Modeling
m
A = 0.035 m
w = 11.31 rad/s
m = 15 kg
x(t)
y = Asin(wt)
The amplitude of the oscillation of the backpack as well as the frequency are based on typical
values of human motion while walking at a good pace. The mass of the backpack was given in
the problem provided.
Analysis
The magnitude of the dynamic force the backpack exerts on the user is dependent on the
motion of the user.
𝐹 = 𝑚𝐴𝜔!
The amplitude of the force of the backpack on the user is F = 67.16 N.
The backpack on the shoulders of the users can be simplified to the system shown below.
y = Ysin(wt)
Y = 0.035 m
w = 11.31 rad/s
m = 15 kg
k = ? N/m
𝜁 =?
k
c
Type equation here.
The design process calls for the assignment of a stiffness and a damping ratio. The natural
frequency of the backpack is defined as:
𝜔" = &
𝑘
𝑚
Using this system, the displacement transmissibility (X/Y) can be found using the following
equation:
𝑋
1 + (2𝜁𝑟)!
=&
,
(1 − 𝑟 ! )! + (2𝜁𝑟)!
𝑌
𝑟=
𝑤
𝑤"
The force transmissibility is the same as the displacement because the backpack follows the
motions of the user. In the simplified model, we can say that the mass moves along with the
base (the backpack follows the motion of the user).
%
!
𝐹#
1 + (2𝜁𝑟)!
=4
5
(1 − 𝑟 ! )! + (2𝜁𝑟)!
𝐹$
The goal of the design process is to reduce the dynamic force by 90%. Alternatively, the newly
designed backpack dynamic force should be 10% of the original dynamic force. By rearranging
the equation above, a quadratic equation for the k value can be seen below:
(𝑇& − 1)𝑘 ! + (2𝜔! 𝑚7−𝑇&! + 2𝜁 ! (𝑇&! − 1)8𝑘 + 𝑇&! 𝜔' 𝑚! = 0
Results
The damping ratio chosen for the backpack is 0.088. Thus, leading to a spring constant of 150
N/m. These design parameters lead to an exact 90% reduction in force. However, the backpack
will be used in a variety of settings.
If the mass of the load is increased to 30 kg, the backpack will still perform within the desired
force transmission limits. The transmission ratio is 0.05443.
If the user begins to run, their frequency of vertical oscillation would increase. With the base
excitation increasing, the transmission of the force of the backpack on the wearer continues to
decrease which means it stays below 0.1. If it is said that running leads to double the frequency,
then the relative displacement becomes 0.0201.
Discussion
There are a variety of damping coefficients that allow the backpack to operate within desired
limits. That means that are a variety of spring constants that allow the backpack to operate
within the desired limits as well. However, there are various issues that could arise with a
design like this. A potential issue that could take place if the product is not field tested is that
the springs selected may have a maximum load. This means that the backpack would be
essentially free to oscillate within its domain with no restoring force. In theory the lower spring
constant designs allow for greater force reduction to the user, but some of these may not be
feasible in a real design.
In conclusions, it is possible that the design works under given parameters. However, hiking is
not often steady and there is often quick changes in pace that could lead to spikes in force
transmissibility. Certain users walking at a pace near resonance would suffer from the backpack
rather than benefit, and that is the main drawback of this product.