SDOF - Lesson 3

Question 1

Solutions for underdamped case
X(t)=e^(-XiOmega_nt) [(B1sin(Omega_dt) + B2cos(Omega_d*t)]

B1=XiOmega_nX0+V0)/Omega_n;
B2=X0

X(t)=

Answer 1

e^(-XiOmega_nt) [(XiOmega_nX0+V0)/Omega_nsin(Omega_dt) + X0cos(Omega_d*t)]

Question 2

X(t)=e^(-XiOmega_nt) [(B1sin(Omega_dt) + B2cos(Omega_d*t)]

C=sqrt(B1^2+B2^2);
rho=arctan(B1/B2);

final equation x(t)=

Answer 2

Ce^(-XiOmega_n*t) cos(omega_dt-rho)

Question 3

Periodic delay motion
where X0 and V0 have the highest values and the amplitude of the sinus curves is decreasing over time

Question 4

Root- Locus diagram for a viscously damped system

ASYMPTOTICALLY STABLE when 0<Xi<1(amplitude of the sinus curve decreasing over time, starting big), Xi=1 (one short wave) and Xi>1(one wave but stretched over time;

MERELY STABLE when Xi=0 (ideal sinus curve) and Xi<0 (amplitude of the sinus curve increasing over time, staring small);

UNSTABLE ROOTS Re(s)>0;

Question 5

ROOT LOCUS CONDITIONS:

Omega_n fixed;
s1 and s2 change as functions of Xi;

IN GENERAL: s1,2= XiOmega_n PlusMinus sqrt(Xi^2-1)Omega_n

UNDER DAMPED CASE: s1,2= XiOmega_n PlusMinus sqrt(1-Xi^2)Omega_n

Question 6

NATURAL FREQUENCY

abs(s1)=sqrt(Xi^2Omega_n^2 +(1-Xi^2)Omega_n)=

Answer 6

Omega_n