Spring Mass Systems
What is the constant for gravity in both the metric and imperial system?
9.8 m/s^2, 32 ft/s^2
Draw a rough sketch of what happens to the solution of a damped and unforced spring mass system over time.
(explanation of what your drawing should look like) The system will oscillate for a short duration of time and then come to rest due to the damping.
What is necessary for a differential equation to have:
a) Resonance
b) Beats
Neither of them can have damping.
a) The natural frequency of the system equals the frequency of the external force.
e.g. y'' + 4y = cos(2t)
b) The natural frequency of the system and the frequency of the external force are "close".
e.g. y'' + 4y = cos(2.1t)
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) Spiral Sink
c) Source
d) Center
a) Unstable
b) Asymptotically Stable
c) Unstable
d) Stable
An object weighing 24lb stretches a spring 16in, what is the spring constant?
18 lb/ft
The solution to a differential equation is
y = -2cos(4t) -7sin(4t).
Without any calculations, what can you say about the phase shift?
a) δ = 0
b) 3pi/2 < δ < 2pi
c) 0 < δ < pi/2
d) pi < δ < 3pi/2
d), because c1 and c2 are negative, which implies that δ is in the third quadrant.
Find the general solution to the differential equation:
y'' + 4y' + 4y = e^(t)
y = c1e^(-2t) + c2te^(-2t) + 1/9e^(t)
Write the general solution for the differential equation:
y'' - 121y = 0
c1e^(11t) + c2e^(-11t)
Match the terms 1-3 with their corresponding answers.
1) Critically damped
2) Under-damped
3) Over-damped
a) No solution
b) Two complex eigenvalues
c) One real repeated eigenvalue
d) Two real distinct eigenvalues
Critically damped <> One real repeated eigenvalue
Under-damped <> Two complex eigenvalues
Over-damped <> Two real eigenvalues
A spring has a viscous damper that exerts a force of 15lb when the velocity is 100 ft/minute. What is the damping constant of the spring?
9 lb/(ft/s)
Construct a second order differential equation with the general formula:
y = c1e^(2t)cos(2t) + c2e^(2t)sin(2t)
y'' - 4y' + 8y = 0
Guess an appropriate Y(t) for the differential equation:
y'' + 2y' + y = t^2 + 4 - e^(-t)
At^2 + Bt + C + De^(-t)*t^2
Note: There are multiple acceptable answers and different ways to guess in these problems. Your answer may not exactly match mine, but should follow the principle of having a different constant in front of each term.
Solve:
y'' - y' - 2y = 0
y'(0) = 1
y(0) = 1
y = 2/3*e^(2t) + 1/3*e^(-t)
Given the differential equation:
t(4 - t^2)y'' + 2ty' - tan(t)y = 0
Find the largest interval where we are guaranteed to have a unique solution if the initial conditions are y(pi) = 1, y'(pi) = 1
(2, 3pi/2)
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 variables).
a) my'' + γy' + ky = F(t)
b) 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.
(explanation of what your drawing should look like)
It will sort of go crazy early on, but then as the damping "calms" the system, the crazy fluctuation will die off and the system will get into a rhythmic cycle and oscillate due to the external force keeping the system in motion.
Construct a second order differential equation with the general solution:
y = c1e^(t) + c2e^(4t) + 4sin(2t)
y'' - 5y' + 4y = -40cos(2t)
Solve:
y'' + 16y = 0
y(0) = 0
y'(0) = 1
y = 1/4*sin(4t)
Given y = c1cos(t) + c2sin(t), y(0) = 3, y'(0) = 4;
a) What is the natural frequency?
b) What is the period?
c) What is amplitude?
d) What is the phase angle in radians to four decimal places?
e) Rewrite in phase amplitude form
a) 1
b) 2pi
c) 5
d) 0.9273 (radians)
e) 5cos(t - 0.9273)
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. The system is kept in motion by an external force of 4cos(t). Model a second order differential equation representing this system.
2y'' + 0.6y' + 30y = 4cos(t)
y(0) = -0.05 m
y'(0) = 0.1 m/s
(y'' + 0.3y' + 15y = 2cos(t) is also perfectly 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
Guess an appropriate Y for the differential equation:
y'' + 3y' = e^(-3t) + e^(t)cos(t) + t^2 + 2t + 5
Ate^(-3t) + Be^(t)cos(t) + Ce^(t)sin(t) + Dt^3 + Et^2 + Ft + G
Note: There are multiple acceptable answers and different ways to guess in these problems. Your answer may not exactly match mine, but should follow the principle of having a different constant in front of each term.
Solve:
y'' +6y' + 9y = 0
y'(0) = 2
y(0) = 0
2te^(-3t)
The Wronskian of f and g is t^(2)e^(t). If g = t, find f.
-te^(t) + tc