1
Linear spring element
Fs
Free length
l
Fs
s
static deflection
F
Applied force on spring
deflection
K=dF/d
Fs
s
Notes:
a) Spring element is regarded as massless
b) K = stiffness coefficient is constant
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Linear and Nonlinear Spring Elements: Equations of Motion and Static Equilibrium, Study notes of Physics
Information on the dynamics and statics of linear and nonlinear spring elements. It covers the calculation of static deflection, the equation of motion, and the free body diagram for various spring configurations. The document also includes notes on the assumptions made and the significance of the static equilibrium position.
Typology: Study notes
2021/2022
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1
Linear spring element
F
s
Free length
l
F
s
s static deflection
F
Applied force on spring
deflection K =d
F /d
F
s
s
Notes: a)
Spring element is regarded as massless b)
K
= stiffness coefficient is constant
2
Linear spring + added weight
W
Free length
l
W
s staticdeflection
F
static deflection
K =d
F /d
W
s
Notes: a)
Block has weight
W= Mg
b)
Block is regarded as a point mass
W=F
s^
= K
s
Balance of static forces
W Fs
Reaction force from spring
4
Equation of motion
F
spring
= K
( X+
s^ )
Notes:
Motions from SEP
Freelength l
W+F
s^ +X
SEP (t)
FBD:
X
W+F Fspring
M
M Ax = W+F - F
spring
M Ax = W+F – K(X+
s^ )
M Ax = ( W-K
s^ ) +F – KX
M Ax = +F – KXM Ax + K X= F
(t)
2
2
X
d^
X
A^
X
d t
^
^
1
Nonlinear spring element
F
s
Free length
l
F
s
s static deflection
F
Applied force on spring
static deflection
F
s
s
Notes: a)
Spring element is massless b)
Force vs. deflection curve is NONlinear c)
K
1 and
K
3 are material parameters
3
1
3
F^
K^
K ^
^
3
1
3
F
K
K ^
3
Nonlinear spring + weight + force
F
(t)
W
Free length
l
W+F
s^ +X
SEP (t)
F
Reaction force from spring
deflection
W+F
s
X+
s
Notes: a)
Coordinate
X
describing motion has
origin at
Static Equilibrium Position
(SEP)
b)
For free body diagram, assume stateof motion, for example
X
(t)^ >
c)
Then,
state
Newton’s equation of
motion d)
Assume no lateral (side motions)
Free Body diagram
W+F Fspring
F
(t)
M
M Ax = W+F - F
spring
acceleration
4
Equation of motion Notes: a)
Motions from SEP
Freelength l
W+F
s^ +X
SEP (t)
FBD:
X
W+F Fspring
M
2 2 X
d^
X
A^
X
d t ^
^
^
^
^
(^3)
1
3
spring
s^
s
F^
K^
X^
K^
X
^
^
^
( ) t^
spring
M X
F
W
F
^
^
^
^
( )^
1
3
t^
s^
s
M X
F^
W
K^
X^
K
X
^
^
^
^
^
Expand RHS:
^
^
^
3
2
2
3
( )^
1
3
t^
s^
s^
s^
s
M X
F^
W
K^
X^
K^
X^
X^
X
^
^
^
^
^
^
^
^
^
^
^
^
^
^
3
2
2
3
( )
1
3
1
3
3
3
3
t^
s^
s^
s^
s
M X
F^
W
K
K
K
K
X^
K^
X^
X
^
^
^
^
^
^
^
^
Cancel terms from force balance at SEP to get
^
^
^
2
2
3
1
3
3
( )
s^
s^
t
M X
K
K
X
K
X
X
F
6
Linear equation of motion Freelength l
W+F
s^ +X
(t)
SEP FBD:
X
W+F Fspring
M
( )
e^
t
M X
K
X
F
^
2
1
3 3
e^
s
K
K
K
^
3
1
3
2
1
3 3
s
spring
e^
s
d^
K^
K
dF
K^
K^
K
d^
d
^
^
^
^
^
^
^
F
deflection
W
s^
X
K
e
s
deflection
Stiffness(linear)