Idealized spring-mass systems have numerous applications throughout engineering. Fig.1 shows an arrangement of three masses and four springs which, after is released, is pulled downward by the force of gravity. The springs are made of the same homogeneous, isotropic, linear elastic solid material. This means that the springs are identical  and follow Hooke's law (the strain-stress constitutive curve is a straight line of slope ). The system of ordinary differential equations satisfied by this system follows from Newton's second law of motion: 

where is the gravitational acceleration, are the masses, is the spring constant of elasticity (also known as the Young's modulus), and are the displacements of the masses.