Question

Consider the swing with friction \(\bar{x}+\alpha(t) \dot{x}+\omega^{2}(t) x=0\). Show that asymptotic stability is impossible if the coefficient of friction is negative \((\alpha(t)<0 \mathrm{Vt})\).

Short answer

Answer: No, asymptotic stability is not possible if the coefficient of friction is negative for all values of time.

Step-by-step solution

Write the given differential equation in standard form

The given equation can be written as: $$\ddot{x} + \alpha(t)\dot{x} + \omega^2(t)x = 0.$$ This is a second-order, linear, homogeneous differential equation with time-dependent coefficients.

Determine the general solution to the differential equation

Let us assume that the solution to the given equation is in the form \(x(t) = e^{rt}\). We can find the first and second derivatives: $$\dot{x}(t) = re^{rt}, \qquad \ddot{x}(t) = r^2e^{rt}.$$ Now, substitute these expressions back into the original differential equation: $$r^2e^{rt} + \alpha(t)re^{rt} + \omega^2(t)e^{rt} = 0.$$ Since \(e^{rt}\) is non-zero, we can divide the entire equation by it, resulting in: $$r^2 + \alpha(t)r + \omega^2(t) = 0.$$

Analyze stability conditions

For asymptotic stability, we need the real parts of the roots to be negative. In this case, we have a quadratic equation \((r^2 + \alpha(t)r + \omega^2(t) = 0)\), with the discriminant given by: $$\Delta = (\alpha(t))^2 - 4\omega^2(t).$$ If \(\Delta > 0\), we have two distinct real roots, and the system is asymptotically stable if both roots have negative real parts. If \(\Delta = 0\), we have a double real root, and the system is asymptotically stable if the root has a negative real part. If \(\Delta < 0\), we have complex conjugate roots, and the system is asymptotically stable if the real parts of the complex roots are negative. Now, let us analyze the given condition, \(\alpha(t) < 0 \ \forall t.\)

Demonstrate the impossibility of asymptotic stability

If \(\alpha(t) < 0\), then \(\Delta = (\alpha(t))^2 - 4\omega^2(t) \geq 0\), because \(\alpha(t)\) is always negative and \(\omega^2(t)\) is always positive. Therefore, we have either two distinct real roots or a double real root. Now, let's say \(r_1\) and \(r_2\) are the roots of the quadratic equation. By Vieta's formulas, we have: $$r_1 + r_2 = -\alpha(t) > 0.$$ This means that the sum of the two real roots is positive. Since \(r_1\) and \(r_2\) are the roots, at least one of them must be positive. Therefore, the system cannot be asymptotically stable as the real part of at least one root is positive. Hence, we conclude that asymptotic stability is impossible if the coefficient of friction is negative \((\alpha(t) < 0 \ \forall t)\).

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations that involve functions and their derivatives. They express a relationship between the rate of change of a quantity and the quantity itself. ODEs are categorized by their order, which indicates the highest derivative present in the equation.

For instance, a first-order ODE contains only the first derivative of the function, while a second-order ODE includes terms with the second derivative. The equation presented in the exercise, \(\ddot{x} + \alpha(t)\dot{x} + \omega^2(t)x = 0\), is an example of a second-order linear homogeneous differential equation with time-dependent coefficients, highlighting the dependence of the equation's coefficients on time.

These equations are foundational in modeling many natural phenomena, including mechanical vibrations, electrical circuits, and population dynamics. In our case, _x(t)_ represents the displacement of a swing with friction, and the coefficients involve friction (\(_\alpha(t)\_)) and natural frequency (\(_\omega^2(t)\_)).

Stability Conditions in Differential Equations

Stability in differential equations refers to the behavior of solutions as time progresses, particularly in response to small perturbations or changes in initial conditions. For a system described by an ODE, we often want to know if its solutions stay bounded (stable), or if they decay to zero over time (asymptotic stability).

The condition for asymptotic stability is that all the solutions of the differential equation approach zero as time goes to infinity. Mathematically, this translates to the requirement that the real parts of all the roots of the characteristic equation, derived from the ODE, must be negative.

In our case, the characteristic equation \(r^2 + \alpha(t)r + \omega^2(t) = 0\) represents such a stability condition. If the coefficients of the equation were constants, we could precisely determine the roots and assess stability. However, the time-varying nature of these coefficients in our expression makes the analysis more complex, necessitating a closer examination of the discriminant and the sum of roots, as shown in steps 3 and 4 of the solution.

Second-Order Linear Homogeneous Differential Equations

Second-order linear homogeneous differential equations, such as \(\ddot{x} + \alpha(t)\dot{x} + \omega^2(t)x = 0\), are essential tools in the study of systems that can be modeled by second derivatives — such as acceleration in mechanics. Homogeneity refers to the absence of a forcing function or external input, indicating that the system is self-contained.

For these types of equations, solutions are generally a combination of exponential functions, trigonometric functions, or polynomials, depending on the characteristic roots. If the roots are real and distinct, solutions are exponential functions of time. If the roots are complex, solutions involve sinusoidal functions that reflect the oscillatory nature of the system.

The stability of such systems is determined by the sign of the characteristic roots. As the solution provides, using Vieta’s formulas, we can assess the sum of roots and their implications for system stability. If the coefficients are constant, we would seek roots of the characteristic polynomial, but with time-dependent coefficients as in our exercise, we must analyze these roots over time to draw conclusions about stability.