Two-Degree-of-Freedom Systems
By Tom Irvine
SECTION 1
Grounded Two-DOF System, Equation of Motion
Figure 1.1. Two-Degree-of-Freedom System, Spring-Mass Model

Newton’s law can be applied to the system in Figure 1.1 to derive the equations of motion. The steps are omitted for brevity. The resulting pair of ordinary differential equations can be represented in matrix form and are coupled via the damping and stiffness matrices.

(1.1)

A shorthand form is

(1.2)

The matrices and vectors are

(1.3)

Consider the undamped, homogeneous form of equation (1.2).

(1.4)
SECTION 2
Grounded Two-DOF System, Eigenvalues & Vectors

Seek a harmonic solution of the form

(2.1)
(2.2)
(2.3)

The vector is the generalized coordinate vector. The undamped, homogeneous equation is transformed through substitution and algebraic manipulation into the generalized eigenvalue problem.

(2.4)

TThe eigenvalues can be found by setting the determinant equal to zero.

(2.5)

There is an eigenvalue for each degree-of-freedom. Each angular natural frequency is then calculated from the square root of the respective eigenvalue. The corresponding eigenvectors represent orthogonal mode shapes. The eigenvector for each mode is found via

(2.6)

An eigenvector matrix can be formed. The eigenvectors are inserted in column format.

(2.7)

The coefficient represents the modal displacement of mass i for mode j.

Each eigenvector can be multiplied by an arbitrary scale factor. A mass-normalized eigenvector matrix can be calculated such that the following orthogonality relations are obtained.

(2.8)
(2.9)

The superscript T represents matrix transpose. The identity matrix is I, a diagonal matrix of ones. The diagonal matrix of eigenvalues Ω.

SECTION 3
Grounded Two-DOF System Example, Free Vibration

Now define a modal coordinate in terms of the normalized eigenvector matrix such that the displacement vector is

(3.1)

Substitute equation (3.1) into equation (1.2).

(3.2)

Premultiply by the transpose of the normalized eigenvector matrix.

(3.3)

The orthogonality relationships yield

(3.4)

Furthermore, the following simplifying assumption is made for the damping matrix.

(3.5)

The equation of motion can now be written as

(3.6)

The two equations are now uncoupled in terms of the modal coordinates. The modal displacement for free vibration response to initial conditions is found via Laplace transforms as shown in Reference [3]. The modal displacement for dof i is

(3.7)

The damped angular natural frequency is

(3.8)

The physical displacement can then be found by substituting equation (3.7) into (3.1). The physical displacements can be expressed in terms of the eigenvector coefficients and modal coordinates as

(3.9)
(3.10)

The modal displacement initial conditions are required for the complete solution. These can be found from the following transformations.

(3.11)
(3.12)
SECTION 4
Grounded Two-DOF System, Participation Factors & Effective Modal Mass

The participation factors and effective modal masses are indicators of how excitable the modes are given uniform base excitation to the given system. Some modes are more readily excited than others, and some cannot be excited at all for this excitation type.

The system’s generalized mass matrix is given by

(4.1)

Again, the generalized mass will be the identity matrix if the eigenvectors are mass normalized. Let be the influence vector which represents the displacements of the masses resulting from static application of a unit ground displacement. Define a coefficient vector as

(4.2)

The modal participation factor matrix for mode i is

(4.3)

The effective modal mass for mode i is

(4.4)
SECTION 5
Two-DOF System Example, Response to Initial Displacement

The system in Figure 5.1 has the following parameters.

Table 5.1. Grounded Two-DOF System, Parameters
Variable Value
2 lbm
1 lbm
150 lbf/in
100 lbf/in
0.01 in
0.005 in

Assume 5% damping for each mode. The initial velocity is zero for each mass.

Apply the mass and stiffness values from Table 1.1 into equation (1.1) to form the following matrices.

(5.1)

The 1/386 factor is needed to convert lbm to lbf sec^2/in.

The eigenvalues and vectors are found by inserting these matrices into equations (2.5) and (2.6). The results are shown in the following table.

Table 5.2. Grounded Two-DOF System Modal Parameters
Mode Angular Natural Frequency (rad/sec) Natural Frequency (Hz) Participation Factor Modal Mass Ratio
1 125.25 20.0 0.085 0.935
2 266.72 42.5 -0.022 0.065

The mass-normalized mode shapes in matrix format are

(5.2)

The response time histories for the initial conditions are shown in Figure 5.2.

Figure 5.2 Grounded, Two-DOF System, Response to Initial Displacement
SECTION 6
Semi-definite, Two-DOF System
Semi-definite, Two-DOF System Equation of Motion
Figure 6.1. Ungrounded, Two-DOF System

Consider the system in the above figure. This system could represent a simple, two-node finite element model of a rod’s longitudinal vibration. A characteristic of this system is that the fundamental mode is a rigid-body mode at zero frequency. Both masses move in unison for the rigid-body mode. Newton’s law can be applied to the system to derive the equations of motion. The steps are omitted for brevity. The resulting pair of ordinary differential equations can be represented in matrix form.

(6.1)

The eigenvalues and vectors are found using the method shown in Section 6.1.2. Each angular natural frequency is then calculated from the square root of the respective eigenvalue.

(6.2)
(6.3)

The mass-normalized eigenvectors, representing the mode shapes, are

(6.4)
(6.5)

Again, the two masses move in unison for the first mode. They move 180 degrees-out-phase for the second mode. The semi-definite system is revisited in Section 12.3.3.

Semi-definite, Two-DOF System Example

Consider the system in Figure 6.3 with the values in the following table.

Table 6.1. Semi-definite, Two-DOF System, Parameters
Variable Value
2 lbm
1 lbm
k 2000 lbf/in

The mass and stiffness matrices are

(6.6)

The 1/386 factor is needed to convert lbm to lbf sec^2/in.

The eigenvalues and vectors are found by inserting these matrices into equations (2.5) and (2.6). The angular natural frequencies are

(6.7)
(6.8)

The mass-normalized eigenvectors in column format are

(6.9)

The total mass sum is 3 lbm (0.00777 lbf sec^2/in) which is also the effective modal mass of the rigid-body modes. The second mode has zero effective modal mass.

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