Dr. Wenbin Yu
School of Aerospace Engineering
Georgia Institute of Technology
Email: wenbin.yu@ae.gatech.edu
URL: www.ae.gatech.edu/~wyu

Outline of the Course
• Introductions
• Theory of resonance
• Introduction to DYMORE
• Blade dynamics
• The rotor as a filter, airframe dynamic response and coupled blade-fuselage response
• Vibration control devices
• Typical instabilities
– Ground resonance
– Pitch-lag instability
– Pitch-flap instability
– Flap-lag instability

Introduction
• Rotorcraft are dynamic machinery. The dynamic problem are very important
• Some dynamic problem are detrimental to the vehicle performance. If not dealt properly, they could cause catastrophic tragedies
• Three categories of rotorcraft vibration
– Vibrations due to rotor excitation. The frequencies are integral multiples of the rotor rotation speed
– Vibrations due to random aerodynamic excitation. The frequencies are the natural frequencies of the structure
– Self-excited vibrations, such as flutter and ground resonances. Negative damping could cause divergent oscillations

Theory of Resonance
• A single DOF dynamic system
• Natural frequency
• Forced vibration of the system without damping
• The importance of natural frequency for design
• Vibration with damping
• Mathematica example
• Flapping blade
• Lagging blade
10
2
4
6
8
0.5
1 1.5
2

Finite Element Based Formulation for Nonlinear Multibody Systems
• Model configurations of arbitrary topology:
– Assemble basic components chosen from an extensive library of structural and constraint elements
• Avoids modal expansion
• This approach is that of the finite element method which has enjoyed, for this very reason, an explosive growth
• This analysis concept leads to simulation software tools that are modular and expandable
• Elements of the library can be validated independently

Rotor as a Nonlinear Multibody System

Transmission as a Nonlinear Multibody
System

Simulation of Rotor on Ship Board
• The complete model involves:
– 17 beam elements,
– 5 prescribed displacements,
– 1 prismatic joint,
– 1 relative displacement,
– 21 rigid bodies,
– 12 revolute joints,
– 12 relative rotation,
– 3 spherical joints,
– 1 universal joints,
• For a total of 950 degrees of freedom .

Element Library: Structural Elements
• Rigid bodies
• Flexible joints : linear and torsional springs and damper
• Cable element
• Beam elements: geometrically exact, shear deformable.
Capable of modeling all the elastic coupling effects arising from the use of advanced laminated composite materials
• Shell elements: geometrically exact, shear deformable, modeling of composite material effects
The finite element formulation is used for all elements, no modal reduction is performed

Element Library: Beam Elements
• Geometrically exact beam elements. Six degrees of freedom (three displacements, three rotations) per node
• Accounts for
– Shearing deformation effects
– Offsets of the center of mass, shear center, and centroid
– All elastic couplings that can arise from the use of laminated composite materials (Fully coupled 6x6 stiffness matrix)
– Material viscous dissipation

Element Library: Shell Elements
• Geometrically exact shell elements. Five degrees of freedom (three displacements, two rotations) per node.
Locking free element is achieved using the mixed interpolations of strains tensorial components
• Accounts for
– Shearing deformation effects
– Offsets of the center of mass
– All elastic couplings that can arise from the use of laminated composite materials (Fully coupled 8x8 stiffness matrix)
– Material viscous dissipation

The Six Lower Pairs

Blade Dynamics
• Blade dynamics is important because
– High blade vibratory response results in high stresses
– High blade vibratory response leads to high fuselage vibration levels
– Blade resonances and mode shapes are important in stability analysis of rotor systems
• DYMORE example for a single blade
• DYMORE example for a complete rotor (ITU LCH)
• FAN plot for ITU LCH
• Changing frequencies by playing with weight

DYMORE Rotor Model
The DYMORE model for the ITU LCH Rotor

Fan Plot of Frequencies for the Rotor
4
3.5
3
2.5
2
1.5
1
FLAP1
L-L1
FEATHERING1
FLAP2
1P
2P
3P
0.5
0
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1.1
1.2
1.3
1.4
Fan plot in Vacuum for the ITU LCH Rotor (Verifying the Auto-Trim concept)

Dynamic Responses - Displacements
0.4
0.2
-0.2
0
-0.4
-0.6
-0.8
0 0.5
1 1.5
Time Sec
2 2.5
3
Time history of blade tip displacement red: axial displacement; green: in-plane displacement; blue: out-of-plane displacement

0.08
Dynamic Responses
0.06
0.04
0.02
0
-0.02
0 0.5
1 1.5
Time Sec
2 2.5
Time history of blade tip rotations red: pitching; green: flapwise direction; blue: chordwise direction
3

2500
2000
1500
1000
500
0
0
Dynamic Responses - Forces
0.5
1 1.5
Time Sec
2 2.5
Time history of forces at different locations red: at flex root; green: at flex tip
3

1000
Dynamic Responses - Moments
500
-500
0
0 0.5
1 1.5
Time Sec
2 2.5
Time history of moments at different locations red: at flex root; green: at flex tip
3

Pitch-lag Instability

Pitch-flap Instability

Flap-lag Instability
–Ground resonance

Conclusions
• Dynamic problem are very important for rotorcraft. A good design must come from a good understanding to dynamic behavior of the vehicle
• Locating the natural frequencies of the system is the key to avoid resonance
• DYMORE is a handy tool to deal with rotorcraft dynamics
• Either passive (damping) or active devices (vibration absorbers) can be used to reduce the resonance or shift the natural frequencies
• Dynamic instabilities should and can be avoided by design tradeoffs