Formulation of the total energy and power flows into a discrete system as dissipated power and Lagrangian energy

The real part of the total power flow into a system is theoretically equivalent to the dissipated power in the system, while the imaginary part divided by angular frequency is equal to the Lagrangian energy which is the potential energy subtracted from the kinetic energy. On the other hand, the imaginary part of the total energy flow into a system is also exactly equal to the dissipated power divided by angular frequency, while the real part is equal to the sign-inverted Lagrangian energy. These relationships considerably facilitate the calculation of the power dissipation (and Lagrangian energy), because no system internal information, such as material property and internal motions, is required, unlike the kinetic and potential energies. In order to calculate the power flow, force (and/or moment) is required to obtain. However, the direct measurement of force is often difficult, since the sensor has to be installed in series with a proper modification of the system. Therefore, an indirect method is usually employed instead. We derive the matrix inverse method from the equations of motion (Newton-Euler equations), and show that it is an indirect yet exact method to obtain interfacial force and moment. A two degree-of-freedom system is used to analytically confirm the derived relationships, and a finite element plate-beam model is further utilized to computationally verify the formulas. The objectives of this poster presentation are 1. Formulate the complex energy and power flows as dissipated power and Lagrangian energy. 2. Derive the matrix inverse method as an indirect exact method to calculate interfacial force and moment. 3. Demonstrate and verify the derived relationships by using the two discrete systems. The scope is limited to linear time-invariant discrete systems in the steady state under harmonic excitations. The analysis is restricted to the frequency domain.