Development of Differential Equations for Mechanical Models

Methods for the Development of Differential Equations for Mechanical Models 

As with the simple models from Chapter 3, each different mechanical model can be described by a differential equation. The differential equation governing the response for any mechanical model may be obtained by considering the constitutive equations for each element as well as the overall equilibrium and kinematic constraints of the network. Once the differential equation is obtained, the response of the model to any desired loading can be examined by solving the differential equation for that particular loading. The solution for simple creep or relaxation loading will provide the creep compliance or the relaxation modulus for the given model. In this 

As a first example, consider the three-parameter model (sometimes known as the Voigt-Kelvin model) shown in Fig. 5.1. This model is best approached as a combination of a spring and a Kelvin model acting in series. The three sets of equations then become, where the subscripts 0 or s indicate the value of quantities in the free spring, the subscripts 1 or k indicate the value of quantities in the Kelvin element, and unsubscripted a and E are the remote values of stress and strain (the total stress and strain carried by the three-parameter solid). 

The objective is to find the constitutive equation (governing differential equation) for the three-parameter model. The kinematic equation for the three-parameter solid is, 

From equilibrium, the stress in the free spring, Os, and the stress in the Kelvin element, Ok, are the same as the remote stress, a. To find the differential equation it is convenient to write the Kelvin constitutive equation 

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