Vibrations and Dynamics - 1
HyperLoop (Fig. 1) is a concept that the Founder of SpaceX has introduced to create effective and
fast transportation between different cities in the Country. Such an idea involves a series of
pressurized capsules pushed through a low pressure tube interlinking the origin and destination.
One idea is for the capsules to be afloat on a film of high pressure air, created by induction motors
and high pressure air compressors. Therefore, the friction force between the inner surface of the
tube and the capsule will be negligible.
To bring the capsule to stop, a “bumper” mechanism is used, which can be modeled as a parallel
spring and damper, as shown in Figure 2. The bumper absorbs the kinematic energy of the capsule
upon contact. The bumper is commonly referred to as the brake mechanism since its sole purpose
is to safely bring the capsule to stop.
The capsules are designed such that their fully loaded mass is 450 kg. Tests have shown that a
“safe” contact velocity at the end of the tube when the capsule is approaching the station is 12 m/s.
This is the velocity at which the capsule makes contact with the brake mechanism. Once the
capsule makes contact with the brake system, it does not separate from it, i.e., the capsule and the
brake system behave like a single-degree-of-freedom system.
If c = 1.75×104 Ns/m and k = 3.5×106 N/m, then answer the following questions for when the
capsule makes contact with the brake mechanism:
a) What is the system’s natural frequency, damped frequency, and damping ratio? (30 points)
b) Using your answer in (a), is the system underdamped or overdamped? Justify your answer. (10
points)
c) Choose the damping constant (c) that makes the system critically damped and plot the deflection
of the capsule from the point of contact to when the capsule comes to full stop (i.e., deflection
versus time). Note, you are required to show the trend, not the actual values on your plot. (20
points)
d) During braking, the brake system deflection can be expressed by:
𝑥(𝑡) = 𝑒 −𝜁𝜔𝑛𝑡 (
𝑣0 + 𝜁𝜔𝑛 𝑥0
𝑠𝑖𝑛𝜔𝑑 𝑡 + 𝑥0 𝑐𝑜𝑠𝜔𝑑 𝑡)
𝜔𝑑
Estimate the maximum deflection of the brake system, if the capsule makes contact at a velocity
of 12 m/s. Use the damping constant c and stiffness constant k given above for your answer.
Hint: do not take the derivative of the above equation. Instead, assume 𝜔𝑛 𝑡 =
equation for its maximum value. (30 points)
𝜋
2
and solve the
e) How many seconds after the moment of contact will it take for the bumper to reach its maximum
deflection? (10 points)
Fall 2016
Figure 1. HyperLoop 2
[from: oooOOC CC BY-NC-SA 2.0; Credit: Hyperloop Transportation Technologies (HTT);
License URL: https://creativecommons.org/licenses/by-nc-sa/2.0/legalcode]
Figure 2. Free Body Diagram of a HyperLoop capsule as it reaches the terminal
Fall 2016