PinnedBeam_abg.tex

MTT command:

mtt PinnedBeam abg tex

Figure 2.1: System PinnedBeam: acausal bond graph

The acausal bond graph of system PinnedBeam is displayed in Figure 2.1 (on page ) and its label file is listed in Section 2.1.1 (on page ). The subsystems are listed in Section 2.1.2 (on page ).

This example represents the dynamics of a uniform beam with two pinned ends. The left-hand end is driven by a torque input and the corresponding collocated angular velocity is measured. The beam is approximated by 20 equal lumps using the Bernoulli-Euler. Because the two end lumps have different causality to the rest of the beam lumps, they are represented seperately. The system has 40 states (20 modes of vibration), 1 input and 1 output.

Table 2.1: Beam parameters

Name	Value
Beam Length, $L$	0.60 m
Beam Width $w$	0.05 m
Beam Thickness $t_b$	0.003
Young's Modulus $E$	$68.94 \times 10^9$
Density $\rho$	2712.8
Derived quantities
$EI$	7.76
$\rho A$	0.40692

The beam was made of aluminium with physical dimensions and constants given in Table 2.1. The derived beam constants are given by the formulae:

$$\begin{align}EI &= E \times w \frac{1}{12} t_b^3\\ \rho A &= \rho \times w t_b \end{align}$$

The system parameters are also given in Section (on page ).

Table 2.2: Resonant and anti-resonant frequencies (Hz)

Index	$f_r$ (theory)	$f_r$ (model)	$f_a$ (theory)	$f_a$ (model)
1	19.05	19.01	29.72	31.28
2	76.24	75.57	96.50	100.80
3	171.58	168.29	200.73	208.20
4	304.76	294.89	344.13	350.88
5	476.34	452.25	524.98	525.23

Standard modal analysis give the theoretical system resonant frequencies $f_r$ (based on the Bernoulli-Euler beam with the same values of $EI$ and $\rho A$). The system anti-resonances $f_a$ correspond to those of the inverse system with reversed causality, that the driven pinned end is replaced by a clamped end; again modal analysis of the inverse system gives the system anti resonances. The model and theoretical values are compared in Table 2.2 for the first 5 modes. (This table was generated using the script MakeFreqTable.m)