FREE RESPONSE OF SDOF - Lesson 2

Question 1

x_dot_dot+2XIOmega_nx_dot+Omega^2_nx=0

Solution of EOM (Equation Of Motion)
x(t)=Ae^(st)
A,s_ 2 constant parameters

Answer 1

Ae^(st)(s^2+2XIOmega_ns+Omega^2_n)=0

If A=0: No motion. TRIVIAL solution

else: Characteristic equation polynomial (s^2+2XIOmega_ns+Omega^2_n)=0
s_1,2=XIOmega_n plusminus Omega_n*sqrt(Xi^2-1)

SYSTEM RESPONSE: x(t)= A_1e^(s_1t) + A_2e^(s_2t)

Question 2

Constants ex A_1and A_2 depending on initial conditions (i.c)

2 i.c:
DISPLACEMENT at t=0: x(t)=x_0
VELOCITY at t=0: x_dot(t): x_dot(t)=V_0

      X_0=A_1+A_2
      V_0=s_1*A_1+s_2*A_2

A_1=?
A_2=?

Solutions s_1 and s_2 depends on sign of (XI^2-1)

Answer 2

A_1=(V_0-s_2x_0)/s_1-s_2
A_2=s_1x_0-V_0)/s_1-s_2

Question 3

if XI>1: what happens to s_1,2?

Answer 3

s_1,2:
Real and negative
OVERDAMPED

Question 4

if XI=1: what happens to s_1,2?

Answer 4

s_1=s_2=-Omega_n
Real, negative coincident
CRITICALLY DAMPED

Question 5

if 0<XI<1: what happens to s_1,2?

Answer 5

s_1,2=-XIOmega_n plusminus iOmega_n*sqrt(1-XI^2)
complex conjugates
UNDERDAMPED

Question 6

Angular frequency of the damped vibrations
Omega_d =

Answer 6

Omegaa_n*(1-Xi^2)^1/2

Question 7

EXAMPLE:
1DOF SYSTEM WITH 0<Xi<1

Amplitude of oscillations decreases with time (dissipation of vibration energy in the damper)

How to measure Omega_d from the system free response?

Answer 7

f_d=1/T_d;
Omega_d=2pif_d