Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 28

Making linear approximations to trigonometric functions is not the only consideration we have in creating linear differential equations of motion. Many times the physical properties of structural elements in the system are nonlinear. A rubber suspension element is a good example. Depending on how it is designed, it can be made to get softer or harder as it deflects. Note that “softer” and “harder” are non‐technical words relating to the stiffness of the element. Figure 1.6 shows the characteristics of a “hardening” spring where the element gets stiffer as it deflects. The stiffness is measured by the local tangent to the curve.

The equilibrium solution to the nonlinear equation of motion will place the system in equilibrium at (the point labeled operating point on Figure 1.6). Once there, we consider motions away from the operating point and use a Taylor's Series expansion for the nonlinear function

(1.64)

For small values of , we can neglect the higher order terms and write

(1.65)

Figure 1.6 Nonlinear structural element – Linearization and effective stiffness.

where we can see that the linear term involves only the first derivative of the function. This derivative is the local tangent to the curve and is the effective stiffness of the element for this operating point. Linearization of functions about operating points is therefore an exercise in finding the local slope of the function and assuming that small deviations away from the operating point can be approximated by points lying on this straight line. The linear approximation is shown graphically in Figure 1.6.

The constant value of the force at the operating point is the force acting at equilibrium and will enter the equation of motion in such a way that it and constant forces in other elements in the system will sum to zero. We consider these constant forces to be preloads on the elements and will quickly fall into the habit of leaving them out of the analysis because they always add up to zero.