Understanding Equations of Motion

A flexible body dynamics is commonly expressed as follows:

Mxdd + (C + Cr + Cf) xd + Kx + Fg= F(t) …………………………………….(Eq. 1)

where M, C, Cr, Cf, and K respectively denote the inertia, damping, Coriolis, Centrifugal, and stiffness matrices, whereas x, xd, and xdd denote the displacement, velocity, and acceleration vectors. Fg and F(t) denote the gravitational and external forces, respectively. On the left hand side, each term indicates, in order, the inertial force, the damping force, the Coriolis force, the centrifugal force, the elastic force, and the gravity force. Each term in both the left and right hand sides, except the gravity force, in general consists of translational and rotational components. Hence, the term “force” here actually includes both force and torques. This is the most general expression of a dynamic system. This equation is commonly called the equations of motion. Why equations (with s in the end of the word)? Because in general Eq. 1 consists of several algebraic equations, which can be presented compactly by using matrix and vector notation as in Eq. 1.

When the system is assumed to be undamped, the damping component should be omitted. In modal analysis, the damping component is typically omitted since the damping does not affect much the modal analysis result. The Coriolis and centrifugal components only exist in a moving (rotating) system, i.e. in a mechanism. Hence, the Coriolis and centrifugal components are omitted in a stationary system, i.e. in a structure. The elastic force only exists in a system having flexibility (compliance). If a system is assumed to be completely rigid, the elastic force vanishes. Please notice that the term “mechanism” means one or more bodies capable to make finite displacement (also called rigid body motion), whereas the term “structure” means one or more bodies incapable to make finite motion. The only motion a structure can undergo is infinite displacement; this happens if the structure has flexibility. If the gravity effect is neglected, the gravity force vanishes.

The following cases are some special cases of Eq. 1:

• In a structure, the Coriolis and Centrifugal components vanish. If the structure is considered undamped, the damping component vanishes. If the structure is assumed to be rigid, the elastic force vanishes.
• In a mechanism, the Coriolis and Centrifugal components in general exist. If the mechanism is considered rigid and undamped, the elastic and damping forces vanish. If there is one or more flexible components in the mechanism, the elastic force should be retained.

Regarding the dependency on time, the external force in a dynamic system is time-dependent. Sometimes, the inertia matrix can also be time-varying, such as the inertia matrix of a rocket with varying fuel mass. Sometimes, the stiffness matrix can also be time-varying, such as the stiffness matrix of a system with variable stiffness.

Equation 1 is a second-order differential equation. Integrating Eq. 1 once will give us the velocity vector xdd. Integrating Eq. 1 twice will give us the displacement x. In other words, integrating Eq. 1 means solving for the velocity and displacement vectors. For a complex system, Eq. 1 typically cannot be solved analytically. In such a case, a numerical integration is performed. In this case, the second-order differential equation is first transformed to a state space (first-order) differential equation, and subsequently a numerical integration algorithm, such as ODE45, is run. Since Eq. 1 is a second-order differential equation, its solution (i.e. displacement) is in general oscillatory.