This paper extends the response of uncertain nonlinear vibration systems to vectorvaluedand matrixvaluedfunctions. Random variables and system derivatives are conveniently arranged into 2Dmatrices. The methodis based on a second order expansion of the governing equations and matrix calculus, Kronecker algebra are usedin the mathematical development. The results derived are easily amenable to computational procedures.

The first passage of uncertain single degree-of-freedom nonlinear oscillators subjected to random excitation is examined. Statistical fourthmoment method is developed to determine first four moments of system response and state function. Distribution function of the systemstate function is approximately determined by the standard normal distribution functions using Edgeworth series technique and the reliabilityis obtained

The dynamic equations of the Jeffcott rotor with imbalance are concerned. The reliabilityof rotor systems with rubbing is examined. A statistical fourth moment method is developedto determine the first four moments of the system response and the state function.The distribution function of the system state function is approximately determined byusing the Edgeworth series technique. Its reliability is obtained. The effect on the reliabilityof the shaft stiffness, the external damping coefficient, the degree of imbalance andthe stator stiffness is studied. Numerical results are also presented and discussed.

The coil tube spring, a new type of spring, has been recently used in engineering. Thispaper presents a method for reliability-based analysis of a coil tube spring. A technique, using theperturbation method and reliability-based design theory, is employed to present a practical andeffective method for the reliability-based analysis of a coil tube spring. Theoretical formulae of thereliability of a coil tube spring are obtained. The given approach can be used to determine thereliability of a coil tube spring accurately and quickly.