āļāļĢāļąāļāļāļēāļāļēāļĢāļĻāļķāļāļĐāļē Philosophy of Education Weerachat Martluplao
Â
āđāļāļāļŠāļēāļĢāļāļĢāļ°āļāļāļāļāļēāļĢāļāđāļāļāļ§āđāļē āļāļĢāļąāļāļāļēāļāļēāļĢāļĻāļķāļāļĐāļē Philosophy of Education āđāļāļ·āđāļāđāļāđāļāļŠāđāļ§āļāļŦāļāļķāđāļāđāļāļāļēāļĢāļāđāļēāļāļāļīāļāļāļīāļĒāļēāļĄāđāļĨāļ°āļāļ§āļēāļĄāļŦāļĄāļēāļĒāļāļāļāļāļĢāļąāļāļāļēāļāļēāļĢāļĻāļķāļāļĐāļē Philosophy of Education
āļāļĢāļąāļāļāļēāļāļēāļĢāļĻāļķāļāļĐāļē Philosophy of Education Weerachat Martluplao
Â
āđāļāļāļŠāļēāļĢāļāļĢāļ°āļāļāļāļāļēāļĢāļāđāļāļāļ§āđāļē āļāļĢāļąāļāļāļēāļāļēāļĢāļĻāļķāļāļĐāļē Philosophy of Education āđāļāļ·āđāļāđāļāđāļāļŠāđāļ§āļāļŦāļāļķāđāļāđāļāļāļēāļĢāļāđāļēāļāļāļīāļāļāļīāļĒāļēāļĄāđāļĨāļ°āļāļ§āļēāļĄāļŦāļĄāļēāļĒāļāļāļāļāļĢāļąāļāļāļēāļāļēāļĢāļĻāļķāļāļĐāļē Philosophy of Education
The document discusses the stability of equilibrium for mechanical systems. It defines three types of equilibrium:
1) Stable equilibrium occurs when a small displacement causes the system to return to its original position with potential energy at a minimum.
2) Neutral equilibrium occurs when a small displacement does not change the potential energy, which remains constant.
3) Unstable equilibrium occurs when a small displacement causes the system to move farther away from the original position with potential energy at a maximum.
The stability of equilibrium depends on the potential energy function and its derivatives evaluated at the equilibrium position. A system is stable if the second derivative of the potential energy is positive and unstable if it is negative.
6161103 11.3 principle of virtual work for a system of connected rigid bodiesetcenterrbru
Â
The document discusses using the principle of virtual work to solve for equilibrium in systems of connected rigid bodies. It explains that the number of degrees of freedom must first be determined by specifying independent coordinates. Virtual displacements are then related to these coordinates. Equating the virtual work done by external forces and couples to zero provides equations to solve for unknowns like force magnitudes or equilibrium positions. Examples show applying this process to determine values like joint angles or reaction forces.
This chapter introduces the principle of virtual work and how it can be used to determine the equilibrium configuration of connected rigid bodies. It defines work and virtual work, and establishes the potential energy function. The chapter outlines applying the principle of virtual work to particles, rigid bodies, and connected systems to investigate equilibrium and stability, using the potential energy criterion.
Mass moment of inertia measures an object's resistance to angular acceleration and is defined as the integral of the second moment of mass elements about an axis. It depends on the axis chosen and the distribution of mass. For composite objects, the total mass moment of inertia is found by summing the contributions of each component. The parallel axis theorem allows calculating mass moments of inertia about parallel axes.
6161103 10.8 mohrâs circle for moments of inertiaetcenterrbru
Â
The document describes Mohr's circle, which is used to analyze the principal moments of inertia for a given cross-sectional area. It presents equations to determine the radius and center of the Mohr's circle based on the area's moments of inertia (Ix, Iy) and product of inertia (Ixy). An example problem is shown where these values are used to construct the circle and determine the maximum and minimum moments of inertia and their corresponding principal axes.
6161103 10.7 moments of inertia for an area about inclined axesetcenterrbru
Â
This document discusses calculating moments of inertia for an area about inclined axes. It provides transformation equations to relate moments of inertia with respect to x-y axes to moments with respect to inclined u-v axes. It also describes how to determine the principal axes, which are the orientations that produce the maximum and minimum moments of inertia. An example is provided to illustrate finding the principal moments of inertia for a beam cross-section.
Moment of inertia and product of inertia are properties that depend on the orientation of the axes they are calculated around. The product of inertia, Ixy, of an area can be positive, negative, or zero depending on the location and orientation of the x and y axes. Ixy is calculated by taking the double integral of xy over the entire area or using the parallel axis theorem which relates Ixy to the product of inertia about the centroidal axes, Ix'y', plus the cross product of the distance from the centroidal axes to the x and y axes.
6161103 10.5 moments of inertia for composite areasetcenterrbru
Â
1) Moments of inertia for composite areas can be determined by dividing the area into its composite parts, finding the moment of inertia of each part about its centroidal axis and the reference axis using the parallel axis theorem, and taking the algebraic sum.
2) The procedure was demonstrated by calculating the moment of inertia of a composite area made of a rectangle and circle, and another made of three rectangles.
3) For the second example, the cross-sectional area was divided into three rectangles, the moment of inertia of each was found about the x and y axes using the parallel axis theorem, and summed to find the total moment of inertia.
6161103 10.4 moments of inertia for an area by integrationetcenterrbru
Â
This document discusses calculating moments of inertia for planar areas using integration. It describes:
1) Choosing a differential element for integration that has size in only one direction to simplify the calculation.
2) The procedure involves specifying a rectangular differential element and orienting it parallel or perpendicular to the axis of rotation.
3) Moments of inertia are calculated through single or double integration, depending on whether the element has thickness in one or two directions.
6161103 10.10 chapter summary and reviewetcenterrbru
Â
The document summarizes key concepts relating to area moment of inertia including that it represents the second moment of area about an axis and is used in structural strength equations. It can be determined by integration for irregular shapes or found in tables for common shapes. The parallel axis theorem allows calculating it about other axes or for composite shapes. Product of inertia and principal moments of inertia can also be determined using formulas, Mohr's circle, or the parallel axis theorem. Mass moment of inertia measures rotational resistance and can be found using disk or shell elements for axially symmetric bodies.
6161103 9.2 center of gravity and center of mass and centroid for a bodyetcenterrbru
Â
This document discusses the center of gravity, center of mass, and centroid of rigid bodies. It defines these terms and presents methods to calculate them using integrals of differential elements. Examples are provided to demonstrate calculating the centroid for areas and lines using appropriate coordinate systems and differential elements. Centroids are found by taking moments of these elements and the document outlines the general procedure to perform these calculations.
1) Pascal's law states that pressure in a fluid at rest acts equally in all directions. The pressure (Ï) at a point depends on the specific weight (Îģ) or density (Ï) of the fluid and the depth (z) of the point, where Ï = Îģz or Ïgz.
2) For a submerged plate, the pressure at different points depends on the depth of each point. The resultant force on the plate acts through its center of pressure, not the plate's centroid.
3) For curved or variable width plates, integration is used to determine the total pressure distribution and locate the center of pressure.
This document discusses analyzing the center of gravity of composite bodies made up of simpler shapes. It provides the procedure for doing so which includes: [1] Dividing the body into composite parts; [2] Determining the coordinates of each part's center of gravity; [3] Using equations to calculate the total center of gravity by taking weight-weighted sums of the coordinate positions. Examples are provided to demonstrate locating the center of gravity for areas and volumes made of multiple components.
6161103 9.7 chapter summary and reviewetcenterrbru
Â
This document summarizes key concepts from a chapter including: center of gravity and centroid determination using moment balances; calculating body properties like surface area and volume using theorems of Pappus and Guldinus; fluid pressure distribution and determining resultants passing through the loading diagram's centroid.
6161103 8.4 frictional forces on screwsetcenterrbru
Â
Screws are commonly used as fasteners to transmit power or motion between machine parts. A screw can be thought of as an inclined plane wrapped around a cylinder. When a screw is rotated, the nut moves along the inclined plane of the screw thread. The distance the nut moves in one revolution is called the lead of the screw. Large axial loads on a screw generate significant frictional forces on the threads. The moment required to turn a screw under an axial load depends on the frictional forces and can be calculated using force and moment equilibrium equations. A self-locking screw will remain locked in place through frictional forces alone when the applied moment is removed.
Wedges are simple machines that transform applied forces into larger forces directed at approximately right angles. They are used to give small adjustments to heavy loads by applying a smaller force. To analyze wedges, force and moment equilibrium equations are used along with frictional force equations, with unknown normal and frictional forces. Wedges can be self-locking if the frictional forces alone are enough to hold the load without any applied force needed.
This document discusses problems involving dry friction. It describes three types of friction problems: equilibrium, impending motion at all points, and impending motion at some points. Examples are provided to demonstrate how to set up and solve static friction problems using free body diagrams, equilibrium equations, and frictional equations. The key steps are to determine the number of unknowns and equations, draw free body diagrams, apply the appropriate equilibrium and frictional equations, and solve the system of equations.
