3/31/ Product of Inertia. Sample Problem Sample Problem 10.6 (continue)

Transcription

1 /1/ Product of Inertia Product of Inertia: I xy = xy da When the x axis, the y axis, or both are an axis of symmetry, the product of inertia is zero. Parallel axis theorem for products of inertia: I xy = I xy + xya 9-1 Sample Problem 10.6 Determine the product of inertia of the right triangle (a) with respect to the x and y axes and (b) with respect to centroidal axes parallel to the x and y axes. Determine the product of inertia using direct integration with the parallel axis theorem on vertical differential area strips Apply the parallel axis theorem to evaluate the product of inertia with respect to the centroidal axes. 9 - Sample Problem 10.6 (continue) Determine the product of inertia using direct integration with the parallel axis theorem on vertical differential area strips x x y = h 1 da = y dx = h 1 dx b b x x = x y = 1 y = 1 el el h 1 b Integrating di x from x = 0 to x = b, b x Ixy dixy xel yelda x( 1 = = = ) h dx 1 0 b b b x x x x x x = h + dx = h b + b b 0 8b 0 I 1 xy = b h 9-1

2 /1/01 Sample Problem 10.6 (continue) Apply the parallel axis theorem to evaluate the product of inertia with respect to the centroidal axes. x = 1 b y = 1 h With the results from part a, Ixy = Ix + xya I 1 x y = b h ( 1b)( 1 h)( 1 bh) Ix y = 1 b h Moments of Inertia about inclined axis In structural and mechanical design, it is sometimes necessary to calculate the moment of inertia with respect to a set of inclined u, v, axes when the values of θ, I x, I y, I xy are known. To do this we will use transformation equations which relates the x, y, and x, y coordinates. From Figure, these equations are: Note: x = x cosθ + y sinθ = y cosθ xsinθ 9-5 Given Moments of Inertia about inclined axis,, continue I x = y da I xy = xy da I y = x da we wish to determine moments and product of inertia with respect to new axes x and y. The change of axes yields = + cosθ I xy sin θ I = cosθ + I xy sin θ I x I y = sin θ + I xy cosθ By adding the equations for I x and I x we can show that the polar moment of inertia about z axis passing through point O is independent of the orientation of x and y; J o = I x + I y = I x + I y These equations show that I x, I y and I x y depend on the angle of the inclination, θ, of the x, y axes. 9-6

3 /1/01 Principal Axes and Principal Moments of Inertia We will now determine the orientation of these axes about which I x, I y are maximum and minimum. This particular axes are called principal axes By differentiating the first of Eqs with respect to θ and setting the result to zero. Thus; di x' dθ Therefore, at θ = θp ; By substituting for θ in I x, I y and I x y equations and simplifying, we obtain; 9-7 Summary,,,,, = + cosθ I xy sin θ I = cosθ + I xy sin θ I x I y = sin θ + I xy cosθ (a) (a) (a) Principal Axes and Principal Moments of Inertia Squaring Eqs. 10-9a and 10-9c and adding, it is found that; ( I I ) x ave + = R where ; I x + I y I ave =, I x I y R = + I xy At the points A and B, I x y = 0 and I x is a maximum and minimum, respectively. Imax,min = Iave ± R I xy tan θ m = I x I y The equation for θ m defines two angles, 90 o apart which correspond to the principal axes of the area about O. I max and I min are the principal moments of inertia of the area about O. 9-9

4 /1/ Mohr s Circle for Moments and Products of Inertia The moments and product of inertia for an area are plotted as shown and used to construct Mohr s circle, I x + I y I x I y I ave = R = + I xy Mohr s circle may be used to graphically or analytically determine the moments and product of inertia for any other rectangular axes including the principal axes and principal moments and products of inertia Sample Problem 10.7 For the section shown, the moments of inertia with respect to the x and y axes are I x = 10.8 in and I y = 6.97 in. Determine (a) the orientation of the principal axes of the section about O, and (b) the values of the principal moments of inertia about O. Compute the product of inertia with respect to the xy axes by dividing the section into three rectangles and applying the parallel axis theorem to each. Determine the orientation of the principal axes (Eq. 9.5) and the principal moments of inertia (Eq. 9. 7) Sample Problem 10.7 Compute the product of inertia with respect to the xy axes by dividing the section into three rectangles. Apply the parallel axis theorem to each rectangle, I xy = ( Ix + xya) Note that the product of inertia with respect to centroidal axes parallel to the xy axes is zero for each rectangle. Rectangle I II III Area, in x, in y, in xya,in xya = 6.56 = A = 6.56 in xy I xy 9-1

5 /1/01 Sample Problem 10.7 Determine the orientation of the principal axes (Eq. 9.5) and the principal moments of inertia (Eq. 9. 7). Ixy ( 6.56) tan θ m = = = +.85 Ix I y θ m = 75. and 55. θ m = 7.7 and θm = Ix = 10.8 in I y = 6.97 in I xy = 6.56 in Ix + I y Ix I y Imax,min = ± + Ixy = ± + ( 6.56) Ia = Imax = 15.5 in Ib = Imin = in 9-1 Sample Problem 10.8 The moments and product of inertia with respect to the x and y axes are I x = 7.x106 mm, I y =.61x106 mm, and I xy = -.5x10 6 mm. Using Mohr s circle, determine (a) the principal axes about O, (b) the values of the principal moments about O, and (c) the values of the moments and product of inertia about the x and y axes Plot the points (I x, I xy ) and (I y,-i xy ). Construct Mohr s circle based on the circle diameter between the points. Based on the circle, determine the orientation of the principal axes and the principal moments of inertia. Based on the circle, evaluate the moments and product of inertia with respect to the x y axes. 9-1 Sample Problem 10.8 Plot the points (I x, I xy ) and (I y,-i xy ). Construct Mohr s circle based on the circle diameter between the points. OC = I 1 ave = ( I ) x + I y =.95 1 CD = ( I ) =.15 x I y R = ( CD) + ( DX ) =.7 I x = 7. I y =.61 I xy =.5 Based on the circle, determine the orientation of the principal axes and the principal moments of inertia. DX tan θ m = = θ m = 7. 6 θ m =. 8 CD Imax = OA = Iave + R Imin = OB = Iave R Imax = 8.6 Imin =

6 /1/01 Sample Problem 10.8 Based on the circle, evaluate the moments and product of inertia with respect to the x y axes. The points X and Y corresponding to the x and y axes are obtained by rotating CX and CY counterclockwise through an angle Θ = (60 o ) = 10 o. The angle that CX forms with the x axes is φ = 10 o o = 7. o. o I x' = OF = OC + CX cosϕ = Iave + Rcos 7. I = mm x o I y' = OG = OC CY cosϕ = Iave Rcos7. I = mm OC = Iave =.95 R =.7 o y' = FX = CY sinϕ = Rsin 7. I 6 mm x =