Forced harmonic vibration

Question 1

forcing for harmonic excitation

Answer 1

F0cos(omega.t) or F0sin(omega.t)

f0 = F0/m = F0/k .omega_n^2
u_st = F0/k

-PDE now linear and inhomogeneousso the solutions has a particular part which is added to the homogeneous solution

Question 2

undamped

Answer 2

-ansantz for particular: up = A0cos(omega.t)

A0 = f0/(omega_n^2 - omgea^2)
A1 = u0 - A0
A2 = v0/omega_n
- A1/A2 is from the homogenous solution

Question 3

transient vs steady state

Answer 3

-transient depends on the initial conditions, so the homogeneous part
-steady-state doesnt depend on them so the particular part

considering on the SS part:

R = amplification factor = 1/|1 - beta^2|
u = u_st.R.sin(omega.t - phi)
- phi is zero for omega < omega_n and pi when its greater
-resonance when omega = omega_n
* for u0=v0=0

u = f0/2.omega_n .t.sin(omega_n.t)

*amplitude of the sin func. grows linearly therefore as t->inf. so does u

Question 4

damped

Answer 4

-transient part only important at the start then it dies down
*must consider the accuracy of initial conditions and loading applied

-particular ansantz: up = A3cos(omega.t) + A4sin(omega.t)

A3 = f0(omega_n^2 - omega^2)/[(omega_n^2 - omega^2)^2 + (2.squiggle.omega_n.omega)^2]

A4 = (2.squiggle.omega_n.omega)/[(omega_n^2 - omega^2)^2 + (2.squiggle.omega_n.omega)^2]

-resonance is no longer a special case, u doesnt tend to infinity

A3 = 0
A4 = f0/2.squiggle.omega_n^2

Question 5

damped dynamic amplification factor

Answer 5

R(omega) = [(1-beta^2)^2 + (2.squiggle.beta^2)^2]^-1/2
b

R_max = 1/[2.squiggle.sqrt(1-2.squiggle^2)]

beta = omega/omega_n
- «1 is stiffness controlled
*damping unimportant and a slow variation in excitation
* phi = 0 so motion and forcing are in phase
-&raquo_space; 1 is mass controlled, large inertial F
*damping unimportant and quick variation in excitation
*phi = 180 degrees
-approx. 1 is damping controlled
*stiffness and damping F at similar magnitude but out of phase
*phi = 90 degrees so there is no displacement at maximum excitation F

Question 6

response spectra

Answer 6

plots the system response against the natural angular f for a given damping ration
-provides direct info on the response without considering temporal characteristics
-can be max disp., acc. or vel.

Rv(omega) = w/w_n.R(w)

Question 7

response spectra

Answer 7

plots the system response against the natural angular f for a given damping ration
-provides direct info on the response without considering temporal characteristics
-can be max disp., acc. or vel.

Rv(omega) = w/w_n.R(w)
Ra(omega) = w^2/w_n^2.R(w)
*derived from up/(F0/k) = R(w)cos(omega.t - phi) and normalised by omega_n or omega_n^2