Dynamics 2

Question 1

EOM for a linear undamped oscillating system

Answer 1

m x’’ + k x = 0

Question 2

EOM for a rotational undamped oscillating system

Answer 2

I_G θ’’ + k_θ θ = 0

Question 3

EOM for linear damped oscillating system

Answer 3

m x’’ + c x’ + k x = 0

Question 4

EOM for linear damped forced oscillating system

Answer 4

m x’’ + c x’ + k x = F(t)

Question 5

Frequency ratio r

Answer 5

r = ω / ω_n

Question 6

Maximum amplitude response frequency

Answer 6

ω_max = ω_n sqrt( 1 - 2ζ² )

For low damping, basically the same as ω_n

Question 7

Out of Balance Oscillating Excitation Force

Answer 7

F_exi = m’ r ω² sin(ωt)

Question 8

Forces for Phasor Diagrams

Answer 8

Inertia: m ω² x₀ (-90 from damping)
Damping: c ω x₀ (-90 from spring)
Restoring spring: k x₀ (-φ from force)

Question 9

Force transmitted to ground

Answer 9

F_T = k x+ c x’
(Spring and damping, not inertia)

Question 10

Max Force on Ground

Answer 10

F_T,max = x₀ sqrt( k² + (cω)² )

Question 11

Stiffness Matrix, K

Answer 11

k1 + k2 -k2
-k2 k2

Question 12

Mass Matrix, M

Answer 12

m1 0
0 m2

Question 13

Solving (K - Mω²)x for natural frequencies

Answer 13

Determinant of K - Mω² = 0

Question 14

Assumed harmonics solutions to differentiate

Answer 14

x₁(t) = x_1,0 Sin(ωt - φ1)
Where x is one of the degrees of freedom (can be rotational)