free vibration analysis of micro-satellite...

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CHAPTER 5

FREE VIBRATION ANALYSIS OF MICRO-SATELLITE

STRUCTURE

5.1 NORMAL MODE ANALYSIS

Normal modes analysis computes the natural frequencies and mode

shapes of a structure. The natural frequencies are the frequencies at which a

structure tends to vibrate if subjected to a disturbance. The deformed shape of

the structure at a specific natural frequency of vibration is termed as normal

mode of vibration. Some other terms used to describe the normal mode are

mode shape, characteristic shape, and fundamental shape. Each mode shape is

associated with a specific natural frequency. Natural frequencies and mode

shapes are functions of the structural properties and boundary conditions. If

the structural properties change, the natural frequencies change, but the mode

shapes may not necessarily change. If the boundary conditions change, then

both the natural frequencies and mode shapes change.

Normal mode analysis is also called the real Eigen value analysis.

Normal mode analysis forms the foundation for a thorough understanding of

the dynamic characteristics of the structure. Normal mode analysis is

performed for the following reasons:

1. Assessing the dynamic interaction between a component and

its supporting structure, if the natural frequency of the

supporting structure is close to an operating frequency of the

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component then there can be a significant dynamic

amplification of the loads.

2. Assessing the effects of design changes on the dynamic

characteristics.

3. Using the modes in the subsequent forced response analysis.

4. Using the natural frequencies as a guide to select the proper

time or frequency step for a transient and frequency response

analysis.

5. Assessing the degree of correlation between modal test data

and analytical results.

6. In normal modes analysis the Eigen values and Eigen vectors

of the model are determined. For each Eigen value, which is

proportional to a natural frequency, there is a corresponding

Eigen vector, or mode shape.

Normal modes analysis for the undamped free vibrations is as

follows:

[K – liM] { fi }=0

Where ‘K’ is the system stiffness matrix, ‘M’ is the system mass matrix and

‘li’ and ‘fi’ are to be computed. ‘li’are the Eigen values and ‘fi’ are the Eigen

vectors or mode shapes. The Eigen values are related to the natural

frequencies as follows:

fi = li/2

Eventhough number of methods is available for Eigen value

extraction, Lanczos method is used for extracting Eigen values for the

dynamic analysis of the micro-satellite

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5.1.1 Lanczos Method

The Eigen value extraction for a particular model depends on four

factors (i) the size of the model (the total number of degrees-of-freedom as

well as the number of dynamic degrees-of-freedom), (ii) the number of Eigen

values desired, (iii) the available real memory of computer, and (iv) the

conditioning of the mass matrix (whether there are mass less degrees-of-

freedom).

The Lanczos method overcomes the limitations and combines the

best features of the other methods. It requires that the mass matrix be positive

semi definite and the stiffness matrix be symmetric. Like the transformation

methods, it does not miss roots, but has the efficiency of the tracking

methods, because it only makes the calculations necessary to find the roots

requested by the user. This method computes accurate Eigen values and Eigen

vectors. The Lanczos method is the preferred method for the most medium to

large-sized problems, since it has a performance advantage over other

methods.

In general, the Lanczos method is the most reliable and efficient,

and is the recommended choice of the FEA software MSC NASTRAN.

5.2 FREE VIBRATION ANALYSIS

The natural frequencies of the satellite are of special interest as the

sensitive electronic instrumentation and onboard computers should not be

affected due to the vibrations developed on the satellite structure. The

frequencies of vibration of launch vehicle at the time of launching need to be

considered to avoid conditions of resonance occurring in the satellite during

the launch. In the current problem the frequencies of vibration of launch

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vehicle during launch are specified. The aim is to design the satellite

structure or modify the given structure such that resonance would not occur.

In general there are four methods for analysis and dynamic control

of satellite which are:

1) Using passive or active damping: This method use materials

with high damping coefficient or in active case, the actuators

are used.

2) Controlling and decreasing the dynamic response of structure

in sensitive sections: In this method, actuators are used in

specified sections, which are very sensitive to vibration and

displacement. For instance, in the satellite, the camera location

is very important; the method is applied for reducing the

vibration at that specified region.

3) Controlling the applied load function: This method try to

control the applied load not to happen in the same frequency

as that of the system’s first natural frequency.

4) Frequency separation: This method is one of the most

common approaches used for vibration control. In this method

the designer try to shift the natural frequency of the structure

to a safe range far from the excitation loads frequency.

By considering the characteristics of the micro-satellite and the

above four suggested methods, it could be realized that the best-fitted method

is third method. The first method needs viscous-elastic or vibration damping

system that increases the cost and weight of the satellite. The second approach

is not applicable due to lack of sensitive section in the micro-satellite based

on its mission (like camera). In the third approach, no control over the loads

caused by launcher, it was decided to take into account the launcher

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specification as design constraints. Therefore, as a unique solution the

geometrical parameters has to be changed. Actually this can decrease the

system response to excitations loads and prevents from resonance

phenomenon. To avoid dynamic coupling between the low frequency vehicle

and satellite modes, based on the Launcher loads and frequency requirements

discussed earlier, the satellite should be designed with a structural stiffness

which ensures that:

Longitudinal frequency > 90Hz

Lateral frequencies > 45 Hz

5.2.1 The Finite Element Model

The finite element model of the satellite was constructed using the

pre and post-processing package, MSC PATRAN. The interface ring made of

aluminium alloy was modeled by shell elements made of an isotropic

material. The sandwich panels were modeled by laminate shell elements made

of composite material which use the properties of Aluminum alloy for the

faces and the properties of the aluminum honeycomb for the core. The

subsystems were modeled as lumped masses positioned at its center of gravity

and possessing both, translational and rotational inertia properties. The

subsystems were connected to the sandwich panels by beam elements which

are shown in Figure 5.2. Interconnections of panels are modeled by rigid

elements (NASTRAN’s RBE2 element) which are shown in Figure 5.3.

Boundary conditions were simulated by single point constraints located at the

attachment points of the cylindrical adapter to the rocket which is shown in

Figure 5.4.

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5.2.2 Convergence study

The objective of convergence study is to determine the optimum

number of nodes required for carrying out a finite element analysis and to

minimize the memory requirements for the analysis without any compromise

on the accuracy of the obtained results. Convergence study usually involves

plotting of a particular result parameter against mesh size. The value of mesh

size at which the result parameter almost becomes a constant is the ideal one.

A mesh size finer than this indicates the wastage of memory without any

additional increase in accuracy, while a coarser mesh size than this indicates

inaccurate results. The finite element model of the micro-satellite structure without

solar panels and subsystems was generated for various mesh sizes (element size in

m), namely 0.2, 0.15, 0.1, 0.07, 0.04 and 0.01. The natural frequency for the first

three modes of vibration in each case was determined.

Figure 5.1 Convergence curves

From the Figure 5.1, it can be seen that natural frequencies for the firstthree modes converge to a steady value for a mesh size of about .01m and after

12000 nodes. Hence for all analyses to be carried out henceforth a mesh size same as

or finer than this will be used. The analysis has to be carried out after modeling the

satellite structure with solar panel substrates and the subsystems attached. The

summary of the finite element model of the micro-satellite is shown in Figure

5.5 is given below.

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Number of grid points = 36715

Number of CBAR elements = 1215

Number of CONM2 elements = 46

Number of CQUAD4 elements = 34152

Number of CTRIA3 elements = 8

Number of RBE2 elements = 107

Figure 5.2 Equipment deck showing subsystems modeled as lumped

masses

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Figure 5.3 RBE2 elements for connecting panels

Figure 5.4 SPC applied at the attachment point of

the adapter to the launcher

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Figure 5.5 Finite element model of the micro-satellite

5.2.3 Subsystem modeling

The micro-satellite considered have several subsystems that are

generally mounted to the bottom deck, middle deck, top deck, cross webs and

vertical webs. The micro-satellite structural panels are designed to withstand

the dynamic loads. Frequencies of these excitations range up to 100 Hz and

therefore the frequencies of interest for structural design are often limited to

the spacecraft modes ranging up to 120 Hz. Theoretically the natural

frequency of the micro-satellite is determined using the finite element analysis

software MSC PATRAN/MSC NASTRAN. The determination of the natural

modes of the micro-satellite structural panels involves modeling of the large

number of subsystems also.

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The Finite element model of the subsystem for determining thenatural modes of the structural panels can be created by the followingmethods.

1. Modeling of subsystems in detail. The subsystems aregenerally designed to have their fundamental naturalfrequencies to be well above 120 Hz to reduce the dynamicloads. The frequencies of the fundamental modes of thesubsystem panels are expected to be around 30 – 40 Hz.Hence for the prediction of the first few natural modes there isno need to have the detailed FE model of the subsystemswhich make the FE model to be very huge, but simplemathematical models should be sufficient.

2. Distributing of mass uniformly over the panels or deck plates.The distribution of mass on the deck plates is not sensitive indetermining the more number of out of plane bending mode.

3. The mass of the subsystem lumped at its C.G and connected tothe deck by using RBE2 elements or RBE3 elements. RBE2and RBE3 elements are built in multi-point constraintequations in the software and it acts as rigid link between thedependent and independent nodes. Dependent nodes are thenodes taken at the locations where the subsystem is connectedto the deck plates and the independent nodes are taken at thenode where the mass is lumped. The rigid elements connectedto the deck add unnecessary stiffness to the deck.

4. Mass of the subsystem lumped at its C.G and connected to thedeck at the interface points through beam or rod elements.Both these elements can simulate the inertial characteristics ofthe subsystems and can also determine large number of out ofplane bending modes of the structural panels. The beam androd element do not add unnecessary stiffness to the deck plateand can best predict the theoretical modes of the deck platesand finally the modes of the micro-satellite.

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For the estimation of the modes of the deck plates or the micro-satellite it is not necessary to model the subsystems in detail and hence it isrequired to lump the mass of the subsystem at its C.G and attach the lumpedmass element to the deck plate at its interface points through beam elements.

5.3 SIZE OPTIMIZATION OF MICRO-SATELLITE

STRUCTURE

The optimization of the micro-satellite structures like bottom deck,top deck, and middle deck, cross webs, vertical webs and solar panels whichwere all made by sandwich panel structure can be done by

Altering the face sheet thickness

Altering the honey comb core thickness

Increasing the face sheet thickness is not desirable since theincrease in mass due to slight change in thickness will be high for the facesheet owing to its high density (2700 kgm-3) when compared to that ofhoneycomb (32 kgm-3) and decreasing the face sheet thickness is not possibleas the minimum available face thickness is 0.20 mm.

Figure 5.6 Effect of increasing panel core thickness

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From the Figure 5.6, tc2 > tc1 (where ‘tc1’ and ‘tc2’ are the core

thickness of the deck plates) and hence I2 > I1 (where ‘I2’ and ‘I1’ are the

Moment of Inertia values). Thus increasing the thickness increases stiffness

(since K I) and natural frequency.

Based on the available aluminium core thickness of 15mm, 20mm,

30mm and 40mm size, optimization study was done to identify the micro-

satellite structure with less mass and more stiffness values to satisfy the

frequency constraints. The optimization was started with the core thickness

and mass values as shown in Table 5.1.

Table 5.1 Core thickness values before optimization

Core thickness of Micro-satellite structure (facesheet thickness keptconstant) in mm

Mass

(kg)

Natural Frequency

LateralX (Hz)

LateralY (Hz)

LongitudinalZ (Hz)

Bottom Deck = 30

Middle Deck = 30

Top Deck = 30

Vertical Web= 30

Cross Web = 30

Solar Panel = 15

38.56 46.86 47.86 128.94

Altair Optistruct the optimization software was used and the

constraints given were to identify the structure with less mass having lateral

frequencies greater than 50 Hz and the longitudinal frequency greater than

100Hz.The results obtained were shown in Table 5.2 and percentage decrease

in mass and percentage increase in natural frequency is shown in Table 5.3.

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Table 5.2 Core thickness values after optimization

Core thicknessvalues arrived afteroptimization (mm)

Mass

(kg)

Natural Frequency

LateralX (Hz)

LateralY (Hz)

Longitudinal

Z (Hz)

Bottom Deck =29.8

Middle Deck = 15

Top Deck = 15

Vertical Web= 15

Cross Web =15

Solar Panel = 15

33.98 50.7 52.0 132.92

Table 5.3 Percentage variation in mass and frequencies

Percentagedecrease inmass (%)

Percentage increase in NaturalFrequency

LateralX (%)

LateralY (%)

LongitudinalZ (%)

12 7.5 8 3

Since the huge interface ring was attached to the bottom deck and

the subsystems are placed on both sides of the bottom deck and middle deck

the structure needs extra thickness for the inserts and fasteners. Hence it was

decided to keep the micro-satellite structure with the following specifications

as shown below.

Bottom deck - 40mm thick core, Face sheets 0.25mm thickTop deck - 15mm thick core, Face sheets 0.20mm thick

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Middle deck - 20mm thick core, Face sheets 0.25mm thickCross web - 15mm thick core, Face sheets 0.20mm thickVertical web - 15mm thick core, Face sheets 0.20mm thickSolar panel - 15mm thick core, Face sheets 0.20mm thickTotal mass = 35 kg

The results of the analysis for free vibration analysis of the micro-

satellite with above dimensions of the primary structure with all the

subsystems, solar panel substrates, antennas and interface ring are given in

Table 5.4.

Table 5.4 Free vibration analysis results

Mode Number Frequency(Hz) Mode shape

1 26.99 Antenna mode








9 56.96 Lateral XX mode

10 58.11 Lateral YY mode

11 111.19 Torsion mode

12 135.84 Longitudinal mode

From the Table 5.4 it is evident that the first eight modes of the

analysis show the antenna mode and one of the antenna modes is shown in

Figure 5.7. The ninth and tenth modes show the lateral modes which are

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shown in Figure 5.8 and Figure 5.9. The eleventh mode shows the torsion

mode which is shown in Figure 5.10 and the twelfth mode shows the

longitudinal mode which is shown in Figure 5.11. The above obtained natural

frequency values satisfy the launcher requirements.

From the optimization study the following information wereobtained

1. The natural frequencies increase with increase in bottom deckcore thickness which was verified by strain energy analysis.

2. The natural frequencies decrease with increase in top deckcore thickness.

3. The natural frequencies decrease with increase in middle deckcore thickness.

Figure 5.7 Antenna mode of vibration of the satellite

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Figure 5.8 Lateral vibration of the satellite in the XX direction

Figure 5.9 Lateral vibration of the satellite in the YY direction

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Figure 5.10 Torsion mode of vibration of the satellite

Figure 5.11 Longitudinal vibration of the satellite in the ZZ direction

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5.4 MODAL STRAIN ENERGY

Modal strain energy is a useful parameter for identifying the

elements of the finite element model for design changes to meet the frequency

requirements. Elements with large values of strain energy in a mode indicate

the location of large elastic deformation (energy). These elements are those

which directly affect the deformation in a mode mostly. Therefore, changing

the properties of these elements with large strain energy should have more

effect on the natural frequencies and mode shapes than if elements with low

strain energy were changed. The Figure 5.12 shows the element strain energy

values of the micro-satellite structure and Figure 5.13 shows the element

strain energy values of the bottom deck. The Table 5.5 shows results of the

modal strain energy analysis of the micro-satellite.

Figure 5.12 Element strain energy values of micro-satellite structures

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Figure 5.13 Element strain energy values of bottom deck

Table 5.5 Modal strain energy results

Micro-satellite structure Strain energy values (Nm)

Interface of bottom deck and crossweb

9.50*10-3

Bottom deck 3.28*10-3

Interface of ring and bottom deck 3.28*10-3

Middle deck 1.78*10-3

Top deck 1.29*10-3

Solar panel 0.185*10-3

Vertical web 0.381*10-3

From the Table 5.5 the element strain energy values are more in theinterface of bottom deck and the cross web and in the bottom deck. Increasingthe thickness in and around that element will increase the natural frequencyvalues. Hence increasing the thickness of the bottom deck will have moreeffect in increasing the frequency values compared to increase in the thicknessof any other structural members of the micro-satellite.

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5.5 FREE VIBRATION ANALYSIS OF SUBSYSTEMS

The subsystems are designed to have their fundamental naturalfrequencies to be well above 120 Hz to reduce the dynamic loads. A greatmany creative, innovative and highly useful electronic products have failed tobe profitable because of low reliability, premature failure or fragilepackaging. Failure of satellites can be of mechanical rather than electricalmalfunction. The launch environment usually provides the most severe shockand vibration loads on the satellite and subsystem assembly that will ever beseen during normal operation. Each subsystem design varies according to itsfunctional requirements and it is necessary to do its intended function at anysevere load environment. All the subsystems that are to be accommodatedinside the micro-satellite considered were made of machined aluminiumcomponents. The proper stiffening members are provided in the sidewalls andin top and bottom of the subsystems. Cutouts are provided for the connectorsand sufficient numbers of insert locations were identified to make thesubsystem stiffer. The printed circuit boards are kept inside the subsystem insuch a way that any kind of load does not affect its functioning.

Figures 5.14, 5.17, 5.20 and 5.23 show the CAD model of thesubsystems payload baseband, power electronics, bus electronics and powerdistribution respectively. Figures 5.15, 5.18, 5.21 and 5.24 show the finiteelement model of payload baseband, power electronics, bus electronics andpower distribution respectively created using MSC/PATRAN. Figures 5.16,5.19, 5.22 and 5.25 show the free vibration analysis results of payloadbaseband, power electronics, bus electronics and power distributionrespectively. The Table 5.6 shows the fundamental frequency values of largesubsystems whose values are well above the launcher constraint.

Table 5.6 Fundamental frequency values of subsystems

Subsystem Fundamental frequency (Hz)

Payload baseband 481Power electronics 942.5Bus electronics 424Power distribution 703

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Figure 5.14 Payload baseband

Figure 5.15 Finite element model of Payload baseband

Figure 5.16 Free vibration analysis result of Payload baseband

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Figure 5.17 Power Electronics

Figure 5.18 Finite element model of Power Electronics

Figure 5.19 Free vibration analysis result of Power Electronics

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Figure 5.20 Bus Electronics

Figure 5.21 Finite element model of Bus Electronics

Figure 5.22 Free vibration analysis results of Bus Electronics

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Figure 5.23 Power Distribution

Figure 5.24 Finite element model of Power Distribution

Figure 5.25 Free vibration analysis result of Power Distribution

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