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

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 2

MTU Friedrichshafen GmbHShort introduction

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 3

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

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 4

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.

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 5

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

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 6

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.

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 7

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

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 8

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

)(

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 9

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).

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 10

Risk analysis and risk assessment

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*

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 11

Risk analysis and risk assessment

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*

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 12

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 .

Risk analysis and risk assessment

PfN

Pf

Pf

Pf

1

Pf

1.0

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

001.0Pf 100000N

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 13

Risk analysis and risk assessment

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!

Christian Wetzel, MTU Friedrichshafen GmbH 20. Mai 2010 Seite 14

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.