Modal analysis determines the natural vibration characteristics of a structure. Natural frequency depends on mass, stiffness, and boundary conditions, and is important to understand possible resonance. Resonance occurs when natural frequency coincides with excitation frequency, and can cause excessive deformation. The document provides an example modal analysis of a simply supported aluminum plate, calculating its natural frequencies. Finite element analysis is used to model the system and structures are substantiated to have sufficient margin of safety under limit loads.

1. FINITE ELEMENT ANALYSIS
Prepared by:
KARTHIKEYAN.K
Associate Manager, Siemens Gamesa Renewable Energy.

2. 1. MODAL ANALYSIS
 What is Modal Analysis?
• Structures are connected to the vibrating systems
• Excessive wear of bearings
• Noise
• Study of the dynamic properties of linear structures
• Helps to determine the vibration characteristics
• Natural frequency

3. 1. MODAL ANALYSIS
 Why is Natural Frequency (Eigenfrequency)
• Frequency of the system when there is no force or damping factor
• Functions of mass, stiffness and boundary conditions
 Why important?
• To understand possible resonance occurring
• Input data for transient dynamic analysis, harmonic analysis, random vibration etc.
 Why is Resonance
• Oscillation of the structure at high amplitude with specific frequency
• Natural frequency coincides with the excitation frequency
• Can cause excessive deformation

4. 1. MODAL ANALYSIS
 Geometry and Boundary Conditions
b=1m
a=2m
x
y
Formula of Natural Frequency Simply Supported Plate
ω𝑚𝑛 =
π2
D
ρ𝑚
h
m2
a2 +
n2
b2
2
Properties of Aluminum
E = 70 GPa
ρ = 2700 kg/m3
h = 2 mm
ν = 0.33
ω𝑚𝑛 = Natural circular frequency in radians per unit time
f𝑚𝑛 = Natural frequency of vibration in cycles per second (Hertz)
m = half wave no. of long edge
n = half wave no. of short edge
D = flexural stiffness
ρ𝑚 = density
h = thickness of plate
a = length of long edge
b = length of short edge
f𝑚𝑛 =
ω𝑚𝑛
2𝜋
Meaning of the Modes (m,n) (Mode Shapes)
b
a
y
z
n
x
z m
t
Example of
Mode (3,1)
D =
Eh3
12(1−ν2
)
Calculation Natural Frequency of the Plate
f11 =6.115 Hz
f21 =9.78 Hz
f31 = 15.9 Hz
f32 = 20.79 Hz
w

5. • Margin-Of-Safety
The Margin-Of-Safety(MOS) is defined as,
MOS = (Design Strength/ Design load) - 1 = FOS - 1
1. STRUCTURAL SUBSTANTION

6. The substantiation of the installations will be carried out for flight condition limit loads with
pressurized fuel tank conditions. The MOS based on yield/ultimate strength have been presented
for substantiation and the MOS limit adopted is presented.
Analysis Failure Strength MOS limit
Flight condition
(limit loads)
Yield 0.15
Ultimate 0.50
Emergency crash
loading condition
Ultimate 0.00
MOS limit for different analysis & failure strength criteria
1. STRUCTURAL SUBSTANTION

7. The host structure of the installations are assumed to be made up of ___________
aluminum and having 5% MOS. Therefore, under flight condition limit loads ,the host
structur is substantiated when stresses are shown to be less than 5% of the yield
strength of _______ aluminum.
2. SUBSTANTION OF HOST STRUCTURE

8. • Bolts
The specification of the bolts ,i.e. the nominal diameter and minimum tensile
strength is obtained from the manual and consequently, the shear strength for
bolts (Ref. Bruhn )is taken as 1/√3 of the tensile strength. The ultimate single shear
load and ultimate tensile load of threaded fasteners are obtained from the
reference manual respectively .However, in the case of fastener where the ultimate
tensile & shear loads were not available in the reference doc the same was taken as
a product of minimum tensile/shear strength and effective cross-sectional area of
the respective fastener.
3. SUBSTANTION OF FASTENERS

9. The MOS for the bolt is expressed as,
Where
The fitting factor is taken as 1.15
3. SUBSTANTION OF FASTENERS
Ref. Bruhn. Analysis and Design of Flight Vehicle Structures.
Max shear load

10. • Unless otherwise specified a factor of safety of 1.5 must be applied to the limit load:
 Ultimate load = 1.5 x limit load
 The structure must be able to support ultimate loads without failure.
 The structure must be able to support limit loads without detrimental permanent deformation.
3. SUBSTANTION OF FASTENERS
• The general procedure is to design a structure to zero margin. The Margin of Safety (MOS/MS) for the
stress analysis is equal to zero or greater, but is never a negative.

11. 3. SUBSTANTION OF FASTENERS

12. • Rivets
The specification of the rivets, i.e. the nominal diameter and minimum shear
strength is obtained from the references manual. Consequently, the allowable
tensile load is taken as 40% of the allowable shear load. The MOS is computed as
follows,
3. SUBSTANTION OF FASTENERS

13. The inertial loading for missile as per MIL-STD-8591, Appendix C shall be adopted
for the analysis. The load calculation equation and missile parameters are
mentioned below.
Case Condition
Load Factor (g)
FOS
nx ny nz
FC1
Cruise phase
- - 2.5
1.5
FC2 - - 1.5
FC3 - -1.5 -2.5
FC4 Boost phase 0.5/0.85 0.5/0.85
4. FLIGHT CONDITION LIMIT LOADS

14. The material properties have been taken from reference manual and their
parameters of the materials used are listed below.
Material
E
(Gpa)
µ
ρ
(kg/m3)
σy
(MPa)
σu
(MPa)
σby
(MPa)
σbu
(MPa)
AL 6061 T652 68.3 0.33 2713 241.32 262 372.32 420.58
5. MATERIAL PROPERTIES

15. LRU mass factor of 1.25 has been applied in the analysis that will account for the
wire bundle, miscellaneous mass and for any inaccuracy between the actual and
assumed CG. The CG for the LRU’s has been assumed to be at the geometric center
of the equipment.
6. LRU MASS FACTOR

16. • The finite element modelling has been performed using HyperMesh® and analysis using MSC
Nastran®. The post processing have been made using MSC Patran® (Stress plot) and HperView®
(Mode shapes and vibration response).
• In the finite element modelling, the mesh quality parameters used meshing is listed in Table 1. The
rivet/bolt holes have been modelled by having two rings of elements (eight elements in each) around
the hole (Figure 1), where the diameter of the ring is maintained at two times the diameter of the
hole.
• The bolts and rivets have been modelled as CBUSH elements (Figure 2) with translational stiffness of
106 along X, Y and Z direction and rotational stiffness of 108 about X, Y and Z directions, respectively.
• The LRU units are modelled as lumped masses at the LRU C.G. and element connectivity to the
structure is provided using RBE2 elements to their rivet/bolt connections (Figure 3 & 4).
• The installations meshed with QUAD4 & TRIA3 elements and the percentage ratio of TRIA3 to QUAD4
is maintained below 5%. The element size used for meshing is mentioned in individual sections of the
installations.
7. FINITE ELEMENT ANALYSIS

17. Parameters Value
Warpage <5
Aspect Ratio <5
Skewness 60
Jacobian >0.7
Triangular Element
Min. Angle (deg) >20
Max. Angle (deg) <120
Quadrilateral Element
Min. Angle (deg) >45
Max. Angle (deg) <135
Table 1: Adopted Mesh Quality Parameters
7. FINITE ELEMENT ANALYSIS

18. 7. FINITE ELEMENT ANALYSIS

19. 8. PRE-TENSION LOAD ANALYSIS
• External loads (tensile) tend to separate members, bolt force cannot increase much unless the members
separate, the higher the preload the less likely the members are to separate.
• For external loads tending to shear the bolt, the higher the preload the greater the friction force resisting the
relative motion in shear.
• Higher preload reduces the dynamic load on the bolt because the effective area of the clamped members is
larger.
• Higher preload results in maximum protection against overloads, which can cause joint separation, and
provides protection against thread loosening.
P
P
P
P

20. 8. PRE-TENSION LOAD ANALYSIS
• Tightening torque related to preload and bolt diameter. The constant value, 0.2, remains approximately
the same regardless of the bolt size.
• For critical applications a torque wrench should be used to apply the proper preload.
• Initial tension (preload):
Ai – Tensile stress area; Sp – Proof strength
• Bolt should not be reused if tightened to 90% of the proof load, yielding may have occurred.
 For static loads and permanent connection, tighten to 90% of the proof load.
 For fatigue loading and non-permanent connections (reused fastener) tighten to 75% of the
proof load.
T = K*Pi*d
Pi = 0.75*Sp*Ai
K
(nut factor/friction factor/tightening factor)
Type of bolt
0.2 Steel bolts (with no plating)
0.15 Steel bolts with Cadmium plating
0.28 Steel bolt with Zinc plating
0.18 Steel bolts with lubrication

21. 8. PRE-TENSION LOAD ANALYSIS
• Calculate bolt preload as
σy – yield strength of the bolt material; F – external load applied on the bolt
• Equivalent stress
Stress by torsion is relaxed after tightening
equivalent stress should remain below the yield strength (linear elastic)
Pi = {[σy*(πD^2/4)]/FOS} - F
F
F
F
F
P
F+P
Total load
on the bolt
External load
on the flanges
σe = sqrt(σt^2 + 3τ^2)

22. 8. TYPES OF CONNECTIONS
• Mechanically fastened joints are conveniently classified according to the type of
forces to which the fasteners are subjected. These classes are (1) shear, (2) tension,
and (3) combined tension and shear. Under category 1 the fasteners are loaded
either in axial or eccentric shear. If the line of action of the applied load passes
through the centroid of the fasteners group, then the fasteners are loaded in axial
shear. In eccentric shear the shear force does not pass through the centroid of the
fastener group. This results in a torsional moment on the fastener group that increases
the fastener shear stresses. This loading condition is referred to as eccentric
shear.

23. • A system level approach is used in modeling fastener fixities as it is impractical to model actual fasteners.
• This approach assumes:
 Fasteners and member interface is infinitely rigid
 Shear and tensile loads are transmitted through the fastener joint
 Contact between members is not modeled
 In the max stress locations(cross piece) reacting fastener forces are mainly acting in the fastener tensile
direction therefore accounting for member friction will not greatly reduce resulting stress.
• It should be noted that stresses at the fixities will be unrealistic due to the nodal fixity and may causes an
artificially induced stress concentration (singularity). Unless stress values are below desired material
durability limits, the stresses in these areas are ignored and stress values are sampled away from the
singularity where stress values are not influenced by the singularity. Extracting detailed joint stresses
usually entails a detailed fastener analysis or a detailed such model.
9. SYSTEM LEVEL APPROACH