[{"data":1,"prerenderedAt":36},["ShallowReactive",2],{"topic-en-strength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fproduct-of-inertia-and-axis-rotation":3},{"topic":4,"trail":21,"children":31,"tasks":32,"alternates":33},{"id":5,"name":6,"locale":7,"path":8,"seo_title":9,"seo_description":10,"seo_text":11,"content_html":12,"content_chunks":13},122,"Product of Inertia and Axis Rotation","en","strength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fproduct-of-inertia-and-axis-rotation","Product of Inertia and Rotation of Area Axes","Product of inertia Ixy, transformation of Ix, Iy and Ixy under axis rotation, invariance of Ix + Iy, and the condition for principal axes.","This topic explains the product of inertia Ixy and how second moments of area depend on coordinate-axis orientation. It presents the transformation equations for Ix, Iy, and Ixy, the invariant Ix + Iy, and the condition used to determine principal centroidal axes.","\u003Cp>The \u003Cstrong>product of inertia $I_{xy}$\u003C\u002Fstrong> characterizes the combined distribution of area relative to two mutually perpendicular axes. Unlike $I_x$ and $I_y$, it may be positive, negative, or zero.\u003C\u002Fp>\u003Cdiv class=\"formula-chunk\">\u003Cp>The second moments of area about the x and y axes are:\u003C\u002Fp>\u003Cp>$$I_x=\\int_A y^2\\,dA,\\qquad I_y=\\int_A x^2\\,dA.$$\u003C\u002Fp>\u003Cp>The product moment of area is:\u003C\u002Fp>\u003Cp>$$I_{xy}=\\int_A xy\\,dA.$$\u003C\u002Fp>\u003Cp>The polar second moment of area about the intersection of orthogonal axes is:\u003C\u002Fp>\u003Cp>$$J_p=\\int_A(x^2+y^2)dA=I_x+I_y.$$\u003C\u002Fp>\u003Cp>Second and polar moments of area have dimensions of length to the fourth power, for example mm⁴.\u003C\u002Fp>\u003C\u002Fdiv>\u003Ch2>Symmetry\u003C\u002Fh2>\u003Cp>If one centroidal axis is an axis of symmetry, the product of inertia with respect to that axis and the perpendicular centroidal axis is zero. However, $I_{xy}=0$ by itself does not necessarily imply geometric symmetry.\u003C\u002Fp>\u003Ch2>Axis rotation\u003C\u002Fh2>\u003Cdiv class=\"formula-chunk\">\u003Cp>When orthogonal axes are rotated through an angle θ, the second moments and product of inertia transform as:\u003C\u002Fp>\u003Cp>$$I_{x'}=\\frac{I_x+I_y}{2}+\\frac{I_x-I_y}{2}\\cos2\\theta-I_{xy}\\sin2\\theta,$$\u003C\u002Fp>\u003Cp>$$I_{y'}=\\frac{I_x+I_y}{2}-\\frac{I_x-I_y}{2}\\cos2\\theta+I_{xy}\\sin2\\theta,$$\u003C\u002Fp>\u003Cp>$$I_{x'y'}=\\frac{I_x-I_y}{2}\\sin2\\theta+I_{xy}\\cos2\\theta.$$\u003C\u002Fp>\u003Cp>Principal centroidal axes satisfy $I_{x'y'}=0$. Their orientation follows from:\u003C\u002Fp>\u003Cp>$$\\tan2\\theta_p=-\\frac{2I_{xy}}{I_x-I_y},$$\u003C\u002Fp>\u003Cp>with the quadrant and the adopted positive direction of angle measured consistently.\u003C\u002Fp>\u003C\u002Fdiv>\u003Cp>Under rotation of axes, the sum $I_x+I_y$ remains unchanged. This invariant is useful for checking calculations.\u003C\u002Fp>\u003Ch2>Principal axes\u003C\u002Fh2>\u003Cp>Orientations for which $I_{xy}=0$ and $I_x$ and $I_y$ take extreme values are called principal axes of inertia. If they pass through the centroid, they are principal centroidal axes.\u003C\u002Fp>\u003Ch2>Engineering significance\u003C\u002Fh2>\u003Cp>For unsymmetrical sections, principal axes are required in unsymmetrical bending and other problems where the loading direction does not coincide with convenient geometric axes.\u003C\u002Fp>",[14,18],{"id":15,"code":16,"type":17,"locale":7},88,"area-second-moments","formula",{"id":19,"code":20,"type":17,"locale":7},120,"area-inertia-axis-rotation",[22,26,30],{"id":23,"name":24,"path":25},45,"Strength of Materials","strength-of-materials",{"id":27,"name":28,"path":29},59,"Geometric Properties of Plane Areas","strength-of-materials\u002Fgeometric-properties-of-plane-areas",{"id":5,"name":6,"path":8},[],[],{"en":34,"uk":35},"https:\u002F\u002Fmechclassroom.com\u002Fen\u002Ftopics\u002Fstrength-of-materials\u002Fgeometric-properties-of-plane-areas\u002Fproduct-of-inertia-and-axis-rotation","https:\u002F\u002Fmechclassroom.com\u002Ftopics\u002Fopir-materialiv\u002Fheometrychni-kharakterystyky-pereriziv\u002Fvidtsentrovyi-moment-inertsii-ta-povorot-osei",1787712539973]