Lagrangian Dynamics

Abstract

In the preceding chapter, the differential equations of motion of single degree of freedom systems were developed using the vector approach of Newtonian mechanics. Another alternative for developing the system differential equations of motion from scalar quantities is the Lagrangian approach where scalars such as the kinetic energy, strain energy, and virtual work are used. In this chapter, the use of Lagrange’s equation to formulate the dynamic differential equations of motion is discussed. The use of Lagrange’s equation is convenient in developing the dynamic relationships of multi-degree of freedom systems. Important concepts and definitions, however, have to be first introduced. In the first section of this chapter, we introduce the concept of the system generalized coordinates. In Section 2 the virtual work is used to develop the generalized forces associated with the system generalized coordinates. The concepts and definitions presented in the first two sections are then used in Sections 3-5 to develop Lagrange’s equation of motion for multi-degree of freedom systems in terms of scalar quantities such as the kinetic energy, strain energy, and virtual work. An alternate approach for deriving the dynamic equations of motion using scalar quantities is Hamilton’s principle which is discussed in Section 6. Hamilton’s principle can be used to derive Lagrange’s equation and, consequently, both techniques lead to the same results. The use of conservation of energy to obtain the differential equations of motion of conservative systems is discussed in Section 7. In this section, general conservation theorems are developed and their use is demonstrated using simple examples.