, there is a need to answer the above problem: why the result from the Euler backward scheme is so different from the others. Theoretically, the Euler backward is of first order accuracy while the other four schemes have higher order accuracy. Therefore, a trial with a smaller time step size is made to study the result of the Euler backward scheme again. For this time, time step size is chosen as 0.01s resulting 3000 time steps for the duration of the external loading. This time step size is 1/10 of the original one used. The result of a smaller time step size using Euler backward scheme is shown below:

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Fig 17 Top point displacement in X direction

Compare Fig 12 and 17, it is easily shown that reducing the time step size generates a better result that is more in consistence with the other schemes for the Euler backward one. The sacrifice is of course the apparently increased amount of time for the computation. Also, compare Fig 17 and Fig 13(a), 14(a), 15(a) and 16(a), it can be seen that the amplitude of the Euler backward result is generally smaller than that of the other four schemes. The reason is probably that Euler backward scheme has a larger numerical damping on the results. After answering the characteristics of different time integration schemes, nonlinear dynamic analysis will be performed in the following. As stated above, for simplicity, only New Mark scheme will be used. 6) Nonlinear analysis without damping using New Mark scheme:

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Fig 18(a) Top point displacement in X direction

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Fig 18(b) Bottom reaction force in X direction

Fig 18(c) Von Mises stress at time of maximum response (t=5s)

Through Fig 18, it can be seen that even nonlinear analysis function is applied (including physical and geometrical nonlinear considerations), the result is almost the same as the result from linear analysis (Fig 13). Therefore, in order to give a better understanding of the nonlinear characteristics, damping is applied for the following section. 7) Nonlinear analysis with damping using New Mark scheme: Damping parameters: In the transient dynamic analysis, it is necessary to apply Rayleigh damping. Rayleigh damping is characterized by the two constants a and b, whereas the damping coefficient of the steel ς is about 0.025. So now the question is how to determine the coefficients a and b for givenς . If a modal analysis would be used, the damping coefficients 1ς and 2ς

for the two lowest frequencies 1ω and 2ω will equal ς if

1 22a ωω β= 2b β= where

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For this wind turbine tower, based on eigen mode analysis stated previously, the lowest two frequencies 1ω and 2ω are 2.902 rad/s and 10.864 rad/s respectively. Therefore, the Rayleigh damping constants a and b are computed as:

0.115a = 0.00363b =

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Fig 19(a) Top point displacement in X direction

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Fig 19(b) Bottom reaction force in X direction

Fig 19(c) Von Mises stress at time of maximum response (t=5s)

From the above results, it is easily seen that the damping influences the response of the structure to the loading to a large degree. Amplitudes of the structural response are reduced and the inner force of the structure is also decreased. And as time goes by, the damping has more positive effects on the structure. In addition, comparing Fig 17 and 19(a), it can be seen the nonlinear analysis with damping result is somehow similar to the linear analysis result of Euler backward scheme. This again shows that Euler backward scheme introduces much more numerical damping than other time integration schemes. 5.2 Dynamic wave only As for dynamic wave load, a similar transient analysis could also be performed in DIANA. Similar to the static load analysis, the loaded area of the dynamic wave load is also different from that of the dynamic wind load. The time load shape depends on the local wave environment. An example of the dynamic wave load can be:

Fig 20 Sample wave load at MSL

Again, for simplicity reason, dynamic wave analysis will not be performed again here. 6. Simulated earthquake spectral response analysis After the above dynamic analysis, a spectral response analysis is performed also. In order to perform the spectral response analysis, first an earthquake spectrum is needed. In this analysis, the following spectrum will be used:

Fig 21 Simulated earthquake spectrum

Resulting structural response of the above earthquake loading is shown below. Results of two different rules are both presented here. ABS rule

Fig 22(a) X direction deformation of ABS rule spectral response

Fig 22(b) Von Mises stress of ABS rule spectral response

SRSS rule

Fig 23(a) X direction deformation of SRSS rule spectral response

Fig 23(b) Von Mises stress of SRSS rule spectral response

Conclusion: Based on the above results, it can be seen that ABS rule gives a much larger structural response than the SRSS rule. Under the ABS rule, the maximum structural deformation is in the order of 620 mm and the maximum stress is about 0.0820 kN/mm2, given the condition of the earthquake spectrum of Fig 21. The structural response is in the linear stage which means if the simulated earthquake spectrum is the same as the in situ external force, then the structural would be safe from failure. 7. Fatigue analysis Fatigue analysis of the structure is performed in this part. For fatigue analysis, a so-called Wöhler diagram is required. According to the steel material of the structure, the following material data will be used: SWOEHL 0.4 kN/mm2 NWEOHL 10000000

Based on the above material properties and under the load of self weight only, the result is shown as following:

Fig 24 Number of load cycles to fatigue failure

From the above result, the number of load cycles to fatigue failure range from 6.54×106 to 9.98×106. The weakest part is at the toe of the structure and the strongest part is the upper tower of the structure. 8. Coupled wind-wave load analysis After analysis of the wind and wave load on the tower separately, the coupled wind-wave load analysis will be performed in this section. According to Laurens B. Savenije (December 2009), tower top oscillations into the wind of fixed turbines are aerodynamically damped, for which reason the worst case wind-wave misalignment is 90 degree, when there is no aerodynamic damping. Therefore, wind-wave misalignment of 90 degree will be analyzed here, more specifically, the wind direction is X direction and the wave direction is Y direction.

8.1 Static coupled wind-wave load analysis Wind load parameters Wind speed: V=13m/s (Beaufort scale 7 wind speed) Impact area: surface area formed by the rotors (round surface with diameter D=82.4m) Drag coefficient: Cd=2.0 Resulted force magnitude: F=1163.79kN Force direction: X direction Loaded area: tower top Wave load parameters For wave load, a sample of the loading condition is shown below. For static load, the magnitude of the force will be approximated via equalizing the amplitude of the wave force of this fluctuating loading situation, that is to say, the static wave force will be set as about 2000kN which is the amplitude of the dynamic wave load in Fig 25.

Fig 25 Sample wave force at MSL

Impact area: section of tower submerged in water Resulted force magnitude: P=2kPa Loaded area: tower surface submerged in water

Results:

Fig 26(a) Y direction structural deformation under static wind-wave load

Fig 26(b) Total absolute structural deformation under static wind-wave load

Fig 26(c) Von Mises stress under static wind-wave load

The above shows that structural deformation in the Y direction due to the static wave force is less than the structural deformation in the X direction due to wind force. However, the combined wind-wave force would give the maximum deformation at the top of the tower a displacement around 280 mm. The maximum von Mises stress in this situation is about 0.0318 kN/mm2 at the bottom of the tower. The value is about 10% higher than the situation of the wind load alone (Fig 7(b)). 8.2 Dynamic coupled wind-wave load analysis For the dynamic coupled wind-wave load analysis, the following load conditions are applied simultaneously on the structure. Rayleigh damping is also included for this analysis. Wind load condition Wind load is simulated as periodical force acting on the top of the tower in the X direction. The phase and amplitude is determined through the wind gust condition in the following figure. This results in a force with amplitude of about 1076 kN.

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Fig 27 Wind gust condition

Wave load condition For wave load, the force is in the same phase as the wave elevation variation in the figure below, which means the wave load is approximated as a sine wave with period of 10 s. The amplitude of the load is 2 kPa on the section of the tower submerged in water. The load is acting in Y direction.

Fig 28 Wave elevation condition

For the two different dynamic loads above, they have different amplitudes, periods and they also act in different parts of the tower in perpendicular directions. The results of the structural response to this complex loading condition are shown as below:

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Fig 29 Top point X direction displacement under coupled dynamic wind-wave load

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Fig 30 Top point Y direction displacement under coupled dynamic wind-wave load

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Fig 31 Top point absolute displacement under coupled dynamic wind-wave load

Fig 32 Von Mises stress under coupled dynamic wind-wave load at T=5s

From the above results, it is shown that the eigen modes in the structural response to the fluctuating wave load in Y direction are more damped out than the situation of the wind load. Due to the mild wave load applied here, the structural response in Y direction is quite minor compared with that in X direction and thus the total absolute deformation situation at the tower top is more or less similar to the result of that in X direction only. The maximum deformation at the tower top is about 370 mm at T=5s (Fig 31). At the time of maximum structural response, the von Mises stress condition of the structure is given in Fig 32. The situation is not far from that under the dynamic wind load condition only (Fig 19(c)). Yet, the resulting stress is about 10% larger in this case, due to the presence of the wave load. The structural response to the coupled wind-wave load is more or less the combination of the response to each load condition, with some nonlinear coupling effects present. Analysis of different structural response characteristics under dynamic wind and wave loads computed in DIANA According to the above analysis, there are some interesting characteristics about the results computed in DIANA. Compare both the dynamic wind and wave load conditions (Fig 27 & 28), although there are many differences, there are also some common points. Both loads have a period of 10 s, which is simplified in the input data in this analysis. The wind gust load is purely positive because the wind gust direction does not change during the loading period. For the dynamic wave load, it is simplified as a sine wave in this analysis with changing forcing directions. However, the period of the wave load is the same as the period of the wind gust load. Compare structural responses to the wind and wave loadings in Fig 29 & 30, interestingly, even the loading conditions share the common characteristics as stated above, the results are somehow different. Both results are using the New Mark time integration scheme with the same scheme parameters and time step size, but the response to the wind load has an apparently larger influence of the structural eigen period while the response to the wave load has only some slight hints of the structural eigen period. For both cases, the structural response influence from the eigen modes is damped more and more with the time goes by. For the duration between 25s and 30s, there is almost no hint of the eigen mode which is explained by the Rayleigh damping applied in this analysis. In order to provide a more consolidated reference for the above issue, another transient analysis with a much smaller time step size is performed. The time step size is now 0.001s for a period of 10s. New Mark scheme is still used. The structural responses to the wind and wave loads in two different directions are shown below. From Fig 33 & 34, it can be seen that even the time step size is reduced to a much more conservative level, the result is still the same case as what is got in the above situation. Still, the response to the wind gust load is subject to a larger influence from the structural eigen mode and the response to the idealized sine wave load has a much smaller influence from the structural eigen mode.

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Fig 33 Top point X direction displacement under coupled dynamic wind-wave load (∆t=0.01s)

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Fig 34 Top point Y direction displacement under coupled dynamic wind-wave load (∆t=0.01s)

For the problem stated above, the theoretical answer is that the structural response to the wind load has a larger influence of the structural eigen mode because the dynamic wind gust load still has some minor components of lower periods which are closer to the

structural eigen period even if it looks like to have a period of 10s. The wave load is idealized as a sine wave with a period of 10s and that’s why it doesn’t show much hint from the structural eigen mode. On the other side, for the situation of forced motion of damped system, the result is composed of two parts: damped free vibration and damped forced vibration. As time goes by, the damped free vibration will be faded away while the damped forced vibration will dominant the result and this is why the influence of the structural eigen mode will be damped out as manifested in the above analysis results.