Centroid And Moment Of Inertia Solved Problems Pdf

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Published: 30.03.2021  We equivalently represent the system of forces by single force acting at a specific point. The centroid coincides with the center of mass or the center of gravity only if the material of the body is homogenous density or specific weight is constant throughout the body. Centroids and centers of gravity steven vukazich san jose state university.

Method of Composite Parts for Moments of Inertia and the Parallel Axis Theorem

As an alternative to integration, both area and mass moments of inertia can be calculated via the method of composite parts, similar to what we did with centroids. In this method we will break down a complex shape into simple parts, look up the moments of inertia for these parts in a table, adjust the moments of inertia for position, and finally add the adjusted values together to find the overall moment of inertia.

This method is known as the method of composite parts. A key part to this process that was not present in centroid calculations is the adjustment for position. As discussed on the previous pages, the area and mass moments of inertia are dependent upon the chosen axis of rotation.

Moments of inertia for the parts of the body can only be added when they are taken about the same axis. The moments of inertia in the table are generally listed relative to that shape's centroid though.

Because each part has its own individual centroid coordinate, we cannot simply add these numbers. We will use something called the Parallel Axis Theorem to adjust the moments of inertia so that they are all taken about some standard axis or point.

Once the moments of inertia are adjusted with the Parallel Axis Theorem, then we can add them together using the method of composite parts. When we calculated the area and mass moments of inertia via integration, one of the first things we had to do was to select a point or axis we were going to take the moment of inertia about. We then measured all distances from that point or axis, where the distances were the moment arms in our moment integrals.

Because the centroid of a shape is the geometric center of an area or volume, the average distance to any one point in a body is at a minimum. If we pick a different point or axis to take the moment of inertia about, then on average all the distances in our moment integral will be a little bit bigger. Specifically, the further we move from the centroid, the larger the average distances become. Though this complicates our analysis, the nice thing is that the change in the moment of inertia is predicable.

It will always be at a minimum when we take the moment of inertia about the centroid, or an axis going through the centroid. This minimum, which we will call I c is the value we will look up in our moment of inertia table. From this minimum, or unadjusted value, we can find the moment of inertia value about any point I p by adding an an adjustment factor equal to the area times distance squared for area moments of inertia, or mass times distance squared for mass moments of inertia.

This adjustment process with the equations above is the parallel axis theorem. The area or mass terms simply represent the area or mass of the part you are looking at, while the distance r represents the distance we are moving the axis we are taking the moment of inertia about.

This may be a vertical distance, a horizontal distance, or a diagonal depending on the axis the moment or inertia is about. Say we are trying to find the moments of inertia of the rectangle above about point P.

We would start by looking up I xx , I yy , and J zz about the centroid of the rectangle C in the moment of inertia table. Then we would add on an area times distance squared term to each to find the adjusted moments of inertia about P. The distance we are moving the x axis for I xx is the vertical distance r x , the distance we are moving the y axis for I yy is the horizontal distance r y , and the distance we would move the z axis which is pointing out of the page for J zz is the diagonal r z.

Center of mass adjustments follow a similar logic, using mass times distance squared, where the distance represents how far you are moving the axis of rotation in three dimensional space. To find the moment of inertia of a body using the method of composite parts, you need to start by breaking your area or volume down into simple shapes. Make sure each individual shape is available in the moment of inertia table, and you can treat holes or cutouts as negative area or mass.

Next you are going to create a table to keep track of values. Devote a row to each part that your numbered earlier, and include a final "total" row that will be used for some values. Most of the work of the method of composite parts is filling in this table The columns will vary slightly with what you are looking for, but you will generally need the following. The overall moment of inertia of your composite body is simply the sum of all of the adjusted moments of inertia for the pieces, which will be the sum of the values in the last column or columns if you are finding the moments of inertia about more than one axis.

Use the parallel axis theorem to find the mass moment of inertia of this slender rod with mass m and length L about the z axis at its end point. A beam is made by connecting two 2" x 4" beams in a T pattern with the cross section as shown below.

Determine the location of the centroid of this combined cross section and then find the rectangular area moment of inertia about the x axis through the centroid point. A dumbbell consists of two. Determine the mass moment of inertia of the dumbbell about the y axis shown in the diagram.

Method of Composite Parts for Moments of Inertia and the Parallel Axis Theorem As an alternative to integration, both area and mass moments of inertia can be calculated via the method of composite parts, similar to what we did with centroids.

The Parallel Axis Theorem When we calculated the area and mass moments of inertia via integration, one of the first things we had to do was to select a point or axis we were going to take the moment of inertia about. The distances used in our moment integrals depends on the point or axis chosen.

These distances will be at a minimum at the centroid and will get larger as we start further from the centroid. The distance r in the Parallel Axis Theorem represents the distance we are moving the axis we are taking the moment or intent about. Using the Method of Composite Parts to Find the Moment of Inertia To find the moment of inertia of a body using the method of composite parts, you need to start by breaking your area or volume down into simple shapes.

Start by breaking down your area or volume into simple parts and number those shapes. Holes or cutouts will count as negative areas or masses. Most work in the method of composite parts will revolve around filling out a table such as this one.

This table contains the rows and columns necessary to find the rectangular area moments of inertia I xx and I yy for this composite body. The area or mass for each piece area for area moments of inertia or mass for mass moments of inertia. Remember cutouts should be listed as negative areas or masses.

The centroid or center of mass locations X, Y and possibly Z coordinates. Most of the time, we will be finding the moment of inertia about centroid of the composite shape, and if that is not explicitly given to you, you will need to find that before going further. The moment of inertia values about each shape's centroid.

To find these values you will plug numbers for height, radius, mass, etc. Do not use these formulas blindly though as you may need to mentally rotate the body, and thus switch equations, if the orientation of the shape in the table does not match the orientation of the shape in your diagram. The adjustment distances r for each shape. For this value you will want to determine how far the x-axis, y-axis, or z-axis moves to go from the centroid of the piece to the overall centroid, or point you are taking the moment of inertia about.

To calculate these values generally, you will be finding the horizontal, vertical, or diagonal distances between piece centroids and the overall centroids that you have listed earlier in the table.

See the parallel axis theorem section of this page earlier for more details. Finally, you will have a column of the adjusted moments of inertia. Take the original moment of inertia about the centroid, then simply add your area times r squared term or mass times r squared term for this adjusted value.

Video Lecture. Worked Problems: Question 1: Use the parallel axis theorem to find the mass moment of inertia of this slender rod with mass m and length L about the z axis at its end point. PDF Solution. Question 2: A beam is made by connecting two 2" x 4" beams in a T pattern with the cross section as shown below. Question 3: A dumbbell consists of two. Practice Problems: Practice Problem Practice Problem Method of Composite Parts for Moments of Inertia and the Parallel Axis Theorem

As an alternative to integration, both area and mass moments of inertia can be calculated via the method of composite parts, similar to what we did with centroids. In this method we will break down a complex shape into simple parts, look up the moments of inertia for these parts in a table, adjust the moments of inertia for position, and finally add the adjusted values together to find the overall moment of inertia. This method is known as the method of composite parts. A key part to this process that was not present in centroid calculations is the adjustment for position. As discussed on the previous pages, the area and mass moments of inertia are dependent upon the chosen axis of rotation.

The moment of inertia also called the second moment is a physical quantity which measures the rotational inertia of an object. The moment of inertia can be thought as the rotational analogue of mass in the linear motion. In general case, finding the moment of inertia requires double integration or triple integration. However, in some special cases, the problem can be solved using single integrals. First we determine the moment of inertia of the rod about the axis passing through the center of gravity. Suppose now that the rod is rotated about the axis passing through one of the ends. P07_ Centroid and Moment of Inertia Calculations. 5. An Example. ○ Lets start with an example problem and see how this develops. 1. 1 n. i i i n i i x A x. A.

Second moment of area

Try our cross section builder to create and analyze custom cross sections. The centroid of a shape represents the point about which the area of the section is evenly distributed. If the area is doubly symmetric about two orthogonal axes, the centroid lies at the intersection of those axes. If the area is symmetric about only one axis, then the centroid lies somewhere along that axis the other coordinate will need to be calculated.

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