Second Order Dynamical System

Dynamical behavior of a mass-spring system which is subject to an external force, Fin, and dry friction force, Ff, is expressed in form of a second order differential equation as,

Mx = Fin — Kx — Ff S (x) (A.1)

Where x is the position, m is the mass, K is the spring stiffness, and S(.) is the sign function. The system of (A.1) can be expressed and analyzed as a piecewise linear (a. k.a. hybrid) system. However, to avoid unnecessary and complicated mathematics involved, the linear system approach is preferred here. A ring-down is the response of an autonomous dynamical system (i. e., Fin = 0) to a nonzero initial condition. Therefore, the ring-down of system in (A.1) is expressed as,

 (A.2)

X = u2n (x + 8S(x)), x0 = 0

Where <Jn = K/m and 8 = Ff /K. By choosing,

 (A.3) (A.4)

Yi = un (x + 8S(x))

P2 = x

The state space representation for Eq. (A.2) becomes,

 Y 1 0 ^n Yi Y2 — Un 0 V2
 (A.5)

Which is the canonical representation of a linear oscillator that oscillates at an angular velocity of un. Origin is the equilibrium of system (A.5). The equilibrium is the center­point of a circle which, depending on the initial condition, defines the system’s phase portrait. Therefore, there are two equilibria to the system (A.2) which are calculated as,

 (A.6)

(xe = +8, Xe = 0) X < 0 (xe = —8, xe = 0) x > 0

Figure A.1 shows ring-down phase portrait and figure A.2 depicts the time-domain position of a second order mass-spring system with dry friction. Both figures indicate that the oscillation envelope is a line that can be expressed as,

 (A.7)

, 4 A ( 2un Ff’

Xenv = mxt + xo = -— t + xo =————————- — t + xo

 T

K

Where mx is the slope of the envelope to the position signal. Similarly the slope of the

TOC o "1-5" h z Figure A.1: Ring-down phase portrait of a second order mass-spring system with dry friction. For the simulated system, m = 1 kg, K =1 kN/m, and Ff = 50 N.

Envelope to the velocity signal, mx, and the slope of the acceleration signal envelope, mx,

Are given by,

46

Mx №nmx №n T (A.8)

M = №l mx = —№l T (A.9)

Therefore, observation of a linear envelope in the position, velocity, or acceleration signal during a ring-down assessment test indicates existence of dry friction. Magnitude of the friction force is estimated based on the slope of this envelope given that either the spring stiffness or the mass of the ringing system is known. Consequently, the power loss associated with the estimated friction force for a certain frequency and displacement, can be calculated by the corresponding expression given in Table 5.1.

Figure A.2: Time-domain position for the ring-down of a second order mass-spring system with dry friction. For the simulated system, m =1 kg, K = 10 kN/m, and Ff = 50 N.

Viscous Friction

Figure A.3 depicts the time-domain position of a second order mass-spring system with viscous friction. The oscillation envelope expressed as,

_idt, . ,

Xenv = Xoe 2 m (A.10)

Where D is the friction factor and m is the oscillating mass. Therefore, observation of an exponential envelope in the position, velocity, or acceleration signal during a ring-down

Assessment test indicates existence of viscous friction. Friction factor, D, is estimated based

On the following calculation given that either the spring stiffness, K, or the mass of the

Figure A.3: Time-domain position for the ring-down of a second order mass-spring system with viscous friction. For the simulated system, m =1 kg, K = 10 kN/m, and D = 6 Ns/m.

Ringing system, m, is known.

(A.11)

Where xPi and xP2 correspond to the position (also velocity or acceleration) of the ringing mass at two arbitrary peaks, and At is the time delay between the peaks which, of course, is a multiple of ringing period, T. These parameters are shown on figure A.3 as well. Needless to mention that,

2^ I K

(A.12)

T

Consequently, the power loss associated with the estimated viscous friction for a certain frequency and displacement, can be calculated by the corresponding expression given in

Table 5.1. Lastly, the quality factor, Q, of this system is calculated as,

 (A.13)

D/m