Question

Consider a damped mass/spring system with spring constant, mass and damping coefficient being, respectively.

κ = 37 , m = 4 and γ = 4

a. Verify that this is an underdamped system.

b. Find the decay coefficient α , the angular quasi-frequency ω , the quasi-period p and the quasi-frequency ν of this system.

Answer 1

To verify if a damped mass-spring system is underdamped, we compare the angular frequency of the undamped spring to the damping coefficient. To find the decay coefficient α, angular quasi-frequency ω, quasi-period p, and quasi-frequency ν, we use formulas involving the spring constant κ, mass m, and damping coefficient γ.

Step-by-step explanation:

A damped mass-spring system is considered underdamped when the angular frequency of the undamped spring (√k/m) is greater than the damping coefficient (b/2m). In this case, we have √37/4 > 4a/2(4a), which simplifies to 37/4 > 2a. Since we do not have a specific value for a, we cannot verify if this system is underdamped.

To find the decay coefficient α, angular quasi-frequency ω, quasi-period p, and quasi-frequency ν of the system, we use the following formulas:

α = b/2m = 4a/8 = a/2

ω = √(κ/m - α²) = √(37/4 - (a/2)²)

p = 2π/ω

ν = 1/p