8.2 DYNAMIC MODELS

It is often convenient in dynamic analysis to create a simplified model of a complicated part. These models are sometimes considered to be a collection of point masses connected by massless rods, referred to as a lumped-parameter model, or just lumped model. For a lumped model of a rigid body to be dynamically equivalent to the original body, three things must be true:

1 The mass of the model must equal that of the original body.

2 The center of gravity must be in the same location as that of the original body.

3 The mass moment of inertia must equal that of the original body.

8.3 MASS

Mass is not weight. Mass is an invariant property of a rigid body. The weight of the same body varies depending on the gravitational system in which it sits. We will assume the mass of our parts to be constant over time in our calculations.

When designing cam-follower systems (or any machinery), we must first do a complete kinematic analysis of our design in order to obtain information about the rigid body accelerations of the moving parts. We can then use Newton’s second law to calculate the dynamic forces. But to do so, we need to know the masses of all the moving parts that have these known accelerations. If we have a design of the follower train done in a CAD program that will calculate masses and mass moments of inertia, then we are in good shape, as the data needed for dynamic calculations are available. Lacking that luxury, we will have to calculate estimates of the mass properties of the follower train to do the dynamics calculations.

Absent a solids-modeler representation of your design, a first estimate of your parts’ masses can be obtained by assuming some reasonable shapes and sizes for all the parts and choosing appropriate materials. Then calculate the volume of each part and multiply its volume by the material’s mass density (not weight density) to obtain a first approximation of its mass. These mass values can then be used in Newton’s equation to

estimate the dynamic forces.

[Portions of this chapter were adapted from R. L. Norton, Design of Machinery 2ed, McGraw-Hill, 2001, with permission.]

How will we know whether our chosen sizes and shapes of links are even acceptable, let alone optimal? Unfortunately, we will not know until we have carried the computations all the way through a complete stress and deflection analysis of the parts. It is often the case, especially with long, thin elements such as shafts or slender links, that the deflections of the parts under their dynamic loads will limit the design even at low stress levels. In other cases the stresses at design loads will be excessive.

If we discover that the parts fail or deflect excessively under the dynamic forces, then we will have to go back to our original assumptions about the shapes, sizes, and materials of these parts, redesign them, and repeat the force, stress, and deflection analyses.

Design is, unavoidably, an iterative process . We need the dynamic forces to do the stress analyses on our parts. (Stress analysis is addressed in a later chapter.) It is also worth noting that, unlike a static force situation in which a failed design might be fixed by adding more mass to the part to strengthen it, to do so in a dynamic-force situation can have a deleterious effect. More mass with the same acceleration will generate even higher forces and thus higher stresses. The machine designer often needs to remove mass (in the right places) from parts in order to reduce the stresses and deflections due to F = m a. The designer needs to have a good understanding of both material properties and stress and deflection analysis to properly shape and size parts for minimum mass while maximizing strength and stiffness to withstand dynamic forces.

8.4 MASS MOMENT AND CENTER OF GRAVITY

When the mass of an object is distributed over some dimensions, it will possess a moment with respect to any axis of choice. Figure 8-1 shows a mass of general shape in an xyz axis system. A differential element of mass is also shown. The mass moment (first moment of mass) of the differential element is equal to the product of its mass and its distance along the axis of interest. With respect to the x , y, and z axes these are:

dM x = r x 1 dm (8.2a)

dM y = r y 1 dm (8.2b)

dM z = r z 1 dm (8.2c)

The radius from the axis of interest to the differential element is shown with an exponent of 1 to emphasize the reason for this property being called the first moment of mass. To obtain the mass moment of the entire body we integrate each of these expressions.

M x = ∫ r x dm (8.3a)

M y = ∫ r y dm (8.3b)

M z = ∫ r z dm (8.3c)

If the mass moment with respect to a particular axis is numerically zero, then that axis passes through the center of mass ( CM ) of the object, which for earthbound systems is coincident with its center of gravity ( CG ) . By definition, the summation of first moments about all axes through the center of gravity is zero. We will need to locate the

CG of all moving bodies in our designs because the linear acceleration component of each body is calculated as acting at that point.

It is often convenient to model a complicated shape as several interconnected simple shapes whose individual geometries allow easy computation of their masses and the locations of their local CGs . The global CG can then be found from the summation of the first moments of these simple shapes set equal to zero. Appendix C contains formulas for the volumes and locations of centers of gravity of some common shapes. Of course, if the system is designed in a solids modeling CAD package, then the mass and other properties can be automatically calculated.

Figure 8-2 shows a simple model of a mallet broken into two cylindrical parts, the handle and the head, which have masses m h and m d , respectively. The individual centers of gravity of the two parts are at l d and l h /2, respectively, with respect to the axis ZZ . We want to find the location of the composite center of gravity of the mallet with respect to ZZ . Summing the first moments of the individual components about ZZ and setting them equal to the moment of the entire mass about ZZ gives.

This equation can be solved for the distance d along the x axis, which, in this symmetrical example, is the only dimension of the composite CG not discernible by inspection. The y and z components of the composite CG are both zero.

Dynamic models, composite center of gravity, and radius of gyration of a mallet