Lower and upper solutions method to the fully elastic cantilever beam equation with support

The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, $$ \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=g(u(1)), \end{cases} $$ { u ( 4 ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) , u ‴ ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ′ ( 0 ) = 0 , u ″ ( 1 ) = 0 , u ‴ ( 1 ) = g ( u ( 1 ) ) , where $f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}$ f : [ 0 , 1 ] × R 4 → R , $g: \mathbb{R}\rightarrow \mathbb{R}$ g : R → R are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.

An Investigation of Tip Force Characteristics of Brush Seals

Thanks to their compliant nature and superior leakage performance over conventional labyrinth seals, brush seals found increasing use in turbomachinery. Utilizing high temperature super-alloy fibers and their compliance capability these seals maintain contact with the rotor for a wide range of operating conditions leaving minimal passage for parasitic leakage flow. Consequently, the contact force/pressure generated at seal rotor interface is of importance for sustained seal performance and longevity of its service life. Although some analytical and numerical models have been developed to estimate bristle tip pressures, they simply rely on linear beam equation calculations and other such assumptions for loading cases. In this paper, previously available analytical and/or numerical models for bristle tip force/pressure have been modified and enhanced. The nonlinear cantilever beam equation has been solved and results are compared to a linear cantilever beam equation solution to establish application boundaries for both methods. The results are also compared to experimental data. With the support of testing, an empirical model has been developed for tip forces under operating conditions.