Spring Mass Systems
What is the constant for gravity in BOTH the metric and impirical system?
9.8 m/s^2, 32 ft/s^2
What is used to find the eigenvalues of a second order differential equation?
The quadratic formula
What are the units for force in both metric and impirical?
Newtons (N) and pounds (lb)
What is the general formula for a second order differential equation with two complex eigenvalues?
For eigenvalues in the form λ = µ ± vi
y = c1e^(µt)cos(vt) + c2e^(µt)sin(vt)
State the stability of the following origins:
a) Saddle
b) Sink
c) Source
a) Unstable
b) Asymptotically Stable
c) Unstable
An object weighing 24lb stretches a spring 16in, what is the spring constant?
18 lb/ft
Draw a rough sketch of what happens to the solution of a damped and unforced spring mass system over time.
The system will oscillate for a short period of time and then come to rest due to the damping.
What are the units for a spring constant in both metric and impirical?
N/m OR kg/s^2, and lb/ft
Given that a second order differential equation has one real repeated eigenvalue of 11, write the general solution for this DE.
c1e^(11t) + c2te^(11t)
An nth order differential equation can be rewritten as a system containing n number of first order differential equations.
a) False
b) True
True
A spring has a viscous damper that exerts a force of 15lb when the velocity is 100 yards/minute. What is the damping constant of the spring?
27 lb/(ft/s)
Construct a differential equation with the general formula:
y = c1e^(2t)cos(2t) + c2e^(2t)sin(2t)
y'' - 4y' + 8y = 0
What are the units for damping in both metric and impirical?
N/(m/s) and lb/(ft/s)
Solve:
y'' - y' - 2y = 0
y'(0) = 1
y(0) = 1
y = 2/3*e^(2t) + 1/3*e^(-t)
Select the answer that makes this statement true. A damped and forced spring mass system is:
a) Non-homogeneous
b) Linear
c) Non-autonomous
d) Homogeneous
e) Autonomous
Non-homogeneous, because F(t) is nonzero.
a) Write the most general form of a second order differential equation representing a spring mass system
b) State what each component represents (not including the y's).
my'' + γy' + ky = F(t)
m represents a mass constant, γ represents a damping constant, k represents a spring constant, and F(t) represents an external force acting on the system.
Draw a rough sketch of what happens to a damped and forced spring mass system over time.
It will fluctuate a lot early on, but then as the damping "calms" the system, the fluctuation will die off and the system will oscillate due to the external force.
What are the units for mass in both metric and impirical?
kg and lb/(ft/s^2)
Solve:
y'' + 16y = 0
y(0) = 0
y'(0) = 1
y = 1/4*sin(4t)
To be an improper node, a differential equation must have:
a) No solution
b) Two complex eigenvalues
c) One real repeated eigenvalue
d) Infinitely many solutions
e) Two real distinct eigenvalues
One real repeated eigenvalue
A spring is stretched 10cm by a force of 3N. A mass of 2kg is hung from the spring and is also attached to a viscous damper that exerts a force of 3N when the velocity of the mass is 5m/s. The mass is pulled down 5cm below its equilibrium position and given an initial upward velocity of 10cm/s. Model a second order differential equation representing this system.
2y'' + 0.6y' + 30y = 0
y(0) = -0.05 m
y'(0) = 0.1 m/s
(y'' + 0.3y' + 15y = 0 is also acceptable)
Find all values for p such that the second order differential equation is a source:
y'' - (2p-3)y' + p(p-3)y = 0
p > 3
A second order differential equation representing a spring mass system has what units? Answer in both metric and impirical.
Newtons (N) and pounds (lb)
Solve:
y'' +6y' + 9y = 0
y'(0) = 2
y(0) = 0
2e^(3t) - 6te^(3t)
The Wronskian of f and g is t^(2)e^(t). If g = t, find f.
-te^(t) + tc