Simscape modeling of thin bar vibration

The Simscape Foundation Mechanical domain has components that are sufficient to model the transverse vibration of thin bars.

Much of this work was done by Carter J. Childs as part of his PhD research work at Penn State University.

The Simscape Foundation Mechanical domain has components that are sufficient to model the transverse vibration of thin bars. The theory behind these components is in the paper by Childs and Thompson that is included in this submission. Figure below is an example of the Simscape components described in the theory. The figure shows a bar with three segments. Each segment is a bar-spring element surrounded in cascade by mass elements. Material parameters for these parts are described in the referenced paper. Each bar-apring element has two Simscape conserving ports at each end. The upper port at each end in the figure is a Translational Mechanical domain port that models displacement normal to the top surface of the bar. The lower port at each end is a Rotational Mechanical domain part that models the rotation due to bending of the bar.The bar-aprings are connected in cascade so that the vertical displacement and rotation angle are both continuous across the boundary. An external driving force could be added at any of the mass connections.

The file bar10_example.slx is a 10-segment bar as described above. In addition to the bar, this model includes a driving force, and a set of measurement components that allow the bar motion to be recorded. The driving force is a sinusoidal force applied to the second mass from the clamped left end. Measurement components are included to record the position of each of the masses along the bar.

The first calculation performed is a frequency domain calculation of the bar displacement to obtain the mechanical resonance frequencies of the clamped-free bar. The chosen example is an aluminum bar that is 1 cm wide, 5 mm wide and 50 cm total length. Figure belowshows the frequency response of the magnitude of the driving point displacement. The first three resonance frequencies are calculates as 16.3 Hz, 102 Hz and 289 Hz, which match the analytically calculated frequencies to three significant figures.

The next calculation is to drive the bar at one of its resonance frequencies and calculate the displacement of the bar. The result for the first three modes is shown in the Figure below Note that the Simscape model calculates the displacement only at the nodes in the model. The figure uses straight line interpolation between the nodes, which is clear at least in the second and third mode shapes. A better interpolation function would be a curve that matches the displacement and slope at the two ends of the segment. That upgrade will be made at a future time.

The included paper describes the means to upgrade the model to that of the Timoshenko bar, and to include internal longitudinal stress. The first would provide a more accurate model and one that can be expanded to a two dimensional thin plate. The second would allow an especially thin bar to model a thin tensioned wire like that on a guitar or piano, and include a small amount flexural rigidity. All of those are features and models are anticipated in the future.