Mechanical Vibrations

The subject of mechanical vibrations deals with the oscillating response of elastic bodies to disturbances, such as an external force or other perturbation of the system from its equilibrium position. All bodies that possess mass and have finite stiffness are capable of vibrations.

Let’s extend our free vibration analysis from Chap. 2 to include two degrees of freedom in the model. This would make sense, for example, if we completed a measurement to determine the frequency response function (FRF) for a system and saw that there were obviously two modes of vibration within the frequency range of interest; see Fig. 4.1.

In Chaps. 1–5, we assumed a model and then used that model to determine the system response in the time or frequency domain (or both). More often, however, we have an actual dynamic system and would like to build a model that we can use to represent its vibratory behavior in response to some external excitation. For example, in milling operations, the flexibility of the cutting tool–holder–spindle–machine structure (and sometimes the workpiece) determines the limiting axial depth of cut to avoid chatter, a self-excited vibration (Schmitz and Smith 2009). In this case, the dynamic response at the free end of the tool (and/or at the cutting location on the workpiece) is measured. Using this measured response, a model in the form of modal parameters can be developed for use in a time-domain simulation¹ of the milling process. How can we work this “backward problem” of starting with a measurement and developing a model? To begin, we need to determine the modal mass, stiffness, and damping values from the measured frequency response function (FRF).

In Chap. 6, we solved the “backward problem” of starting with frequency response function (FRF) measurements and developing a model. However, we did not describe the measurement procedure. The basic hardware required to measure FRFs is: a mechanism for known force input across the desired frequency range (or bandwidth) a transducer for vibration measurement, again with the required bandwidth a dynamic signal analyzer to record the time-domain force and vibration inputs and convert these into the desired FRF.

In Chaps. 1 through 5 we discussed the solution of discrete, lumped-parameter models. For multiple degree of freedom systems, we employed modal analysis to enable us to transform the coupled equations of motion in local (model) coordinates into modal coordinates. In this coordinate frame, the equations of motion were uncoupled and we could apply single degree of freedom solution techniques. In Chap. 6 we shifted our attention to the “backwards problem,” which is representative of a common task for vibration engineers. In this problem, we begin with measurements of an existing structure and use this information to develop a model. We again used discrete models to describe the system behavior.

In Chap. 1 through 8 we discussed both discrete and continuous beam models that can be used to describe the behavior of vibrating systems. We also detailed experimental techniques that we can use to identify these models. In this chapter, we will introduce an approach to combine models or measurements of individual components in order to predict the assembly’s frequency response function (FRF). This method is referred to as receptance coupling (Bishop and Johnson 1960); recall from Sect. 7.1 that a receptance is a type of FRF.