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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rB
rJ
rC
Bearing
Journal
Link i
Link j
Y
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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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Noise
pd
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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
A
C
B
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𝜃
𝑠
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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
𝑅2
𝑅3
𝜃
𝑠
1e6 Simulations•
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s
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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.