stochastic assessment of crank-drive component …...christian wetzel, mtu friedrichshafen gmbh 20....

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 1

Stochastic assessmentof crank-drivecomponent toleranceswith respect to the massbalancing of an 8Vdiesel-engine

Christian Wetzel, Frank Schmidt

MTU Friedrichshafen GmbH

Dynamiksimulation in der Fahrzeugentwicklung

St. Valentin, 20.05.2010


MTU Friedrichshafen GmbHShort introduction


Outline

• Introduction and objective of the probabilistic assessment

• Principles of mass-balancing computations

• Stochastic variables and probability density functions

• Failure criterion and limit state function

• Risk analysis and risk assessment

• Summary and outlook


Introduction and objective of the probabilistic analysis

Introduction

• Correct mass-balancing of high-speed reciprocating engines is very important forthe vibration behavior of the engine.

• The mass-tolerances of the engine parts: piston, conrod and crankshaft areleading to unbalanced mass-forces and mass-moments, which are the cause ofundesirable vibrations of the engine.

• Worst-case computations are leading to unnecessary tight mass-tolerances.

Objective of the probabilistic assessment

• Stochastic rating of the mass-tolerances by taking the mass distributions of thepiston, conrod and crankshaft into account.

• Probabilistically verified specification of the mass-tolerances.


Principles of mass-balancing computations

• The whole crankshaft is split into single crank-drives.

• The determination of the mass-forces and mass-moments are based on a simplesummation of the inertia-forces of all single crank-drives.

• System is linear with respect to the masses of the engine parts.

single V-crank-drive


Stochastic variables and probability density functions

• The weights of the engine parts: piston, conrod (oscillational and rotational part)and counter-weights are defined as stochastic variables.

• Hence for an 8V-crank-drive there are about 4*8=32 stochastic variables altogether.

Probability density functions (PDF)

• The mass-distributions of the piston, conrod and counter-weights are usually notknown precisely. Measurements lack of too few independent samples.

• An uniform-distribution between the lower and upper limits of tolerances woulddefinitely be a worst-case distribution.

• A truncated normal-distribution with the mean at the nominal value is not always agood assumption.


Stochastic variables and probability density functions

Example: probability density function of conrod

fitted normal-distribution

Empirical PDF ofmeasured weights

Possible PDF:• Truncated normal-distribution• Mean is also a stochastic variable and varies• STD is defined as ¾ * tolerance

centered mean PDF shifted mean PDF


Failure criterion and limit state function

• The used failure criterion is the effective vibration velocity at the front and the aft ofthe rigid engine.

• The effective vibration velocity is a function of the stochastic variables and istherefore also stochastic.

• The vibration velocity is computed by an integration of Newton’s second law.

front

aft

x y

z

2

ˆ...ˆˆ 222

21 n

eff

vvvv

Periodical signal:

effv

dm

Fvd

v

v 0 0

)(


Risk analysis and risk assessment

Typical steps in a risk analysis and risk assessment

• Definition of stochastic variables of the system and the corresponding PDFs.(already done)

• Definition of the failure criterion and the limit state function. (already done)

• Transformation of the physical stochastic variables (masses) to dimensionlessstandard-normally distributed variables x=T(z).

• Computation of the probability of failure by means of stochastic algorithms(Monte-Carlo Simulation, approximation methods e.g. FORM)

• Comparing the probability of failure with a predefined limit probability, which mustnot be exceeded (Plimit = 0,0013 in our case, 3-sigma level).

• Taking counter-measures if the limit probability has been exceeded (tighten thetolerances in our case).



Results of an 8V-diesel-engine:reference state

000Mz*

[-]

000My*

[-]

001Fz*

[-]

000Fy*

[-]

2.

ord.

1.

ord.

all

ord.

engine speed n = 1900 [1/min]

Fz*Fy*

My* Mz*



Results of an 8V-diesel-engine:typical tolerance state

00,060,08Mz*

[-]

00,10,09My*

[-]

00,401,41Fz*

[-]

0,090,600,65Fy*

[-]

2.

ord.

1.

ord.

all

ord.

engine speed n = 1900 [1/min]

Fz*Fy*

My* Mz*


Coefficient of variation as a measure of efficiency:

Example:

100000 independent simulation runs are necessary to get a STD of 10% of the failureprobability.

One simulation lasts ~0.1[s] 2h50min computation time

Efficient Monte-Carlo Simulations with variance reduction are better suited to solvethis problem (Importance Sampling, Subset Simulation).

Brute force Monte-Carlo Simulation is not efficient for low probabilities of failure .


PfN

Pf

Pf

Pf

1

Pf

1.0

Results of an 8V-diesel-engine: probability of failure

001.0Pf 100000N



Probability of failure

1,00E-12

1,00E-11

1,00E-10

1,00E-09

1,00E-08

1,00E-07

1,00E-06

1,00E-05

0,85 0,95 1,05 1,15

normalized tolerances

Pf Pf mean

Pf 3-sigma

Results of an 8V-diesel-engine: probability of failure

Very efficient Monte-Carlo Simulation with variance reduction

Example: different conrod tolerances n=1900 [1/min]

Considerable advantages:

Conrod – tolerances increased from0,85 to 1,15!

Less expensive manufacturing!


Summary and outlook

Summary

• Implementation of a mass-balance program in a Matlab environment

• Profound knowledge of mass-distributions of engine parts is essential.

• Efficient Monte-Carlo Simulation is obligatory for low computation times.

• Probabilistic assessment gives deeper insight into the system.

• Probabilistic specification of mass-tolerances is possible.

Outlook

• Probabilistic based optimization of mass-tolerances with respect tonormalized costs under the condition, that a predefined probability offailure must not be exceeded.

Thank you for your attention

I’m looking forward to answering yourquestions.

stochastic assessment of crank-drive component …...christian wetzel, mtu friedrichshafen gmbh 20....

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