ENGR/MTH/PHYS 25
Problem 4.22 Tutorial
Problem 4.22 – Dual-Stage Spring Platform & DROPPED Pkg
The diagram at the bottom of the page shows a mass-spring applied-physics model used to
design a package handling systems. The center spring is “soft” to accommodate light-weight
packages (Case I). The two “stiff” side springs provide additional resistance if the weight, W, is
too heavy for the center spring alone to support (Case-II).
We need to analyze the situation where a package of weight, W, is dropped onto the platform
from a height, h, as shown in the diagram. When the package drops onto the platform the
platform depresses (sinks) to a maximum value, Xm, as a function of the spring compression.
For the current design the spring forces are described by:
Fs1  k1 L1 
n
with n  1
Fs2  k2 L2 

The Center support is a LINEAR Spring where n = 1,
which follows Hooke’s Law
Also a LINEAR Spring; i.e., the exterior springs follow
Hooke’s Law (note that there are TWO secondary
springs)
Where
o Fs1,2 ≡ Spring Force (Newtons)
o k1 ≡ Spring-1 constant (Newtons/metern)
o k2 ≡ Spring-2 constant (Newtons/meter)
o ΔL1,2 ≡ Spring compression distance (meters)
Fs1
Fs2
Fs2
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A preliminary design has been completed where h, d, k1, k2, and n have all been selected by
the mechanical design engineer. As the system design engineer you need to find mathematical
relations that describe Xm as function of the physical parameters for the two cases. That is,
you need to find:
X m  f1 k1 , h,W1  ; 0  X m  d
X m  f 2 k1 , k2 , d , h,W2  ; X m  d
Assume that the Platform THICKNESS and WEIGHT are negligible
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The FUNCTION FILE

CAUTION: May contain booby-trap(s) that students must Find & Fix
The PROGRAM/SCRIPT FILE

CAUTION: May contain booby-trap(s) that students must Find & Fix
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The PLOT OutPut
Spring Compression for a Dropped Package
180
160
Spring Compression, x (mm)
140
120
100
80
60
40
20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Drop Height, h (m)
Print Date/Time = 29-May-16/03:58
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