Linear analysis of a rod under dynamic loading

The first step for all of the linear dynamics procedures is to calculate the eigenvalues and eigenvectors of the system. The mass matrix of element type T3D2 is lumped; therefore, the mass matrix of this three truss system is

The stiffness matrix of the system is

The three eigenvalues and the corresponding eigenvectors using the default normalization method are given in the following table:

Abaqus also calculates the modal participation factors, ${\mathrm{\Gamma }}_i$, the generalized mass, $m_i$, and the effective mass for each eigenvector (see Variables associated with the natural modes of a model for definitions). The values in this case are:

This analysis is performed for three types of systems, described below.

The displacement response spectra shown in Figure 1 are used in the next analysis. Spectra are defined in the figure for no damping and for 10% of critical damping in each mode. In this example 2% of critical damping is used so that the logarithmic interpolation gives a magnitude of 1.7411 for the maximum displacement for each mode. The analysis is done for two cases: absolute summation of the contributions from each mode and the square root of the sum of the squares (SRSS) summation. Since frequencies are well separated in this case, the use of the ten-percent summation method will give results that are identical to the SRSS method, the complete quadratic combination response will differ only by a small amount from SRSS (because of very small cross-correlation factors between the modes), and the Naval Research Laboratory summation method will calculate results that are very close to the absolute summation. For a comparison of all five summation rules, see Response spectra of a three-dimensional frame building. Absolute summation means that the peak displacement response is estimated as

where $u^k$ is the displacement at degree of freedom k, ${\mathrm{\Phi }}_i^k$ is the i^th eigenmode in degree of freedom k, $q_i^{\max }$ is the maximum value for the amplitude in the i^th mode, and $q_i$ is found from the appropriate spectrum definition S given in the input. In this case S is represented by displacement spectrum $S^D$, applied in the global x-direction. SRSS summation estimates the peak displacement response as

The steady-state analysis procedure is verified by exciting the model over a range of frequencies. A load of the form

where $\mathrm{\Omega }$ is the forcing frequency and $F_0=$5, is applied to node 4 in the x-direction.

Two kinds of damping are available for this type of analysis. One is modal damping, which defines the damping term for a mode as

where ${\xi }_i$ is the fraction of critical damping. The other is structural damping, for which the damping force is defined as

where $i=\sqrt{-1}$ and $s_j$ is the structural damping factor.

Abaqus provides output as the response amplitude, $\left|q_i\right|$, and phase angle, ${\psi }_i$, for the i^th mode. For this example, with only the real loads applied, the exact solution—with both modal and structural damping present—is

and

where $f_i{}_0=\sum _N{\mathrm{\Phi }}_i^N{F}_0^N$ is the amplitude of the forcing function, $F_0$, projected onto the i^th mode.

The input file rodlindynamic_ssdynamics.inp requests a steady-state dynamic analysis for the forcing frequency range from 0.01 to 10 cycles/time. All three mode shapes are extracted with a frequency step and are used throughout the steady-state analysis, as indicated by the modal damping definition, where the damping value is defined to be 10% of critical damping in each mode.

The same rod model with structural damping present is now exposed to nondeterministic loading. The case we consider is uncorrelated white noise applied to all nodes. The exact solution for the cross-spectral density matrix of the modal amplitudes (the generalized coordinates) as a function of frequency, $S_{\alpha \beta }^q\left(\mathrm{\Omega }\right)$, for continuously distributed white noise is

where

is the complex frequency response function for mode $\alpha$, with $m_{\alpha }$ the generalized mass for the mode, ${\omega }_{\alpha }$ the frequency of the mode, and $s_{\alpha }$ the structural damping used with the mode; $H_{\alpha }^{\ast }$ is the complex conjugate of $H_{\alpha }$; and $S^F({\mathbf{x}}_1,{\mathbf{x}}_2,\mathrm{\Omega }$) is the cross-spectral density matrix of the external loading. Abaqus assumes that the integrated projection of the cross-spectral density matrix onto the eigenmodes can be expressed as a matrix between the loaded nodal degrees of freedom projected onto the eigenmodes, so

$S_{NM}^F$ is defined by applying nodal loads, $F_N^I$ (where N refers to a degree of freedom in the model and I refers to the load case number) and giving a matrix of scaling factors, ${\mathrm{\Psi }}_{NM}^{IJ}$, and corresponding frequency functions, $P^J\left(\mathrm{\Omega }\right)$, for each load case. Here J refers to the matrix of scaling factors ${\mathrm{\Psi }}_{NM}^{IJ}$ by which to scale $P^J$ in load case I. $S_{NM}^F$ is then defined as

In this case we need only one load case, $I=$1, and one frequency function and associated matrix of scaling factors, $J=$1. (See Random response to jet noise excitation for a problem in which several frequency functions and scaling factor matrices are needed to define the cross-spectral density matrix of the loading.) Since white noise is assumed to be uncorrelated, ${\mathrm{\Psi }}_{NM}^{11}$ is defined as a diagonal matrix: ${\mathrm{\Psi }}_{NM}^{11}=$0 for $N\ne M.$ Uncorrelated loadings are specified using a correlation definition, where ${\mathrm{\Psi }}_{NM}^{IJ}$ is defined. We choose a unit magnitude for the scaling factors so that ${\mathrm{\Psi }}_{NM}^{11}$ becomes a unit matrix. Since the diagonal terms of the cross-spectral density matrix are the power spectral density functions of the loading, the cross-spectral density matrix will be a real diagonal matrix. Therefore, imaginary frequency functions and scaling factors need not be considered here. As a result, the power spectral definition is a reference power spectral density function (rather than a general frequency function), $P^1\left(\mathrm{\Omega }\right)$, which is scaled by the product of load magnitudes, $F_N^1{F}_M^1$ (and by ${\mathrm{\Psi }}_{NM}^{11}$, but ${\mathrm{\Psi }}_{NM}^{11}$ is a unit matrix). We apply loads $F_N^1$ of 10 to each of nodes 2 and 3 and a load of 5 to node 4, corresponding to a unit load distributed continuously along the rod.

At a frequency of 0.1 cycles/time $S_{\alpha \beta }^q$ is, therefore,

The cross-spectral density matrices for the displacements, velocities, and accelerations of the nodes can be calculated directly from $S_{\alpha \beta }^q$. For example, the cross-spectral density matrix of the displacements is