Multi-Degree-of-Freedom Systems

The system we have been using for illustration is known as a single-degree-of – freedom system since only the value of a single variable, x, is needed to describe its instantaneous position. Aircraft—and, indeed, most other dynamic systems—have

more than a single degree of freedom. For example, an airplane has six degrees of freedom, representing three linear displacements—forward, upward, and side­ward—and three angular displacements in pitch, roll, and yaw. A helicopter has at least three more, representing rotor coning, longitudinal flapping, and lateral flapping. For illustration of the principles, we will deal with a two-degree-of – freedom system obtained by adding one more spring, weight, and damper to our original system. Figure 9-4 shows the new system.

To define the instantaneous position of this system, we must know both x and y, so this is a two-degree-of-freedom system, and there are two simultaneous equations of equilibrium representing all the forces acting on both weights.

kxx — kyy + cxx — Cyj + mxx = 0
k у + cyy + my{x + у) = 0

Using the substitution,

* = *(j)e"
y=y(i)e"

the two equations of equilibrium become:

(kx + scx + s2mx)x(s) + {~ky – scy)y(s) = 0
s2myx(s) + (ky + scy + s2my)y(s) = 0

or in matrix form:

The roots can be found by expanding the determinant and setting it to zero:

sAmxmy + sb[cymx + my(cx + с,)] + s2[kymx + cycx + my(kx + ^)]

+ skycx + k+ kxky = 0

This is known as the characteristic equation of the system, and its roots are sometimes called eigen values. There is no simple analytical method for solving the quartic equation as there is for the quadratic that represents the single-degree-of-freedom system; but after the values of the spring, damper, and mass constants are inserted, it can be solved by numerical methods. Depending on the values of the constants, the four roots can be either four real numbers, two real numbers and one pair of complex numbers, or two pairs of complex numbers. The existence of two pairs of complex roots indicates that the system has two separate natural frequencies. The time histories for the displacement of either of the masses can independently have any of the characteristics of the six time histories shown in Figure 9.1.