This repository contains two python scripts; one to preform singular value decomposition (SVD) on a matrix and the other to solve relevant equations of a spring-mass system.
To clone this repository:
$ git clone https://github.com/jthet/SVD-Spring-Mass-System.git
To run the scripts (detailed below):
$ python3 svd.py
OR
$ python3 spring-mass.py
The terminal will then prompt you for input.
This script prefors SVD on a given matrix. Solving,
The SVD method decomposes a matrix into three other matrices: U, Sigma, and V. U represents the left singular vector matrix, V is the right singular vector matrix, and Sigma is a diagonal matrix containing the singular values, or the square root of the eigenvalues of AAT and ATA, which are equal. This script also compares the found SVD values to the numpy.linalg.svd() blackbox function.
The 3 most notable functions from this script are:
svd(A): Perform Singular Value Decomposition on matrix A; returning U, Sigma, and V.
get_condition(A): Compute the condition number of matrix A using its singular values to find the spectral norm of the matrix.
get_Ainv(A): Calculate the inverse of matrix A, if it exists. (If a singular value = 0, the inverse does not exist)
This code is written to take user input when executed directly.
Example (for a non-singular matrix):
jthet$ python3 svd.py
Enter the number of rows: 2
Enter the number of columns: 3
Enter element at position 1, 1: 3
Enter element at position 1, 2: 2
Enter element at position 1, 3: 2
Enter element at position 2, 1: 2
Enter element at position 2, 2: 3
Enter element at position 2, 3: -2
----------------------------------------
Input Matrix:
[[ 3. 2. 2.]
[ 2. 3. -2.]]
----------------------------------------
U Matrix:
[[-0.70711 0.70711]
[-0.70711 -0.70711]]
----------------------------------------
Sigma Matrix:
[[5. 0. 0.]
[0. 3. 0.]]
----------------------------------------
V.T Matrix:
[[-0.70711 -0.70711 -0. ]
[ 0.2357 -0.2357 0.94281]
[-0.66667 0.66667 0.33333]]
----------------------------------------
U @ Sigma @ V.T (should be A) Matrix:
[[ 3. 2. 2.]
[ 2. 3. -2.]]
----------------------------------------
Condition Number: 1.66667
----------------------------------------
A Inverse Matrix:
[[0. 0.]
[0. 0.]
[0. 0.]]
----------------------------------------
Blackbox Input Matrix:
[[ 3. 2. 2.]
[ 2. 3. -2.]]
----------------------------------------
Blackbox U Matrix:
[[-0.70711 -0.70711]
[-0.70711 0.70711]]
----------------------------------------
Blackbox Sigma Vector:
[5. 3.]
----------------------------------------
Blackbox V Matrix:
[[-0.70711 -0.70711 -0. ]
[-0.2357 0.2357 -0.94281]
[-0.66667 0.66667 0.33333]]
Example comparing my code to the blackbox for an invertible matrix:
Using the matrix, A = [[1, 0], [1, 0]]
jthet$ python3 svd.py
Traceback (most recent call last):
File ".../coe352/project_1/svd.py", line 138, in get_Ainv
raise Exception(
Exception: Error: Input matrix is singular and cannot be inverted
Jacksons-MBP:project_1 jacksonthetford$ python3 svd.py
----------------------------------------
Blackbox Input Matrix:
[[1 0]
[1 0]]
----------------------------------------
Blackbox U Matrix:
[[-0.70711 -0.70711]
[-0.70711 0.70711]]
----------------------------------------
Blackbox Sigma Vector:
[1.41421 0. ]
----------------------------------------
Blackbox V Matrix:
[[-1. -0.]
[ 0. 1.]]
My svd function will catch the singular nature of the matrix and raise an exception, however the numpy.linalg.svd function will not, and will still provide an answer that is incorrect.
This script accepts user input for a spring mass system and then solves the system by calculating the equilibrium displacements, the internal stresses and the elongations of the spring/mass system. To solve this system, the svd function from my svd.py is used to invert the stiffness matrix to solve
Where f is the force vector, K is the stiffness matrix, and u is the equilibrium displacements.
The code allows for user input for the number of springs/masses, the spring constants for each spring, the masses, and which boundary condition to apply (either fixed at both ends or just one).
This script solves the system by the following method, using the solve_system function which takes the user input: num_springs, num_masses, spring_constants, masses, bc.
First, the A, or difference, matrix is found as it represents the relationship between the displacments and elongations of the spring as
Then, the C matrix is found; it is a diagonal matrix of the user-inputted spring constants. The K matrix is then found through the identity,
The f vector, the gravitational force on the masses, is the mi*g, given the user-input for the masses and the svd.get_Ainv function is utilized to find the equilibrium displacement, u, such that as f = K u, u can be found by K-1 f = u. From this, u is known so the elongations can be found (from e = A u), then the internal stresses can be found by w = C e.
The matrix conditionn number, κ, is also calculated. This takes advantage of the svd.get_condition function from my svd.py script. This uses the l2-condition number, found from the product of the spectral norms of the A and A-1 matricies. Where
κ = ||A||2||A-1||2, and ||A||2 = σmax(A) and ||A-1||2 = 1/σmin(A)
An example of running this script using a 4 spring, 3 mass fixed/fixed system:
Note: It is assumed that spring constants are in units N/m, g = -9.81 m/s2, masses are in units of kg, forces are in Newtons, and the displacement and elongation vectors are in units of meters. The K matrix is in units of N/m. The u vector measures displacements such that (+) is down and the internal stress vector measures (+) as tension.
jthet$ python3 spring_mass.py
Enter the number of springs: 4
Enter the number of masses: 3
Enter the spring constant for spring 1: 1
Enter the spring constant for spring 2: 1
Enter the spring constant for spring 3: 1
Enter the spring constant for spring 4: 1
Enter the mass for mass 1: 1
Enter the mass for mass 2: 1
Enter the mass for mass 3: 1
Enter the type of boundary condition (1 or 2 fixed ends):
Note: 1 means 'fixed/fixed' system, 2 means 'fixed/free' system
1
----------------------------------------
Number of Springs: 4
Number of Masses: 3
Spring Constants: [1.0, 1.0, 1.0, 1.0]
Masses: [1.0, 1.0, 1.0]
Boundary Condition: 1
----------------------------------------
A Matrix:
[[ 1. 0. 0.]
[-1. 1. 0.]
[ 0. -1. 1.]
[ 0. 0. -1.]]
----------------------------------------
C Matrix:
[[1. 0. 0. 0.]
[0. 1. 0. 0.]
[0. 0. 1. 0.]
[0. 0. 0. 1.]]
----------------------------------------
K Matrix:
[[ 2. -1. 0.]
[-1. 2. -1.]
[ 0. -1. 2.]]
----------------------------------------
F Vector:
[-9.81 -9.81 -9.81]
----------------------------------------
Equilibrium Displacement Vector:
[14.715 19.62 14.715]
----------------------------------------
Internal Stress Vector:
[ 14.715 4.905 -4.905 -14.715]
----------------------------------------
Elongation Vector:
[ 14.715 4.905 -4.905 -14.715]
----------------------------------------
Condition Number: 5.82843
----------------------------------------
It should be notted that when the number of springs is 1 more than the number of masses, the system is always fixed at both ends. When the number of springs is equal to the number of masses, the system is always free at one end. No other configuration of springs and masses is physically possible. In the case where both ends are free, for out case of a vertical system, this would mean that the system at a whole would be in free-fall and that in a vacuum the internal forces would be zero and there would be no displacement, elongation, or internal stresses. In other words, the system is numerically indeterminate as there is simply not enough equations of equilibrium to determine the reactions at the supports. Also, this situation is impossible to achieve as there were always be some external force acting on the system (normal force, pressure, etc.) unless one went to space and let the spring mass float around.
Note: The user input and the A, C, K matricies and the F vector output have been omited.
jthet$ python3 spring_mass.py
Enter the number of springs: 3
Enter the number of masses: 3
Enter the spring constant for spring 1: 1
Enter the spring constant for spring 2: 1
Enter the spring constant for spring 3: 1
Enter the mass for mass 1: 1
Enter the mass for mass 2: 1
Enter the mass for mass 3: 1
Enter the type of boundary condition (1 or 2 fixed ends):
Note: 1 means 'fixed/fixed' system, 2 means 'fixed/free' system
2
----------------------------------------
Number of Springs: 3
Number of Masses: 3
Spring Constants: [1.0, 1.0, 1.0]
Masses: [1.0, 1.0, 1.0]
Boundary Condition: 2
----------------------------------------
A Matrix:
[[ 1. 0. 0.]
[-1. 1. 0.]
[ 0. -1. 1.]]
----------------------------------------
C Matrix:
[[1. 0. 0.]
[0. 1. 0.]
[0. 0. 1.]]
----------------------------------------
K Matrix:
[[ 2. -1. 0.]
[-1. 2. -1.]
[ 0. -1. 1.]]
----------------------------------------
F Vector:
[-9.81 -9.81 -9.81]
----------------------------------------
Equilibrium Displacement Vector:
[29.43 49.05 58.86]
----------------------------------------
Internal Stress Vector:
[29.43 19.62 9.81]
----------------------------------------
Elongation Vector:
[29.43 19.62 9.81]
----------------------------------------
Condition Number: 16.39373
----------------------------------------
Created by Jackson Thetford 10/5/2023