Chapter 3

LOAD DETERMINATION

77

 

 

mentum about the CG. The left sides of these equations respectively sum all the forces and moments that act on the body, whether from known applied forces or from interconnections with adjacent bodies in the system.

For a three-dimensional system of connected rigid bodies, this vector equation for the linear forces can be written as three scalar equations involving orthogonal components taken along a local x, y, z axis system with its origin at the CG of the body: 3

 

F =

=

=

x max

Fy may

Fz maz

(3.1 b)

If the x, y, z axes are chosen coincident with the principal axes of inertia of the body,*

the angular momentum of the body is defined as

ˆ

ˆ

H =

G Ix x i + Iy y j + Iz z kˆ

(3.1 c)

where Ix, Iy, and Iz are the principal centroidal mass moments of inertia (second moments of mass) about the principal axes. This vector equation can be substituted into equation 3.1 a to yield the three scalar equations known as Euler’s equations:

M =  − − ) 

x Ix x ( Iy Iz

y z

M =  − − ) 

(3.1 d)

y Iy y ( Iz Ix

z x

M =  − − ) 

z Iz z ( Ix Iy

x y

 

where Mx, My, Mz are moments about those axes and  x,  y,  z are the angular accelerations about the axes. This assumes that the inertia terms remain constant with time, i.e., the mass distribution about the axes is constant.

 

NEWTON’S THIRD LAW states that when two particles interact, a pair of equal and opposite reaction forces will exist at their contact point. This force pair will have the same magnitude and act along the same direction line, but have opposite sense.

 

We will need to apply this relationship as well as applying the second law in order to solve for the forces on assemblies of elements that act upon one another. The six equations in equations 3.1 b and 3.1 d can be written for each rigid body in a 3-D system. In addition, as many (third-law) reaction force equations as are necessary will be written and the resulting set of equations solved simultaneously for the forces and moments. The number of second-law equations will be up to six times the number of individual parts in a three-dimensional system (plus the reaction equations), meaning that even simple systems result in large sets of simultaneous equations. A computer is needed to solve these equations, though high-end pocket calculators will solve large sets of simultaneous equations also. The reaction (third-law) equations are often substituted into the second-law equations to reduce the total number of equations to be solved simultaneously.

 

* This is a convenient choice for

 

symmetric bodies but may be less

Two-Dimensional Analysis

convenient for other shapes. See

F. P. Beer and E. R. Johnson,

Vector Mechanics for Engineers,

All real machines exist in three dimensions but many three-dimensional systems can be 3rd ed., 1977, McGraw-Hill, New

York, Chap. 18, “Kinetics of Rigid

analyzed two dimensionally if their motions exist only in one plane or in parallel planes.

Bodies in Three Dimensions.”

Image 190

 

78

MACHINE DESIGN -

An Integrated Approach

 

 

Euler’s equations 3.1 d show that if the rotational motions (, ) and applied moments or couples exist about only one axis (say the z axis), then that set of three equations reduces to one equation,

 

M = 

z

Iz z

(3.2 a)

3

because the  and  terms about the x and y axes are now zero. Equation 3.1 b is reduced to

 

F =

=

x max

Fy may

(3.2 b)

Equations 3.2 can be written for all the connected bodies in a two-dimensional system and the entire set solved simultaneously for forces and moments. The number of second-law equations will now be up to three times the number of elements in the system plus the necessary reaction equations at connecting points, again resulting in large systems of equations for even simple systems. Note that even though all motion is about one ( z) axis in a 2-D system, there may still be loading components in the z direction due to external forces or couples.

 

Static Load Analysis

The difference between a dynamic loading situation and a static one is the presence or absence of accelerations. If the accelerations in equations 3.1 and 3.2 are all zero, then for the three-dimensional case these equations reduce to

 

F =

 =

 =

x 0

Fy 0

Fz 0

(3.3 a)

M =

 =

 =

x

0

My 0

Mz 0

 

 

and for the two-dimensional case,

 

 

 

 

F =

 =

 =

x 0

Fy 0

Mz 0

(3.3 b)

Thus, we can see that the static loading situation is just a special case of the dynamic loading one, in which the accelerations happen to be zero. A solution approach based on the dynamic case will then also satisfy the static one with appropriate substitutions of zero values for the absent accelerations.

 

 

3.4

TWO-DIMENSIONAL, STATIC LOADING CASE STUDIES

This section presents a series of three case studies of increasing complexity, all limited to two-dimensional static loading situations. A bicycle handbrake lever, a crimping tool, and a scissors jack are the systems analyzed. These case studies provide examples of the simplest form of force analysis, having no significant accelerations and having forces acting in only two dimensions.

 

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