## Receptance Inertance Mobility Mass Examples

where

0 1 " |
, B = |
0 |

L | ||

L Jo J |

In structural and control engineering, the frequency response function is graphically plotted as log \H(s)\ and arg (H(s)) against log(w), which is called the Bode diagram or Bode plot.

A few observations can be made. Examination of Eq. (G.5) reveals that the natural frequency of a SDOF system with input and output configuration as selected, is equal to

Furthermore, it is clear from Eq. (G.6) that the DC-gain (w = 0) is equal to L 2/c, while for high frequencies (w ^ wn) the asymptote has a slope of -2 on a log-log

Consider a SDOF system consisting of a rigid body with length L = 9 m, distance from the center of mass to point O yb2j = 5 m, mass m = 1000 kg, centroidal mass moment of inertia Jo = 50000 kgm2, and torsional spring stiffness c = 250000 N/m. The receptance, mobility and inertance (or accelerance) Bode plot of this system are shown in Fig. G.2. The receptance plot displays the ratio between a harmonic displacement response and the harmonic input force, the mobility plot displays the ratio between the velocity response and input force, while the inertance (or accelerance) plot displays the ratio of acceleration response and input force.

The low-frequency asymptote (indicated by the dashed horizontal line in the receptance plot) intersects the high-frequency asymptote (indicated by the dashed line with a slope of -2 in the receptance plot) at a point corresponding to the natural frequency of the system. This is to be expected if we recall that spring force and inertia force cancel when the system is oscillating at its natural frequency (w = wn). From this figure it can be concluded that the inertia and stiffness properties always appear as straight lines in a Bode plot. In addition, the mobility plot is symmetrical about a vertical line passing through the resonance frequency (this is approximately true for lightly damped SDOF systems).

Receptance

Mobility

Receptance

Mobility

Inertance

lm2- L2/c | |||||||||||||

l2/J0 |
/V | ||||||||||||

Frequency [rad/s] Frequency [rad/s] Frequency [rad/s] Inertance Figure G.2: Bode diagrams (receptance, mobility and inertance) of an undamped, single degree of freedom system (example rigid body with length L = 9 m, distance from the center of mass to point O yb2j = 5 m, mass m = 1000 [kg], centroidal mass moment of inertia Jc = 50000 [kgm2], and torsional spring stiffness c = 250000 [N/m].) Hence, if Fig. G.2 represented a Bode diagram of experimental data, it would be possible to derive the torsional spring constant (provided that the length is known), and the mass moment of inertia about O of a SDOF model for the system under analysis. Damping characteristics can be obtained as well, as will be explained further on. This reasoning forms the basis of grey-box system identification techniques which aim at deriving the dynamic characteristics from experimental data using a physical model structure. If we add a viscous damper to the SDOF system at the pin joint (by specifying a coefficient k of viscous damping), the system transfer function from force to displacement becomes H (s) = °(S)L = L 2/Jo (G 12) H(s) = F(s) s 2 + k/Jo s + c/Jo (G12) and the receptance FRF becomes Observe that the receptance FRF is now (as opposed to the undamped case) a complex valued function, containing both phase and frequency information. The real and imaginary parts of the receptance, mobility and inertance FRF of the viscously damped SDOF system are shown in Fig. G.3. The coefficient k is selected such that the system has a damping ratio of 1 % (i.e. lightly damped). It is interesting to note that the phase change trough the resonance region is characterized by a sign change in one part accompanied by a peak (either a minimum or a maximum) in the other part. From this figure it can be concluded that the receptance and inertance frequency response function are, at resonance, purely imaginary, while the mobility FRF is at resonance purely real. This fact can be used to check the viscously damped assumption for lightly damped systems. x 10-3 Receptance x 10 x 10 Frequency [rad/s] Frequency [rad/s] x 10-3 Mobility x 10-3 Mobility x 10 x 10 Frequency [rad/s] Frequency [rad/s] 3 Inertance x 10
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