Chapter 3

LOAD DETERMINATION

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Table 3-1

Load Classes Constant Loads

Time-Varying Loads

 

 

Stationary Elements

Class 1

Class 2

Moving Elements

Class 3

Class 4

 

3

 

Class 2 describes a stationary system with time-varying loads. An example is a bridge which, though essentially stationary, is subjected to changing loads as vehicles drive over it and wind impinges on its structure. Class 3 defines a moving system with constant loads. Even though the applied external loads may be constant, any significant accelerations of the moving members can create time-varying reaction forces. An example might be a powered rotary lawn mower. Except for the case of mowing the occasional rock, the blades experience a nearly constant external load from mowing the grass. However, the accelerations of the spinning blades can create high loads at their fastenings. A dynamic load analysis is necessary for Classes 2 and 3.

Note however that, if the motions of a Class 3 system are so slow as to generate negligible accelerations on its members, it could qualify as a Class 1 system and then would be called quasi-static. An automobile scissors jack (see Figure 3-5, p. 88) can be considered to be a Class 1 system since the external load (when used) is essentially constant, and the motions of the links are slow with negligible accelerations. The only complexity introduced by the motions of the elements in this example is that of determining in which position the internal loads on the jack’s elements will be maximal, since they vary as the jack is raised, despite the essentially constant external load.

Class 4 describes the general case of a rapidly moving system subjected to time-varying loads. Note that even if the applied external loads are essentially constant in a given case, the dynamic loads developed on the elements from their accelerations will still vary with time. Most machinery, especially if powered by a motor or engine, will be in Class 4. An example of such a system is the engine in your car. The internal parts (crankshaft, connecting rods, pistons, etc.) are subjected to time-varying loads from the gasoline explosions, and also experience time-varying inertial loads from their own accelerations. A dynamic load analysis is necessary for Class 4.

 

 

3.2

 

 

FREE-BODY DIAGRAMS

 

In order to correctly identify all potential forces and moments on a system, it is necessary to draw accurate free-body diagrams (FBDs) of each member of the system. These

* While it is not a requirement

FBDs should show a general shape of the part and display all the forces and moments that the local coordinate system

that are acting on it. There may be external forces and moments applied to the part from for each element be located at its

CG, this approach provides

outside the system, and there will be interconnection forces and/or moments where each consistency and simplifies the

part joins or contacts adjacent parts in the assembly or system.

dynamic calculations. Further,

most solid modeling CAD/CAE

systems will automatically

In addition to the known and unknown forces and couples shown on the FBD, the calculate the mass properties of

dimensions and angles of the elements in the system are defined with respect to local parts with respect to their CGs.

The approach taken here is to

coordinate systems located at the centers of gravity (CG) of each element.* For a dy-apply a consistent method that

works for both static and dynamic

namic load analysis, the kinematic accelerations, both angular and linear (at the CG), problems and that is also

need to be known or calculated for each element prior to doing the load analysis.

amenable to computer solution.

Image 188

 

76

MACHINE DESIGN -

An Integrated Approach

 

 

3.3

LOAD ANALYSIS

This section presents a brief review of Newton’s laws and Euler’s equations as applied to dynamically loaded and statically loaded systems in both 3-D and 2-D. The method of solution presented here may be somewhat different than that used in your previous statics and dynamics courses. The approach taken here in setting up the equations for 3

force and moment analysis is designed to facilitate computer programming of the solution.

This approach assumes all unknown forces and moments on the system to be positive in sign, regardless of what one’s intuition or an inspection of the free-body diagram might indicate as to their probable directions. However, all known force components are given their proper signs to define their directions. The simultaneous solution of the set of equations that results will cause all the unknown components to have the proper signs when the solution is complete. This is ultimately a simpler approach than the one often taught in statics and dynamics courses which requires that the student assume directions for all unknown forces and moments (a practice that does help the student develop some intuition, however). Even with that traditional approach, an incorrect assumption of direction results in a sign reversal on that component in the solution. Assuming all unknown forces and moments to be positive allows the resulting computer program to be simpler than would otherwise be the case. The simultaneous equation solution method used is extremely simple in concept, though it requires the aid of a computer to solve. Software is provided with the text to solve the simultaneous equations.

See program MATRIX on the CD-ROM.

Real dynamic systems are three dimensional and thus must be analyzed as such.

However, many 3-D systems can be analyzed by simpler 2-D methods. Accordingly, we will investigate both approaches.

 

Three-Dimensional Analysis

Since three of the four cases potentially require dynamic load analysis, and because a static force analysis is really just a variation on the dynamic analysis, it makes sense to start with the dynamic case. Dynamic load analysis can be done by any of several methods, but the one that gives the most information about internal forces is the Newtonian approach based on Newton’s laws.

NEWTON’S FIRST LAW A body at rest tends to remain at rest and a body in motion at constant velocity will tend to maintain that velocity unless acted upon by an external force.

NEWTON’S SECOND LAW The time rate of change of momentum of a body is equal to the magnitude of the applied force and acts in the direction of the force.

Newton’s second law can be written for a rigid body in two forms, one for linear forces and one for moments or torques:

 

F = ma

M =

G H

˙ G

(3.1 a)

where F = force, m = mass, a = acceleration, M G = moment about the center of gravity, and H

˙ G = the time rate of change of the moment of momentum, or the angular mo-

 

Image 189