The moment of inertia is one of the most important properties of the structural element, which largely determines the engineer’s choice of section. It is also known as the second moment of the field.
The word inertia has meanings such as inertia and inertia. The moment of inertia of a body corresponds to the resistance of the engineering element against stresses such as torsion and bending in the geometric design. In fact, most people think of the following picture when it comes to the moment of inertia. Comparison of the resistance of a beam against the same load that is influenced by its design in two different positions.
As shown in the diagram, some shapes are better at resisting bending than others. Obviously, the orientation of the shape also affects the bending. In a way, it roughly shows why the long part of the beam should be placed vertically.
Columns, beams, and most other structural elements have cross-sections similar to I, H, C shapes. This situation brings with it some question marks:
- Why aren’t these elements made of fully filled simple geometries (square, circle, etc.)?
- Which principles are used to decide on the geometry to be chosen in the design of a structural element?
In order to find answers to these questions, it will be useful to explain the event numerically. So let’s examine the three different sections below:
Three Different Sections to Explain the Importance of Moment of Inertia in Engineering Designs
- Let’s consider three different sections for the RS beam as above, A, B, and C. If we calculate that they are all made of the same material, the total cross-section areas are equal. The masses of the objects per unit length are also equal.
- Which section shape is used for the beam with vertical force P, shown on the right, causes less internal stress and deflection?
- The answer is related to the beam’s Moment of Inertia around the x-axis. Since most of the areas in section A are further away from the x-axis, this section has the greatest moment of inertia. Therefore, less stress and deflection occur.
Moment of inertia, the second moment of a name field, although it may change section according to specifications, we think of the engineering construction element for rectangular cross- bh 3 /12 is one of the proofs and calculation methods.
Moment of Particle Inertia
Before discussing the moment of inertia of a rigid object , first, study the moment of inertia of a particle. in this case, do not think of the particle as a very small object. There is actually no set size limit for said particles. So the use of the term particle is only to facilitate discussion of the motion, where the position of an object is described as the position of a point. The concept of this article that we use in discussing the motion of objects on the topic of Kinematics (Straight Motion, Parabolic Motion, Circular Motion) and Dynamics (Newton’s Law). So things are considered as particles.
The concept of particles is different from the concept of rigid bodies. In straight motion and parabolic motion, for example, we think of objects as particles, because when they move, each part of the object has the same velocity (meaning linear velocity). When a car is moving, for example, the front and back of the car have the same speed. So we can think of a car as a particle alias at a point.
When an object rotates, the linear velocity of each part of the object is different. The part of the object that is near the axis of rotation moves slower (small linear velocity), while the part of the object that is on the edge moves faster (the linear velocity is greater). So, we cannot think of objects as particles because the linear velocity of each part of the object varies when it rotates. the angular velocity of all the parts is the same. Regarding this, it has been explained in Rotation Kinematics.
So on this occasion, first we review the Moment of Inertia of a particle that performs a rotational motion. This is intended to help us understand the concept of a moment of inertia. After discussing the moment of inertia of particles, we will get acquainted with the moment of inertia of a rigid object. Rigid objects come in various shapes and sizes. So to help us understand the moment of inertia of objects that have different shapes and sizes, we first need to understand the moment of inertia of particles. After all, each of these objects could be thought of as being composed of particles.
Now let’s review a particle in rotation.
For example, a particle with mass m is given a force F so that it performs a rotational motion about the axis O. The particle is r from the axis of rotation. at first, the particle is at rest (velocity = 0). After being given a force F, the particle moves at a certain linear velocity. At first, the particles are at rest, then they move (experience
change in linear velocity) after applying a force. In this case, the object is experiencing tangential acceleration. Tangential acceleration = the linear acceleration of the particle as it rotates.
We can state the relationship between force (F), mass (m), and tangential acceleration (at), with the equation for Newton’s Second Law:
F = ma tan
Since the particle is rotating, it must have angular acceleration. The relationship between tangential acceleration and angular acceleration is expressed by the equation:
a tan = r .α
Now we input a tangential into the equation above:
F = ma tan → a tan = r α
F = mr α
Multiply the left and right sides by r:
rF = r ( mr α)
rF = mr 2
Pay attention to the left side. rF = Torque, for the force whose direction is perpendicular to the axis (comparison with the image above). This equation can be written as:
τ = ( mr 2 ) α
mr 2 is the moment of inertia of the particle with mass m, which rotates as far r as the axis of rotation. This equation also states the relationship between torque, moment of inertia and angular acceleration of the particle carrying out the rotational motion. Cool term, this is Newton’s Second Law equation for a particle in rotation.
So the moment of inertia of a particle is the product of the mass of the particle (m) and the square of the vertical distance s from the axis of rotation to the particle (r 2 ). For simplicity, compare it with the image above. Mathematically, the moment of inertia of a particle is formulated as follows:
I = mr 2
Description: I = moment of inertia
m = mass of particles
r = distance of the particle from the axis of rotation
The Moment of Inertia Firmness
In general, the Moment of Inertia of each rigid object can be stated as follows:
We can think of a rigid body as composed of many particles that are scattered all over the object. Each of these particles has mass and, of course, has a distance of r from the axis of rotation. so the moment of inertia of each object is the total number of moments of inertia of each particle that makes up that object.
This is just a general equation. However to determine the Moment of Inertia of a rigid body, we need to examine the rigid body as it rotates. Even though the shape and size of two objects are the same, if the two objects rotate on a different axis, aka the axis, then the Moment of Inertia is also different.