Probabilistic Mechanism Analysis

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Outline

• Uncertainty in mechanisms• Why consider uncertainty• Basics of uncertainty• Probabilistic mechanism analysis• Examples• Probabilistic mechanism synthesis• Conclusions

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Uncertainty in Mechanisms• Lengths , and are random variables due to

manufacture imprecision

Uncertainty in Mechanisms• The joint clearances at A, B, C, and D are also

random due to manufacture imprecision and installation errors.

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rJ

rC

Bearing

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Uncertainty in Mechanisms• Loads and material properties are also

random.

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Impact of Uncertainty• If the above mechanism generates a

functional relationship and the required motion output is , then the motion error is , where )

• is also a random variable.• Required motion output may not be realized

with certainty.

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Why Consider Uncertainty?

• We know the true solution.• We know the effect of uncertainty.• We can design mechanisms whose

performance is not sensitive to uncertainty• We can make more reliable decisions.

In this presentation, we focus on uncertainty only in R.

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• We use probability distributions to model parameters with uncertainty.

How Do We Model Uncertainty?

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Probabilistic Design

Baseline Design

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Probability Distribution

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• With random samples, we can draw a histogram.

• If y-axis is frequency and the number of samples is infinity, we get a probability density function (PDF) for random variable

• The probability of is

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Normal Distribution

• : cumulative distribution function (CDF)

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More about Standard Deviation (std)

• It indicates how data spread around the mean.• It is always non-negative.• High std means

– High dispersion– High uncertainty– High risk

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More Than One Random Variables

• If all lengths are normally distributed

– are independent

– are constants.• Then

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Mechanism Reliability• Let the required motion error be .• Mechanism reliability is • Probability of failure

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First Order Second Moment Method (FOSM)• Assume lengths are independent• First order Taylor expansion for motion output =)

• Motion error • , ,

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Monte Carlo Simulation (MCS)*A sampling-based simulation method

*This topic is optional.

Step 3: Statistic Analysis on model outputExtracting probabilistic information

Step 3: Statistic Analysis on model outputExtracting probabilistic information

Step 2: Numerical ExperimentationEvaluating performance function

Step 2: Numerical ExperimentationEvaluating performance function

Step 1: Sampling of random variablesGenerating samples of random variables

Step 1: Sampling of random variablesGenerating samples of random variables

Analysis Model

Samples of input variables

Samples of input variables

Samples of output variables

Samples of output variables

Probabilistic characteristics of output variables

Probabilistic characteristics of output variables

Distributions of input variables

Distributions of input variables

Analysis Model

Y=g(X)

X Y

Step 1: Sampling on random variables

• Generate samples of input random variables according to their distributions.

• For example, for , samples can be generated by Matlab.– Matlab normrnd(, ) produces a row vector of

random samples.– Excel can also be used.

Step 2: Obtain Samples of Output

• Suppose N sets of random variables have been generated

, is the number of simulations

• Then samples of output s are calculated a

Step 3: Statistic Analysis on output

• Mean

• Standard deviation

• The probability of failure is the number of failures.

= number of <0,

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FORM vs MCS

• FORM is more efficient• FORM may not be accurate when a limit-state

function is highly nonlinear• MCS is very accurate if the sample size is

sufficiently large• MCS is not efficient

Example - FOSM

• mm, mm, mm; they are independent.• Requirement: mm when (The motion output here is

displacement instead of an angle .)• Allowable motion error: mm

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𝑅1

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Example - FOSM• 287.2261 mm• 0.6478

where

Example - FOSM• mm•

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Example - MCS

• mm, mm, mm; they are independent.• Requirement: mm when • Allowable motion error: mm

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𝑅1

A

C

B

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1e6 Simulations•

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Reliability–Based Mechanism Synthesis

• Find: Design variables (average mechanism dimensions)

• Minimize: average motion error or other objective

• Subject to reliability constraint where is the required reliability

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Conclusions

• For important mechanisms in important applications, it is imperative to consider reliability.

• Uncertainty can be modeled probabilistically.• Reliability can be estimated by FOSM and

MCS.• Same methodologies can also be used for

cams, gears, and other mechanisms.