Q2. An inverted pendulum consisting of a uniform bar of mass m has a torsional spring of torsional stiffness k at the pivot, with rest angle equal to 0 (no torque when the pendulum is vertical). There is no damping. Figure below.
Q2. An inverted pendulum consisting of a uniform bar of mass m has a torsional spring of torsional stiffness k at the pivot, with rest angle equal to 0 (no torque when the pendulum is vertical). There is no damping. Figure below.
Elements Of Electromagnetics
7th Edition
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Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
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Please help. This problem involves vibrations. Thank you.
![Q2. An inverted pendulum consisting of a uniform bar of mass m has a torsional spring of torsional
stiffness k at the pivot, with rest angle equal to 0 (no torque when the pendulum is vertical). There
is no damping. Figure below.
a) As in HW8, show that the equations of motion for the pendulum is:
mL?
mgL
sin 0 – k0.
3
b) By using the small-angle approximation sin 0 z 0, convert the above equation into the form:
að + b0 = 0. Determine a and b in terms of mass, length, gravity, stiffness, etc. This is called
'linearization' (the original equation from part-a is nonlinear because sin(e) is nonlinear). More
generally, one uses Taylor series for linearization if about any other 0 value.
c) Determine the natural frequency wn of the system. Also, for what value of stiffness k is the
natural frequency 0? For what values of stiffness k is the natural frequency real-valued? (It turns
out that the equilibrium becomes unstable when the natural frequency of this system becomes
complex valued).
Here's a fun video related to this Q2 problem (a bri-cycle):
https://www.youtube.com/watch?v=rNQdSfg|DNM
d) Should you increase or decrease the torsional stiffness so that the torsional stiffness, so that the
natural frequency is very high.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8735dee1-ec31-4346-b70c-e4d10f64e04c%2Ff4ffe359-22c3-436c-8d45-eb418f907a72%2Fdu4ogr_processed.png&w=3840&q=75)
Transcribed Image Text:Q2. An inverted pendulum consisting of a uniform bar of mass m has a torsional spring of torsional
stiffness k at the pivot, with rest angle equal to 0 (no torque when the pendulum is vertical). There
is no damping. Figure below.
a) As in HW8, show that the equations of motion for the pendulum is:
mL?
mgL
sin 0 – k0.
3
b) By using the small-angle approximation sin 0 z 0, convert the above equation into the form:
að + b0 = 0. Determine a and b in terms of mass, length, gravity, stiffness, etc. This is called
'linearization' (the original equation from part-a is nonlinear because sin(e) is nonlinear). More
generally, one uses Taylor series for linearization if about any other 0 value.
c) Determine the natural frequency wn of the system. Also, for what value of stiffness k is the
natural frequency 0? For what values of stiffness k is the natural frequency real-valued? (It turns
out that the equilibrium becomes unstable when the natural frequency of this system becomes
complex valued).
Here's a fun video related to this Q2 problem (a bri-cycle):
https://www.youtube.com/watch?v=rNQdSfg|DNM
d) Should you increase or decrease the torsional stiffness so that the torsional stiffness, so that the
natural frequency is very high.
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