This problem is solved following the five-step linearization process introduced earler in this section.
Step 1. The mathematical model of the system is a second-order nonlinear differential equation:
m\ddot{x}+b\dot{x}+ f_{NL}(x)=F_i , (2.69)
m\ddot{x}+b\dot{x}+2.5\sqrt{x} =F_i . (2.70)
Step 2. The normal operating point is defined by the given constant input force \overline{F}_i=0.1N and by the corresponding displacement \overline{x}. One can find the unknown value of \overline{x} from model equation (2.70) by settingF_i=\overline{F}_i and x = \overline{x} which yields
m\ddot{\overline{x} } +b\dot{\overline{x} }+2.5\sqrt{\overline{x} } =\overline{F}_i . (2.71)
The first two terms of Eq. (2.71) drop out because\overline{x} is, by definition, a constant, and does not vary with time.Hence
2.5\sqrt{x} =\overline{F}_i, (2.72)
\overline{x}=\left(\frac{0.1}{2.5} \right)^2=0.0016 m. (2.73)
Thus the normal operating point corresponding to the constant portion of the input force of 0.1 N is a deflection of 0.0016 m.
Step 3. Introduce the incremental variables by substituting x=\overline{x}+\hat{x} \space and \space F_i=\overline{F}_i+\hat{F}_i into Eq. (2.71):
m\ddot{\hat{x}}+b\dot{\hat{x}}+f_{NL}(\overline{x}+\hat{x})=\overline{F}_i+\hat{F} _i . (2.74)
Step 4. The nonlinear term in Eq. (2.74) is approximated by the first two terms of the Taylor’s series expansion:
f_{NL}(\overline{x}+\hat{x} )\thickapprox 2.5\sqrt{\overline{x} }+\left(\frac{2.5}{2\sqrt{\overline{x} } } \right)\hat{x}=0.1+31.25\hat{x}. (2.75)
Step 5. Substitute the linear approximation in Eq. (2.75) into (2.74) to get the following result:
m\ddot{\hat{x} }+b \dot{\hat{x} }+0.1+31.25\hat{x}=\overline{F}_i+\hat{F}_i (2.76)
Note that the constant portion of the input force appears on both sides of Eq. (2.76) and therefore cancels out, leaving a linear second-order ordinary differential equation (ODE):
m\ddot{\hat{x} }+b \dot{\hat{x} }+31.25\hat{x}=\hat{F}_i . (2.77)
Note that the coefficient of the incremental displacement term occupies the position where one normally finds a spring constant. In point of fact, this is the incremental stiffness k_{inc} of the NLS in the vicinity near the normal operating position established by the force \overline{F}_i = 0.1N:
m\ddot{\hat{x} }+b \dot{\hat{x} }+k_{inc}\hat{x} =\hat{F}_i (2.78)
where
k_{inc}=31.25N/m. (2.79)
To further illustrate this important concept, both the linearized and nonlinear spring characteristics can be plotted by use of MATLAB. Appendix 3 is a tutorial on the MATLAB environment, and readers not well versed in its usage are encouraged to review the tutorial before proceeding with the book material.
The nonlinear function to be plotted is given by Eq. (2.68). The linearized model of the spring is represented by a straight line tangent to the nonlinear function at the normal operating point (\overline{x} = 0.0016 m). The slope of the straight line is the incremental stiffness, 31.25 N/m and its y intercept is 0.05 N, which can be found by substitution of the coordinates of the normal operating point into a general equation for the tangent line. Thus the approximated linearized spring force equation is
F_{NLS}\approx 31.25x +0.05. (2.80)
The following MATLAB commands can be used to generate the plots:
