Hiroshi Yabuno - Linear and Nonlinear Instabilities in Mechanical Systems: Analysis, Control and Application

We consider a smooth function of with . is expanded at by Taylor series as

(A.1)

or equivalently

where . As shown in expresses the general nonlinear spring force with elongation , i.e.

where , , , and , are linear, quadratic nonlinear, cubic nonlinear, and th-order nonlinear stiffness, respectively, with being an integer.

If there are only odd powers of , i.e. , the nonlinearity of is called symmetric and is rewritten as

Spring with natural length . (a) The origin O of the coordinate is located at the left end of the spring. (b) The elongation of the spring is .

Cubic nonlinear characteristics of a spring with positive linear stiffness: solid line denotes the nonlinear force. The dashed line tangent to the solid line at the origin is the linear component. (a) Hardening cubic nonlinearity: slope of solid line is larger than that of dashed line. (b) Softening cubic nonlinearity: slope of solid line is less than that of dashed line.

where is odd integer. On the other hand, if there are even powers of , i.e. , the nonlinearity of is called nonsymmetric. The characteristic of a nonlinear spring is described as the solid line in are hardening and softening, respectively.

In this section, we consider two spring-mass systems in which the mass is supported by a spring with a symmetric cubic nonlinear characteristic that is expressed as

(A.5)

where higher order terms than quintic nonlinearity in . The equation of motion of the first system is expressed as

Nonlinear characteristics of a spring with negative linear stiffness (): (a) hardening (); (b) softening ().

Spring-mass system: (a) without excitation; (b) with excitation.

where it is accounted that the displacement of the mass is equal to the elongation of the spring