Al-Khwarizmi 

Engineering   
Journal 

Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 

 

 

An Experimental Study of the Effect of Vortex Shedding on Solar 

Collector Performance 
 

Alaulddin Abdulqader Kadim 
Department of Engineering Affair/ University of Baghdad 

Email:aldina81@yahoo.com 

 
(Received 24 March 2014; accepted 13 July 2014) 

 
 

 

Abstract 

 
In this work, the effect of vortex shedding on the solar collector performance of the parabolic trough solar collector 

(PTSC) was estimated experimentally. The effect of structure oscillations due to wind vortex shedding on solar 

collector performance degradation was estimated. The performance of PTSC is evaluated by using the useful heat gain 

and the thermal instantaneous efficiency. Experimental work to simulate the vortex shedding excitation was done. The 

useful heat gain and the thermal efficiency of the parabolic trough collector were calculated from experimental 

measurements with and without vortex loading. The prototype of the collector was fabricated for this purpose. The 

effect of vortex shedding at different operation conditions was examined. The variation of angles of attack and wind 

velocity leads to different values of vortex loading coefficients and shedding frequencies. The relation between the 

dynamic characteristics and solar collector performance was evaluated. The finite element method was used to estimate 

the dynamic characteristic of the solar collector in addition to experimental work to evaluate the relation between the 

dynamic behavior of the collector and its performance.  

 

Keywords: Vortex shedding, solar collector efficiency, useful heat gain.  
 

 

1. Introduction 
 

In order to deliver high temperatures with good 

efficiency, a high performance solar collector is 

required. Systems with light structures and low 

cost technology for process heat applications up to 

400 ̊C could be obtained with parabolic through 
solar collectors (PTSC). PTSC can effectively 

produce heat at temperatures between 50 and 400 

̊C [1]. Most of the solar power plants installed 
with parabolic collectors are on flat terrain and 

they may be subjected to some environmental 

problems. One of problems for large parabolic 

collector is their stability to track the sun with 

respect to time very accurately. Any small off 

tracking as well as the collector structure stability 

will be affected by strong wind blowing for the 

regions where the wind velocity is high [2]. Wind 

loading excites the structures to vibrate with 

specific deformation pattern called wind-induced 

vibration. This excitation induced due to many 

flow phenomenons such as vortex shedding, 

flutter, buffeting and fluid- structure interaction 

[3]. This flow phenomenon can affect the 

performance of solar collector significantly [4]. 

Holmes et al. [5] studied flow pattern around a 

microwave dish. They studied wind drag forces 

on microwave dish antennas and their supporting 

lattice tower for different angles of attack. They 

found that, drag coefficient for an isolated dish is 

maximum with direction normal to the plane of 

the dish, but does not change in an angular 

window within 30˚ to the normal of the dish. 

Raddu and Axinte [6] studied the wind effect on 

solar flat-plate collectors. They found that when 

the wind direction is normal to the plate the drag 

coefficient is maximum. Chevalien and Norton [7] 

studied rows of solar panels on a model building 

in a wind tunnel and their sheltering effects. they 

founded that the number of rows and presence of 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

26 

 

sheltering lead to significant decreasing in wind 

loading. Kopp and Surry [8] performed an 

experimental research on the total system torque 

measured at the main drive gear box used for 

panel’s solar tracking system.   

       In this work, the effect of vortex shedding on 

collector performance was investigated 

experimentally. The experimental vortex shedding 

excitation was determined according to hybrid 

model and applied to the parabolic trough with 

specific magnitudes and frequencies. Different 

values of vortex excitation magnitude and vortex 

excitation frequency are examined. Then, the 

collection efficiency of parabolic trough solar 

collector was measured experimentally first 

without presence of vortex excitation and second 

with presence of vortex excitation. Different 

angles of attack of wind and wind velocities are 

investigated. The tests were done by using special 

test rig fabricated for this purpose. The 

experimental tests have been carried out in 

Baghdad (AL-Jaderiah) with climatic conditions 

(latitude of 33.3˚ N, longitude of 44.4˚ E and a 

standard time meridian of 45˚) during selected 
days of the months of August, September and 

October. 

 

 

2. Vortex Shedding Model 
 

For a bluff body in uniform cross flow, the 

wake behind the body is not regular but contains 

distinct vortices. The vortices are shed alternately 

from each side of the body in a regular manner 

and give rise to an alternating force on the body. 

Experimental studies have shown that the 

frequency, in Hertz, of the alternating lift force is 

expressed as [9]: 

fs =
S  U

D
                                     ... (1)  

where S: The dimensionless number called the 

Strouhal number,  

U: wind velocity (m/s)  

D:  is the width of the object perpendicular to the 

flow (m) 

The Strouhal number is determined only by 

experimental work. For solar collector, it depends 

on angle of attack of wind (θ). The angles of 

attack examined in current study are (30˚, 45˚, 
60˚, 75˚, 90˚). These angles of attack are shown 
in figure (1). The values of Strouhal versus these 

angles of attack are shown in table (1) [10]. The 

oscillating lift force imposed on a structure of 

length L and width D, in a uniform cross flow of 

velocity U, due to vortex shedding is given by 

[11]: 

F =
1

2
 ρU2 CL  DL sin 2πfs t  = Fo sin (2πfs t)                                                                                                     

                                                                       … (2) 

where 

Fo= 
1

2
 ρU2 CL  DL 

t: time (s) 

The lift coefficient of solar collector is also 

depends on the angle of attack. The lift coefficient 

determined only by experimental work. The 

values of lift coefficient corresponding to the 

angles of attack are obtained by using boundary 

layer wind tunnel tests for the solar collector with 

specification shown in figure (2) [12]. These 

values are listed in table (2) [12]. The validity of 

boundary-layer wind tunnel testing for wind loads 

on structures is based on similarity arguments and 

on model-to-full-scale test comparison [12]. In 

current work, 1/4 model- to-full- scale was  used 

and the specification of prototype shown in Figure 

(2).  

 

 

   

 

 
 

Fig. 1. The angles of attack. 

 

 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

27 

 

 
 

Fig. 2. Specification of full scale solar collector and prototype used in current work. 

 

 
Table 1, 

Strouhal number versus the angle of attack [10]. 

θ 30˚ 45˚ 60˚ 75˚ 90˚ 

S 0.23 0.216 0.201 0.196 0.182 

 
Table 2, 

Lift coefficient versus the angle of attack [12]. 

θ 30˚ 45˚ 60˚ 75˚ 90˚ 

CL 1.1 1.5 1.9 0.9 0.2 

 

The oscillating lift force imposed on a 

parabolic trough can be computed for each angle 

of attack and wind velocity. 

 

3. Experimental Representation of Vortex 
Shedding Excitation 

 
The oscillating force due to vortex shedding 

can be simulated experimentally. The magnitude 

and frequency are calculated from equations 1 and 

2. The experimental apparatus used to simulate 

the oscillating force is consist of: 
 

1. Function generator used supply wave signal 
with specific type, amplitude, and frequency 

2. Power amplifier used to amplify the signal 
which come from function generator. 

3. Electro-mechanical shaker used to supply 
excitation force to the structure. 

4. Nylon stinger connects the electro-mechanical 
shaker to the structure.   



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

28 

 

The apparatus can be used to supply oscillating 

force (harmonic or repeated) in specific 

magnitude and frequency. Figure 3 shows the 

schematic diagram of the system of application of 

oscillating force. 

 

 
 

Fig. 3. Schematic diagram of oscillating force supply system. 

 

 

4. Solar Collector Setup 
 

The prototypes used in this study are fabricated 

by using special mold. A special composite 

material [polyester resin+ E-glass fiber] was 
applied to obtain the final product of parabolic 

trough solar collector. Before applying the 

polyester and E-glass fiber layers, a layer of resin 

which called (gel coat) is applied. This layer will 

make the interior surface of trough very smooth. 

Fine brushing was applied on the mixture of 

polyester resin and E-glass fiber (chopped fiber) 

to ensure isotropic properties for the material. 

Then the interior surface coated by reflective 

material (mirror film) as shown in Figure (4).  

  

  
 

Fig. 4. The parabolic trough with reflective material 

and support frame. 

 

 

5. Solar Radiation Absorption and 
measuring System  

 
The first component of solar radiation 

absorption system is the reflective surface which 

was obtained by using reflective mirrors film as in 

Figure (4) which has been introduced to 

concentrate solar technology as an alternative to 

mirrored glass reflectors. The second component 

is the copper pipe which is placed parallel to the 

rotation axis along the focal line. The wall 

thickness of the pipe is small (less than 0.5mm) to 

ensure quick transfer of energy from solar 

radiation to the working fluid which flow inside 

the pipe. Black paint is used to cover the outside 

surface of the pipe to increase the absorbance of 

the incident solar irradiance and reduces the 

reflectance. The interior diameter of the pipe is 

(12 mm). Pump of (0.1 kg/s) flow rate is used for 

circulating the water through a closed circuit 

consist of the absorbance pipe, flexible tube, 

connections and insulating storage tank. The 

water temperatures at inlet (Ti) and outlet (To) of 

the absorber tube, water mean flow rate (mw), 

ambient temperature (Tamb ) and solar radiation 

intensity (qs) are measured using two calibrated 

K-type thermocouples probes. The solar radiation 

intensity was measured by using a pyranometer. 

The schematic diagram of the experimental 

apparatus is shown in picture (5).  

The useful heat can be defined on the base of fluid 

difference temperature as [13] 

Q =  mw . Cw . (𝑇𝑒 − 𝑇𝑖  )                                 … (3) 

and the solar collector efficiency is:  

ηc =  
m w .C w .(Te −Ti  )

q s .A
                                      … (4) 

 

 

 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

29 

 

 

Fig. 5. The test rig.  

 

6. Natural Frequencies and Deflection 
Measuring System 

 
Natural frequencies was measured by using 

experimental modal analysis technique. the 

system of measuring included the excitation part 

and the response measuring part. The excitation 

part as previous included function generator, 

amplifier, electromechanical shaker, and nylon 

stinger. The response measuring system included 

piezoelectric accelerometer, charge amplifier and 

digital storage oscilloscope. The experimental 

results of natural frequencies are obtained by 

changing the excitation frequency and recording 

the behavior of the response of the structure to the 

changing of excitation frequency until the 

response of the structure reach to one of the peak 

values. The value of excitation frequency 

corresponding to one peak response equal to one 

value of natural frequencies. The response 

(deflection ) of structure to dynamic excitation are 

also measured and recorded by using piezoelectric 

accelerometer, charge amplifier, and digital 

storage oscilloscope.        

7. Numerical Simulation 
 

The numerical simulation was used in current 

study to perform two types of analysis which are: 

1- Modal analysis to estimate the natural 

frequencies for the structure. The estimation of 

natural frequencies was done to avoid applying 

excitation frequency close to any of these natural 

frequencies. Applying excitation frequencies close 

to any natural frequency will lead to undesired 

response for the structure. 

2- Harmonic analysis to estimate the response of 

the structure to harmonic excitation which will 

apply to structure to simulate the wind loading 

due to vortex shedding. 

The numerical simulation was done by using 

software package (ANSYS Mechanical APDL). 

Simulation parameters such as model geometry, 

boundary conditions, material properties, element 

type, and governing equation solution scheme are 

chosen carefully to give the optimum results of 

natural frequencies and the response to harmonic 

excitation. The numerical results of natural 

frequencies were verified with corresponding 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

30 

 

experimental results. . The detailed information 

about simulation parameters and the figures of 

element type and meshing model are shown in 

appendix 1. 

 

 

8. Results and Discussion 
 

       First, the first five natural frequencies are 

estimated numerically and experimentally. Table 

3 lists the value of the first five natural 

frequencies.

Table 3, 

First five natural frequencies. 

Natural frequency Numerical(Hz) Experimental(Hz) Error percentage 

First 54 49 9.26 % 

Second 63 61 3.17% 

Third 78 71 8.97% 

Fourth 94 88 6.383% 

Fifth 112 103 8.03% 

 

The deformation of parabolic trough solar 

collector due to vortex shedding excitation will be 

introduced. The magnitude and frequency of 

vortex shedding excitation are computed and 

listed in table (4) for each angle of attack and 

wind velocity. The deflection of the parabolic 

trough is evaluated numerically and 

experimentally. Figure (6 and 7) shows the 

deflection of the trough at each wind velocity and 

angle of attack. It is clear from these results that 

both angle of attack and wind velocity have an 

effect on deflection pattern of trough but the 

effect of velocity is larger than the effect of angle 

of attack. This estimation established because the 

difference in deflection pattern when the angle of 

attack is constant and wind velocity changed is 

larger than the difference in deflection when the 

velocity was constant and angle of attack 

changed.  

 

Table 4, 

 Magnitude and frequency of wind excitation force. 

θ = 30˚ Velocity (m/s) fs (Hz) Fo (N) 

 
5 0.75 4.175 

 10 1.5 33.287 

 15 2.24 92.688 

θ = 45˚    

 5 0.7 5.314 

 10 1.4 42.4 

 15 2.1 142.366 

θ = 60˚    

 5 0.652 6 

 10 1.3 49.88 

 15 1.96 168.5 

θ = 75˚    

 5 0.636 2.897 

 10 1.272 23.12 

 15 1.9 77.4 

θ = 90˚    

 5 0.59 0.59 

 10 1.18 4.76 

 15 1.772 16 

 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

31 

 

Figure (8) shows the maximum value of 

deflection of trough versus the angles of attack 

and wind velocity. It is observed that the 

deflection of trough has maximum value at angle 

of attack (θ = 60˚) and minimum value at angle of 

attack (θ = 90˚). This observation was recorded 

for all wind velocity. Also, the value of deflection 

increases when wind velocity increases. This 

result is normal because the increasing of wind 

velocity will increase both excitation force 

magnitude and excitation frequency 

. 

 
 

Fig. 6. Deflection of collector surface at ϴ=30̊ and v=5m/s (dimension in mm). 

 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

32 

 

 

Fig.7. The deflection of trough at each wind velocity and angle of attack. 

  



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

33 

 

 
 

Fig. 8. The maximum deflection of trough versus angle of attack and wind velocity. 

 

 

9. Solar Collector Performance 
Parameters 

 
All parameters necessary for the calculation of 

collector performance are measured during 

different days of months August, September and 

October. These parameters are measured and 

collected first without vortex induced vibration 

and second with vortex induced vibration. Then 

the performance of trough for specific angle of 

attack and amount of solar radiation with vortex 

induced vibration was compared with the 

performance of trough for the same angle of 

attack and amount of solar radiation without 

vortex induced vibration. Table (5) lists the 

typical data necessary for performance analysis 

during the day from October month. Then the 

thermal efficiency of trough are calculated for 

both cases and compared. Figure (9) shows the 

thermal efficiency of solar collector for five 

angles of attack for period of time between 9:00 

A.M. to 2:00 P.M. with and without vortex 

induced vibration. Many observations can be 

drawn from this figure. The first observation is 

that the thermal efficiency of solar collector has 

different trend with angles of attack. 

For (θ =30˚ and 45˚), the thermal efficiency 

was noticed to be high during early hours of the 

day and then decreases around the mid noon and 

then increases after mid noon. This behavior of 

thermal efficiency is noticed for solar collector 

with and without applying vortex induced 

vibration. The reduction in thermal efficiency of 

solar collector around the mid noon can be 

attributed to the inertia of solar collector to 

transfer all solar energy to heat when the solar 

radiation reach to its maximum value at mid noon. 

The maximum value of thermal efficiency for 

both these angles of attack without vortex induced 

vibration is approximately (0.47) while the 

minimum value is (0.35) for (θ=30˚) and (0.33) 

for (θ=45˚). The applying of vortex induced 

vibration on solar collector leads to decreasing the 

value of thermal efficiency significantly. 

Increasing of the wind velocity leads to more 

decreasing in the value of thermal efficiency. The 

larger decreasing in thermal efficiency occurs at 

wind velocity (15m/s). This reduction in thermal 

efficiency attributed to deviation of sun light 

because the changes in collector profile due to 

wind excitation. The amount of these changes in 

collector profile depends on the amount of 

external excitation and the vibrational 

characteristics of the structure. It is clear that the 

increasing of wind velocity leads to excite the 

structure to vibrate in high mode as shown in 

figure (6) because the increasing of wind velocity 

leads to increase the vortex shedding frequency. 

High mode of vibration means more concaves in 

collector surface and more deviations of sun light. 

It is clear from figure (6) that the increasing of 

wind velocity leads to increase the number of 

concaves on collector surface for all angles of 

attack.          

For (θ=60˚, 75˚ and 90˚) without vortex 

excitation, the thermal efficiency was noticed to 

exhibit little fluctuating around specific value 

along the test period. Also, the thermal efficiency 

of solar collector with vortex excitation is noticed 

to be less than that without wind velocity and has 

fluctuated values around specific value except that 

for (θ=60˚) and vortex excitation 15m/s where 

large reduction in thermal efficiency is appear. 

This reduction increases as the local time increase 

until reach to maximum reduction at 12:00 

o’clock. This reduction can be attributed to the 

effect of vortex excitation because that excitation 

was shown to be maximum at angle of attack 

(θ=60˚).  diffusion        

                                              



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

34 

 

Table 5, 

Typical performance data of solar collector. 

Time (hr) Tamb (c˚) Tin (c˚) Tout (c˚) qS=Id+Ib 
W/m

2 
Id 
W/m

2 
Ib 
W/m

2
 

Q (W) ηsys 
 

9:00 33 43.1 44.3 462 77 385 504 0.425 

9:30 34.3 45.2 46.5 515 80 435 546 0.407 

10:00 35 48.7 50.1 571 81 490 588 0.390 

10:30 35.5 54.3 55.8 629 84 545 630 0.375 

11:00 36.1 60.5 62 678 85 593 630 0.345 

11:30 36.7 64 65.6 728 88 640 672 0.340 

12:00 37.3 68.4 70.1 775 91 684 714 0.338 

12:30 37.5 72.4 73.7 701 86 615 546 0.288 

1:00 37.9 77 78.3 688 85 603 546 0.294 

1:30 38.5 80.2 81.6 672 82 590 588 0.323 

2:00 38.9 84.3 85.7 660 79 581 588 0.328 

 

 

Also, the useful heat gain is computed for each 

angle of attack without vortex induced vibration 

and with vortex induced vibration. Figure (10) 

shows the polynomial curveting of the results of 

useful heat gain without and with vortex induced 

vibration for two typical angles of attack (60˚ and 

75˚). It is shown from this figure that the vortex 

induced vibration leads to remarkable decreasing 

in useful heat gain. This decreasing in useful heat 

gain increases with wind velocity at early time but 

become closely at after noon. Also, the presence 

of vortex excitation does not change the trend of 

the gain where the maximum value of gain is 

recorded at about 12:00 o'clock when the vortex 

excitation is applied or not.         

 
 
 
 
 
 
 
 
 
 
 
 
 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

35 

 

             
                                           

θ = 30˚                                                               θ = 45˚ 

 

                       

                                           θ = 60˚                                                                   θ = 75˚ 

 

          θ = 90˚ 

Fig. 9. Thermal efficiency of solar collector at each angle of attack with and without wind induced vibration. 

 



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

36 

 

 

Fig. 10. Useful heat gain for two angles of attack with and without wind induced vibration. 

 

 

10. Conclusions 
 

1- The presence of vortex induced vibration leads 
to decreasing both thermal efficiency and 

useful heat gain but does not change the trend 

of variation with local time. 

2- The increasing of wind velocity leads to more 
decreasing of thermal efficiency and useful 

heat gain.  

3- The maximum decreasing of thermal 
efficiency with vortex excitation is noticed at 

angle of attack (60˚ and 75˚).  

4- The excitation of the structure to vibrate at 
high modes of vibration will increase the 

degradation of the collector performance. 

 

 

Notation 

 
A Collector aperture area [m

2
] 

CL Lift coefficient 

Cw Water specific heat [kJ/kg.K] 

D Object diameter [m] 

F Lift force [N] 

Fs Vortex shedding frequency [Hz] 

mw Water mass flow rate [kg/s] 

PTSC Parabolic trough solar collector 

Q Heat gain [W] 

Ib Beam solar intensity [W/m
2
] 

Id Diffusion solar intensity [W/m
2
] 

qs Total intensity of solar radiation 

[W/m
2
] 

S Strohal number 

Ti Inlet temperature [K] 

Te Outlet temperature [K] 

Tamb Ambient temperature [K] 

U Wind velocity [m/s] 

 

Greek letters 
     

ϴ Angle of attack [degree] 

Ρ Fluid density [kg/m
3
] 

ηc Collector effeciency 

 

 

11. References 
 

[1] Eckhard, et. al, " Eurotrough design issues 
and prototype testing at PSA",  Proceedings 

of Solar Forum, Washington, 2001.   

[2] Yaghoubi M, Sefidbakht S, Kenary A. 
Performance variation of 250kW solar 

thermal power plant for various locations in 

Iran. In: Proceedings of second international 

conference on energy research & 

development, State of Kuwait, 2002. p. 765–

75. 

[3] Paidoussis M.P., “Fluid-Structure Interaction 
Vol.2: Slender Structures and Axial Flow”, 

Elsevier Academic Press, London, 2004. 

[4] Mukund R. Patel “Wind and Solar Power 
systems: Design, Analysis, and Operation”, 

Taylor & Francis, second edition, 2006. 

[5] Holmes JD, Banks RW, Roberts G. Drag and 
aerodynamic interference on microwave dish 

antennas and their supporting towers. J Wind 

Eng Ind Aerodyn 1993;50:263–9.  

[6] [6]Raddu A, Axinte E. Wind forces on 
structures supporting solar collectors. J Wind 

Eng Ind Aerodyn 1989;32:93–100. 

[7] L. Chevalien, J. Norton, Wind loads on solar 
collector panels and support structure, 

Aerospace Engineering Department, Texas 

A&M University (1979).    



Alaulddin Abdulqader                       Al-Khwarizmi Engineering Journal, Vol. 10, No. 3, P.P. 25- 38 (2014) 
 

37 

 

[8] G.A. Kopp, D. Surry, Wind loads on solar 
array, Wind and Structures. 5, (2002) 393-

406. 

[9] Alan G. Davenport, Milos Novak, “Vibration 
of structures induced by wind”, McGraw-Hill 

Book Company, Inc., New York, 1966. 

[10] R. K. Foran, “ Vibration and flow field 
characteristics of a solar concentrator”, 

M.Sc. thesis, Texas Tech University, 1984. 

[11] R. D. Blevins, “Flow-induced vibration”, 
second edition., Kreiger, Malibar, 1994. 

[12] J.A. Peterka, N. Hosoya, R.C. Gee and D. 
Kearney,” Wind Tunnel Tests of Parabolic 

Trough Solar Collectors” National 

Renewable Energy Laboratory, NREL/SR-

550-32282, May 2008. 

[13] Garg. H. P. and Prakash J., "Solar Energy 
Fundamentals and Applications", Tata 

McGraw-Hill Publishing Company 

Limited, Tenth reprint, 2008. 

 

 

Appedix1: Details of  Numerical 

Simulation Parameters 

  
Model geometry: Parabolic trough solar 

collector 

Boundary conditions: clumped-clumped for 

curved edges and free-free for straight edges. 

Material: isotropic composite material with 

density (1455kg/m
3
) and poisons ratio (0.38). 

Governing equations: the equation of motion for 

free and forced vibration 

1- free vibration   𝑚  𝑥  + [𝑐] 𝑥  + [𝑘] 𝑥 = 0 
2- forced vibration  𝑚  𝑥  + [𝑐] 𝑥  + [𝑘] 𝑥 =
𝐹𝑜 𝑠𝑖𝑛𝜔𝑡 
where 

[m] : mass matrix 
 𝑥   : acceleration vector 
[c] : damping matrix 
 𝑥  : velocity vector 
[k]: stiffness matrix 

{x}: displacement matrix. 

The SHELL281 was used in current study. 

This element is suitable for analyzing thin to 

moderately-thick shell structures. The element has 

eight nodes with six degrees of freedom at each 

node: translations in the x, y, and z axes, and 

rotations about the x, y, and z-axes. 

 

 

 
 

Fig. E1.1. Showing element type. 
 

 

 
 

Fig. E1.2. Showing mashing model. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



 (2014) 25- 38، صفحت 3، العذد10دجلت الخىارزهي الهنذسيت الوجلم                                             عالء الذين عبذ القادر            

38 

 

 

 

 

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 خاوعح تغذاد / ي الهٍذسٍحولسه الشؤ

aldina81@yahoo.com : االلىرشوًٌ الثشٌذ  

 

 

 

 الخالصت

ذٍاولد الذساسح ذأثٍش ذزتزب الهٍىك . أداا الىدىم الشىسً الىمىه تشىك المقم الىىافف عىمٍاًا   دساسح ذأثٍش اٌثثاق الذواواخ عمى أخشٌد, الثحثفً هزا 

لذود و .الذساسح حساب أداا الىدىم تاسرخذان الىسة الحشاسي والىفااج الحشاسٌحوذضىٍد . الٍاذح عً اٌثثاق الذواواخ عمى ٌمماي وفااج أداا الىدىم

 ذأثٍشذضىً الثحث حساب الىسة الحشاسي والىفااج الحشاسٌح لمىدىم الشىسً وً لٍاساخ عىمٍح تذوي . الٍاذدح عً اٌثثاق الذواواخ عىمٍاًا  إلثاسجاوحاواج 

ذأثٍش اٌثثاق الذواواخ عٍذ  اٌضا ذٍاولد الذساسحو. ٌىىرج وً الىدىم الشىسً لهزا الغشضاذه ذمٍٍم وىا و. اٌثثاق الذواواخ ذأثٍشاٌثثاق الذواواخ وتىخىد 

 ذه اٌداد. الى لٍه وخرمفح فً وعاوالخ الحىك الٍاذح عً الذواواخ ووزله ذشدد اٌثثالها اياي ذغٍٍش صواٌا الهدىن وسشعح الشٌاذ ذؤدي. ظشوف ذشغٍك وخرمفح

طشٌمح العٍاصش الىحذدج لذساسح الىىاصفاخ الذٌٍاوٍىٍح لمىدىم  اسرخذودوزله . الىدىم الشىسً وأدااالعاللح تًٍ الخىاص الذٌٍاوٍىٍح لمىدىم الشىسً 

 .  ئهن العاللح تًٍ السمىن الذٌٍاوٍىً لمىدىم وأداويالداٌة العىمً لرك فضال عًالشىسً 

 

mailto:aldina81@yahoo.comالبريد
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