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MEC5881 Practice Questions

Question 1

A system is expected to produce a response overtime, y(t), defined by the ordinary differential equation

dt/dy = −1.5y

At time t = 0, the response y = 4.7.  Use the forward Euler’s method to find the solution at time t  = 5, using a time interval Δt  = 1.

Question 2

The analytical solution to the response in Question 1 is

y(t) = 4.7 exp(−1.5t).

Tabulate the absolute error between the estimates of y obtained from the time integration in Question 1, and the true solution, |y − yn|.  From t = 1 onward, does the error increase or decrease?  Would this correspond to astable or unstable time integration?

Question 3

A different system is expected to produce a response overtime, y(t), defined by the ordinary differential equation

dt/dy = −3.2y

At timest = −0.4, −0.2, and 0, the responses are respectively yn−2 = 343.8387578, yn−1 = 181.3035721 and yn  = 95.6.

a)   The forward Euler method time integrates via

yn+1  = yn  + Δtfn.

Simplify this equation for this problem and rewrite it for yn+1 .

b)   The third-order backwards differentiation method time integrates via

6/11yn+1  = 3yn  −2/3yn−1 + 3/1yn + Δt fn+1 .

Simplify this equation for this problem and rewrite it for yn+1 .

c)    Using a time step size Δt  = 0.2, integrate the solution forward in time from t  = 0 to t  =  1 using each of the forward Euler and 3rd  order backwards differentiation methods.  Which result to you expect will be more accurate, and why?

Question 4

The probability of failure of a system comprising two interacting components is

Psys = P1P2.

Draw the fault tree corresponding to this expression (include any low-level input events and any pertinent logic gates).

Question 5

A small system has the following fault tree diagram.

Find an expression for the failure probability of this system Psys, given the failure probabilities of low-level components 1, 2 and 3 being P1, P2  and P3, respectively.

Question 6

The probability of failure of a system comprising three interacting components is

Psys  = P1 ( 1 − (1 − P2)(1 − P3)).

Draw the fault tree corresponding to this expression (include any low-level input events and any pertinent logic gates).

Question 7

A system has the following fault tree diagram.

Find an expression for the failure probability of this system Psys, given the failure probabilities of low-level components 1, 2, 3 and 4 being P1, P2, P3  and P4, respectively.

Question 8

The probability of failure of a system comprising six interacting components is

Psys  = 1 − (1 − (1 − P1)(1 − P3)) ( 1 − (1 − (1 − P2)(1 − (1 − P4)(1 − P5)))) (1 − P6).

Draw the fault tree corresponding to this expression (include any low-level input events and any pertinent logic gates).

Question 9

The NASA Space Shuttle employed a redundant avionics system comprising four (4) computers.  If a single computer fails, the system integrity remains intact because the other three computers produce consistent output.  If a second computer fails, the remaining two sustain operational performance.  The Shuttle would be lost in the event of a third computer failure, as the system would not know which of the last two should be relied upon.  If the probability of failure of a single computer on a given mission is 0.0016, what would the probability of avionics system failure be for the four-computer redundant system?

Question 10

A mechanical component has a failure density function

where time τ is measured in weeks.

a)    Find the function expressing the probability of failure after time t.

b)   Find the reliability probability as a function of time.

c)    Find the hazard function.

Question 11

An electronics component manufacturer develops a new capacitor for application in high humidity environments. The manufacturer seeks to determine the failure rate for the component at its most extreme operating conditions.  One thousand (1000) prototypes are produced and are simultaneously   tested. The test put the capacitors into operation while exposed to the specified environment. The test ran for 90 days.  Every 10 days, the number of failed capacitors was counted and recorded.  The results are tabulated below.  Based on these results, calculate the failure rate (failures per day) within each 10 day block.

Question 12

An automotive manufacturer seeks to stress-test a new engine they are seeking to install in a newline of petrol-powered utility vehicles.  The test involves taking 8 prototypes of the new engine, and running each of them until failure.  The time to failure for each engine is recorded.  Estimate the failure rate for the new engine.

Estimate the meantime between failure for this new engine under the stress test conditions.

Question 13

An aerospace manufacturer is designing a control system for an unmanned aerial vehicle (UAV).  To provide some redundancy, two avionics computers (Avionics 1 and Avionics 2) are arranged in parallel. The avionics computer(s) supply commands to an actuator-controlled rudder.  Two design alternatives are being considered.

In Option A (sketched below), the actuator is duplicated, so that Avionics 1 commands Actuator 1, while Avionics 2 commands Actuator 2.  In normal operation, both actuators will control the rudder in unison.

In the event of either an avionics or actuator failure, the surviving operational path continues to control the UAV.

In Option B (sketched below), the avionics computer signals are fed to a single actuator.

If the avionics computers have a failure probability of 0.1% per flight hour, while the actuators have a failure probability of 0.01% per flight hour, which of Options A or B has the greater reliability?