Unit Checks

There has been a lot of problems with my units. I came up with the strain equation back in March, but I realized that the units weren't matching up and my strain was at least to the 5th power.

Looking back at the Transverse Vibration of Beam notes I realized that

w(x,t) = SUM[ W_i(x)* N_i(t)]

where

W_i = ith natural mode

N_i = time dependent function

I already have W_i(x), so I need to solve for N_i(t).

Two equations I can use are:

N_i(0) = INTEGRAL(0,L) of p*A(x)*W_i(x)*w_o(x) dx

d N_i(0) = INTEGRAL(0,L) of p*A(x)*W_i(x)*w_o(x) dx

I need w(x,t), when t=0 it becomes w_o(x)

NEED: initial deflection of beam under distributed load of intensity f_o => use the static deflection curve.

ASSUMPTION: initial velocity of beam assumed zero

d w_o(x)=0

The static deflection curve can be found by double integrating the equation

d^2 y/ dx^2 = -Px/EI + PL/EI

If P=f and with the I.C. being y(0)=0, y'(0)=0 (fixed bottom)

y(x) = -f/EI * (x^3/3 + x^2*L/4)

I will email my professors with this.

Goals:

• Apply this information to find my N_i(x) equation
• Check my units from previous applications (and on MATLAB files)

Updates for Ch 3

Below are the four mode shapes used:

In order to find the maximum curvature position the third derivative of the W(x) equation was equaled to zero. All four derivatives of each mode shape is graphed below:

The first mode shape was chosen to analyze since the it is the most critical. The maximum curvature for this takes place at x = l, or in this case, x=10 feet.

Current research emphasis:

• Design a column that would be typical in a popular earthquake area. Use this column's new characteristics to update W(x) equation.
• Find out the stresses the column experiences at x=10. (Refer to Mechanics of Materials)
• Apply this information to the PZ strain sensor equations.

Things to do:

• Make a Research Poster for Student Research Week competition
• Make a PP presentation for Research presentation (April 7th, 2011 CE 222)

MATLAB debugger

Met with Hurlebaus and fixed small errors in MATLAB .m file.

tan(B_n*l)+tanh(B_n*l) = 0

tanx+tanhx=0

x=B_n*l

Meeting with Barroso tomorrow to discuss MRD's.

Will review Buc Wen equations.

Matlab program and transverse vibration of column

The transverse equation of the column has been derived.

A matlab .m file has been created to check the mode shape results of the equation.

The results make sense. Meeting with Hurlebaus in the afternoon today to check and to move towards applying the second derivative of the equation to the PZ equation...

With Barroso:

• reading Maryam's MRD paper
• reading references
• MRD constants
• starting to write matlab file for MRD

Chapter 2 of thesis due Monday. Outline shown below.

CH II: METHOD

• II.1 System setup/Motivation (detailed?)
• II.2 Equation of column
• Virtual work
• Vibrations using specific boundary conditions
• II.3 Strain-Acceleration Equation of Column
• II.4 Piezoelectric strain sensor equation
• II.5 Matlab/Simulink program
• II.6 Buc Wen Analysis

CH III: RESULTS AND ANALYSIS

CH IV: CONCLUSIONS

Notes before Meeting

Derivation of the Curvature Equation

The compatibility equation is as follows:

The slope at the top ends are taken from the Beam Deflection Formula tables. For the first analysis shown, a simple force load at the end is transmitted to the column.

The second analysis shows a 1 k-ft moment at the tip of the column which is multiplied by the real Moment at "B." This take usage of flexural (beam) theory.

The total degree change when added up is zero. This allows for the calculation of the real moment at the top of the column.

The moment diagram of each system was created. The sum of the moment equations of each gave the moment of the column as a function of x-- x being the vertical axis. The curvature equation was calculated in this manner.

Research Schedule

Week 11/15 - 11/19

• check formula
• finish 1.1-1.3 of Ch. 1

Week 11/22-11/26

• Finish Ch. 1 & send draft to Barroso/Hurlebaus
• Read Ch. 7: Modelling of Smart Structures (Antisymmetric Configuration) and apply the curvature equation.
• Re-read "Adaptive Control to Mitigate Damage Impact on Structural Response" (Bitaraf, Maryam) to refresh on the Bouc-Wen Model
• Start writing the Bouc-Wen Model on Matlab/Simulink*
  • Assume MR is 10kN for now

Current Problems

Need to focus on Flexibility Method for indeterminate structures.

Quick review:

• Sufficient redundants are removed from an indeterminate structure to produce a stable, determinate released structure.
• The design loads and the redundants are then applied to the released structure.
• Then analyze the determinate released structure for the applied loads and redundants
• The structure is assumed to behave elastically, so these individual analyses can be combined (superimposed) to produce an analysis that includes the effect of all forces and redundants.
• To solve for redundants, the deflections are summed at each point and set equal to the known value of deflection.
• This produces a set of compatibility equations equal in number to the redundants.

In my case, choose two redundants = 2 compatibility equations.

2 redundants: Mtop & By (top)

structure = (P -> | )+ ( | 1 mom)*Mtop + ( | 1k force down)* Ry