Springs
ME 512 –Vibration Engineering
Spring Elements
• A spring is a mechanical link,
which in most applications is
assumed to have a negligible
mass and damping.
• A spring is defined as an elastic
body, whose function is to
distort when loaded and to
recover its original shape when
the load is removed.
Spring Elements
• The most common type of
spring is the helical-coil spring
used in retractable pens and
pencils, staplers, and
suspensions of freight trucks
and other vehicles.
Spring Elements
• Several other types of
springs can be
identified in
engineering
applications.
• Conical and Volute
Springs
Spring Elements
• Several other types of
springs can be
identified in
engineering
applications.
• Torsion Springs
Spring Elements
• Several other types of
springs can be
identified in
engineering
applications.
• Laminated or Leaf
Springs
Spring Elements
• In fact, any elastic or
deformable body or
member, can be
considered as a spring.
Deformation of a spring
Spring Element
• A spring is said to be linear if the elongation or reduction in length x
is related to the applied force F as:
𝐹 = 𝑘𝑥
• Where:
• F-Force
• k-spring constant (force per unit length, lb/ft, N/m)
• x-elongation/reduction in length
Deformation of a spring
Work done in deforming a spring
• The work done (U) in deforming a spring is stored as strain or
potential energy in the spring, and it is given by:
1 2
𝑈 = 𝑘𝑥
2
Where:
𝑈 − 𝑤𝑜𝑟𝑘 (𝑘𝐽, 𝑙𝑏 − 𝑓𝑡)
𝑙𝑏 𝑘𝑁
𝑘 − 𝑠𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
,
𝑓𝑡 𝑚
Combination of Springs
• Case 1: Springs in Parallel.
𝒌𝒆𝒒 = 𝒌𝟏 + 𝒌𝟐 + ⋯ 𝒌𝒏
Combination of Springs
• Case 2: Springs in Series.
𝟏
𝟏
𝟏
𝟏
=
+
+⋯
𝒌
𝒌
𝒌
𝒌
Equivalent Spring Constants (𝑘𝑒𝑞 )
• Rod under axial load
𝑘𝑒𝑞
𝐸𝐴
=
𝐿
Where:
𝐸 − 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
𝐴 − 𝑐𝑟𝑜𝑠𝑠 𝑠𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑟𝑒𝑎
𝐿 − 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑅𝑜𝑑
L
Equivalent Spring Constants (𝑘𝑒𝑞 )
• Tapered Rod under Axial Load
𝑘𝑒𝑞
𝜋𝐸𝐷𝑑
=
4𝐿
Where:
𝐸 − 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
𝐷, 𝑑 − 𝑒𝑛𝑑 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟𝑠
𝐿 − 𝐿𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑅𝑜𝑑
Equivalent Spring Constants (𝑘𝑒𝑞 )
• Helical Spring Under Axial Load
𝑘𝑒𝑞
𝐺𝑑4
=
8𝑛𝐷3
Where:
𝐺 − 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠
𝐷 − 𝑚𝑒𝑎𝑛 𝑐𝑜𝑖𝑙 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
𝑛 − 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑐𝑡𝑖𝑣𝑒 𝑐𝑜𝑖𝑙𝑠
𝑑 − 𝑤𝑖𝑟𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
Helical Spring
• Deflection of Round-wire Helical Springs:
𝑇𝐿
𝜃=
𝐽𝐺
𝐹𝐷𝑚
𝑇=
;
2
𝐿 = 𝜋𝐷𝑚 𝑁𝑐 ;
4
𝜋𝐷𝑤
𝐽=
32
3𝑁
3𝑁
𝜃𝐷𝑚 8𝐹𝐷𝑚
8𝐹𝐶
𝑐
𝑐
𝛿=
=
=
4
2
𝐺𝐷𝑤
𝐺𝐷𝑤
Helical Springs:
• End Connections for Compression Helical Springs:
Helical Springs Table AT 16 – pg 589
Equivalent Spring Constants (𝑘𝑒𝑞 )
• Fixed-Fixed Beam with load at the middle
192𝐸𝐼
𝑘𝑒𝑞 =
𝐿3
• Cantilever beam with end load
3𝐸𝐼
𝑘𝑒𝑞 = 3
𝐿
• Simply supported beam with load at the middle
48𝐸𝐼
𝑘𝑒𝑞 = 3
𝐿
Where:
𝐸 − 𝑚𝑜𝑑𝑢𝑙𝑢𝑠 𝑜𝑓 𝑒𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
𝐼 − 𝑎𝑟𝑒𝑎 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑖𝑛𝑒𝑟𝑡𝑖𝑎
𝐿 − 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑒𝑎𝑚
Table AT 1 – Properties of Sections @ pg 563
T-Beam: Determine Area Moment of Inertia
(I)
• Determine the location of NA
(Neutral Axis) based on the
bottom of the I beam:
𝐴1 𝑦1 + 𝐴2 𝑦2
𝑦=
𝐴1 + 𝐴2
• Determine the Moment of
Inertia:
𝐼=
𝐼𝑖 + 𝐴𝑖 𝑦𝑖 − 𝑦
𝑏𝑖 ℎ𝑖3
𝐼𝑖 =
12
2
Equivalent Spring Constants (𝑘𝑒𝑞 )
• Hollow shaft under torsion
𝑘𝑒𝑞
𝜋𝐺
=
(𝐷4 − 𝑑4 )
32𝐿
Where:
𝐺 − 𝑠ℎ𝑒𝑎𝑟 𝑚𝑜𝑑𝑢𝑙𝑢𝑠
𝐷 − 𝑜𝑢𝑡𝑒𝑟/𝑜𝑢𝑡𝑠𝑖𝑑𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
𝑑 − 𝑖𝑛𝑛𝑒𝑟/𝑖𝑛𝑠𝑖𝑑𝑒 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟
𝐿 − 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠ℎ𝑎𝑓𝑡
Example 1:
• The figure below shows the suspension system of a freight truck with a
parallel-spring arrangement. Find the equivalent spring constant of the
suspension if each of the three helical springs is made of steel with a
shear modulus 𝑮 = 𝟖𝟎𝒙𝟏𝟎𝟗 𝑵/𝒎𝟐 and has five effective turns, mean
coil diameter 𝑫 = 𝟐𝟎 𝒄𝒎, and wire diameter 𝒅 = 𝟐 𝒄𝒎.