MEEN 5314 Notes Schedule for rest of Semester: Today: Ch 5 & Chapter 11 – Dynamics April 14 FEA Model & Ch 5 HW Due, Finish Ch 11, SRIs April 19 No class, work on project during class time. April 21 Exam 3 Review April 26 Exam 3, meet at 9 am

HW - 11.2, 6 (do a, and use NENastran) due. April 29, 8:30 am Project Presentations May 2 Last day to ask questions about project May 4, 5 pm Turn in Final Project Report to Roxanne before this

time. It must have the time stamped on it.

Linear Elements

Quadratic Elements

What about odd situations, can use a triangular “quad”

General Meshing Tips: 1) Watch odd shapes, 2) Sometimes elements don’t “fill” up model space 3) Holes may not be modeled correctly 4) Type or shape of element is important 5) Translation of geometry is important, what to keep, what to not model 6) Small features may or may not be important

Questions:

1) How would you make meshing automatic? 2) When do you not mesh small features? 3) Which types of elements are best? 4) What about welds on an engine mount? 5) What about geometry with small radii?

Chapter 11 – Dynamics – Natural Frequencies, Mode Shapes Homework problems 11.2, 4 due Nov 23, 3:20 pm

Formulation:

What about a Single Degree of Freedom System?

Two DOF Problem:

Multiple Degrees of Freedom (MDOF):

Example

Use NeNastran and notes from web page to demonstrate Natural Frequencies.

Problem 11.4

h 0.025:= b 0.075:= rho 7850:=

Ae h b⋅:= Ae 1.875 10 3−×= Le .8:= i b h3⋅12

:= i 9.766 10 8−×= Ee 200 109×:=

K1 Ee i⋅

Le3

12

6 Le⋅

12−

6 Le⋅

6 Le⋅

4 Le2⋅

6− Le⋅

2 Le2⋅

12−

6− Le⋅

12

6− Le⋅

6 Le⋅

2 Le2⋅

6− Le⋅

4 Le2⋅

⋅:= K1

4.578 105×

1.831 105×

4.578− 105×

1.831 105×

1.831 105×

9.766 104×

1.831− 105×

4.883 104×

4.578− 105×

1.831− 105×

4.578 105×

1.831− 105×

1.831 105×

4.883 104×

1.831− 105×

9.766 104×

=

Me

4.374

0.493

1.514

0.292−

0.493

0.072

0.292

0.054−

1.514

0.292

4.374

0.493−

0.292−

0.054−

0.493−

0.072

=Me rho Ae⋅Le420⋅

156

22 Le⋅

54

13− Le⋅

22 Le⋅

4 Le2⋅

13 Le⋅

3− Le2⋅

54

13 Le⋅

156

22− Le⋅

13− Le⋅

3− Le2⋅

22− Le⋅

4 Le2⋅

⋅:=

Applying Boundary Conditions, eliminating the first and 3rd rows for q1 and q3:

KglobalK11 1,

K13 1,

K11 3,

K13 3,

:= Kglobal

9.766 104×

4.883 104×

4.883 104×

9.766 104×

=

MglobalMe1 1,

Me3 1,

Me1 3,

Me3 3,

:= Mglobal

0.072

0.054−

0.054−

0.072

=

Natural Frequencies, Getting general Eigen values:

lambda genvals Kglobal Mglobal,( ):= lambda8.164 106×

3.888 105×

=

f1lambda02 3.14159⋅

:= f1 454.748= f2lambda12 3.14159⋅

:= f2 99.234=

Mode Shapes:

Shapes genvecs Kglobal Mglobal,( ):= Shapes0.707

0.707

0.707

0.707−

=

Problem 11.4, b and h switched, bending in plane

b 0.025:= h 0.075:= rho 7850:=

Ae h b⋅:= Ae 1.875 10 3−×= Le .8:= i b h3⋅12

:= i 8.789 10 7−×= Ee 200 109×:=

K1 Ee i⋅

Le3

12

6 Le⋅

12−

6 Le⋅

6 Le⋅

4 Le2⋅

6− Le⋅

2 Le2⋅

12−

6− Le⋅

12

6− Le⋅

6 Le⋅

2 Le2⋅

6− Le⋅

4 Le2⋅

⋅:= K1

4.12 106×

1.648 106×

4.12− 106×

1.648 106×

1.648 106×

8.789 105×

1.648− 106×

4.395 105×

4.12− 106×

1.648− 106×

4.12 106×

1.648− 106×

1.648 106×

4.395 105×

1.648− 106×

8.789 105×

=

Me

4.374

0.493

1.514

0.292−

0.493

0.072

0.292

0.054−

1.514

0.292

4.374

0.493−

0.292−

0.054−

0.493−

0.072

=Me rho Ae⋅Le420⋅

156

22 Le⋅

54

13− Le⋅

22 Le⋅

4 Le2⋅

13 Le⋅

3− Le2⋅

54

13 Le⋅

156

22− Le⋅

13− Le⋅

3− Le2⋅

22− Le⋅

4 Le2⋅

⋅:=

Applying Boundary Conditions, eliminating the first and 3rd rows for q1 and q3:

KglobalK11 1,

K13 1,

K11 3,

K13 3,

:= Kglobal

8.789 105×

4.395 105×

4.395 105×

8.789 105×

=

MglobalMe1 1,

Me3 1,

Me1 3,

Me3 3,

:= Mglobal

0.072

0.054−

0.054−

0.072

=

Natural Frequencies, Getting general Eigen values:

lambda genvals Kglobal Mglobal,( ):= lambda7.348 107×

3.499 106×

=

f1lambda02 3.14159⋅

:= f1 1.364 103×= f2lambda12 3.14159⋅

:= f2 297.702=

Mode Shapes:

Shapes genvecs Kglobal Mglobal,( ):= Shapes0.707

0.707

0.707

0.707−

=