1 MODAL ANALYSIS. 2 Tacoma Narrows Galloping Gertie.

1

MODAL ANALYSIS

2Tacoma Narrows Galloping Gertie

3Flutter of Boeing 747 wings

4

B52 parked B52 flying

Note deflection of wings

51DOF.SLDASM

6

Every structure has its preferred frequencies of vibration, called resonant frequencies. Each such frequency is characterized by a specific shape of vibration. When excited with a resonant frequency, a structure will vibrate in this shape, which is called a mode of vibration.

Recall that structural static analysis calculates nodal displacements as the primary unknowns:

where [K] is known as the stiffness matrix, d is unknown vector of nodal displacements and F is the known vector of nodal loads.

In dynamic analysis we additionally have to consider damping [C] and mass [M]

[K]d = F

In a modal analysis, which is the simplest type of dynamic analysis we investigate the free vibrations in the absence of damping and in the absence of excitations forces. Therefore, the above equation reduces to:

7

Non-zero solutions of a free undamped vibration present the eigenvalue problem:

Solutions provide with eigenvalues and associated modal shapes of vibration

2i i

2i

i

K - ω M = 0

ω eigenvalue (square of circular frequency)

eigenvector (shape)

ω circular frequency [rad/s]

ωf = frequency [Hz]

2π1

f =T

8

In resonance, inertial stiffness subtracts from elastic stiffness and, in effect, the structure loses its stiffness.

The only factor controlling the vibration amplitude in resonance is damping. If damping is most often low, therefore, the amplitude of may reach dangerous levels.

Note that even though any real structure has infinite number of degrees of freedom it has distinct modes of vibration. This is because the cancellation of elastic forces with inertial forces requires a unique combination of vibration frequency and vibration mode (shape).

Note, that the equation of free undamped vibrations can be re-written to show explicitly that in resonance inertial forces cancel out with elastic forces.

2ii i

K = ω M

Elastic forces

Inertial forces

9

Material density must be defined in units derived from the unit of force and the unit of length.

[mm] [N]unit of mass tonneunit of mass density tonne/mm3

for aluminum 2.794x10-9

[m] [N]unit of mass kgunit of mass density kg / m3

for aluminum 2794

[in] [lb]

unit of mass lbf = slug/12unit of mass density slug/12/in3

lbf s2/in4

For aluminum 2.614x10-4

Notice that the erroneous mass density definition (kg / m3 instead of tonne / mm3) will results in part mass being one trillion (1e12) times higher.

10

U SHAPE BRACKET

model file U BRACKET

model type shell

material alloy steel

thickness 2mm

restraints hinge

load none

objective

• demonstrate modal analysis

• study convergence of natural frequencies

• defining supports for shell element model

• properties of lower and higher modes of vibration

hinge support(no translations)

hinge support(no translations)

U BRACKET

SAE models

11

cantilever beam.SLDPRT

04 models modal

12

TUNNING FORK

Chapter 6

PLASTIC PART

Chapter 6

13

truck.SLDPRT

04 models modal

car.SLDPRT

04 models modal

14

EXERCISE helicopter blade

Model file ROTOR

Model type solid

Material 1060 Alloy

Supports fixed to the I.D.

sym B.C. to hub

Loads centrifugal load due to 300RPM

Units mm, N, s

Objectives

• Modal analysis without pre-stress

• Modal analysis with pre-stress

Analysis is conducted on one blade only.

ROTOR

CHAPTER 21

15

pendulum 02.SLDPRT

04 models modal