Course introduction

Question 1

How is C given in Rayleigh damping?

Answer 1

C=alphaM+betaK

Question 2

What does LTI stand for?

Answer 2

Linear time-invariant

Question 3

What is the conditions for a system be a LTI system?

Answer 3

• Stable (input results in a bounded response)
• Linear (material behave linear elastically with small displacements, linear damping)
• No contact problems
• Time-invariant (mass, damping and stiffness matrices are not functions of time)

Question 4

When is a Matrix positive definite?

Answer 4

When for all x execpt x=0, is X^T A x > 0

Question 5

What is the spatially discrete formulation expression for an LTI system without damping?

Answer 5

Md(t).. + Kd(t) = B_2*u(t)

Question 6

when is the motion for a system free motion/eigenmotion?

Answer 6

When for all time is the input 0, ∀t:u=0

Question 7

How is any given matrix A converted to the corresponding modal matrix?

Answer 7

A_tilde = Phi^T A Phi

Question 8

What is the spatially discrete formulation expression for an LTI system with damping?

Answer 8

Md(t)..+Cd.(t) + Kd(t) = Bu(t)

Question 9

Which variables are used to determine the
Rayleigh’s damping coefficients?

Answer 9

damping ratio and omega

Question 10

What is an important property of a system where the damping matrix C is positive definite?

Answer 10

The system is stable

Question 11

What is the condition for the existence of classical damping?

Answer 11

M^-1 K and M^-1 C commute

Question 12

What is the expression for the linear viscous damping model

Answer 12

f_d=C*d.(t)

Question 13

What are the three steps in the Galerkin’s method

Answer 13

1. Discretization
2. Pre-multiplication with S_e^T
3. Integration over the element

Question 14

What is the strong formulation of the undamped LTI system

Answer 14

rho*phi..=nabla^T sigma_c+b_c

Question 15

What is the elemental expression for the damped LTI system with Galerkin’s method applied and evaulated?

Answer 15

M_e d.._e + C_e d_e. + K_e d_e = f_e

Question 16

How many DOFs are there in a 2d beam element?

Answer 16

6

Question 17

What is the DOF of a beam element?

Answer 17

d_1^x, d_1^y, theta_1, d_2^x, d_2^y, theta_2

Question 18

How is a rigid BC applied?

Answer 18

Multiplying the associated DOF of the mass, damping and stiffness matrix with zero

Question 19

When the mass matrix is mass-normalized (M=I), then K_tilde=Lambda, how is Lambda defined?

Answer 19

Lambda=diag(omega_j^2)

Question 20

When using the ‘[Phi,Lam]=eig(K,M)’ function in Matlab, What is Phi?

Answer 20

Mode shapes

Question 21

When using the ‘[Phi,Lam]=eig(K,M)’ function in Matlab, how is the eigenfrequency determined from Lam?

Answer 21

omega=diag(sqrt(Lam))