Euler Bernoulli Beam 4th order ODE -Balance of Units

In summary, the conversation discusses the balance of units for an equation involving modulus of elasticity, second moment of area, elastic foundation modulus, deflection, and distributed transverse load. The units are calculated to be Nm^3 + N/m = N/m, and it is shown mathematically that the second derivatives have units of 1/m^2. The concept of derivatives and their units is also briefly explained.
  • #1
bugatti79
794
1
Folks,

I am trying to understand the balance of units for this eqn

## \displaystyle \frac{d^2}{dx^2}(E(x)I(x) \frac{d^2 w(x)}{dx^2})+c_f(x)w(x)=q(x)##

where ##E## is the modulus of Elasticity, ##I## is the second moment of area, ##c_f## is the elastic foundation modulus, ##w## is deflection and ##q## is the distributed transverse load.

Based on the above I calculate the units to be

## \displaystyle \frac{d^2}{dx^2}[\frac{N}{m^2} m^4 \frac{d^2 m}{dx^2}]+\frac{N}{m^2} m=\frac{N}{m}##

gives

##\displaystyle {Nm^3} +\frac{N}{m}=\frac{N}{m}##

##LHS \ne RHS##...?
 
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  • #2
The derivatives ##\frac{d^2}{dx^2}## have units of ##1/m^2##.
 
  • #3
Mute said:
The derivatives ##\frac{d^2}{dx^2}## have units of ##1/m^2##.

Ok, I see how they balance now. The question I have is how is this shown mathematically that the 2nd derivatives have ##1/m^2## units?

##f(x)= f(units in meters)##
##f'(x)= f(units in meters)##
##f''(x)= f(units in meters)##...?
 
  • #4
df/dx is defined as [itex]\lim_{h\to 0} (f(x+h)- f(x))/h[/itex]. The numerator is in what ever units h has. The denominator is in whatever unis x has- "meters" in your case- so the derivative has the units of f divided by the units of x and the second derivative has units of units of f divided by the units of x, squared.

Surely you learned this in basic Calculus? if f(t) is a distance function, with units "meters" and t is time, in "seconds", then df/dt is a speed with units "meters per second" and d2f/dt2 is an acceleration with units of "meters per second squared".
 
  • #5


Hello,

The balance of units in this equation can be confusing at first, but let me clarify it for you. First, let's break down each term and its units:

- E(x) is the modulus of elasticity, which has units of N/m^2. This represents the stiffness of the beam.
- I(x) is the second moment of area, which has units of m^4. This represents the cross-sectional shape of the beam.
- c_f(x) is the elastic foundation modulus, which also has units of N/m^2. This represents the stiffness of the foundation on which the beam rests.
- w(x) is the deflection of the beam, which has units of m.
- q(x) is the distributed transverse load, which has units of N/m.

Now, let's look at the units of the left-hand side of the equation:

- The first term, ##\frac{d^2}{dx^2}(E(x)I(x) \frac{d^2 w(x)}{dx^2})##, has units of N/m^3. This is because it is the product of the modulus of elasticity, the second moment of area, and the second derivative of the deflection, all of which have units of N/m^2, m^4, and m^-2, respectively.
- The second term, ##c_f(x)w(x)##, has units of N/m. This is because it is the product of the elastic foundation modulus and the deflection, both of which have units of N/m^2 and m, respectively.

Therefore, the units on the left-hand side of the equation are consistent and equal to N/m^3 + N/m = N/m, which matches the units on the right-hand side of the equation.

I hope this helps to clarify the balance of units in the Euler Bernoulli Beam 4th order ODE. Let me know if you have any further questions.
 

1. What is Euler Bernoulli Beam 4th order ODE?

Euler Bernoulli Beam 4th order ODE is a mathematical equation used to model the behavior of a beam under different loading conditions. It takes into account the bending, shear, and axial forces acting on the beam and describes how the beam will deform.

2. How is the Euler Bernoulli Beam 4th order ODE derived?

The Euler Bernoulli Beam 4th order ODE is derived from the Euler-Bernoulli beam theory, which is based on the assumptions that the beam is long and slender, the deflections are small, and the material is homogenous and isotropic. By applying the principles of statics and differential equations, the 4th order ODE is derived.

3. What is the significance of the 4th order in the Euler Bernoulli Beam 4th order ODE?

The 4th order in the Euler Bernoulli Beam 4th order ODE represents the number of derivatives of the beam deflection. This means that the equation takes into account the 1st, 2nd, 3rd, and 4th derivatives of the deflection, making it more accurate in predicting the behavior of the beam compared to lower order ODEs.

4. How is the balance of units maintained in the Euler Bernoulli Beam 4th order ODE?

The balance of units in the Euler Bernoulli Beam 4th order ODE is maintained by ensuring that all terms in the equation have the same units. This is important in order for the equation to accurately describe the physical behavior of the beam and for the solution to make sense in terms of units.

5. What are the applications of the Euler Bernoulli Beam 4th order ODE?

The Euler Bernoulli Beam 4th order ODE has many applications in engineering and physics, particularly in the design and analysis of structures such as bridges, buildings, and aircraft. It is also used in the study of vibrations and resonance in beams and other mechanical systems.

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