Ae611 Final Project2013(2)

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Figure 1. Geometry of the wing box structure of a tapered unswept wing.

Figure 2. Exploded view of the wing box structure showing spars, ribs, and top and bottom cover panels.

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thickness of 5 mm. The wing structure has three spars, one each at the leading edge, center and trailingedge of the wing-box. The shear webs of the spars have a constant thickness along the length of the wing.Their thickness is 3.5 mm. The wing-box has three spars and three ribs equally spaced along the chordwiseand spanwise directions, respectively. The thicknesses of the ribs can be selected among 2 possible options:t1 = 2.5mm and t2 = 4mm (see below for the explanation). At the top and bottom edges of the shearwebs there are spar caps. Model them as ROD elements (cross section area A = 5cm2). Model the wingtop and bottom panels, the shear webs of the spars and the ribs with 4-noded quadrilateral (QUAD4) shellelements. Assume clamped boundary conditions at the root of the wing. The material properties ofall the

wing components are the following: E= 70.0 GPa, = 0.33, = 2700 Kg/m3. Eis the elastic modulus, is Poissonss ratio and is the density.

1. Model the geometry of the wing with FEMAP (suggestion: find the coordinates of the points thatidentify the surfaces and create the geometry; please note that the caps are not created at this stage.You can also take advantage of the symmetry and mesh only half wing box and mirror the mesh. Ifyou do so please be careful to avoid coincident nodes and coincident ROD elements in the central sparcaps.)

2. Define the material properties in FEMAP. Define a plate property that is used to model theribs withthickness t1. Define a plate property that is used to model the ribs with thickness t2. Define a plateproperty that is used to model the top skin panels. Define a plate property that is used to model thebottom skin panels. Define a plate property that is used to model the leading edge spar. Define a

plate property that is used to model the central spar. Define a plate property that is used to modelthetrailing edge spar. Define a ROD property that is used to model the spar caps.

3. Mesh the spars, top skin, bottom skin and spar caps.Do not meshthe ribs (note that if the ribs are notmeshed they are not considered in the analysis). Please verify thatno coincident nodes are present(in FEMAP this is done with the command ToolsCheckCoincident Nodes). Do not imposethe boundary conditions at the root of the wing (so for this analysis the wing is an unconstrainedstructure).

Calculate the first 15 natural modes and frequencies and comment about the first six modes.

Identify the shape and frequency relative to the first elastic mode.

Identify the shape and frequency relative to the second elastic mode.

Refine the mesh and perform a convergence test (which you have to document).

4. Impose the boundary conditions at the root of the wing (all the analyses that follow must be performedwith the cantilevered wing so the root is always constrained).

Calculate the first 15 natural modes and frequencies and comment about the first six modes.

Identify the shape and frequency relative to the first bending mode.

Identify the shape and frequency relative to the first torsional mode.

Refine the mesh and perform a convergence test (which you have to document).

5. Select the thickness t1 for the rib near the root (Rib # 1) of the wing. In addition to the existingmesh, mesh this rib and verify that in your FEM model there are no coincident nodes. Calculate themodes and natural frequencies and discuss the differences you find with the previous findings (case of

wing box with no ribs). What happens to the first torsional mode? What happens to the first bendingmode? What happens to the high frequency modes? What is the effect of this rib? Comment thefindings.

6. Change the thickness of therib near the root (Rib # 1) and select the value t2. Calculate the modesand natural frequencies and discuss the differences you find with the previous findings. What happensto the first torsional mode? What happens to the first bending mode? What happens to the highfrequency modes?

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Part # 2

Program in MATLAB (no other language will be accepted) Newmarks Method for the integration ofthe equations of motion for a multi-degree-of-freedom system problem subjected to an external vector ofapplied loads (the components change with time according to several assigned laws assumed by the user).The program must be able to integrate in time equations of the following type:

M q +C q +K q = Q (1)

Where Q is a vector of generalized forces, q is a vector of generalized coordinates, Mis the mass matrix, Cis the damping matrix and Kis the stiffness matrix.The program must have the following features:

The user selects the number of degrees of freedom and assign the mass, stiffness and damping matrices(via m file).

The user selects the function used to describe how each component of the vector Q changes with theparameter t indicating the time. For simplicity, assume that each component Qi of the generalizedforce vector can either be a constant(in that case the user needs to provide the constant) or harmonic[in that case the user needs to provide the amplitude Q0i and frequency i and the component isQi= Q0icos (it)]

The user selects the size of time step t.

Your code must be validated according to the following instructions:

1. You have to prove that the approximated numerical solution for the free vibration response of one-degree-of-freedom system is in good agreement with the analytical exact solution of the same problem.AssumeM= 10[Kg ], K= 2250[N/m] and C= variable [N s/m]. M is the mass, Kis the stiffnessand C is the damping coefficient. The initial conditions are q(0) = 2.4[m]; q(0) = 0.5[m/s]. Theanalysis is performed fromt = 0 to a meaningful time (which you have to select) tmax. You have topresent the following studies:

Analytical derivations and solution of the differential equation.

Convergence study for different sizes of the time step t.

Perform the analysis for the following cases:

C= 0 [N s/m].

C= 30 [N s/m]

C= 300 [N s/m]

C= 3000 [N s/m]

2. You have to prove that the approximated numerical solution for the forced vibration response ofone-degree-of-freedom system is in good agreement with the analytical exact solution of the sameproblem. Assume M = 10[Kg ], K= 2250[N/m], C= variable [N s/m] and Q = 8sin(15t). M isthe mass, Kis the stiffness and Cis the damping coefficient. The initial conditions are q(0) = 2.4[m];q(0) = 0.5[m/s]. The analysis is performed from t = 0 to a meaningful time (which you have to

select)tmax. You have to present the following studies:

Analytical derivations and solution of the differential equation (both the complementary andparticular solutions must be considered).

Convergence study for different sizes of the time step t.

Perform the analysis for the following cases:

C= 0 [N s/m].

C= 30 [N s/m]

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C= 300 [N s/m]

C= 3000 [N s/m]

3. You have to prove that the approximated numerical solution for the free vibration response of two-degree-of-freedom system is in good agreement with the analytical exact solution of the same problem.Assume the following mass matrix:

M= 20 10.5

10.5 40 (2)

Assume the following stiffness matrix:

K=

2600 1600

1600 4800

(3)

Assume the following initial conditions:

q =

10

5

; q =

5

2

(4)

The analysis is performed from t = 0 to a meaningful time (which you have to select) tmax. Youhave to present the following studies:

Analytical derivations and solution of the system of differential equations. The analytical deriva-tions will be performed as follows:

The natural frequencies and modes are calculated

The normal modes are determined

The equations of motion are uncoupled by using the normal mode matrix, and solved withan appropriate set of initial conditions determined from the given initial conditions.

Convergence study, of each generalized coordinate, for different sizes of the time step t.

4. Consider a two-degree-of-freedom system subjected to no external excitations. Assume the followingmass matrix:

M= 20 10.510.5 40

(5)Assume the following stiffness matrix:

K=

2900 1500

1500 4900

(6)

Assume the following damping matrix:

C=

35 10

10 47

(7)


q =

10

5

; q =

5

2

(8)




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The light damping approximation is considered



5. Consider a forced two-degree-of-freedom system. Assume the following mass matrix:

M=

35 10.5

10.5 45

(9)


K=

2300 200

200 4570

(10)


C=

5 1.1

1.1 7

(11)


q =

1.0

1.5

; q =

1.7

1.9

(12)

Assume the following excitation vector (=

7

15):

Q =

10cos(t)

5cos(t)

(13)

The analysis is performed from t = 0 to a meaningful time (which you have to select) tmax. You

have to present the following studies:







6. Consider a forced two-degree-of-freedom system. Assume the following mass matrix:

M=

25 10.5

10.5 35

(14)


K=

3700 2300

2300 4700

(15)

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C=

5 0.5

0.5 9

(16)


q = 3

5

; q = 7

9

(17)

Assume the following excitation vector (=

4

15):

Q =

10 cos t

7

(18)



The natural frequencies and modes are calculated The normal modes are determined




7. Consider a forced two-degree-of-freedom system. Assume the following mass matrix

M=

2 0

0 5

(19)

Assume the following stiffness matrix

K=

6 2

2 4

(20)

Assume the following damping matrix

C=

0 0

0 0

(21)


q =

0

0

; q =

0

0

(22)

Assume the following constant excitation vector:

Q =

0

7

(23)



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Write a technical report with the discussion of this study (including the convergence tests you performed).Include the key equations of Newmarks method and show the off-line and step-by-step procedure. Presenta flowchart of the code. The quality of the flowchart will be crucial in the determination of your grade.

The technical reportmust comply with the following guidelines:

It has to be written in WORD. Please use the template posted on blackboard (it is the AIAA journaltemplate). The maximum number of pages is 40. Reports exceeding this page constraint or with acharacter size less than 11pt or with wrong page margins will receive a 25% penalty.

An Abstract should be present.

An Introduction should be present.

Technical discussions and results must be presented. These are crucial parts in the determination of

the final grade. A section with the conclusions must be included.

Provide citations to all reference you use at the end of your report.

Attach the MATLAB code as an appendix to the document. The MATLAB code of the appendixdoesnot count in the 40-page limit.

An electronic copy of the MATLAB code (all the m files) must be sent to the instructor. Please notethat the code must have the capability to handle systems with more than 2 degrees of freedom.

An electronic copy of the report must be sent to the instructor.

Part # 3Consider a two-degree-of-freedom system subjected to no external excitations. Assume the following

mass matrix:

M=

20 10.5

10.5 40

(24)


K=

2900 1500

1500 4900

(25)


C=

35 1010 47

(26)

Note that this problem was also analyzed in Part # 2. You have to present the following studies:

Writing of the equations of the free vibration problem in state space form.

Calculate the damped modes.

Calculate the damped modal masses and frequencies.

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Write a technical report with the discussion of this study. Include the key equations of the damped modes.Present a flowchart of the code. The quality of the flowchart will be crucial in the determination ofyour grade.

The technical reportmust comply with the following guidelines:

It has to be written in WORD. Please use the template posted on blackboard (it is the AIAA journaltemplate). The maximum number of pages is 10. Reports exceeding this page constraint or with acharacter size less than 11pt or with wrong page margins will receive a 25% penalty.

An Abstract should be present.

An Introduction should be present.

Technical discussions and results must be presented. These are crucial parts in the determination ofthe final grade.

A section with the conclusions must be included.

Provide citations to all reference you use at the end of your report.

Attach the MATLAB code as an appendix to the document. The MATLAB code of the appendixdoesnot count in the 10-page limit.

An electronic copy of the MATLAB code (all the m files) must be sent to the instructor. Please notethat the code must have the capability to handle systems with more than 2 degrees of freedom.

An electronic copy of the report must be sent to the instructor.

How this exam will be graded

The grades will be assigned according to the following criteria:

Correctness and completeness of the results/responses.

Quality of the presentation of the results (e.g., clear, easy to follow, etc.).

This project is graded on a 100 point basis. However, exceptional works can receive up to 110

points.

Important Information

The exam must be given to the instructor in both electronic format (WORD document or PDF; pleasenote that TWO reports must be written: see Part 1 and Part 2) and hardcopy by December 13th 4:00PM.This is an exam and no exceptions will be allowed. It is the students responsibility to make sure that theexam is received by the instructor before the deadline.This is a group project. The students will createthree three-people groups. Each group must select aleader. The leader will supervise the work of the team members. Moreover,the leader will write a separatereport with the following information:

Tasks performed by each student (including the leader).

The leader will haveN 20 points available (Nis the total number of students of the group) to giveto the students of the group. The points will be assigned by the leader on a meritcriteria and will beused by the instructor to properly grade each single student. The following rules apply:

The points cannotbe equally subdivided among the team members.

The points mustbe assigned to all students of the group.

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The difference between the grades of the students must be at least5 points. For example, supposethat a group has 3 students: A,B, and C. In that case the leader will have 60 available points.The subdivision A = 30, B = 20, and C = 10 is acceptable. The subdivisionA = 25, B = 15,and C= 20 is also acceptable. On the contrary, the subdivision A = 18, B = 22, and C= 20 isnotacceptable because the difference between the grades of students A,B , andCis smaller than5.

The members (besides the leader) of the team will also prepare a report in which they report their

activities performed to complete this project. If even one single student of a selected groupdoes not agree with the point subdivision provided by the leader, an ORAL EXAM will takeplace for all the involved students of that group. The oral exam will be only about this finalproject and will be used to correctly distribute the points according to the actual merit of thestudents. The oral exam for the groups that did not find a unique subdivision of the points willtake place on Friday December 13th 10:00AM in the instructors office.Please note that the instructorwill not follow all the indications of the leader in the subdivisionof the points to each student. They have only an indication purpose

Academic honesty

All students admitted to SDSU have signed a statement of academic honesty committing themselves tobe honest in all academic work and understanding that failure to comply with this commitment will result

in disciplinary action. This statement is a reminder to uphold your obligation as a student at SDSU and tobe honest in all work submitted and exams taken in this class and all others.

Plagiarism

Plagiarism consists of passing off as yours the work that belongs to someone else. As such, you willbe committing plagiarism if you present someone elses work as your own, even with the other personsconsent.

Miscellaneous

.... No late homework will be accepted. Submitting copy of a homework prepared by somebody else

is considered cheating.Submitting a project which is copied in full or in part from other students work is considered

plagiarism.Collaboration outside the group is not allowed.

The due date is December 13th (4:00 PM).

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Ae611 Final Project2013(2)