A Bracket Design Optimization in Random Vibration Environment with ...

A BRACKET DESIGN OPTIMIZATION IN RANDOM VIBRATION ENVIRONMENT WITH FEA AND ROBUST ENGINEERING METHODS

Ahmet T. Becene Vibrations Test and Analysis Engineer Delphi Automotive Systems 5500 West Henrietta Road MC 146.HEN.540 Rochester, NY 14602

ABSTRACT

ANOA4 Analysis of Means FEA Finite Element Analysis FRF Frequency Response Function

Engine mounted components experience high levels of vibration. Depending on the number of cylinders, the majority of the vibration energy is transferred to components in a random nature. Therefore, design optimization in a random vibration environment becomes an important problem. In this paper, Finite Element Analysis and Robust Engineering methods are used to investigate the effects of various parameters on the design of a simple bracket.

1. SPECTRAL

ANALYSIS

WITH FEA

The power spectral density (PSD) of stresses is required for a fatigue analysis with FEA. This information is generally calculated in two steps, the first step being to calculate the frequency response function (FRF) of the structure. This can be performed in two ways:

NOMENCLATURE

1 I ii, b’} 2, FFT IFFT PSD PDF

. .

Mass Matrix Stiffness matrix Damping Matrix Modal Mass Matrix Identity Matrix Natural Frequencies

The Modal Method is used in this paper. This is the preferred method for structures with many degrees of freedom or analyses requiring a large number of frequency steps. 1 .I Modal Frequency Response

Eigenvectors

The undamped natural frequencies of a structure are given as

Displacements Modal Participation Coefficient Fast Fourier Transform Inverse Fast Fourier Transform Power Spectral Density Probability Density Function Moments of PSD

?A

PSD Amplitudes

/”

Frequency Moment Arm

&‘I &‘I @I Y N,

Expected Number of Zeros Expected Number of Peaks Cumulated Damage Irregularity Factor Number of Allowable Stress Ranges

S, T

Stress Ranges Time

Direct Method Modal Method

[-co:M+K~$+}i=O.

o, and mode shapes {0},

r=l,.,.,, n.

(1)

It is a property of the eigenvectors that they are orthogonal with respect to the mass matrix, yielding

bl’ b&l =[II

(2)

Combining Eqs. (1) and (2) gives

bl’kI~1=l-d2 bl’[~I41~ Assuming that any arbitrary displacement as a linear combination of normal modes

368

(3) {D} can be written

{D)=~b)izi=bb).

q(o)=

(4)

The displacement, velocity, and acceleration terms in the harmonic response equation can be written in terms of {z} as k’) = bb)

Pa)

9

(W

Q}=bIbII Q}=[4](i). By substituting these equation, the result is [MI&)+

Ri’

- mp* + ibp + ki

.

(15)

Equation (15) gives the solution of a single degree of freedom system for every row of Equation (12). Figure 1 shows the representation of this single degree of freedom system.

(5c) terms

into the harmonic

3

response -4

[CI&)+

kI&)

(6)

= {R).

Figure 1 - Single degree of freedom system Multiplying

Eq. (6) by [+I’ gives the following:

[~l’[~I~~~}+[~l’[Cbl{~)+blr[~I~l{~)=

bl’{R)=

In order to determine the displacements, the rows of Eq. (12) are solved at any forcing frequency w The displacements for the structure are then calculated as

k* )a (7)

where the damping term is taken as

k’) = bh)

M=&l+PM.

(8)

The final result is

bl’kI41=&r kbl+ &ITLd~l =a[oh]+&I= [o&)m]

(9)

Now all coefficient matrices are diagonal. Therefore, the equations can be solved independently as a one degree of freedom system: [m]{~)+[og(o)m~i)+ If the eigenvectors reduces to

are

prnl#= normalized

(R1). on mass,

In modal approach

Eq. (10)

(11)

z = %lw , which reduces Eq. (11) to

-o*[m]{i++io[b~i++[k@J=~},

Once the displacements are known, the calculation of strains and stresses is a straightforward process. If the input for the forcing function is a constant amplitude function between the smallest and the largest forcing frequencies, the plotted results will form the transfer function of the structure. The common approach is to use a 1 g constant amplitude forcing function for the FRF analysis, so that the analyst can use any random vibration spectrum input without any amplitude modification. 1.2 Random Vibration Induced Fatigue

(10)

~4+b&JN4+f&~= cp*}.

(16)

(12)

In MSCJNASTRAN. random response analysis is treated as a data reduction procedure that is applied to the results of a frequency response analysis. First, the frequency response analysis is performed for sinusoidal loading conditions, at a sequence of frequencies wi. Normal data reduction procedures are then applied to the output of the frequency response analysis module. The response quantities may be displacements, velocities, accelerations, internal forces, or stresses. The power spectral densities of the response quantities are calculated by different procedures depending on whether the loading conditions are correlated or uncorrelated. See Reference 2 for more information on this subject.

where bi=w,mig(wi)=wimi25,,

(13a)

r=,gcv

UW

k, = w2im,.

(13c)

The PSD of strain and stress data are the input quantities for spectral fatigue analysis. The most important information extracted from the PSD of strains and stresses are the moments of the PSD. The information gained from these moments is enough to estimate the fatigue damage induced by a random vibration spectrum. The probability density function (PDF) of stress ranges, expected number of zero crossings, and expected number of peaks per second are calculated from the moments of the PSD. The nth moments of the PSD are computed by the function

Individual rows of Eq. (12) are in the form of - w2miri + idpi with the analytical

+ k,zi = R,: 1

(14)

solution of Eq. (14) being

mn=h&f.

(17) 0 The values for the Equation (17) are illustrated in Figure 2

369

range interval expression

ds at

S, is calculated

by the

‘dPbP(S, )ds I

following

(24)

and the total damage is given by &,I=

c"i= ,Ni

Figure 2 - Moments of PSD

kqh

TEb']=+s

b

p(s)&.

(25)

0

2. DESIGN OF EXPERIMENTS

An empirical closed form expression for the PDF of stress ranges was given by Dirlik [1985]. This expression was obtained by extensive computer simulations using the Monte Carlo technique. Dirlik’s method is accurate and efficient to compute the PDF of rainflow ranges with the first four terms of the moments of PSD. Therefore Eq. (17) can be re-written as

m,=k=, %‘G,(fk.

The Design of Experiments (DOE) method is used to evaluate the effect of design parameters on the functionality of the product. The functionality is chosen by the engineer and it can be one of many responses created by the product. The response should relate directly to the main function of the product, be continuous, and easily measurable.

(18)

The theoretical validation of Dirlik’s method was completed by Bishop and Sherratt [1989]. The Dirlik formula is given below.

4e$+D?Ze$+D&$ 3

Q

R2

qG--

e= Wb4-(4R)) 4

_

, R-

Y-X,-~* I-y-Q+Df

(19)

A DOE matrix must be set up to describe the tests or analyses to be run. The design parameters and the response are displayed in columns while the treatment combinations are displayed in rows. The treatment combinations refer to how the parameters will be set for the test or analysis. For complete information on DOE, see References 3 and 4. DOE can be an extremely useful tool during product design since it can reveal the effects of the design parameters on the response and their interactions with each other. This information is used to maximize (or minimize) the response of a product. A robust product design can be achieved by including noise factors in the DOE. The goal becomes choosing design parameters so that the noise effects on the response will be minimized. One example would be to choose emissions of an engine as response and environmental temperature as noise. In this case, the goal would be to minimize the effects of temperature on the emission values. DOE tests must be performed under controlled environments. Depending on the number of design parameters, a mid-size DOE can require 16 individual tests with sensitive control of the test parameters. This means 16 individual prototypes (or one prototype with adjustable parameters). This method can quickly become time consuming and expensive. On the other hand the FEA method is usually faster than testing, not as expensive, and can be very accurate.

(22)

In order to compute the fatigue damage, the number of allowable stress ranges ( Ni ) must be extracted from the S-n diagram.

i

These techniques are incorporated into the MSC/Fatigue software and were used for the example presented in this paper.

i/

p(s)=

1 E[Ph'(si&

N, is given by N, = kS-h

(23) 3. A BRACKET EXAMPLE

The value

of ni for a given

stress

range interval

S to

This section presents a simple bracket geometry optimization using the techniques outlined in above sections. The geometry was chosen for its simplicity.

S + 8s is given by p(S,)dS The number of cycles in a period of T is equal to the number of peaks in the same period and is given by E[Pp The total number of cycles in the stress

370

Figure 3 - Bracket Model

The levels for the design parameters

The bracket model included plate elements, as well as a point mass element, representing a product that the bracket supports. The MSClNastran solver was used for the FEA part of the analysis. First, a FRF analysis was performed, using the modal method. The results of this analysis were exported to MSCIFatigue. the fatigue software. in order to calculate fatigue life. A random vibration spectrum was used to simulate the loading on the bracket. This profile is given below.

Table 3 - Parameter Values Parameter Thickness Height of Rib Corner Radius

Three significant analysis: . . .

design parameters

Level 2 “mm” 3 -i2 5

Table 4 - FEA Life Results

g2lHz 0.02 0.05 0.05 0.001 0.001 were considered

Level 1 “mm” 2 7 2

A total of four FEA models were created for eight analyses. The life results for these analyses were recorded for eight treatment conditions. The FEA results are given below.

Table 1 - Random Vibration Input Spectrum Frequency 10 20 100 800 2000

are given below

for the

Thickness of the sheet metal Radius of the corner bend Height of the stiffening rib 8

2

2

1

2

11

2

22326

These results suggest that increasing the height of the rib decreases the life of the bracket. This may be due to the way that the bracket is mounted, or the mass contribution of the rib resulting in higher stresses in the critical region. A plot of the life contour is given in Figure 5.

Figure 4 - Design Parameters A two level full factorial DOE matrix (Ls 2s) was chosen for the analysis, since a full factorial DOE matrix ensures results without any confounding. Table 2 - DOE Matrix

Figure 5 -Life 371

Contour Plot

An Analysis of Means (ANOM) was performed on the DOE results. The ANOM results are used to scale the effects of the design parameters on the response, in this case the life of the bracket. Since a full factorial DOE matrix was used for the experiment, the ANOM results also show the interaction effects, The graphical representation of main and interaction effects are given in Figure 6 and Figure 7 respectively.

time. The FEA method alone cannot give all the effects of the design parameters in a design concept. Alternately, the DOE method with testing requires many individual tests and many individual prototypes (or a prototype with adjustable parameters). This can be time consuming and expensive. Using both methods together can be the most cost effective way to evaluate design concepts. Clearly, FEA does not require carefully controlled tests or prototypes. With the recent increase of inexpensive computing power, the analyst can create complex models with little simplification. This leads to FEA models with higher accuracy. The example presented in this paper was chosen for its simplicity, so the parameter effects were obvious even before the analysis. This is not the case for most engineering problems. Most automotive parts have very complex geometries with several design parameters, making the parameter effects anything but obvious. Effective usage of both methods presented in this paper can yield valuable information about the product in a relatively short time. This method can also be used with any type of computer simulation, including computational fluid dynamics (CFD), and magnetic modeling.

Figure 6 - Main Effects

REFERENCES [I] N. W. M. Bishop, L. W. Lack, T. Li, S. C. Kerr, Analytical Fatigue Life Assessment of Vibration Induced Fatigue Damage. [Z] David N. Herting, The MacNeal-Schwendler Corporation MSWNastran Advanced Dynamic Analysis Useh Guide, 1997. [3] Robert H. Lochner, Joseph E. Matar, Designing Quality, ISBN O-527-91633-1, ASQC Quality Press.

for

[4] Madhaw S. Phadke, Quality Engineering Using Robust Design, ISBN O-13-745167-9, Prentice Hall PTR.

Figure 7 - Interaction Effects As seen in Figure 6, increasing the thickness of the bracket results in a higher expected life value while increasing either of the other two factors results in a lower expected life value. Therefore, the best design parameter setting for this bracket would be AZBlCI. This corresponds to a bracket with a 3mm sheet metal thickness, a side rib with a 7mm height, and a comer bend with a radius of Zmm.

In order to achieve a robust product, the effects of noise on the response must be minimized. One possible noise factor for this analysis is the surface quality variation due to manufacturing or environmental factors. Minimizing the response variation due to these noise factors becomes an important part of this analysis. However, noise information was not included in this analysis. A second set of analyses will be performed in the near future to account for noise in a full-scale experiment. 3. CONCLUSION Usage of FEA and Robust Engineering techniques can drastically increase a products quality in a relatively short 372

[5] The MacNeal-Schwendler Corporation, User’s Guide, Publication No. 903030.

MSC/Fatigue

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