Cantilever Beam Stiffness Calculation

In summary, we are trying to find the roots of the characteristic equation which are lambda1,2=-dampingRatioxwnatural/sqrt(1-dampingRatio). We can find the damping ratio from the log decrement. The equivalent stiffness of the beam and shaker is k_equivalent=k_beam+k_shaker.
  • #1
kev.thomson96
13
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A cantilever beam at low frequency behaves like an underdamped 1 DOF mass/spring/damper system.

We are trying to find the roots of the Characteristic equation which are lambda1,2 = -dampingRatio x wnatural /sqrt(1-dampingRatio)

Relevant formulas and given values:

damping ratio = c/2sqrt(m/k)

wnatural = sqrt(k/m)

wdamped = wnatural x sqrt(1-dampingRatio^2)

log decrement = (1/n)ln(x1/xn+1) = 2pi(damping ratio)/sqrt(1-dampingRatio)

beam width = 50 mm

beam depth = 3.8 mm

Modulus of elasticity = 200 GPa

density = m/v = 7800 kg/m^3

deflection x = Fl^3/3EI, l is length and I is second moment of area

kequivalent = kbeam + kshaker, where kshaker = 45 N/m.

I've only found I = bh^3/12 = 15.83 x 10^-12 m and from then on I'm stuck
 
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  • #2
kev.thomson96 said:
A cantilever beam at low frequency behaves like an underdamped 1 DOF mass/spring/damper system.
We are trying to find the roots of the Characteristic equation which are lambda1,2 = -dampingRatio x wnatural /sqrt(1-dampingRatio)
##\lambda_{1,2} = -\zeta \omega_0/\sqrt{1-\zeta}## ... you mean like that?

It's a heroic effort but still hard to read. You could just use roman characters ... L=-zw0/√(1-z) because you can define any variable names you like.
Or, you can learn LaTeX markup - which is what the rest of us do:
https://www.physicsforums.com/help/latexhelp/

So:
damping ratio = c/2sqrt(m/k) ##\zeta = c/2\sqrt{km}## https://en.wikipedia.org/wiki/Damping_ratio#Definition
wnatural = sqrt(k/m) ##\omega_0 = \sqrt{k/m}##
wdamped = wnatural x sqrt(1-dampingRatio^2) ##\omega = \omega_0\sqrt{1-\zeta^2}##
log decrement = (1/n)ln(x1/xn+1) = 2pi(damping ratio)/sqrt(1-dampingRatio) ##\delta = \frac{1}{n}\ln|x_1/x_{n+1}|##

beam width = 50 mm

beam depth = 3.8 mm

Modulus of elasticity = 200 GPa

density = m/v = 7800 kg/m^3

deflection x = Fl^3/3EI, l is length and I is second moment of area

kequivalent = kbeam + kshaker, where kshaker = 45 N/m.

I've only found I = bh^3/12 = 15.83 x 10^-12 m and from then on I'm stuck
... I don't follow.
Am I reading that right: ##I=bh^3/12## ?? Where are these numbers coming from?
Please explain your reasoning?
 
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  • #3
Simon Bridge said:
Or, you can learn LaTeX markup - which is what the rest of us do:
Apologies for not using that.
Simon Bridge said:
Am I reading that right: I=bh3/12I=bh3/12I=bh^3/12 ?? Where are these numbers coming from?
The second moment of area of the beam, which is a rectangular prism.
 
  • #4
OK - so how are you thinking of the problem? Please show your best attempt with reasoning.
 
  • #5
We can find the damping ratio from the log decrement.

Shaker and beam have the same ground, therefore they are in parallel in terms of stiffness, so the equivalent stiffness formula is ##k_equivalent = k_beam + k_shaker##

To find k_beam, we need to equate it to ##\frac F x##, and equate ##\frac F x## to ##\frac 3EI l^3##(just rearranging the deflection formula).

To plug I into the deflection formula, we find it by using the appropriate formula for a rectangular prism.

And then I'm stuck.
 
  • #6
kev.thomson96 said:
We can find the damping ratio from the log decrement.

Shaker and beam have the same ground, therefore they are in parallel in terms of stiffness, so the equivalent stiffness formula is ##k_{equivalent} = k_{beam} + k_{shaker}##

To find k_beam, we need to equate it to ##\frac F x##, and equate ##\frac F x## to ##\frac{3EI}{l^3}##(just rearranging the deflection formula).

To plug I into the deflection formula, we find it by using the appropriate formula for a rectangular prism.

And then I'm stuck.
... did you figure it out?
(BTW: a_bcd gets you ##a_bcd## while a_{bcd} gets ##a_{bcd}## same with \frac ab cd vs \frac{ab}{cd} ... but better this time :) )
 

Related to Cantilever Beam Stiffness Calculation

1. What is a cantilever beam problem?

A cantilever beam problem is a type of structural engineering problem that involves analyzing the behavior and stability of a beam that is supported at only one end, while the other end is free to move. This type of problem is commonly encountered in the design of buildings, bridges, and other structures.

2. How is a cantilever beam problem solved?

The solution to a cantilever beam problem involves determining the forces acting on the beam, as well as the internal stresses and displacements. This is typically done using mathematical equations and principles of mechanics, such as the laws of statics and the equations of equilibrium.

3. What are the main factors that affect the behavior of a cantilever beam?

The behavior of a cantilever beam is primarily affected by its length, cross-sectional shape, and the type of material it is made of. Other factors that can influence its behavior include the magnitude and direction of the applied loads, as well as any external supports or constraints.

4. What is the purpose of analyzing a cantilever beam problem?

The analysis of a cantilever beam problem is important for ensuring the structural integrity and safety of a design. By understanding how the beam will behave under different conditions, engineers can make informed decisions about its design and make any necessary adjustments to ensure its stability and strength.

5. Can a cantilever beam problem be solved using computer software?

Yes, there are various computer programs and software packages that can assist in solving cantilever beam problems. These programs use numerical methods and algorithms to analyze and model the behavior of the beam, allowing for more complex and accurate solutions to be obtained.

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