Second order modeling and analysis

Question 1

How are the kinematic relations used in Newmark integration initially reached?

Answer 1

Applying Taylor expansion to d_k+1 and d._k+1

Question 2

What are the parameters in the kinematic relations used in Newmark integration?

Answer 2

beta and gamma

Question 3

What is the function of the parameters in the kinematic parameters used in Newmark integration?

Answer 3

To specify how the system evolves over the time increment, eg. linear, constant, step

Question 4

What is the only value for the current step (k+1) needed to determine d._k+1 and d.._k+1 using the kinematic relations?

Answer 4

d_k+1

Question 5

How can d_k+1 be determined (Newmark integration)

Answer 5

K^eff*d_k+1 = F^eff

Question 6

When is a formulation consistent?

Answer 6

When the truncation error vanishes for lim_\Delta t –> 0

Question 7

If a formulation is both consistent and stable, what does that imply?

Answer 7

It is also convergent

Question 8

What is the general formulation of a Laplace transformation of a function d(t)?

Answer 8

D(s)=L(d(t)=int_0^inf d(t) e^-st dt

Question 9

For a system with zero-valued initial conditions, how may the system formulation be expressed in the Laplace domain and the frequency domain?

Answer 9

(Ms^2+Cs+K)D(s)=B_2U(s)

(-M omega^2+C i omega + K)D(omega)=B_2U(omega)

Question 10

How is the receptance transfer matrix defined in frequency domain?

Answer 10

H(w) = (-Mw^2 + C iw +K)^-1

Question 11

How is the frequency response functions calculated?

Answer 11

h_kl(w) = D_k(w) / F_k(w)

Question 12

How can the D_k(w) function be determined experimentally?

Answer 12

Applying an FFT to the displacement data

Question 13

If assuming classical damping, how may the receptance frequency response matrix be determined analytically?

Answer 13

H(w) = (Phi Phi^T) / 2xi_j w_j^2 i

Question 14

What is the function of the FRF h_kl

Answer 14

It maps the input load at DOF k to the output displacement DOF l ratio