In the last couple of weeks I have been playing with the results of the Fourier Transform and it has quite some interesting properties that initially were not clear to me. In this post I summarize the things I found interesting and the things I’ve learned about the Fourier Transform. Application The Fourier Transformation is applied in engineering to determine the dominant frequencies in a vibration signal. When the dominant frequency of a signal corresponds with the natural frequency of a structure, the occurring vibrations can get amplified due to resonance. Read more

Problem If you want to solve a vibrations problem with a force acting on the system you often need to find the solution in nummerical algorithms. Say you have got a single degree of freedom mass spring system as shown in the figure below. SDOF damped mass spring system The differential equation of this system is: \[ mu'' + cu' + ku = F\] When the force that acts on the system is a function, this problem can be solved with symbolical maths by solving the differential equation. Read more

This post continues where part 1 ended. In order to increase the accuracy of our function solver we are going to use a 4th order Runga Kutta algorithm. The basics are the same as with the Euler method. However the dy part of the 4th order method is more accurately computed. Definition The incremental values of this method are defined as: \[ y_{n+1} = y_{n} + \frac{h}{6}(k_{1} + 2k_{2} +2k_{3} + k_{4})\] \[ t_{n+1} = t_{n} + h \] With the factors k1 - k4 being: Read more

This post continues where part 2 ended. The Runga Kutta algorithm described in last post is only able to solve first order differential equations. The differential equation (de) for a single mass spring vibrations problem is a second order de. \[ mu'' + cu' + ku = F\] Note that in this equation: u” = acceleration a u’ = velocity v u = displacement Before we can solve it with a Runga Kutta algorithm we must rewrite the base equation to a system of two first order ode’s. Read more

Python 1D FEM Example 3. Simple code example for anaStruct. # if using ipython notebook %matplotlib inline from anastruct.fem.system import SystemElements # Create a new system object. ss = SystemElements(EA=15000, EI=5000) # Add beams to the system. ss.add_element(location=[[0, 0], [0, 5]]) ss.add_element(location=[[0, 5], [5, 5]]) ss.add_element(location=[[5, 5], [5, 0]]) # Add a fixed support at node 1. ss.add_support_fixed(node_id=1) # Add a rotational spring at node 4. ss.add_support_spring(node_id=4, translation=3, k=4000) # Add loads. Read more

Example 2: Truss framework Simple code example for anaStruct. # if using ipython notebook %matplotlib inline import math from anastruct.fem.system import SystemElements # Create a new system object. ss = SystemElements(EA=5000) # Add beams to the system. ss.add_truss_element(location=[[0, 0], [0, 5]]) ss.add_truss_element(location=[[0, 5], [5, 5]]) ss.add_truss_element(location=[[5, 5], [5, 0]]) ss.add_truss_element(location=[[0, 0], [5, 5]], EA=5000 * math.sqrt(2)) # get a visual of the element ID's and the node ID's ss.show_structure() # add hinged supports at node ID 1 and node ID 2 ss. Read more

Example 1: Framework Simple code example for anaStruct. # if using ipython notebook %matplotlib inline from anastruct.fem.system import SystemElements # Create a new system object. ss = SystemElements() # Add beams to the system. ss.add_element(location=[[0, 0], [3, 4]], EA=5e9, EI=8000) ss.add_element(location=[[3, 4], [8, 4]], EA=5e9, EI=4000) # get a visual of the element IDs and the node IDs ss.show_structure() # add loads to the element ID 2 ss.q_load(element_id=2, q=-10) # add hinged support to node ID 1 ss. Read more