Question 7.TC.2: Oscillator Damped by Sliding Friction Theoretical Introducti......
Oscillator Damped by Sliding Friction
Theoretical Introduction
In mechanics, one often uses so called phase space, an imaginary space with the axes comprising of coordinates and moments (or velocities) of all the material points of the system. Points of the phase space are called imaging points. Every imaging point determines some state of the system.
When the mechanical system evolves, the corresponding imaging point follows a trajectory in the phase space which is called phase trajectory. One puts an arrow along the phase trajectory to show direction of the evolution.
A set of all possible phase trajectories of a given mechanical system is called a phase portrait of the system. Analysis of this phase portrait allows one to unravel important qualitative properties of dynamics of the system, without solving equations of motion of the system in an explicit form. In many cases, the use of the phase space is the most appropriate method to solve problems in mechanics.
In this problem, we suggest you to use phase space in analyzing some mechanical systems with one degree of freedom, i.e. systems which are described by only one coordinate. In this case, the phase space is a two-dimensional plane. The phase trajectory is a curve on this plane given by a dependence of the momentum on the coordinate of the point, or vice versa, by a dependence of the coordinate of the point on the momentum.
As an example we present a phase trajectory of a free particle moving along x axis in positive direction (Fig. 7 – 2).
(A) Phase portraits
(Al) Make a draw of the phase trajectory of a free material point moving between two parallel absolutely reflective walls located at x=-{\frac{L}{2}} and x={\frac{L}{2}}.
(A2) Investigate the phase trajectory of the harmonic oscillator, i.e. of the material point of mass m affected by Hook’s force F =- kx:
(a) Find the equation of the phase trajectory and its parameters.
(b) Make a draw of the phase trajectory of the harmonic oscillator.
(A3) Consider a material point of mass m on the end of weightless solid rod of length L, another end of which is fixed (strength of gravitational field is g). It is convenient to use the angle ex between the rod and vertical line as a coordinate of the system. The phase plane is the plane with coordinates\left(\alpha,\,{\frac{\mathrm{d}\alpha}{\mathrm{d}t}}\right).Study and make a draw of the phase portrait of this pendulum at arbitrary angle a. How many qualitatively different types of phase trajectories K does this system have? Draw at least one typical trajectory of each type. Find the conditions which determine these different types of phase trajectories. (Do not take the equilibrium points as phase trajectories). Neglect air resistance.
(B) The oscillator damped by sliding friction
When considering resistance to a motion, we usually deal with two types of friction forces. The first type is the friction force, which depends on the velocity (viscous friction), and is defined by F = –\gammav. An example is given by a motion of a solid body in gases or liquids. The second type is the friction force, which does not depend on the magnitude of velocity. It is defined by the value F = μN and direction opposite to the relative velocity of contacting bodies (sliding friction). An example is given by a motion of a solid body on the surface of another solid body.
As a specific example of the second type, consider a solid body on a horizontal surface at the end of a spring, another end of which is fixed. The mass of the body is m, the elasticity coefficient of the spring is k, the friction coefficient between the body and the surface is μ. Assume that the body moves along the straight line with the coordinate x ( x = 0 corresponds to the spring which is not stretched). Assume that static and dynamical friction coefficients are the same. At initial moment the body has a position x = A_{0}, (A_{0}> 0) and zero velocity.
(Bl) Write down equation of motion of the harmonic oscillator damped by the sliding friction.
(B2) Make a draw of the phase trajectory of this oscillator and find the equilibrium points.
(B3) Does the body completely stop at the position where the string is not stretched? If not, determine the length of the region where the body can come to a complete stop.
(B4) Find the decrease of the maximal deviation of the oscillator in positive x direction during one oscillation ΔA. What is the time between two consequent maximal deviations in positive direction? Find the dependence of this maximal deviation A (t_{n} ) where t_{n} is the time of the n-th maximal deviation in positive direction.
(B5) Make a draw of the dependence of coordinate on time, x(t), and estimate the number N of oscillations of the body?
Note:
Equation of the ellipse with semi-axes a and b and centre at the origin has the following form:
(A) Phase portraits
(A1) Let Ox axis be pointed perpendicular to the walls. Since the material point is free and collisions are absolutely elastic then the magnitude of momentum is conserved, while its direction is changed to opposite at the collisions. Hence, the phase trajectory is of the following form (Fig. 7 – 3):
The motion with positive values of the momentum is directed along increasing values of the coordinate. Thus, the phase trajectory is directed clockwise, as indicated in Fig. 7 – 3.
(A2)
(a) For the harmonic oscillator, let us denote the coordinate by x, the momentum by p, and the total energy by E. The energy conservation law is
{\frac{p^{2}}{2m}}+{\frac{k x^{2}}{2}}=E.This expression determines the equation of phase trajectory, for a given E. Dividing both sides of the equation by E, we obtain
{\frac{p^{2}}{2m E}}+{\frac{x^{2}}{\frac{2E}{k}}}=1.This is a canonical form of the equation of ellipse in (x, p). The centre of the ellipse is at (0, 0) and the semiaxes are \sqrt{\frac{2E}{k}} and \sqrt{2E m} respectively.
(b) The phase trajectory is of the following form in Fig. 7 – 4.
The motion with positive values of the momentum is directed along increasing values of the coordinate. Thus, the phase trajectory is directed clockwise, as indicated in Fig. 7 – 3.
(A3) Let us choose the potential energy level at the lowest point of the pendulum (equilibrium state). Taking into account for that linear velocity of the point is v\,=\,L\,{\dot{\alpha}}\,,we write down the total energy of the mathematical pendulum
{\frac{m L^{2}\dot{a}^{2}}{2}}+m g L\left(1-\cos\alpha\right)=E.Analysis of this expression leads to the following: at E < 2mgL the pendulum oscillates about the lower equilibrium position; if E « mgL, the oscillations are harmonic;
at E = 2mgL the pendulum does not oscillate; the pendulum tends to the upper point of equilibrium.
at E > 2mgL the pendulum rotates about fixed point.
The phase trajectory IS shown in Fig. 7 – 5.
There are K = 3 qualitatively different types of the phase trajectories:
oscillations, rotations, and the motion to the upper point of equilibrium ( separatrisse ). (We do not take the equilibrium points as phase trajectories. )
(B) The oscillator damped by sliding friction
(B1) For the sliding friction the ‘magnitude of the friction force does not depend on the magnitude of velocity, but its direction is opposite to the velocity vector of the body. Therefore, the equations of motion should be written separately for the motion to the right and to the left from the “equilibrium” point (the spring is not stretched). Let us choose the x-axis along the direction of motion, and the origin of the coordinate system at the equilibrium point without the friction force. We obtain the equations of motion as follows:
\ddot{x}+\omega_{0}^{2}x=-\,\frac{F_{fr}}{m}\,,~~~~\dot{x}\gt 0\,, (1)
\ddot{x}+\omega_{0}^{2}x=-\,\frac{F_{fr}}{m}\,,~~~~\dot{x}\lt 0\,,
Here, F_{f r}~=~\mu m g is the friction force, {\omega_{0}}^{2}\;=\;\frac{k}{m} is the frequency of oscillations of the pendulum without the friction.
(B2) Introducing the variables x_{1}\,=\,x+{\frac{F_{f r}}{m\omega_{0}^{2}}} and x_{2}\,=\,x-\frac{F_{\it f r}}{m\omega_{\mathrm{0}}^{2}} we can write down the equations of motion in the same form for both the cases,
which coincides with the equation of motion of harmonic oscillator without the friction. The action of the friction force is reduced to a drift of the equilibrium points: for {\dot{x}}\gt 0, it becomes x_{-}=-{\frac{F_{f r}}{m\omega_{0}^{2}}},\;x_{1}=0 and for {\dot{x}}\lt 0 It ecomes x_{+}={\frac{F_{fr}}{m\omega_{0}^{2}}};\;x_{2}=0.
Thus, due to the section (A2) above, the phase trajectory IS a combination of parts of ellipses with centers at the point x_ for an upper half-plane p > 0, and at the point x_{+} for a lower half-plane p < 0. As the result of a continuity of motion these parts of ellipses should comprise a continuous curve by meeting each other at p = 0.
Thus, the phase trajectory is Fig.7-6.
( B3) According to the phase trajectory combination, the body not necessarily stops at the point x = o. It will stop when it falls into the range from x_{-} to x_{+}. This region is called stagnation region. The width of this region is
x_{+}\!-\!\ x_{-}\!=\!\frac{2F_{fr}}{m\omega_{0}^{2}}.(B4) From the definition of equilibrium points and the obtained form of phase trajectory, it is easy to find the decrease of amplitude during one period:
\Delta A=A(t)-A(t+T)=2(x_{+}-x_{-})=\frac{4F_{fr}}{m\omega_{0}^{2}}.This can be rewritten as
A(t)-A(t+T)={\frac{4F_{f r}}{2\pi m\omega_{0}}}T.One can see that, unlike the case of viscous friction, the amplitude decreases in accord to a linear law,A=A_{0}-pt_{n},\,\,\mathrm{where}\, p=\frac{2F_{fr}}{\pi m\omega_{0}}.
(B5) The total number of oscillations depends on the initial amplitude A_{0} and it can be found as
N={\frac{A_{\mathrm{0}}}{2(x_{\mathrm{+}}-x_{-})}}.As the result of the above conclusions the plot of x(t) is of the following form in Fig. 7 – 7.
The frequency is equal to the frequency of free oscillator \omega_{0}^{2}={\frac{k}{m}}. The time between two successive maximal deviations is T={\frac{2\pi}{\omega_{0}}}.
The oscillations do not stop until the amplitude is more than half-width of the stagnation region x_{+}-x_{-}. In real situations, the body stops in random positions within the stagnation region. In Fig. 7 – 7 the point P denotes the point where the body stops.
