The Spring-Mass System module calculates the natural frequencies, mode shapes, and the frequency response of a generic system of discrete springs, dampers, and masses.

A frequency response analysis is a method used to compute structural response to steady-state harmonic excitation. Harmonic excitation is often encountered in engineering systems. For example, it is commonly produced by an unbalance in rotating machinery.

In a frequency response analysis, the excitation forces are explicitly defined in the frequency domain. The applied loads are implicitly sinusoidal in nature. In the simplest case, the loading is defined as having an amplitude at a specific frequency. The steady-state response occurs at the same frequency as the loading. If damping is present in the system, the response peak will not coincide with the peak loading. The shift in response is called a phase shift and is illustrated in Figure 1.

Figure 1. Phase shift between loading and steady-state response

The results obtained from a frequency response analysis include displacements, velocities, accelerations, and forces. The calculated values are complex numbers that can be represented by a magnitude and a phase angle, or by real and imaginary components.

In this module, the steady-state response of the system is computed at discrete excitation frequencies by solving a set of coupled matrix equations using complex algebra. The complex equation of motion for damped forced response is given by:

The steady-state harmonic response is:

Where {x(w)} is a complex displacement vector. Taking the first and second derivatives gives the associated velocity and acceleration vectors:

When the above expressions are substituted into the equation of motion, the following is obtained:

which simplifies to an equation of motion in the frequency domain:

This expression represents a system of equations with complex coefficients. The steady states response vector {x(w)} is computed by inserting the forcing frequency w and multiplying the inverse of the matrix on the left side of the equation by the vector of applied forces.

The Spring-Mass System main input form is shown in Figure 2. The primary input sections are described in detail below.

Figure 2. Module input form

System Definition

Figure 4. System definition input spreadsheet.

Force Input

The steady-state harmonic forcing function is defined in the frequency domain by the complex value:

The complex input force can be input using real and imaginary pairs, or in phasor form using the magnitude and phase angle.  ETBX input requirements for complex numbers are described in Table 1 below.

Damping Input

The user can specify how numerical values entered in the damping column of the input spreadsheet will be interpreted.

If the user specifies direct damping, input values are interpreted as the viscous damping coefficients, C_i, of discrete dampers between degrees of freedom x_i-1 and x_i.  The form of the relationship is:

where F_i-1 and Fi are damping forces at degrees of freedom x_i-1 and x_i.

If the user specifies modal damping, input values are interpreted as the critical damping ratio, z_i, used in the calculation of the generalized response for the ith mode in an uncoupled modal analysis:

Where, for the ith mode, d_i is the generalized displacement, m_i is the generalized mass, k_i is the generalized stiffness, w_i is the undamped natural frequency, and f_i is the generalized force.

Excitation Frequencies

A major consideration when you conduct a frequency response analysis is selecting the frequencies at which the solution is to be performed. The Excitation Frequencies input group provides two ways to define solution frequencies:

• A list of discrete frequencies, separated by blanks spaces or commas.
• A frequency range, defined by an initial frequency, a frequency increment, and the number of frequency increments.

Output Options

The important results obtained from a forced response analysis usually include displacements, velocities, accelerations, and forces. The Output Options group on the main panel lets you select the output type and format. The following types of output are available in either magnitude/phase or real/imaginary format:

• Displacement
• Velocity
• Acceleration
• Spring Force
• Damper Force
• Effective (Dynamic) Spring Rate
• Impedance
• Effective (Dynamic) Mass

The Spring-Mass System module solves for natural frequencies, mode shapes, and the frequency response of a system of discrete springs, dampers, and masses.

Results Report

Figure 5 shows an ETBX results report generated by the Spring-Mass module. The report contains the following elements:

• An input summary that documents the dynamic system that was analyzed
• An eigenvalue summary that lists the calculated eigenvalues (l=w_n²) and the undamped natural frequencies in radians per second (w_n) and in cycles per second (f_n).
• A list of calculated eigenvectors for each mode
• Frequency response calculations

Figure 6 shows an example of an eigenvalue summary for a dynamic system with 4 modes.

Frequency response output is in real/imaginary format or magnitude/phase format. Figure 7 shows frequency response output in magnitude/phase format. Remember that the module will only print output for degrees-of-freedom with a 'y' in the output column of the system input spreadsheet.

Figure 5. Module results report

Figure 6. Eigenvalue and eigenvector output

Figure 7. Frequency response magnitude/phase output

Frequency Response Plots

Frequency response output is plotted using the standard ETBX plot plot shown in Figure 8.

Figure 8. Frequency response plot

System Animations

The Spring-Mass module creates ETBX Animation Panels to display animated mode shapes and forced response shapes. Press the View System button in the System Definition input group to create a new animation panel.

Note that ETBX modules only pass the model and results data to the animation panel when the panel is created. The animation panel therefore only contains a 'snapshot' of the model and any associated results. After the animation panel is created it is NOT updated to reflect subsequent changes to the model or results calculations. This allows multiple models to be compared side-by-side, and avoids any confusion that might be caused by automatic updates. To display the latest updates to your model, simply create a new animation panel. For more details, please refer to the documentation on Animating Results.

Figure 9. Spring-Mass System Animation