Mechanical Multiport System: Elastic Wave Propagation in a Small Aluminum Plate

In this tutorial, the vibrational behavior of a small aluminum plate with four waveguide structures is analyzed. This is an example of a structural component located in any device where elastic waves are propagating, like a smart speaker, an electric motor, or a MEMS device. The plate can be thought of as a mechanical multiport system. The model uses the port boundary condition at the inlet/outlet of the waveguide structures. The port conditions consistently capture and treat the different propagating elastic modes, like longitudinal, transverse, and torsional waves. The transmission and reflection of the various modes is characterized through the scattering matrix of the system, which is automatically computed. This allows the vibrational behavior of the component to be characterized in great detail, for example, for subsequent use in a system simulation.

Figure 1: Model geometry layout, location of the inlet/outlets of the four waveguides, and the port numbers associated with each inlet/outlet.

Model Definition

The model consists of a small 16 mm by 12 mm by 0.5 mm aluminum piece that has four waveguide structures with an inlet/outlet (connection to other components), see Figure 1. The plate is fixed on one surface. The structure is meant to represent a small subcomponent in a device or system where vibrations are generated, for example, by a transducer. In order to control and understand transmission, such as feedback phenomena, it is necessary to understand how elastic waves (vibrations) propagate through the structure when excited. A compact lumped representation is given by the scattering or S-matrix of the system. The scattering matrix relates the incident and reflected mode amplitudes at all the inlet/outlets. Component Sij, represents the amplitude of the outgoing mode at port i when the system is excited at port j. The the system is excited at 10 kHz.

In this model, each inlet/outlet has four features added. Each port describes one specific propagating mode, either the longitudinal, the torsional, the transverse 1, or the transverse 2 mode. The modes are depicted in Figure 2. The scattering matrix elements are automatically computed when solving a model with a port feature; all the elements can be computed when the functionality is used. Using a combination of ports (representing all propagating modes) at an elastic waveguide outlet will also act as a perfect nonreflecting condition that is superior to a perfectly matched layer (PML) configuration.

Figure 2: The four propagating modes captured by the ports located at the four inlet/outlets of the mechanical system.

Results and Discussion

The displacement of the multiport aluminum structure, when the four different modes are excited at inlet/outlet 1, is depicted in Figure 3, Figure 4, Figure 5, and Figure 6.

Figure 3: Displacement when the longitudinal mode is excited at the inlet/outlet 1.

Figure 4: Displacement when the torsional mode is excited at the inlet/outlet 1.

Figure 5: Displacement when the traverse 1 mode is excited at the inlet/outlet 1.

Figure 6: Displacement when the traverse 2 mode is excited at the inlet/outlet 1.

The power transmitted or reflected by the excitation of a given port can be assessed by plotting the absolute value of the scattering parameters squared |Sij|2. Note that the mode shapes are always scaled to unit power in the mechanical port formulation. For the acoustic ports, both amplitude and power scaling options exist.

The power of the various outgoing modes at all inlet/outlets, when the system is excited with a specific port, is depicted in Figure 7. The graphs show how energy is transmitted and reflected in the system and how energy is converted from one propagating mode to another.

Figure 7: Power of the various outgoing modes at all inlet/outlets when the system is excited with a specific port.

Acoustics_Module/Elastic_Waves/mechanical_multiport_system

Modeling Instructions

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New

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Model Wizard

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Global Definitions

Parameters 1

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Browse to the model’s Application Libraries folder and double-click the file mechanical_multiport_system_parameters.txt.

Geometry 1

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Import 1 (imp1)

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Browse to the model’s Application Libraries folder and double-click the file mechanical_multiport_system.mphbin.

Click

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Create selections for the inlet/outlet boundary of the four waveguide structures. On each waveguide inlet/outlet boundary several features will be added, each capturing one specific propagating mode.

Definitions

input/output 1

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In the window for , type input/output 1 in the text field.

Locate the section. From the list, choose .

Select Boundary 1 only.

input/output 2

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In the window for , type input/output 2 in the text field.

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Select Boundary 6 only.

input/output 3

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In the window for , type input/output 3 in the text field.

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Select Boundary 47 only.

input/output 4

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Select Boundary 48 only.

Add Material

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Solid Mechanics (solid)

Fixed Constraint 1

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Select Boundaries 23 and 25 only.

Now, proceed and add four port conditions per input/output boundary. First add the ports that represent the longitudinal modes, group them, and then duplicate them. Then add the ports that represent the torsional and the two translational modes.

Port 1

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Note that per default the first port added has the excitation turned on. We will turn it off here; you can then select any port as source and run the model. Here we will use a port sweep to excite the desired ports. When a port sweep is performed, all ports are (automatically) one by one used as the excitation source. If all ports are excited, the full scattering matrix is computed.

Port 2

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Port 3

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Port 4

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Port 1, Port 2, Port 3, Port 4

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Longitudinal Mode

In the window for , type Longitudinal Mode in the text field.

Torsional Mode

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In the window for , type Torsional Mode in the text field.

Translational Mode 1

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In the window for , type Translational Mode 1 in the text field.

Translational Mode 2

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In the window for , type Translational Mode 2 in the text field.

Torsional Mode

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Translational Mode 1

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Mesh 1

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Size

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Free Tetrahedral 1

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Free Triangular 1

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Copy Face 1

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toggle button.

Select Boundaries 27 and 31 only.

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Swept 1

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Distribution 1

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Before solving the full model, perform a single study to compute the propagating modes and identify their mode wave numbers. Note that the values for this specific setup are already found in the list.

Study 1 - Modes

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Locate the section. Clear the check box.

Boundary Mode Analysis

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Add Predefined Plot

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Results

Arrow Surface 1

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In the text field, type 20.

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Selection 1

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Select Boundary 1 only.

Mode Shape (solid)

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Clear the check box.

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From the list, choose .

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Proceed to do the full system analysis. This requires setting up a study with one per port feature. In each step, it should be ensured that the correct mode is computed (longitudinal, torsional, transverse). The search uses the values of the mode wave numbers computed in the previous study (also found in the list).

For simplicity, one inlet/outlet boundary will be selected as the source of excitations (here the inlet/outlet 1). The desired modes will be activated one at a time using the port sweep functionality. It is activated in the section and ports 1, 5, 9, and 13 are used as sources.

Add Study

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