Beam-Rod Contact Stability Analysis

Axially Translating Beam Contact-Coupled to an Axial Rod with Tip Mass

Axially Translating Beam Contact-Coupled to a Stationary Axial Rod with Tip Mass

This MATLAB code accompanies the study:

“Stability and modal interactions of an axially translating beam contact-coupled to a stationary in-span axial rod with a tip mass.”

The code computes the complex eigenvalue spectrum, mode-tracking results, root-locus diagrams, mode shapes, and beam--rod participation ratios for an axially translating Bernoulli--Euler beam in continuous contact with a stationary axial rod carrying a tip mass.

Physical model

The system consists of:

• a simply supported Bernoulli--Euler beam translating axially at constant dimensionless speed,
• a stationary in-span axial rod located at a fixed beam coordinate,
• a concentrated tip mass attached to the rod,
• continuous frictionless normal contact between the beam and the rod--tip-mass subsystem.

The formulation treats the rod as a distributed-parameter axial member rather than as a lumped oscillator. Therefore, the longitudinal stiffness and inertia of the rod, together with the inertia of the tip mass, enter the coupled dynamics explicitly.

Main features

The code can generate:

1. complex eigenvalue spectra as a function of dimensionless transport speed,
2. tracked imaginary parts of eigenvalues,
3. tracked real parts of eigenvalues,
4. divergence indicators based on positive real roots,
5. flutter indicators based on unstable complex-conjugate roots,
6. root-locus plots in the complex plane,
7. reconstructed beam and rod mode shapes,
8. rod strain distributions,
9. shape-based beam--rod participation ratios.

The participation ratio is used to quantify whether a tracked branch is predominantly beam-dominated, rod-dominated, or mixed.

Numerical method

The released version includes the Galerkin-based formulation. The beam displacement is expanded using simply supported beam modes, while the rod displacement is expanded using axial rod modes satisfying the rod--tip-mass boundary condition.

For each value of the dimensionless transport speed, the coupled quadratic eigenvalue problem is written in first-order state-space form and solved using MATLAB’s standard eigenvalue solver. Selected eigenvalue branches are then followed using MAC-based mode tracking.

FEM version

An independent finite-element implementation of the same corrected formulation will be added in a later update. The FEM version will allow direct cross-verification of the Galerkin results using an independent discretization.

Requirements

The code is written in MATLAB and uses standard built-in functions only.

Required functionality includes:

• eig
• fzero
• optimset
• standard MATLAB plotting tools
• array2table

No external toolbox or third-party package is required.

Default parameter set

The default dimensional parameters are:

m1 = 1.0; % Beam mass per unit length

m2 = 0.1; % Rod mass per unit length

L1 = 1.0; % Beam length

L2 = 1.0; % Rod length

E1I1 = 1.0; % Beam flexural rigidity

E2A2 = 24.0; % Rod axial rigidity E2*A2

M_tip = 0.5; % Tip mass

eta = 0.5; % Contact location

The corresponding nondimensional parameters are:

alpha_m = (m2*L2)/(m1*L1);

alpha_M = M_tip/(m1*L1);

alpha_k = (E2A2/L2)/(E1I1/L1^3);

The default modal truncation is:

N = 100; % Number of beam modes

P = 100; % Number of rod modes

The transport-speed sweep is refined near regions where modal interactions occur.

Running the code

Open MATLAB in the folder containing the script and run:

Galerkin

The plotting and output options are controlled by switches at the beginning of the file, for example:

PLOT_TRACKED_IM = true;

PLOT_TRACKED_RE = true;

PLOT_ROOT_LOCUS_UPPER = false;

PLOT_MODES_AT_CC = false;

PLOT_SHAPE_PARTICIPATION = true;

PRINT_RESULTS_TABLE = true;

Set the desired options to true or false before running the script.

Outputs

Depending on the selected switches, the script can produce:

• tracked eigenfrequency plots,
• tracked real-part plots,
• divergence and flutter indicators,
• root-locus diagrams,
• beam and rod mode-shape plots,
• rod strain plots,
• participation-ratio plots,
• printed tables of selected tracked branches.

Branch tracking

The script uses MAC-based tracking to follow selected eigenvalue branches as the transport speed changes. Near divergence, restabilization, and modal coalescence regions, branch identities may become ambiguous because real and oscillatory branches can exchange modal character.

Some optional switches are therefore included for reproducing the branch-labeling convention used in the associated manuscript figures. These switches do not change the raw eigenvalues; they only affect how tracked branches are labeled or displayed.