Vibrações

Versão em português aqui

Abstract

This folder containts codes made for vibrations course at UnB. Briefly, the dynamic ODE is given by

$$m \ddot{x} + c \dot{x} + k x = f$$

There are the cases:

• Modeling:
  • Get analitic EDO from real problem
• Free vibration (f = 0):
  • Analitic solve this equation.
  • Implement numerical solution
• Harmonic vibration (f = f0*exp(i*w*t)):
  • Analitic solve the equation
  • Implement numerical solution
  • Frequency analisis of system
• Non-oscilatory vibration (f transient):
  • Analitic solve using Laplace transformation
  • Implement numerical solution
• Multi Degrees of freedom system:
  • Modal decomposition
  • Dynamic vibration absorber
• Experimental: Estimate parameters
  • Free vibration experiment
  • Harmonic vibration experiment

Some documents are written in English (indicated with EN) and others are in Portuguese (BR).

Documents

• Homework
  • BR - Homework/1_VigaMassaEquivalente.pdf: Given a beam with bending stiffness EI and linear density mu, we estimate the equivalent mass m and spring constant k of the first modal frequency.
  • BR - Homework/2_EquacoesGovernantes.pdf: Model a cilinder + spring to find system's ODE using the Lagrange mechanics differential formulation.
  • EN - Homework/3_DropMass.ipynb: Model the colision of an free object into another conected into a spring-damper.
  • BR - Homework/4_MassaDesbalanceada.pdf: Model a unbalanced helicopter propeller which rotates with angular speed w.
  • EN - Homework/5_VaribleForce.ipynb: Find the position x of a mass-spring-damper system with force f decomposed in step and ramp using laplace transform.
• Experimental
  • BR - Experimental/first_experiment/: Using a hammer on a cantilever beam, we find vibrational parameters xi and wn from exponential decay response a mesured by an accelerometer. Uses the estimate-exponential-decay.ipynb theory.
  • BR - Experimental/second_experiment/: Using a cantilever beam connected with a oscilating piston at its end, we find parameters m, c and k from the timed graphs of f(force) and a(acceleration) with different frequencies. Uses the estimate-forced-harmonic.ipynb theory.
• EN - dynamic-vibration-absorber.ipynb: Transform a 1 DOF system into a 2 DOF system to minimize the gain X1 of a mass-spring-damper system (m1, c1, k1 fixed) by adding another mass-spring-damper system (m2, c2, k2 variable)
• EN - estimate-exponential-decay.ipynb: Using a 'mesured' (artificial generated noisy data) timed exponential decay response a of 1 DOF mass-sprint-damper system, we find the best parameters xi and wn to fit the curve using non-linear least square method with newton's iteration. Made for the first_experiment.
• EN - estimate-forced-harmonic.ipynb: Using 'mesured' (artificial generated noisy data) force f and acceleration a with different frequencies w of a 1 DOF mass-spring-damper system, we find the best values for m, c and k of this system using least square method to fit the curves. Made for the second_experiment.
• BR - forcamento-harmonico.ipynb: From a 1 DOF mass-spring-damper system with applied harmonic force f0*exp(i*w*t), we compute the analitical and numeric response from given parameters m, c and k and initial conditions x0 and v0.
• BR - sistema-massa-mola.ipynb: Using a free (f=0) mass-spring-damper system with parameters m, c and k and initial conditions x0 and v0, we compute the analitical and numeric response.
• EN - multi-dofs-system.ipynb: Has the theory and the numerical implementation and modal decomposition for a N DOFs system.