Power flow which is force multiplied by velocity at a point in the frequency domain has been widely used to quantify the mechanical power value through the point or in the system. The power or energy quantity is often believed to be a more inclusive measure than the primary measurable quantities, such as acceleration/velocity and force. Here, the mechanical power flow ∏ (J/s) at the point a at the frequency ω (rad/s) is obtained from the velocity V (m/s) and force F (N) as ∏_a (ω)=1/2 Re[V_a~H(ω)F_a(ω)] (1) Here, Re indicates the real value, and V and F are vectors in general: V = (X, Y, Z, Ψ, Θ, Φ)~T and F = (F_x, F_y, F_z, N_x, N_y, N_z)~T, where Ψ, Θ, Φ are angular displacements (rad) about x, y and z axes, respectively, N is moment (Nm), and the superscript T and H indicate transpose and conjugate transpose, respectively. In the complex notation, X(ω)= Xe~(jΦx(ω)), F(ω)= Fe~(jΦF)(ω)) (j = √-1) and so on. Thus, X = jωX. Equation (1) may be called active power flow. The system is assumed to be liner time-invariant, and operated in a steady-state. The power flow estimation in a vibratory system often leads to a confusing result, which may be caused by 1) absence of a necessary degree of freedom, 2) omission of an estimation point for the system, or simply by 3) an error in the interfacial force estimation. Especially, when the estimated power flow indicates a negative value as power dissipation, there may be a difficulty to interpret its physical meaning. In this short article, we intend to clarify the formulation of power flow as dissipated power in mechanical vibrating systems (in the steady state), using general discrete systems composed of mass, spring and damper elements. This defined power flow is theoretically positive (or zero for trivial cases).
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