a study of inertia relief analysis

American Institute of Aeronautics and Astronautics

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A Study of Inertia Relief Analysis

Lin Liao1 Worldwide Aeros Corp., Montebello, CA, 90640

[Abstract] Inertia relief analysis is regarded as an effective technique for the modeling of unconstrained structural systems. In this paper the principle of inertia relief analysis is first described. Inertia relief capability of commercial finite element packages is discussed. The paper studies the implementation of inertia relief techniques into finite element analysis of a variety of structures. Two types of inertia relief methods of MSC/NASTRAN (conventional inertia relief and automatic inertia relief) are also addressed. The application of inertia relief method in the analysis of unbalanced and balanced structural systems is discussed.

I. Introduction The technique of inertia relief has been a well-known approach for the analysis of unsupported systems such as air vehicles in flight, automotives in motion, or satellites in space. The sum of forces and moments are calculated and applied to achieve an equilibrium state in inertia relief analysis. Inertia relief was applied to calculate load redistribution in a helicopter rotor support structure due to flight load imbalances [1]. Inertia relief allowed for the analysis of free structures in space instead of conventional approach of grounding fuselage to landing gears. The finite element model was built with MSC/NASTRAN and the aircraft center of gravity was chosen to be the reference point for inertia relief analysis. Inertia relief method was employed to determine the distribution of non-linear internal forces in aircrafts by counterbalancing rotor hub loads [2]. Inertia relief was also used to estimate impact loads of a space frame structure composed of welded tubular elements [3]. In order to obtain accurate inertia relief calculation, the periods of applied loads should be much greater than the periods of rigid body modes restrained. Inertia relief was used to balance externally applied forces on a free-flying solar sail [4]. The inertia loads were developed under steady-state rigid body acceleration and the center of mass of the solar sail was selected as the reference point for inertia relief calculation. The finite element model was constructed using ABAQUS and geometric nonlinearity was considered. Moreover, inertia relief method was employed to analyze aeroelasticity of non-rigid airships [5]. Airship nonlinearity was introduced due to large deformations and nonlinear material behavior of envelope membranes. Pagaldiptis work showed that inertia relief effect had influence on optimal structural designs [6]. The selected constraints for inertia relief calculation eliminated rigid body motions and didnt generate associated constraint forces while actual structural supports had constraint forces. Thus, the topology optimization was different for the case with inertia relief effects in comparison with the case without inertia relief. The presence of concentrated masses in structural systems with rigid body modes significantly altered load distribution. The implementation of sensitivity correction corresponding to inertia relief load vectors correction is an essential step in optimization procedure. Although inertia relief approach has been widely employed in the simulation of unconstrained aircrafts and space vehicles, the published work has rarely been found. There is still lack of research on inertia relief analysis of diverse types of basic structures and critical structural considerations associated with inertia relief calculation. In this paper, inertia relief method is applied to analyze a variety of structures including spring-mass structures, truss structures, plate structures, and etc. The work is aimed to study various key issues associated with inertia relief analysis, such as conventional inertia relief and automatic inertia relief, the effect of constraints and mass distribution on inertia relief calculation, the accuracy of inertia relief, and critical considerations. Commercial finite element program MSC/NASTRAN is applied to generate numerical results.

II. Principle of Inertia Relief Analysis In inertia relief calculation, the unconstrained structure or system is assumed to be in a state of static equilibrium.

Acceleration is computed to counterbalance the applied loads. A set of translational and rotational accelerations provide distributed body forces over the structure in such a way that the sum of applied forces and the sum of

1 Aeronautical Engineer, PhD, Worldwide Aeros Corp., Montebello, CA, AIAA Senior Member.

52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference 19th4 - 7 April 2011, Denver, Colorado

AIAA 2011-2002

Copyright 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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moments are zero. Since rigid body motions are restrained, conventional static analysis can be performed. Rigid body mass matrix is calculated about the selected reference point. Inertia relief releases the inertia effect and the resulting relative displacements are independent of diverse choices of constraint conditions. Inertia relief method is commonly used in the analysis of unsupported structures/systems, and inertia effects are especially significant in structures having concentrated non-structural masses.

An example is given to illustrate inertia relief method. As shown in Figure 1, two masses (m1 and m2, m2> m1) are connected by a spring having elastic constant k and an external force F is applied to mass m1. Assume that W1+W2


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for result check. As long as the relative displacements and strains/stresses for various selections of constraints are nearly identical, the accuracy is achieved. Automatic inertia relief is a relatively new feature in MSC/NASTRAN. For automatic inertia relief, a reference frame is selected automatically and relevant constraints are distributed to all points with masses. This method resolves the problems of inadequate stiffness of support points and low accuracy of inertial load calculations. It is especially applicable to structures with modeling errors, structural defects, and disjoint parts. Automatic inertia relief removes the difficulty and uncertainty of constraint selection and offers a convenient approach for users without considerable experiences and skills of inertia relief. However, automatic inertia relief is not applicable to iterative solver and certain cases.

In contrast with MSC/NASTRAN, the inertia relief calculation of ANSYS is limited. To carry out inertia relief analysis of ANSYS, constraints must be specified to prevent rigid body motions: three constraints for 2D problems, and six constraints for 3D problems. Exact number of constraints should be selected, and more or fewer prescribed conditions will generate errors. Generally speaking, Hard Points (no deformation or minimum deformation) are selected as constraints. Inertia relief of ANSYS does not allow for nonlinearities and analysis of axisymmetric or generalized plane strain elements. ABAQUS also offers the modulus of inertia relief calculation. Geometric nonlinearity and inertia relief are incorporated in ABAQUS solver. ABAQUS is capable of dealing with the problems with large deformation and nonlinear loading. Detailed discussion of inertia relief of various finite element programs is beyond the scope of this paper, and only MSC/NASTRAN is used in the following examples.

IV. Case Study In this section, inertia relief methods and finite element program

MSC/NASTRAN are applied to the analysis of diverse structures including spring-mass structures, cable-truss structures, plate structures, and etc. Conventional inertia relief method and automatic inertia relief of MSC/NASTRAN are discussed. Although simple structures are given as examples, inertia relief methods can be implemented into the analysis of complex systems, such as flight aircrafts, space vehicles in operation, sailing boats, and etc. A. Spring-mass Structures

A spring-mass structure containing three masses and two springs is shown in Figure 2a (m1=10 lbm, m2=40 lbm, m3=30 lbm, Y1=0 in, Y2=20 in, Y3=40 in). Two spring constants are k1=k2=20 lbf/in. The external loads include applied forces at mass m1 and m3 (F1=20 lbs, F2=40 lbs) and gravity (opposite to direction of F1 and F2 shown in Figure 2a). In the MSC/NASTRAN model, element CELAS1 is used for springs and element CONM2 is used for masses. There are errors generated for automatic inertia relief of MSC/NASTRAN and supports are needed for inertia relief calculation. We study three cases that each mass is constrained separately. The displacements of masses and spring forces are listed in Table 1. It shows that the relative displacements between three masses are the same and the forces in two springs are identical for these three cases. Also, the same downward acceleration is obtained (-96.5 in/s2). This example demonstrates that the selection of constraints doesnt affect inertia relief analysis.

Table 1. Displacements and spring forces.

Displacements (inch) Spring Forces (lbs) Mass 1 (inch) Mass 2 (inch) Mass 3 (inch) Spring 1 (lbs) Spring 2 (lbs)

Constraint m1 0.000 -0.625 0.250 12.500 -17.500 Constraint m2 0.625 0.000 0.875 12.500 -17.500 Constraint m3 -0.250 -0.875 0.000 12.500 -17.500

The next example is spring-mass system having four masses and three springs (see Figure 2b). This system has more degrees of freedom in comparison with last example. The parameters in Figure 2b are as follows: m1=20 lbm, m2=10 lbm, m3=30 lbm, m4=80 lbm, Y1=0 in, Y2=20 in, Y3=40 in, Y4=80 in. The spring constants are k1=k2=20 lbf/in, k3=40 lbf/in. The applied loads are F1=20 lbs, F2=-40 lbs. Displacements and spring forces are solved for four support conditions that each mass is restrained, respectively (see Table 2). Although the displacements of fours

(a) (b) Figure 2. Spring-mass structures.


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masses vary, the relative displacements and spring forces are the same, which shows that the deformation of the spring-mass system is not relevant to constraint selection.

Table 2. Displacements and spring forces.

Displacements (inch) Spring Forces (lbs) Mass 1 Mass 2 Mass 3 Mass 4 Spring1 Spring2 Spring3

Constraint m1 0 -1.143 -2.357 -3.071 22.857 24.286 28.571 Constraint m2 1.143 0 -1.214 -1.928 22.857 24.286 28.571 Constraint m3 2.357 1.214 0 -0.714 22.857 24.286 28.571 Constraint m4 3.071 1.928 0.714 0 22.857 24.286 28.571

B. Cable-truss Structures

The second example is a 3D truss with concentrated masses at some truss joints ( see Figure 3). The nodal coordinates are provided in Table 3. The truss contains 21 truss members (black lines), 12 cables (green lines), and 5 lumped masses (100 lbm at Nodes 1, 5, 9, 10, 11). Cable elements are displayed by green filled circles. In NASTRAN model, truss members are modeled as beam elements (capable of supporting tension, compression, and bending loads) and cables are simulated as tension-only elements. The material and geometry properties of truss beams are E=1.5E7 psi, =0.3, =0.058 lb/in3, A=0.785 in2, Iy=Iz=0.049 in4. Cable properties are E=6.5E5 psi, =0.3, =0.035 lb/in3, A=0.0377 in2. The system is subjected to gravity (negative Y direction) and external forces FY=100 lbs at Node 9, FY=100 lbs at Node 10, FY=50 lbs at Node 11 (Load Case 1). A pretension of 200 lbs is assigned to all cables using thermal loads. Both automatic inertia relief and conventional inertia relief of MSC/NASTRAN are performed. Node 6 or Node 7 are specified as constraints for conventional inertia relief.

Table 3. Coordinate of nodal points.

Node 1 Node 2 Node 3 Node 4 Node 5 Node 6

X (inch) -200.0 -100.0 0.0 100.0 200.0 100.0 Y (inch) 0.0 0.0 0.0 0.0 0.0 0.0 Z (inch) 0.0 100.0 100.0 100.0 0.0 -100.0

Node 7 Node 8 Node 9 Node 10 Node 11 X (inch) 0.0 -100.0 -100.0 0.0 100.0 Y (inch) 0.0 0.0 100.0 100.0 100.0 Z (inch) -100.0 -100.0 0.0 0.0 0.0

As shown in Table 4, the nodal displacements of these three constraint conditions are different. When Node 6 or Node 7 is used as constraints, three degree of freedoms of these nodes are set to zero. Unlike conventional inertia relief method, automatic inertia relief calculation does not generate three zero displacement components in one node. Since it is a 3D structure, it is hard to compare relative displacements as the first example. However, structural behavior can be represented by element forces. Average axial forces in truss beams and cable tension for these three support conditions are the same (presented in Tables 5&6). Additionally, Table 6 shows that cable tension changes in deformed configurations in contrast to pretension, and all cables are still in tension in the deformed state. Another load case is used to study the effect of constraints for different loading. The external loads are changed to FY=30 lbs at Node 9, FY=30 lbs at Node 10, FY=50 lbs at Node 11 (Load Case 2). The nodal displacements and cable tension are presented in Tables 7&8. Similar to the first load case, displacements for three constraint conditions vary while cable tension is the same. These data illustrate that deformation of truss structures are independent of support, and the results of inertia relief analysis does not rely on the selection of supports.

Figure 3. Schematic of a 3D truss.


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Table 4. Displacements of the 3D truss (Load Case 1).

Automatic Constraint Constraint of Node 6 Constraint of Node 7Node X(inch) Y(inch) Z(inch) X(inch) Y(inch) Z (inch) X(inch) Y(inch) Z(inch)

1 1.73E-2 -1.82E-2 0.0 1.84E-2 5.56E-2 4.42E-2 8.73E-3 1.15E-1 1.18E-2 2 1.80E-2 1.51E-2 -1.32E-3 3.43E-2 4.08E-1 2.76E-2 1.68E-2 4.56E-1 3.02E-3 3 1.60E-2 1.01E-1 -3.07E-3 3.23E-2 5.64E-1 1.07E-2 1.48E-2 5.66E-1 -6.15E-3 4 1.41E-2 -3.75E-2 -1.40E-3 3.04E-2 4.97E-1 -2.80E-3 1.29E-2 4.52E-1 -1.19E-2 5 1.51E-2 -8.39E-3 0.0 1.62E-2 3.48E-1 -1.66E-2 6.49E-3 2.23E-1 -1.79E-2 6 1.41E-2 -3.75E-2 1.40E-3 0.0 0.0 0.0 -1.94E-3 -1.13E-1 -9.09E-3 7 1.60E-2 1.01E-1 3.07E-3 1.94E-3 6.73E-2 1.69E-2 0.0 0.0 0.0 8 1.80E-2 1.51E-2 1.32E-3 3.93E-3 -8.89E-2 3.03E-2 1.99E-3 -1.10E-1 5.67E-3 9 -1.33E-2 1.37E-2 0.0 -8.29E-2 1.58E-1 -2.20E-1 -4.66E-2 1.71E-1 -2.79E-1

10 -1.48E-2 9.85E-2 0.0 -8.44E-2 3.14E-1 -2.35E-1 -4.80E-2 2.81E-1 -2.86E-1 11 -1.63E-2 -3.89E-2 0.0 -8.59E-2 2.47E-1 -2.50E-1 -4.95E-2 1.68E-1 -2.93E-1

Table 5. Axial forces in truss beams (Load Case 1).

Structural Member No. S1 S2 S3 S4 S5 S6

Axial Forces (lbs) -37.6 -234.1 -228.8 -23.1 -23.1 -228.8 Structural Member No. S7 S8 S9 S10 S11 S12

Axial Forces (lbs) -234.1 -37.6 72.8 -171.8 -180.9 49.5

Table 6. Cable tension (Load Case 1).

Element No. 27 28 29 30 31 32 Tension (lbs) 195.0 204.0 185.7 213.4 199.5 199.5 Element No. 33 34 35 36 37 38 Tension (lbs) 199.5 199.5 195.1 204.0 213.4 185.7

Table 7. Displacements of the 3D truss (Load Case 2).

Automatic Constraint Constraint of Node 6 Constraint of Node 7Node X(inch) Y(inch) Z(inch) X(inch) Y(inch) Z (inch) X(inch) Y(inch) Z(inch)

1 -5.88E-3 4.23E-3 0.0 4.83E-3 1.24E-1 6.01E-3 -2.45E-3 1.32E-1 -8.98E-3 2 -4.59E-3 -7.68E-3 -1.47E-3 8.60E-3 2.55E-1 2.06E-3 -4.13E-3 2.36E-1 -7.48E-3 3 -6.40E-3 1.78E-2 -3.05E-3 6.79E-3 2.84E-1 -2.00E-3 -5.94E-3 2.51E-1 -6.09E-3 4 -8.23E-3 1.33E-2 -1.43E-3 4.96E-3 2.82E-1 -2.87E-3 -7.77E-3 2.37E-1 -1.51E-3 5 -7.07E-3 3.13E-4 0.0 3.64E-3 1.31E-1 -3.92E-3 -3.64E-3 8.84E-2 2.89E-3 6 -8.23E-3 1.33E-2 1.43E-3 0.0 0.0 0.0 -1.83E-3 -1.43E-2 1.36E-3 7 -6.40E-3 1.78E-2 3.05E-3 1.83E-3 1.78E-3 4.09E-3 0.0 0.0 0.0 8 -4.59E-3 -7.68E-3 1.47E-3 3.64E-3 -2.64E-2 4.99E-3 1.81E-3 -1.56E-2 -4.55E-3 9 7.71E-3 -8.98E-3 0.0 1.57E-2 1.13E-1 -1.37E-1 2.10E-2 1.09E-1 -1.32E-1

10 5.91E-3 1.54E-2 0.0 1.39E-2 1.40E-1 -1.40E-1 1.92E-2 1.23E-1 -1.29E-1 11 4.15E-3 1.21E-2 0.0 1.22E-2 1.40E-1 -1.42E-1 1.74E-2 1.10E-1 -1.26E-1

Table 8. Cable tension (Load Case 2).

Element No. 27 28 29 30 31 32 Tension (lbs) 196.4 202.6 200.2 198.9 199.5 199.5 Element No. 33 34 35 36 37 38 Tension (lbs) 199.5 199.5 196.4 202.6 198.9 200.2


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Figure 4. Inertia relief analysis of a plate.

Figure 5. Inertia relief analysis of a plate (left: Constraint of Node 1; right: Constraint of Node 41).

C. Plate Structures As shown in Figure 4, the next

example is an aluminum plate (E=1.06E7 psi, =0.33, =0.1 lb/in3). The plate has a dimension of 200x50 inches, and a thickness of 1.5 in. The applied loads are FZ=800 lbs at middle Nodes 21 and 431, which have nodal coordinates (100, 0, 0), (100, 50, 0), respectively. Gravity force is applied at the negative Z direction.

The displacements of four corner nodes (Nodes 1, 41, 411, and 451) under four kinds of constraint conditions are presented in Table 9. The displacement components in X and Y directions are zero for the specified loading conditions. It can be seen from Table 9 that absolute displacements of four corner nodes are different and the relative displacements are the same. The displacement distribution for these four cases is shown in Figures 5&6. Displacements display similar distribution for Constraint of Node 1 and Constraint of Node 411, and similar variation for Constraint of Node 41 and Constraint of Node 451. Although displacement distributions vary for these four cases, element stresses are the same (Stresses of Elements 1, 40, 362, 399 are shown in Table 10). Structural deformation, strains, stresses are irrelevant to constraint conditions in inertia relief calculation.

Table 9. Displacements of the plate.

Constraint of

Node 1 Constraint of



Node 451 Node Z(inch) Z (inch) Z(inch) Z(inch)

1 0.0 -1.87E0 7.00E-3 -1.86E0 41 -1.87E0 0.0 -1.86E0 7.00E-3 411 7.00E-3 -1.86E0 0.0 -1.87E0 451 -1.86E0 7.00E-3 -1.87E0 0.0

Table 10. Stresses of the plate.

Constraint of Node 1/41/411/451Element Normal X (psi) Normal Y (psi) Shear XY (psi)

1 0.123 0.122 -8.300 40 0.123 0.122 8.300 362 -5.680 0.122 0.185 399 -5.680 0.122 -0.185


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Figure 6. Inertia relief analysis of a plate (left: Constraint of Node 411; right: Constraint of Node 451).

V. Inertia Relief Analysis of Balanced and Unbalanced Systems Total forces, moments, and accelerations of a balanced system are zero. The sum of external forces and moments

is not zero for unbalanced systems. When inertia relief is used to analyze unbalanced systems, translational/rotational accelerations are calculated from external loads and applied to counteract the inertia effect. Two structural systems consisting of two concentrated masses and beams are used to study inertia relief analysis of balanced and unbalanced structural systems. As shown in Figure 7, fives nodes are connected by beam elements; and m1=3 lbm, m2=3 lbm, F1=5 lbs, F2=5 lbs, L=20 in. The material and geometry properties of beams are E=1.06E7 psi, =0.33, =0.1 lb/in3, A=0.5 in2, Iy=0.042 in4, Iz=0.01 in4. The forces and moments of structural system A are balanced, which includes the contribution from weight of concentrated masses, weight of beams, and applied forces F1 and F2. An additional force F3= 5 lbs is applied at Node 2, which generates an unbalanced system B. Automatic inertia relief method is employed to solve these two systems and no constraints are prescribed.

For balanced system A, the three components of acceleration are zero; for unbalanced system B, the

accelerations in X & Z directions are zero, and the acceleration in Y direction is 515 in/s2. Nodal displacements and maximum and minimum stresses at five nodes are listed in Table 11 and Table 12, respectively. The deformation is different for balanced and unbalanced systems due to load conditions.

Inertia relief methods can be applicable to balanced systems and unbalanced systems. For balanced systems, the resultant acceleration is zero and the no additional inertia forces are applied in inertia relief solver. Unbalanced systems generate nonzero acceleration and inertia forces are applied to the systems to achieve an equivalent equilibrium status, and thus traditional static analysis can be carried out. Inertia relief approach is especially useful for analyzing systems in motion, which usually have a nonzero acceleration. Inertia relief method is able to solve strains and stresses of unbalanced systems without using dynamic analysis.

Table 11. Nodal displacements.

Balanced System Unbalanced System

Node No. X(in) Y(in) Z(in) X(in) Y(in) Z(in) 1 0 1.06E-1 0 0 9.95E-2 0 2 0 -6.95E-3 0 0 -5.81E-3 0 3 0 -5.07E-2 0 0 -4.97E-2 0 4 0 -6.95E-3 0 0 -7.49E-3 0 5 0 1.06E-1 0 0 1.06E-1 0

Figure 7. Schematic of two structural systems (left: A, balanced system; right: B, unbalanced system).


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Figure 8. A plate with lumped masses.

Table 12. Maximum & minimum stresses at nodes.

Balanced System Unbalanced System Node No. Max(psi) Min(psi) Max(psi) Min(psi)

1 0 0 0 0 2 1.08E3 -1.08E3 9.20E2 -9.20E2 3 1.20E3 -1.20E3 1.20E3 -1.20E3 4 1.08E3 -1.08E3 1.12E3 -1.12E3 5 0 0 0 0

VI. Inertia Relief Analysis and Mass Distribution A. Plate structures with lumped masses The next example is used to investigate the influence of mass distribution on inertia relief analysis. The plate has a dimension of 40x40 inches and a thickness of 1 inch (Figure 8). There are four lumped masses 10 lbm located at Nodes 100-103 on the plate (Plate A). The material properties of this plate are E=2.97E7 psi, =0.29, =0.284 lb/in3. Nodal force Fz=125 lbs is applied at each corner node. The plate is also subjected to a gravity load in the negative Z direction. Automatic inertia relief is implemented, and thus no constraints are specified. To study the effect of mass distribution, we change lumped masses by a ratio of 10, and thus each mass has a mass of 100 lbm (Plate B). Figures 9&10 show there are significant difference of displacement distribution and Von Mises stress distribution of these two plates. Increasing the value of concentrated masses changes mass distribution of the plate-mass structure. The variation of mass distribution leads to the change of the distribution of inertia forces and moments, which results in different deformation and stresses. Inertia relief approach is closely related to mass distribution.

Figure 10. Von Mises distribution of two plates with lumped masses (left: Plate A; right: Plate B).

Figure 9. Displacement distribution of two plates with lumped masses (left: Plate A; right: Plate B).


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B. A plate-beam structure

The last example is a plate-beam structure. As shown in Figure 11, a lumped mass is connected to four beams and pressure loads are applied to the plate. In MSC/NASTRAN, weight-mass ratio can be specified to simulate gravity effect. We solve for two cases: the weight-mass ratio is 0.00259 for the first case, and it is increased to 0.0259 for the second case. NASTRAN output of resultant accelerations, strains, and strain energy for these two case are given in Figures 12&13, respectively. It can be seen that the accelerations in Z direction are changed from 0.188 in/s2 to -347.38 in/s2. The weight is significantly increased due to a large weigh-mass ratio, leading to a negative force in Z direction. Thus, a negative acceleration is obtained. Figures 12&13 also show that strains are zero and strain energies have some values. Strain can be used as a parameter to judge the convergence and accuracy of inertia relief analysis. Generally, strain should be zero. The numerical value of strain energy varies case by case and depends on dimension, units, model size, and etc. The magnitude of strain energy for this example has an order of 104. NASTRAN output of displacements, strains, and stresses are the same for these two cases.

Simply increasing weight-mass ratio leads to the increase of total weight and acceleration while the mass distribution is still the same. The resultant deformation, strains, and stresses are the same for the cases of different total weight and same mass distribution. Mass distribution has vital influence on inertia relief analysis. The application of automatic inertia relief method is especially linked to the distribution of concentrated masses since inertia forces used to balance external loads are distributed to all masses.

VII. Considerations of Inertia Relief Analysis Inertia relief techniques have been applied to the modeling and analysis of complex systems including aircrafts,

space vehicles, automotives, ..., and etc. Some critical considerations about the application of inertia relief method are summarized as follows 1. For conventional inertia relief requiring specified constraints, hard points or special points are usually chosen as

constraints. Center of gravity of traditional aircrafts is usually chosen as constraint for inertia relief calculation. For buoyancy air vehicles, a unique type of air vehicles, center of gravity or center of buoyancy are usually used as supports in inertia relief analysis.

Figure 12. NASTRAN output of the plate-beam structure (weight-mass ratio: 0.00259).

Figure 13. NASTRAN output of the plate-beam structure (weight-mass ratio: 0.0259).

Figure 11. A plate-beam structure.


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2. Support conditions up to six degrees of freedom are needed to remove rigid body motion. One degree of freedom is used for 1D problems; three degrees of freedom are used for 2D problem; and six degrees of freedom are used for 3D problems.

3. Strain serves as a criterion for judging solution of inertia relief method. The numerical value of strain is zero while strain energy is not a determined parameter and varies by cases. For conventional inertia relief, resultant forces of single point constraint are zero, which are another way to judge the solution convergence.

4. Automatic inertia relief doesnt need support conditions but is not applicable to iterative solver and some cases. 5. The application of automatic inertia relief method is especially linked to distribution of concentrated masses

since inertia forces used to balance external loads are distributed to all masses. The resultant deformation, strains, and stresses are not affected by the change of total weight when the mass distribution is kept the same. The change of total weight results in different acceleration.

6. In order to apply inertia relief approach, the periods of applied loads should be much greater than the periods of rigid body modes restrained.

VIII. Summary In this paper, published works on inertia relief analysis of unconstrained systems are introduced and the principle

of inertia relief analysis is described. Inertia relief method and finite element analysis are integrated to study a variety of structures using the tool MSC/NASTRAN. Critical considerations in inertia relief analysis are summarized. Conventional inertia relief and automatic inertia relief are studied. Specified constraints up to six degrees of freedom are needed for conventional inertia relief. Supports are automatically prescribed for automatic inertia relief. Inertia relief approach can be applied to analyze balanced and unbalanced structural systems, which generate zero and nonzero accelerations, respectively. Mass distribution has important influence on inertia relief calculation.

References

1Morton, M. H., Kaizoji, A., Effects on Load Distribution in a Helicopter Rotor Support Structure Associated with Various Boundary Configurations, 48th Annual Forum Proceedings of the American Helicopter Society, Vol. 1, 1992, pp. 629-634.

2Smith Jr., F. A., Hopkins, P. M., Non-linear Internal Loads Modeling Methods, Proceedings of AHS International 62nd Annual Forum - Vertical Flight: Leading through Innovation Proceedings, Vol. III, 2006, pp. 1436-1455.

3Nelson, M. F., Wolf Jr., J. A., Use of Inertia Relief to Estimate Impact Loads, Proceeding of International Conference on Vehicle Structural Mechanics, April, 1977, pp. 149-155.

4Sleight, D. W., Muheim, D. M., Parametric Studies of Square Solar Sails Using Finite Element Analysis, Proceedings of 45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Vol. 1, 2004, pp. 85-97.

5Bessert, N., Frederich, O., Nonlinear Airship Aeroelasticity, Journal of Fluids and Structures, Vol. 21, No.8, 2005, pp. 731-742.

6Pagaldipti, N., Shyy, Y.K., Influence of Inertia Relief on Optimal Designs, Proceeding of 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Vol. 1, 2004, pp. 616-621.

7Mahishi, J. M., Nonlinear Static and Multi-axial Fatigue Analysis of Automotive Lower Control Arm Using NEiNASTRAN,http://www.nenastran.com/newnoran/conferencePaper2/10_CPNonlinearStaticMulti-AxialFatigueAnalysisAutomotiveLowerControlArmUsingNEiNastran.pdf

8MSC/NASTRAN Reference Manual 2005

a study of inertia relief analysis

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