Moments and Product of Inertia
Moments and Product of Inertia
Contents·Introduction（绪论）·MomentsofInertiaofanArea（平面图形的惯性矩）·Moments ofInertiaof anAreabyIntegration（积分法求惯性矩）·PolarMoments of Inertia（极惯性矩）·Radius ofGyrationof anArea（惯性半径）·ParallelAxisTheorem（平行移轴定理）·Moments of Inertia of Common Shapes of Areas（常见平面图形的惯性矩）·ProductofInertia（惯性积）·Principal Axes and Principal Moments of Inertia（主惯性轴与主惯性矩）·Mohr'sCircleforMomentsofInertia（惯性矩和惯性积莫尔圆）·PrincipalPoints（主惯性点）2
Contents • Introduction（绪论） • Moments of Inertia of an Area（平面图形的惯性矩） • Moments of Inertia of an Area by Integration（积分法求惯性矩） • Polar Moments of Inertia（极惯性矩） • Radius of Gyration of an Area（惯性半径） • Parallel Axis Theorem（平行移轴定理） • Moments of Inertia of Common Shapes of Areas（常见平面图形 的惯性矩） • Product of Inertia（惯性积） • Principal Axes and Principal Moments of Inertia（主惯性轴与主 惯性矩） • Mohr’s Circle for Moments of Inertia（惯性矩和惯性积莫尔圆） • Principal Points（主惯性点） 2
Lntroduction: Previously considered distributed forces which were proportional to thearea or volume over which they act-The resultant was obtained by summing or integrating over theareas or volumes.- The moment of the resultant about any axis was determined bycomputing the first moments of the areas or volumes about thataxis.: Will now consider forces which are proportional to the area or volumeover which they act but also vary linearly with distance from a given axis- It will be shown that the magnitude of the resultant depends on thefirst moments of the force distribution with respect tothe axis.- The point of application of the resultant depends on the secondmomentsofthedistributionwithrespecttotheaxis.: Current chapter will present methods for computing the moments andproducts of inertia for areas.3
Introduction • Previously considered distributed forces which were proportional to the area or volume over which they act. - The resultant was obtained by summing or integrating over the areas or volumes. - The moment of the resultant about any axis was determined by computing the first moments of the areas or volumes about that axis. • Will now consider forces which are proportional to the area or volume over which they act but also vary linearly with distance from a given axis. - It will be shown that the magnitude of the resultant depends on the first moments of the force distribution with respect to the axis. - The point of application of the resultant depends on the second moments of the distribution with respect to the axis. • Current chapter will present methods for computing the moments and products of inertia for areas. 3
Moments of Inertia of an Area. Consider distributed forces F whose magnitudes areproportional to the elemental areas A on which theyAAact and also vary linearly with the distance of AAF-kyAAfrom a given axis.: Example: Consider a beam subjected to pure bendingInternal forces vary linearly with distancefrom theneutral axis which passes through the section centroidAF = kyAAR=k[ydA=0 JydA=S, = first momentM=k[y’dA[y’dA= second momentExample:Considerthenethydrostaticforceonasubmerged circular gate△F = pAA= pgyAAAF=YyAAR= pgJydAM,=pgJy'dA4
Moments of Inertia of an Area • Consider distributed forces whose magnitudes are proportional to the elemental areas on which they act and also vary linearly with the distance of from a given axis. F A A • Example: Consider a beam subjected to pure bending. Internal forces vary linearly with distance from the neutral axis which passes through the section centroid. 2 2 0 first moment second moment x F ky A R k y dA y dA S M k y dA y dA • Example: Consider the net hydrostatic force on a submerged circular gate. 2 x F p A gy A R g y dA M g y dA 4
Moments of Inertia of an Area by Integration7dA =dxdy: Second moments or moments of inertia ofdxan area with respect to the x and y axes,IdyIx =[y?dAI,=[x?dAdl,=xedadlr=yedaydA=(a-x)dydA=ydx. Evaluation of the integrals is simplified bytchoosing dA to be a thin strip parallel tooneofthe coordinateaxes.ddly=x2dAdlr=y-dA·Forarectangulararea.Ix =[y2dA= y?bdy=1bh3dA-bdy0dy: The formula for rectangular areas may alsobe applied to strips parallel to the axes,dxdlx=-y'dxd,=reydxdl,=x2dA=x2ydx5
Moments of Inertia of an Area by Integration • Second moments or moments of inertia of an area with respect to the x and y axes, I x y dA I y x dA 2 2 • Evaluation of the integrals is simplified by choosing dA to be a thin strip parallel to one of the coordinate axes. • For a rectangular area, 3 3 1 0 2 2 I y dA y bdy bh h x • The formula for rectangular areas may also be applied to strips parallel to the axes, dI y dx dI x dA x y dx x y 3 2 2 3 1 5